High Accuracy Gps Phase Tracking Under Signal

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High Accuracy GPS Phase Tracking Under Signal Distortion A dissertation presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Sai K. Kalyanaraman August 2009 © 2009 Sai K. Kalyanaraman. All Rights Reserved. 2 This dissertation titled High Accuracy GPS Phase Tracking Under Signal Distortion by SAI K. KALYANARAMAN has been approved for the School of Electrical Engineering and Computer Science and the Russ College of Engineering and Technology by Michael S. Braasch Thomas Professor of Electrical Engineering Dennis Irwin Dean, Russ College of Engineering and Technology 3 ABSTRACT KALYANARAMAN, SAI K., Ph.D., August 2009, Electrical Engineering High Accuracy GPS Phase Tracking Under Signal Distortion (139 pp.) Director of Dissertation: Michael S. Braasch Array signal processing is a viable method to provide protection against interference to the desired GPS (Global Positioning System) signal. Adaptive array processing can also be used to provide GPS multipath mitigation. As reported in the literature (Hatke, 1998), adaptive nulling to suppress interference can cause errors in the carrier phase and code phase measurements. The output of multi-element controlled reception pattern antennae (CRPA) may be subject to sizeable variations in phase patterns that arise from the application of non-trivial time varying adaptive array weights (Moelker, 1998). Part of this phase distortion can be compensated in the carrier phase tracking loop of the baseband GPS receiver processing. Classical approaches implement constrained beam steering mechanisms to control the phase of the array in the look direction. An alternative approach, using unconstrained adaptive array processing with compensation for the subsequent phase distortion of the GPS signal is demonstrated by the use of a software radio with results to support the theory. This unconstrained adaptive array implementation does not require attitude information and obviates the need for an attitude sensor such as an inertial reference unit. In addition to addressing carrier phase distortions due to adaptive array processing, this body of work evaluates the carrier phase 4 distortion resulting from GPS signal multipath. Phase multipath characterization is performed for GPS receiver architectures that use typical baseband receiver processing designs (using coherent and non-coherent code tracking loops). Prior to this effort, scant attention had been given to validating carrier-phase multipath theory against GPS data (Brodin, 1996). A comprehensive phase multipath equation (Braasch & Van Dierendonck, 1999) which captures the true nature of carrier multipath is presented. Validation of this equation is carried out for wide and narrow correlator spacing within coherent and non-coherent code tracking architectures by comparing bench test data to theoretical predictions. In addition, data collected for non-zero multipath phase rates are presented. The impact of GPS receiver architecture on the mitigation of phase-rate multipath will be discussed. Approved: _____________________________________________________________ Michael S. Braasch Thomas Professor of Engineering 5 ACKNOWLEDGMENTS I would like to thank my advisor Dr. Michael S. Braasch from the School of Electrical Engineering and Computer Science (EECS) for the support and guidance throughout the course of my graduate program at Ohio University. His expertise and structured approach to engineering has helped me immensely with my research efforts. I would also like to thank Dr. Frank van Graas for taking the time to discuss and explain engineering concepts and ideas that helped me through the course of my research. In addition, I would like to thank Dr. Maarten Uijt de Haag, Dr. Jeffrey Dill and Dr. James Rankin from the School of EECS at Ohio University, Dr. William Kaufman from the Department of Mathematics and Dr. Daniel Phillips from the Department of Physics and Astronomy at Ohio University for taking the time to partake in my dissertation committee. I would like to express my gratitude to Dr. Gary A. McGraw of Rockwell Collins for having taken the time to discuss different aspects of Adaptive Array Processing, reviewing my work and providing insightful feedback on the same. I would also like to thank Dr. Inder J. Gupta of the Ohio State University and Dr. Jade Morton of Miami University for the different discussions that we have had over the years in the area of adaptive array processing as applied to GPS. In addition, I would like to thank Dr. Sanjeev Gunawardena for the support that he had provided towards raw GPS data collection for the purposes of multipath model characterization and validation. I would 6 also like to thank Joseph M. Kelly of the Avionics Engineering Center at Ohio University who provided invaluable high fidelity multipath modeling data towards the validation of GPS carrier phase multipath models. 7 TABLE OF CONTENTS Page ABSTRACT ........................................................................................................................ 3 ACKNOWLEDGMENTS .................................................................................................. 5 LIST OF TABLES .............................................................................................................. 9 LIST OF FIGURES .......................................................................................................... 10 CHAPTER 1: INTRODUCTION ..................................................................................... 15 CHAPTER 2: BACKGROUND ....................................................................................... 19 Adaptive Array ............................................................................................................. 19 Spatial Domain Processing ........................................................................................... 22 Minimum Mean Square Error (MMSE).................................................................... 23 Least Mean Square (LMS) ........................................................................................ 23 Sample Matrix Inversion (SMI) ................................................................................ 26 Minimum Variance Distortion-Free Response (MVDR).......................................... 29 Power Inversion (PI) ................................................................................................. 29 Maximum SINR (MSINR) ....................................................................................... 30 Time and Frequency Domain Processing ..................................................................... 31 Joint Domain Processing .............................................................................................. 31 CHAPTER 3: PREVIOUS RESEARCH .......................................................................... 33 Spatial Domain ............................................................................................................. 33 Power Inversion (PI) and Maximum SINR (MSINR) .............................................. 33 Sample Matrix Inversion (SMI) ................................................................................ 35 Least Mean Square (LMS) ........................................................................................ 35 Joint Domain ................................................................................................................. 36 Space-Frequency Adaptive Processing (SFAP) ........................................................ 36 Space-Time Adaptive Processing (STAP) ................................................................ 36 Blind Beamforming ...................................................................................................... 38 Polarization Rejection ................................................................................................... 40 Digital Beamforming (DBF) ......................................................................................... 42 CHAPTER 4: ADAPTIVE ARRAY PROCESSING ....................................................... 44 8 Impact Of Adaptive Array Techniques On GPS Carrier Phase .................................... 50 Phase Compensation As Applied To The Minimum Variance Algorithm ................... 61 Implementation Results ................................................................................................ 70 CHAPTER 5: MULTIPATH CHARACTERIZATION ................................................... 86 Simplified Models......................................................................................................... 96 Bench Test Setup .......................................................................................................... 99 Non-Coherent Code Tracking Architecture ............................................................ 100 Coherent Code Tracking Architecture .................................................................... 102 Data Analysis and Validation ..................................................................................... 105 Non-Coherent Code Tracking Architecture ............................................................ 105 Coherent Code Tracking Architecture .................................................................... 110 High Fidelity Multipath Modeling .............................................................................. 113 Analysis of Phase Rate Effects ................................................................................... 119 CHAPTER 6: CONCLUSION ....................................................................................... 129 REFERENCES ............................................................................................................... 133 9 LIST OF TABLES Page Table 1:Comparison of Double Difference (DD) standard deviations with and without the ISU under broadband interference (Rosen & Braasch, 1998). ......................................... 41 Table 2: Comparison of Double Difference (DD) standard deviations with and without the ISU under swept CW interference (Rosen & Braasch, 1998). .................................... 42 Table 3:Simulation Parameters for a sample run of the Adaptive Array Processor. ........ 53 Table 4:Attributes of receiver model. ............................................................................. 114 10 LIST OF FIGURES Page Figure 1. Signal flow diagram for the multi-element GPS software adaptive array processor. .......................................................................................................................... 21 Figure 2. N element antenna array with adaptive complex weights applied to signals on each antenna element. The weighed signals are summed and input to the baseband GPS receiver. ............................................................................................................................. 23 Figure 3. Polar plot with locations of the interferers (J1 and J2) and the signal of interest (GPS satellite: SV PRN 12). ............................................................................................. 55 Figure 4. Real and imaginary parts of the adaptive array weights for the MVDR algorithm. “AE” refers to antenna element. ...................................................................... 56 Figure 5. Signal input to and output from the adaptive array. Input to the adaptive array is shown in gray. Output from the adaptive array is shown in black. The MVDR algorithm is implemented here. ......................................................................................................... 56 Figure 6. Signal input to and output from the adaptive array. Zoomed in version of Figure 5. ............................................................................................................................ 57 Figure 7. Real and imaginary parts of the adaptive array weights for the MV algorithm. “AE” refers to antenna element. ....................................................................................... 58 Figure 8. Signal input to and output from the adaptive array. Input to the adaptive array is shown in gray. Output from the adaptive array is shown in black. The MV algorithm is implemented here. ............................................................................................................. 59 Figure 9. Signal input to and output from the adaptive array. Zoomed in version of Figure 8. ............................................................................................................................ 59 Figure 10. Phase of the array in the look direction using the MV algorithm ................... 60 Figure 11. Block diagram of the adaptive array and baseband GPS receiver depicting a control and data path between the adaptive algorithm section and the baseband PLL. .... 62 Figure 12. Baseband GPS receiver carrier lock indicator (Adaptive array algorithm – MVDR) ............................................................................................................................. 73 Figure 13. Baseband GPS receiver code lock indicator (Adaptive array algorithm – MVDR) ............................................................................................................................. 73 Figure 14. Phase discriminator output in units of cycles (Adaptive array algorithm – MVDR) ............................................................................................................................. 74 Figure 15. Normalized code discriminator output (Adaptive array algorithm – MVDR) 74 Figure 16. Carrier to noise ratio (C/No) indicator. (Adaptive array algorithm – MVDR)74 Figure 17. Early, prompt and late power out of the correlators. (Adaptive array algorithm – MVDR) .......................................................................................................................... 76 11 Figure 18. Carrier loop filter acceleration. (Adaptive array algorithm – MVDR) ........... 77 Figure 19. Baseband GPS receiver carrier lock indicator (Adaptive array algorithm – MV) ................................................................................................................................... 77 Figure 20. Baseband GPS receiver code lock indicator (Adaptive array algorithm – MV) ........................................................................................................................................... 78 Figure 21. Phase discriminator output in units of cycles (Adaptive array algorithm – MV) ........................................................................................................................................... 79 Figure 22. Normalized code discriminator output (Adaptive array algorithm – MV) ..... 79 Figure 23. Carrier to noise ratio (C/No) indicator. (Adaptive array algorithm – MV) .... 79 Figure 24. Early, prompt and late power out of the correlators. (Adaptive array algorithm – MV) ................................................................................................................................ 80 Figure 25. Carrier loop filter acceleration. (Adaptive array algorithm – MV)................. 80 Figure 26. Phase compensated baseband GPS receiver carrier lock indicator (Adaptive array algorithm – MV) ...................................................................................................... 81 Figure 27. Phase compensated baseband GPS receiver code lock indicator (Adaptive array algorithm – MV) ...................................................................................................... 82 Figure 28. Phase compensated carrier phase discriminator output in units of cycles (Adaptive array algorithm – MV) ..................................................................................... 82 Figure 29. Phase compensated normalized code discriminator output (Adaptive array algorithm – MV) ............................................................................................................... 82 Figure 30. Carrier to noise ratio (C/No) indicator (Adaptive array algorithm – MV). Phase compensation is applied.......................................................................................... 83 Figure 31. Early, prompt and later power out of the correlators (Adaptive array algorithm – MV). Phase compensation is applied. ............................................................................ 83 Figure 32. Carrier loop filter acceleration (Adaptive array algorithm – MV). Phase compensation is applied. ................................................................................................... 84 Figure 33. Phasor diagram depicts the direct and reflected components along with the phase relation between the direct signal, the multipath θm and the phase tracking error θc of the composite signal (Braasch & Van Dierendonck, 1999). ........................................ 89 Figure 34. Decomposition of the Phasor diagram gives the angle θc subtended by the composite signal with respect to the direct. This is the carrier phase tracking error (Braasch & Van Dierendonck, 1999)................................................................................ 89 Figure 35. Carrier phase multipath error envelope versus relative multipath delay for the standard correlator spacing (1.0 chip) using equation 46, M/D = -3dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. ................................................................................................................. 93 12 Figure 36. Carrier phase multipath error envelope versus relative multipath delay for the standard correlator spacing (1.0 chip) using equation 46, M/D = -10dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. ........................................................................................................ 94 Figure 37. Carrier phase multipath error envelope versus relative multipath delay for the narrow correlator spacing (0.1 chip) using equation 46, M/D = -3dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. ................................................................................................................. 95 Figure 38. Carrier phase multipath error envelope versus relative multipath delay for the narrow correlator spacing (0.1 chip) using equation 46, M/D = -10dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. ................................................................................................................. 96 Figure 39. Carrier phase multipath error envelope versus relative multipath delay using equation 50, M/D = -3dB and -10dB. ............................................................................... 98 Figure 40. Carrier phase multipath error envelope versus relative multipath delay using equations 46 and 50, M/D = -3dB. The dotted lines represent equation 46 while the solid line represents equation 50................................................................................................ 99 Figure 41. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -2dB. ............................................................... 107 Figure 42. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -10dB. ............................................................ 107 Figure 43. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -2dB. .............................................................. 108 Figure 44. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -10dB .............................................................. 108 Figure 45. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -3dB. .................................................... 111 Figure 46. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -10dB. .................................................. 112 Figure 47. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -3dB. .................................................... 112 13 Figure 48. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -10dB. .................................................. 113 Figure 49. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -3dB. ............................................................. 115 Figure 50. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -3dB. Zoomed in version of Figure 49. .............................................................. 116 Figure 51. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -10dB. ........................................................... 117 Figure 52. Zoomed in version of Figure 51. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -10dB. ........................................................... 117 Figure 53. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -3dB. ............................................................. 118 Figure 54. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -3dB. Zoomed in version of Figure 53. .............................................................. 118 Figure 55. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -10dB............................................................ 119 Figure 56. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -10dB. Zoomed in version of Figure 55. ............................................................ 119 Figure 57. Pseudorange multipath error envelope versus relative multipath delay for the standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the code tracking errors obtained for coherent code tracking mode. ............... 121 Figure 58. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the carrier-phase errors obtained for coherent code tracking mode. ............................... 122 Figure 59. Pseudorange multipath error envelope versus relative multipath delay for the narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the code tracking errors obtained for coherent code tracking mode. ............... 122 Figure 60. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the carrier-phase errors obtained for coherent code tracking mode ................................ 123 Figure 61. Plot of pseudorange multipath error versus relative multipath delay for the static and fading multipath scenarios. Standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. ............................................................................................ 124 14 Figure 62. Plot of pseudorange multipath error versus relative multipath delay for the static and fading multipath scenarios. Narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. ............................................................................................ 125 Figure 63. Plot of carrier-phase multipath error versus relative multipath delay for the static and fading multipath scenarios. Standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. ............................................................................................ 127 Figure 64. Plot of carrier-phase multipath error versus relative multipath delay for the static and fading multipath scenarios. Narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. ............................................................................................ 128 15 CHAPTER 1: INTRODUCTION The universal acceptance and use of the Global Positioning System (GPS) has exacerbated the impact of interference to the same. Interference, unintentional or otherwise, to the GPS signal could cause loss of lock at critical phases of GPS use. This may become a threat to mission safety. The effects of interference depend on its type, GPS receiver‟s distance from the source and the propagating medium. Potential sources of GPS interference are Radio Frequency (RF) mobile earth stations, aircraft avionics, ground-based aeronautical emitters, AM, FM, and TV emitters (Pinker, Walker, & Smith, 1999). In addition to other sources of wideband, pulsed, narrow band and Continuous Wave (CW) interferers. "Theory shows that the effect of interference depends on details of receiver design, especially front-end bandwidth and early-late spacing in the discriminator, and that it has a different effect on tracking accuracy than it does on some other aspects of GPS receiver performance" (Betz, 2000). The received power level for the GPS L1 Coarse Acquisition (C/A) code signal is specified at -157dBW (decibels with respect to 1 Watt) which is well below the thermal noise floor of the GPS receiver. It is easily seen that the GPS signal can be subject to interference from sources that transmit very little power in the GPS band of operation. Over the years, various interference mitigation techniques have been investigated to improve the performance of the GPS receivers. These include amplitude, time, frequency, space and polarization domain techniques. Interference suppression or mitigation 16 primarily relies on a technique‟s ability to separate the desired signal from the undesired signal that is seen at the antenna element. The different techniques mentioned above attempt to resolve the desired and undesired components of the received signal into orthogonal spaces as a result of domain transformations. The nature of the interference and the cost function that needs to be minimized would determine if some of these techniques would fare better than others under varying conditions of optimization. Upon application of one or a combination of these domain transform techniques to the incoming signal, it is possible to mitigate the effect of interference. However, there is a penalty to be paid in terms of baseband signal tracking accuracies. Of particular interest are the effects of spatial and spatio-temporal domain based adaptive array techniques on GPS tracking accuracies. These approaches lend themselves quite well to mitigating the effects of multiple continuous wave, narrowband and wideband interferers. The research identifies and focuses on the need to address RF interference mitigation techniques and its subsequent effects on the GPS signal from the perspective of providing continuous carrier-phase tracking. This necessitates the development and implementation of interference mitigation techniques that preclude or take into account the resultant phase variations imposed upon the GPS signal as seen in the baseband tracking domain. Baseband implementation of interference mitigation algorithms integrated with GPS signal processing functions are seen as a viable approach to achieving simultaneous interference mitigation and continuous phase tracking. 17 In addition to possibly losing carrier lock on the satellites and/or being unable to keep accurate track of the carrier phase increments on an epoch by epoch basis under conditions of interference, the other issue that plagues the GPS carrier phase is phase multipath. In light of this, the author has also performed carrier phase multipath model validation for standard and narrow correlator (coherent and non-coherent) code tracking architectures. Data collected for non-zero multipath phase rates are presented. The impact of GPS receiver architecture on the mitigation of phase-rate multipath will be discussed. As a whole, the dissertation focuses on GPS phase tracking under conditions of signal distortion. As indicated above, distortion to the GPS carrier phase can occur due to carrier phase multipath and may also occur due to adaptive array processing. In the case of carrier phase multipath, the distortion is characterized at the carrier phase measurement level. In unconstrained adaptive array processing the carrier phase distortion is evaluated at the baseband tracking loop level. A phase compensation approach is proposed to alleviate the effect of signal phase distortion due to unconstrained adaptive array processing. This enables the baseband tracking loop to maintain continuous carrier phase lock on the GPS signal. This document is organized as follows. Chapter 2 provides relevant background in the area of interference mitigation using adaptive signal processing. Chapter 3 briefly touches upon work performed by other authors in the field of interference mitigation 18 using adaptive signal processing as applied to GPS. Chapter 4 formulates the approach adopted by the author towards addressing interference mitigation and subsequent phase compensation. Relevant results are presented at the end of this chapter. Chapter 5 describes the work performed in the areas of carrier phase multipath validation and code phase multipath characterization under conditions of fading multipath for wide and narrow correlator GPS receiver designs. The distortion imposed upon the GPS carrier phase is examined and validated using bench test data. Chapter 6 presents relevant conclusions. 19 CHAPTER 2: BACKGROUND This chapter provides a description on different adaptive signal processing techniques applicable to GPS interference mitigation. It will concentrate on spatial domain processing and will also highlight temporal and joint domain approaches to interference mitigation as applied to GPS. There is a sizeable body of work that discusses interference mitigation as applicable to GPS (Moelker, 1998; Ward, 2006) (Spilker & Natali, 1996). This rest of this chapter highlights relevant adaptive signal processing approaches to GPS interference mitigation. Adaptive Array A phased array is a term used to describe an array of individual antenna elements. The signals seen on each antenna element are combined to produce the pattern of the phased array. The term “adaptive array” is used to describe a phased array where the weights on each individual antenna element are varied in a dynamic fashion (Godara, 2004). These weights are a function of a prescribed optimization criterion. As the signal operating environment (SOE) varies, the adaptive array weights will change to satisfy the optimization criterion (Compton, 1988). Adaptive GPS arrays can be used to perform both beam steering and null steering. In spatial beam steering, the array weights (henceforth referred to as the weights) are such that they point a beam in space towards the desired signal of interest (SOI). In order 20 to perform beam steering in the context of a GPS signal, the array needs to have knowledge of the direction of arrival of the SOI or certain unique identifying characteristics of the signal which enable recovery of the desired signal from a SOE which contains the SOI and other undesired in-band transmissions (Compton, 1988). On the other hand, spatial null steering refers to the ability of an array to form nulls towards undesired sources of interference. In order to perform this in the context of a GPS signal, it is not necessary to know the direction of arrival of the interference. This is due to the fact that the spread spectrum GPS signal is well below the noise floor and any signal above the ambient noise floor is construed to be a source of interference in the frequency band of interest. Beam steering as applied to GPS typically requires that the direction of the satellite be known a priori. Prior to despreading, the current GPS signal is well below the noise floor and will not be detected by any of the pre-correlation adaptive array processing techniques. However, null steering techniques, such as a power minimization, drive the received power levels of the interferers down by pointing a null towards the direction of undesired sources. If the null steering mechanism drives the power of the interference close to the ambient noise level, the de-spreading process in the downstream GPS receiver will provide additional protection to the desired signal (Kalyanaraman & Braasch, April 2006). 21 Figure 1 (Kalyanaraman & Braasch, April 2006), shows a signal flow diagram which describes the top level system architecture for the adaptive array processor and baseband GPS receiver. The signals from the individual elements of the element antenna array are down-converted, digitized and input to the adaptive array processor. The output of the adaptive array processor is fed to the baseband software GPS receiver (Kalyanaraman & Braasch, April 2006). Note that the adaptive array processing unit can communicate with the baseband receiver via an additional control/data path. A change in the SOE can be detected by the adaptive array processing unit which will convey this information to the baseband software GPS receiver via the control/data path. Figure 1. Signal flow diagram for the multi-element GPS software adaptive array processor. Adaptive array processing can be accomplished in the spatial, temporal, frequency and polarization domains. Processing that exploits a combination of these domains are well suited to GPS interference mitigation but may suffer other limitations such as baseband signal tracking errors. 22 Spatial Domain Processing In case of a spatial domain technique, the sensor utilized to receive the GPS signal is not one, but an array of antenna elements. The number of elements, the inter-element spacing and the dimensions of the array are dependent on the application. In a typical spatial array processing method, the outputs of the antenna array are weighted and summed to produce a single composite signal that is passed along to the rest of the GPS receiver circuitry in order to perform acquisition, tracking and navigation. Figure 2 (Kalyanaraman & Braasch, April 2006) provides a closer look at the adaptive array processing. The signals incident on an 𝑁 element array are shown. Each input, r j (t ) ( j  1 to N ), corresponds to the signal received from one antenna element. The task at hand is to find the best values of the weights that will be applied to the signal as seen on each element of the antenna array such that the criteria for optimization are met. Subject to the criteria, the array performs a cost function minimization. The well-known optimization criteria are (Moelker, 1998): 1) Minimum Mean Squared Error (MMSE) 2) Minimum Variance Distortion-free Response (MVDR) 3) Power Inversion (PI) 4) Maximum Signal to Interference plus Noise Ratio (Maximum SINR) 23 Figure 2. N element antenna array with adaptive complex weights applied to signals on each antenna element. The weighed signals are summed and input to the baseband GPS receiver. Minimum Mean Square Error (MMSE) Least Mean Squares (LMS) and Sample Matrix Inversion (SMI) algorithms fall under the class of MMSE techniques. Least Mean Square (LMS) The LMS algorithm is iterative by design. The algorithm makes successive corrections or updates to the computed weight vector. This is in the direction of the negative of the gradient vector (gradient of the residual error) (Widrow & Hoff, 1960). The solution to the gradient gives the minima and hence the MMSE. Consider an antenna array of 𝑁 elements. From Figure 2, the output of the array at time t is given by: 24 y(t )  w t r (t ) H (1) where, H w is the conjugate transpose of the weight vector w . “ r (t ) ” is the signal impinging on the antenna array. This is given by (Godara, 2004): r (t )  x(t )  i(t )  n(t ) (2) Where, x(t ) is a column array of the desired signal as seen at the elements of the adaptive array, i (t ) is a column array of the interference as seen at the elements of the adaptive array and, n(t ) is a column array of noise as seen at the elements of the adaptive array. Each of these vectors are of size N*1 and each element of the vector represents the value of the vector at the corresponding antenna element. The objective of the algorithm is to infer the desired signal x(t ) in the presence of the undesired interference i (t ) and noise n(t ) . This LMS algorithm is based on Wiener filtering (Brown & Hwang, 1997). The Wiener filter calls for a noiseless copy of the desired signal to be used as a reference to 25 generate the error signal that drives the weights computation mechanism (Brown & Hwang, 1997). The discrete version of the LMS algorithm can be summarized as follows: The output, H y ( n)  w r ( n ) (3) e(n)  d (n)  y(n) (4) w(n  1)  w(n)   r (n)e(n) (5) The error, Weight update, Where, “ d (n) ” is the reference signal used to generate the error signal “ e(n) ”, “  ” is the LMS algorithm's search step size and, "n" is the index of time samples. 26 The convergence of the LMS algorithm is assured, given that the step size  of the LMS algorithm is smaller than twice the inverse of the largest eigenvalue of the correlation matrix (Agamata, 1991): 0 < μ < 2/λmax (6) where, “ max ”is the largest eigenvalue of the correlation vector “ q ” given by q  E[d * ( z )r ( z )] (7) Sample Matrix Inversion (SMI) The LMS algorithm discussed in the previous section is continuously adaptive and has a slow convergence when the eigenvalues of the covariance matrix are widespread. A block adaptive approach could give a better performance than a continuous one (Compton, 1988). One such algorithm is the Sample Matrix Inversion (SMI). It also requires that the number of interferers and their positions remain a constant over the period of the data block. The SMI converges faster in comparison to the LMS as it employs direct matrix inversion. However, this could become computationally intensive (order of N3 with N being the number of antenna elements (Godara, 2004)) and more elegant methods of matrix inversion may be employed. The optimal weights that 27 minimize the input power of the signal are given by the classic Wiener solution (Compton, 1988): w  R 1 q (8) Where, “ R ” is the input signal covariance matrix and is given by: H R  E[r ( z )r ( z )] (9) Since the user does not have perfect knowledge of the signals or the signal ^ ^ environment, “ R ” and “ q ” are estimated by time averaging the input data over a given number of samples (size of the SMI data block). ^ R ^ q 1 K 1 K K  R( p) (10) p 1 K  q( p) (11) p 1 Hence, the estimate of the optimal weights as provided by the Wiener equation (Compton, 1988) is: 28 ^ ^ 1 ^ wR q (12) The success of the SMI approach lies in the choice of the sample size “K”. The required sample size is obtained as the result of computing the expectation of a ratio of SINR‟s with the ratio being (Compton, 1988),  SINRSMI SINROPT (13) where, 𝑆𝐼𝑁𝑅𝑆𝑀𝐼 is the SINR of the system that can be achieved by using the SMI algorithm and, 𝑆𝐼𝑁𝑅𝑂𝑃𝑇 is the SINR achieved by using the optimal weights as per the Wiener filter. The pdf of “  ” is a beta function. If E    0.5, then, K > 2*N, where N is the number of antenna elements (Compton, 1988). In this case, (Compton, 1988) 𝑆𝐼𝑁𝑅𝑂𝑃𝑇 − 𝑆𝐼𝑁𝑅𝑆𝑀𝐼 ≤ 3 𝑑𝐵 (14) This is a rule of thumb measure used with the SMI algorithm. The most important facet of this technique is that the convergence rate of this algorithm does not depend on signal powers, angle of arrival or eigenvalue spread. This technique is better than the LMS approach when the eigenvalue spread is high and the number of antenna elements used in the array architecture is small (Compton, 1988). This scenario is applicable to a small GPS array consisting of two to seven antenna elements arranged in an appropriate 29 fashion. The SMI based approach is better suited to deal with pulsed interferers or rapid platform dynamics. Minimum Variance Distortion-Free Response (MVDR) The MVDR technique places a constraint on the antenna gain in the direction of the signal and minimizes the array power output. The number of elements in the array plays a key role in determining the number of available degrees of freedom, which decreases by one for each additional signal constraint. The technique requires knowledge of the direction of arrival of the desired signal. Input data to the antenna array is the signal plus interference and noise. The array output has minimum noise variance and maximum gain with respect to the target angle of arrival of the desired signal (Agamata, 1991). If the direction of arrival of the signal coincides with direction of arrival of the interference, the system‟s constraint definition may result in the interference not being cancelled. This technique requires more degrees of freedom (read as antenna elements) in comparison to the number of desired signals (Moelker, 1998). Power Inversion (PI) In this technique, the output Signal to Interference Ratio (SIR) is the inverse of the input SIR. The technique suits GPS applications quite well. The algorithm provides good performance when the desired signal is weaker than the interference. However, the quality of the desired signal cannot be guaranteed. The power inversion array is essentially an Applebaum array (Moelker, 1998) with some modifications. The weight on 30 one of the elements is set to unity (unweighted) resulting in the situation with identically zero output power being prevented (Moelker, 1998). Maximum SINR (MSINR) The Maximum SINR criterion, as the name implies, attempts to maximize the Signal to Interference plus Noise Ratio. The SINR for a single antenna element is given by (Moelker, 1998): SINR  PD PN  PI (15) where, 𝑃𝐷 is the power of the desired signal, 𝑃𝑁 is the power of the noise, and 𝑃𝐼 is the power of the interference. The maximum SINR principle attempts to maximize the numerator 𝑃𝐷 , the desired signal power. In terms of the correlation functions 𝑅𝐷 , 𝑅𝑁 , 𝑅𝐼 and the antenna weights 𝑤, the SINR is (Moelker, 1998): wH RD w wH RN wH  wH RI w where, (16) 31 𝑅𝐷 is the covariance matrix of the desired signal, 𝑅𝑁 is the covariance matrix of the noise and 𝑅𝐼 is the covariance matrix of the interference. The maximum SINR criterion is met by maximizing the numerator while the power inversion scheme minimizes the denominator of this equation. Time and Frequency Domain Processing Estimation type filtering and transform domain processing usually dominate processing for interference rejection and mitigation in time and frequency domains. In estimation type filtering the interference is estimated and removed from the incoming signal by using the property of temporal correlation, given the signal‟s recent history (Moelker, 1998). This can be implemented using a linear prediction error filter. In transform domain processing, the time sampled signal is transformed to the frequency domain using a real Fourier transform and the narrow band interference is detected and excised. Accounting for the effects of spectral leakage, the inverse Fourier transform gives the original time sequence minus the interference. Joint Domain Processing Joint domain signal processing offers significant advantages that may not be realizable in either of the individual domains. The independent nature of the spatial and temporal domains offers unique insights into the signal and the interfering process. 32 Signals that are considered wideband in the frequency domain are usually narrowband in the spatial domain. Similarly, narrow band interference that arrives from the same direction as the desired signal can be removed by temporal filtering techniques. Such joint domain processing offers much greater performance in comparison to individual domain processing (Moelker, 1998). 33 CHAPTER 3: PREVIOUS RESEARCH This chapter serves to highlight some of the previous research conducted in the areas of interference mitigation as applied to GPS. In particular, it highlights certain aspects of earlier research efforts that use adaptive array processing as applied to GPS and are relevant to the proposed work. Spatial Domain Power Inversion (PI) and Maximum SINR (MSINR) Earlier research (Moelker, Oct 1996) evaluated the Howells Appelbaum (HA) array (maximum SINR) versus the power inversion technique for SINR improvement, phase lock and array adaptation. A comparison of the performance of the HA array with the power inversion technique was performed for array structures with 2 to 7 elements. The comparisons simulated a maximum of two interferers and lead the author to the following conclusions: a) SINR improvement The power inversion array is about N 1 worse off compared to the maximum N2 SINR criterion implemented by the HA array, where “ N ” is the number of antenna elements. This is the case when the direction of the satellite is known in advance to optimally implement the HA array. However, if the interferer and the signal were from the same direction, the signal would be nullified. The power inversion technique was the 34 best choice as the array pattern variations for the PI technique were minimal in comparison with the HA array (Moelker, Oct 1996). However, the research does not detail interference in terms of its bandwidth characteristics and limits its scope to spatial domain processing. The reality of interference multipath might end up absorbing any remaining degrees of freedom available in the antenna arrays and saturate the systems interference mitigation capability. b) Phase tracking A symmetric array structure is proposed to deal with any deviations in the phase of the signal due to application of weights to the incoming signal. However, this methodology requires that the array is perfectly symmetrical. As a result, the number of available degrees of freedom is reduced by a factor of one half. Half the array would produce the positive part of the complex component of the weights and the other half would produce the negative part of the complex component of the weights and their sum would be a real quantity which results in the array weights modulating the real part (amplitude) of the signal and not the imaginary part which holds the phase information. This results in symmetric 7 and 5 element arrays being equivalent to regular 4 and 3 element arrays in terms of their available degrees of freedom. It is known that the rate of weight adaptation has an influence on the tracking capability of the receiver‟s phase locked loop (PLL) (Moelker, 1998). In a changing SOE, the phase of the array becomes incompatible with the receiver‟s tracking loops. As a result, any phase change may come 35 across as a step change to the PLL, which might not be tracked very well (Moelker, 1998). c) Convergence The convergence of the two algorithms was comparable(Moelker, 1998). However, it was seen that larger array structures converged more slowly than the smaller ones. Sample Matrix Inversion (SMI) Previous work (Progri, 1998) has shown that SMI based beam forming architectures (a Minimum Mean Square approach) can provide 40 dB of protection against multiple broadband interferers (limited by the degrees of freedom). However, the impact of this beam forming technique on the GPS carrier-phase is yet to be studied in complete detail. Given the fact that this is an “open loop” technique, it is very sensitive to both amplitude and phase errors in the array signal paths. Performance of this technique depends on how well the individual channels of the array are matched. Least Mean Square (LMS) Previous research (Gecan & Zoltowski, 1995) has shown that the LMS algorithm is very effective against multiple CW interferers with interference to noise ratios (INR) in the range of 30-50 dB. 36 Joint Domain Research in joint domain processing is primarily split across Space Frequency Adaptive Processing (SFAP) and Space Time Adaptive Processing (STAP). Both techniques exhibit excellent performance in terms of broadband and narrowband interference mitigation. Space-Frequency Adaptive Processing (SFAP) Research efforts indicate that SFAP, when used in conjunction with a windowing process such as a Blackman or Hamming window performs better than a uniform window (Gupta & Moore, 2001). The windowing localizes the narrow band interferers to a few frequency bins, which in turn improves the performance of the SFAP. A seven-element array was used in this study. Three wideband and up to 10 narrowband interferers were used. The Interference to Signal Ratio (ISR) was set at 60 dB (Gupta & Moore, 2001). The interferers spanned the range of azimuth and elevation and had a flat spectral density. (Gupta & Moore, 2001) describe the phase response of the antenna array versus frequency while processing the signal for SV PRN 8. Their work suggests a linear response across the signal bandwidth and , as expected, indicates nulls in the array response for those frequencies in the direction of the interferers. Space-Time Adaptive Processing (STAP) Wideband interference is known to limit adaptive array performance and was the initial motivation behind tapped delay lines. A STAP architecture with “L” antenna 37 elements and “N” taps per element has N*L –1 degrees of freedom. Research (Gupta & Moore, 2003) has shown that the narrow band signal can consume more than one degree of freedom in STAP. It has also been shown (Gupta & Moore, 2003) that the INR at the STAP output oscillates with increasing interference power and bandwidth similar to a conventional array. However, the SINR performance for the STAP does not degrade as much as the conventional array. A wideband RFI source consumes multiple degrees of freedom (DOF) in a STAP depending on interference power and bandwidth. Results indicate that the STAP provides excellent narrow and broadband interference cancellation. Earlier studies (Hatke, 1998) have attempted to identify the presence of wideband interference multipath using the eigenvalue distribution of the interference plus noise array covariance matrix in a STAP. Given the close association with analysis in the time and frequency domains, it is possible that the STAP does not perform any better than the SFAP when it comes to preserving the phase of the desired signal This is more in the case when the tap spacing is equal to the sampling interval (Gupta & Moore, 2001). However, the impact of these architectures on the phase tracking accuracy of the GPS signal had not, until recently, been a point of focus in most STAP implementations. Recent work (O'Brien & Gupta, 2008) presents an algorithmic adaptive array constraint that provides the downstream GPS receiver with bias free GPS range domain measurements. However, this algorithm needs complete knowledge of the antenna array manifold and its front end response. 38 Blind Beamforming Other research efforts have pointed the way to blind beamforming techniques to mitigate the effect of interference in satellite navigation. These algorithms have significant computational complexities. However, the advantage of blind versus conventional beam forming is the fact that precise array calibration is not required and knowledge of the direction of arrival of the desired signal is superfluous. The characteristic used to distinguish the desired signal from interference and noise is knowledge of the cycle frequency of the desired signal (Morton, Liou, Lin, & Tsui, 2004). One of the blind beam forming methods recently explored for its utility in GPS interference mitigation is the Spectral Self-Coherence Restoral (SCORE) technique (Morton et al., 2004). This deals with signals that are spectrally self coherent. The correlation between a signal s(t ) and its frequency-shifted version for some time lag is non-zero. The SCORE algorithm aims at obtaining optimized weight vectors that maximize cyclic components of the receiver input. The cyclic autocorrelation of the input is given by (Morton et al., 2004): Rx ( )  a Rs ( )  b Rv ( )  Rn ( ) 2 2 (17) where, Rx ( ) is the cyclic autocorrelation function of the incoming signal as seen at the antenna, Rs ( ) is the cyclic autocorrelation function of the desired signal, Rv ( ) is the cyclic autocorrelation function of the interference, Rn ( ) is the cyclic autocorrelation function of the noise, and 39 “ a, b ” are steering vectors associated with desired signal and interference respectively. Since there is no spectral self-coherence for the interference and noise, it can be said that Rx ( )  a * Rs ( ) (Morton et al., 2004). The C/A code in GPS has a period of 2 1 millisecond. The set of GPS signals at the input of the GPS receiver are spectrally selfcoherent. The technique utilizes an input signal x(t ) , multiplies it with weights w to generate an output y . It takes a delayed version of the same input x(t  mT ) , multiplies it with weights w' to generate an output y ' . In a least square sense it attempts to minimize the error between y and y ' to generate the optimized weights w' . In controlled field tests, a four-element antenna array was subjected to FM chirp, BPSK and wideband interference at ISR‟s of 20, 30 and 40 dB respectively (Morton et al., 2004). The setup utilized eight satellite signal sources. The algorithm did not generate consistent acquisition results. This was believed by the authors (Morton et al., 2004) to be a consequence of having more satellites than the number of degrees of freedom as provided by a four-element array. A key aspect that needs to be highlighted in this technique is the theoretical capability to clearly distinguish the desired signal from the undesired interfering sources, immaterial of the direction of arrival. The technique, loosely put, transforms the incoming signal into a domain where the desired signal space is rendered orthogonal to the undesired interference and noise, lending some tractability to the problem of interference mitigation. However, the efficacy of this approach to 40 address code phase and carrier phase distortions and ranging errors has not yet been evaluated. Polarization Rejection The next technique is a single antenna interference rejection method introduced here in order to present double difference (DD) carrier-phase residual results that can be compared with DD carrier-phase residual data from other beam forming algorithms. Previous research (Rosen & Braasch, 1998) shows that a low cost, single aperture, interference mitigation capability can be effective against narrowband, CW, swept CW and wideband interference. The technology utilized a dual polarization antenna aperture with an adaptive feed network. Realization of the Interference Suppression Unit (ISU) concept relies on polarization differences between the interferer and the GPS signal (Rosen & Braasch, 1998). The ISU was tested with an array of receivers that were predominantly used in the market, and displayed > 35dB of CW interference suppression and > 20 dB of Wideband interference suppression (Rosen & Braasch, 1998). In addition, the carrier-phase and pseudorange tracking accuracy was analyzed when the ISU was in the loop to mitigate the effect of interference. Table 1 (Rosen & Braasch, 1998) depicts the double difference carrier phase performance in the presence of interference. The data in Table 1(Rosen & Braasch, 1998) applies to the case of broadband interference. The entry "lost lock" refers to losing lock on the GPS satellite when the interference power is increased. 41 Table 1:Comparison of Double Difference (DD) standard deviations with and without the ISU under broadband interference (Rosen & Braasch, 1998). Broadband DD Std Dev (mm) Noise Level Without ISU With ISU -70 Na 2 -60 2 2 -45 7 N/A -30 Lost lock 2 -20 Lost lock 4 (dBm) N/A: Data not collected at this power level The tests were zero baseline tests where two receivers were connected to the same antenna elements and the double differences were computed. Since the antenna phase center was the same for both receivers (for a given signal source), the result is a function of the errors (Rosen & Braasch, 1998). Data in Table 2 (Rosen & Braasch, 1998) applies to the case of swept CW interference. In addition to providing good broadband interference mitigation, the ISU also provided good carrier-phase tracking performance. 42 Table 2: Comparison of Double Difference (DD) standard deviations with and without the ISU under swept CW interference (Rosen & Braasch, 1998). Interference DD Std Dev (mm) level (dBm) Without ISU With ISU Off 3 2 -65 Na 3 -60 4 N/A -52 5 N/A -30 Lost lock 2 -25 Lost lock 2 N/A: Data not collected at this power level It is to be kept in mind that this is not a beam forming or null steering technique. The system architecture does not lend enough degrees of freedom to point multiple spatial beams and/or nulls that would offer the degree of interference rejection and/or multipath mitigation demanded by high accuracy differential GPS applications. The ISU exploits the concept of polarization nulling by taking advantage of the fact that the GPS signal is Right Hand Circularly Polarized (RHCP ) (Rosen & Braasch, 1998). Digital Beamforming (DBF) Multi-element antenna arrays have the capability to mitigate multipath and interference to a good extent. This is a function of the number of degrees of available 43 freedom. In addition, the presence of interference may cause the signal environment to become non-stationary in a statistical sense. Use of imperfect antenna array data in the beam former computation induces errors in the carrier-phase of the signal. Phase center variations in CRPA can amount to a significant fraction of one carrier cycle. However, results indicate that Digital Beam Forming (DBF) offers phase continuity in comparison to nulling under conditions of interference (McGraw, Young, & Reichnauer, 2004). Both DBF and nulling introduce phase offsets in the signals due to array pattern variation. Research (McGraw et al., 2004) indicates that the phase offset is held around the same level under the influence of interference when the DBF architecture is used. Null steering causes a significant jump in the carrier-phase of the signal (McGraw et al., 2004). Controlled zero baseline experimental studies using a 4-element array show that DBF produces carrier-phase double difference residuals that hold about their nominally expected level around zero while there is no interference. Application of the DBF algorithm to mitigate interference results with the carrier-phase double difference residuals being biased around 6 cm, a sizeable fraction of the carrier-cycle. However, beam forming offers a great degree of flexibility and allows the user to point beams in almost any direction and is constrained only by the number of elements in the antenna array. Beam forming is a technique that is applicable to interference rejection as well as GPS multipath mitigation. 44 CHAPTER 4: ADAPTIVE ARRAY PROCESSING GPS adaptive array processing can be used to mitigate the effect of most interfering transmissions in the GPS band. Different classes of algorithms are utilized for this purpose and are a function of the SOE. The work presented focuses upon carrier and code tracking in the presence of interference. Earlier work (Hatke, 1998) has established that adaptive array processing can cause distortion to the spread spectrum ranging signal. Knowledge of platform attitude allows the user to obtain a steering vector towards the satellite of interest and use it to impose a constraint on the look direction of the GPS array (Godara, 2004). However, receivers without access to attitude references are unable to perform constrained adaptive signal processing. As a result, the signal tracked by the GPS receiver has a non-trivial complex phase signature imposed on it by the adaptive array (Moelker, 1998; Moelker & van der Pol, 1996). The work described in this chapter discusses an alternative approach to perform phase compensation at the baseband level to alleviate the effect of the deterministic phase signature variations imposed by unconstrained adaptive array processing on the GPS SOI (Kalyanaraman & Braasch, April 2006). These phase variations have the capability to induce loss of lock in the baseband GPS carrier tracking loops (Moelker, 1998). Based on the background material which surveyed multiple approaches and options to adaptive signal processing as applied to GPS, it is seen that the SMI based array processing approach would yield good interference mitigation immaterial of 45 whether the array electronics has knowledge of platform attitude (in addition to other array and signal parameters) or not. As a result, this work will expand on constrained (needs knowledge of platform attitude in conjunction with other array and signal parameters) and unconstrained adaptive signal processing (needs no a priori knowledge of antenna array platform attitude) as applied to GPS. The impact of choice of adaptive array processing algorithm on the carrier phase tracking of the desired GPS signal will be evaluated. Subsequent to these discussions, an approach to avoid loss of baseband receiver carrier phase lock as a function of adaptive array weights convergence for the case of unconstrained adaptive signal processing will be presented. An overview of how this approach is implemented will also be provided. A comparative performance of GPS baseband signal tracking across architectures using constrained and unconstrained adaptive array processing will be presented. The work presented in this section is based on an article submitted for publication (Kalyanaraman & Braasch, 2009). The desired component of the received signal (desired GPS signal free of interference and noise) X (t ) can be described by the following equation: X (t )  S l * m(t ) * e j 2f ct Where, S l is the steering vector towards the desired signal from source 𝑙, m(t ) is the modulation on the desired signal, and f c is the carrier frequency of the desired signal. (18) 46 𝑆𝑙 is an N-dimensional complex vector. If the array has elements with similar individual responses, the steering vector is given by (Godara, 2004), 𝑆𝑙 = 1, 𝑒 𝑗 2𝜋𝑓𝑐 𝜏 1 𝜂 𝑙 ,𝜈 𝑙 , 𝑒 𝑗 2𝜋𝑓𝑐 𝜏 2 𝜂 𝑙 ,𝜈 𝑙 , … , 𝑒 𝑗 2𝜋𝑓𝑐 𝜏 𝑛 −1 𝜂 𝑙 ,𝜈 𝑙 (19) Where, 𝑙 is the index of the desired source, 𝜂𝑙 is the elevation of the 𝑙‟th source, 𝜈𝑙 is the azimuth of the 𝑙‟th source and, 𝜏𝑛−1 𝜂𝑙 , 𝜈𝑙 is the time delay from the reference element for a plane wave impinging on the 𝑛'th element of the array for source 𝑙, For the case of GPS, the bandwidth of the modulation (~ 20MHz) is quite small compared to the carrier frequency of the signal. For array sizes of the order of a few wavelengths or less, m(t )  m(t   N ) .This approximation is used while characterizing the steering vector in equation 19. It is known from theory (Compton, 1988) that the numbers of degrees of freedom of an 𝑁 element adaptive array with one temporal tap per element are one less than the number of elements that make up the array. These 𝑁 − 1 degrees of freedom provide room in the adaptive algorithm design to point beams or nulls as desired. Some beam steering and null forming mechanisms impose certain constraints on the adaptive 47 algorithm. One such constraint is what is referred to as a point constraint. This effectively states that the signal coming in from a prescribed direction will be passed through the adaptive array without any distortion. This is also termed as a look direction constraint. Such a constraint will utilize one of the array‟s degrees of freedom effectively reducing the number of degrees of freedom available to an array for mitigating interference. One such technique is the Minimum Variance Distortion-less response (MVDR) technique (Godara, 2004). In an optimal scenario, it would be preferable to obtain, individually, the different covariance matrices associated with the interference (𝑅𝐼 ), signal (𝑅𝑆 ) and noise (𝑅𝑁 ) components of the received signal. However, it is impractical and difficult to separately estimate the individual signal, interference and noise components in the received signal r (t ) . As a result, samples of the received signal are used to compute the signal plus noise covariance matrix. In an MVDR approach, the minimum variance optimization criterion is constrained by a unity response in the look direction. The objective is to minimize the total received signal power (Godara, 2004) : H PR  w * R * w subject to (20) 48 H w *Sl 1 (21) Estimates of the weights that meet this criterion are given by (Godara, 2004): wˆ  R 1 * S l S l * R 1 * S l H (22) In the optimal case where the signal environment is well characterized through known distributions, 𝑅 is well known from the statistics. However, in reality, it is not possible to completely characterize the signal operating environment. As a result, 𝑅 is computed using samples of the received signal r (t ) . The estimate of the inverse of the covariance matrix used in equation 22 is determined from the input samples. Subsequent updates to the 𝑅 −1 matrices may be performed using the matrix inversion lemma (Godara, 2004) or using a weighted combination of the existing 𝑅 matrix and samples of the incoming signal as seen at the antenna array (Fante, 2004). This enables the user to compute updates using new data samples while not having to recompute the whole matrix every time. In this Sample Matrix Inversion (SMI) implementation, the weights applied to the incoming signal are computed in an open loop fashion. In general, the adaptive array weight vector applied to the incoming signal is determined as the solution to an optimization problem. However, the optimization criteria vary from one adaptive array technique to another. 49 Another similar but simpler approach is to remove the look direction constraint on the adaptive array mechanism. In this case, 𝑆𝑙 in equation 22 will be replaced by 𝛼 which is given by an 𝑁 element column array with its first element as unity and the rest being zeros (N-1 zeros). Here, the weights are given by (Berefelt, Boberg, Eklöf, & et. al., 2003): R 1 *  w H  * R 1 *  ^ (23) While implementing equation 23, knowledge of the direction of arrival of the GPS signal is no longer required and, as a result, knowledge of the steering vector 𝑆𝑙 is not required. This approach is referred to as the Minimum Variance (MV) algorithm (Godara, 2004). The array correlation matrix for an 𝑁 element antenna array system performing Spatial Adaptive Processing (SAP) has 𝑁 2 elements (𝑁 by 𝑁 matrix). The diagonal elements of the matrix display the auto-correlation of the incident signals as seen at the antenna array. The off diagonal terms of this matrix would typically be dominated by the cross correlation of white gaussian sequences in the absence of an interference signal that rises above the ambient noise floor of the receive array. In the presence of interference, the off diagonal elements will see a deterministic signature caused by an interferer. The interference mitigation algorithm strives to minimize the power seen in the off diagonal terms taking into cognizance the fact that the signature observed on the diagonal terms would be the combination of interference power and ambient receive array noise power. 50 Impact of Adaptive Array Techniques on GPS Carrier Phase The two approaches mentioned in the previous section differ considerably in the manner by which they impact the carrier phase of the GPS signal. For purposes of discussion, the error free GPS signal may be given by: x(t )  A * D(t ) * C (t ) * cos(2f ct   ) (24) Where, “𝐴” is the amplitude, “𝐷(𝑡)” is the 50 Hz data on the GPS signal, “𝐶(𝑡)” is the PRN spreading code on the GPS signal and, “𝛽” is the non-zero phase of the carrier as seen at the analog to digital converter (A/D). Equation 24 presents an idealized version of the output of the adaptive array. If the attitude of the antenna array platform is known then it is possible to compute the geometric steering vector towards the satellite. Attitude information is typically obtained using an inertial sensor. Knowledge of the array manifold may be used to compensate for deviations of the individual element patterns from ideal. This information is obtained through array calibration. As a result, it is possible to point a beam or a distortion-less constraint towards a satellite while interferers from other directions are nulled using the remaining degrees of freedom. Array calibration is a time consuming and difficult procedure when highly accurate carrier phase measurements are desired. 51 Given the look direction constraint of the MVDR mechanism, the phase of the desired GPS signal would not be affected and the GPS signal would pass through the adaptive array undistorted. In the presence of undesired interference, the array weights would adapt to null out the interference. In the case of an MV approach, the lack of a look direction constraint will result in distortion to the desired GPS signal. The complex weights applied to the incoming signal may be considered equivalent to the application of a net complex phase. This results in a net phase distortion. To evaluate the impact of the choice of adaptive algorithms on the GPS signal, the author designed and developed software signal simulators to generate GPS signals, interference and noise as seen at the individual elements of the antenna array. The simulation setup included options to choose the satellite signal levels, interfering signal levels, direction of signal arrival and choice of the number and type of interfering sources. These analyses were performed using MATLABTM. The core capability offered by the simulation involves the generation of SV signals based on user inputs. The user can specify the type of interferer scenario through the duration of the simulation. As depicted in Figure 1 the combination of desired GPS signals, ambient noise and interferer signals are passed through an adaptive antenna processing mechanism, the output of which is passed onto a baseband processing unit. 52 Subsequently, this baseband processing unit performs GPS signal acquisition and tracking. This simulation can be used to assay the impact of adaptive array processing on the baseband GPS processing mechanism in the presence and absence of interference. The GPS signals are simulated at the necessary Doppler offsets. Based on additional user input, varying number and types (CW, pulsed and broadband) of interference inputs are generated. The GPS, interference and independent receive array noise sources are combined to form the input as seen at each individual element of the GNSS antenna array. As a result, the input at each element has the right amount of relative phasing to account for the attitude of the antenna array relative to the desired signal. The input to the antenna array is passed to the adaptive array processing block in Figure 1. Based on user configurable options, the type of adaptive processing mechanism is chosen (Spatial Adaptive Processing: SAP or Space Time Adaptive Processing: STAP). Within each adaptive processing mechanism user configurable options are provided to enable the user to select any combination of elements, number of taps and tap spacing in addition to various options that enable the user to choose between constrained and unconstrained adaptive array processing. As observed in Table 3, the setup introduces the first source of interference into the scenario four seconds after the start of the simulation. Figure 3 depicts the locations 53 of the interferers J1 (CW interferer) and J2 (Broadband interferer) with respect to the satellite of interest (SV PRN 12). Table 3 presents details of a test scenario used to study the impact of adaptive array processing on GPS carrier phase. Current work deals with spatially adaptive arrays. The space-time extensions of these arrays are not covered in this effort. SAP algorithms were implemented to mitigate the impact of interferers. The inter-element spacing for the array is set to a half wavelength as measured at the L1 GPS center frequency (~19 cm at 1575.42 MHz). As observed in Table 3, the setup introduces the first source of interference into the scenario four seconds after the start of the simulation. Figure 3 depicts the locations of the interferers J1 (CW interferer) and J2 (Broadband interferer) with respect to the satellite of interest (SV PRN 12). 54 Table 3:Simulation Parameters for a sample run of the Adaptive Array Processor. Simulation length 6 Seconds Array type 4 element square array Number of CW interferers 1 Interferer to Signal ratio 65 dB (ISR) for CW1 CW1 Azimuth 200 degrees CW1 Elevation 60 degrees CW1 start time 4 seconds into run Number of BB interferers 1 Interference to Noise ratio 37 dB (INR) for BB1 BB1 Azimuth 310 degrees BB1 Elevation 75 degrees BB1 start time 4.3 seconds into run GPS PRN 12 GPS SV Azimuth 45 degrees GPS SV Elevation 45 degrees GPS Signal Dynamics 2g Line of sight acceleration 55 Figure 3. Polar plot with locations of the interferers (J1 and J2) and the signal of interest (GPS satellite: SV PRN 12). The behavior of the adaptive array while implementing the MVDR and the MV approaches are compared in this section. For both approaches, the array processing algorithm is functional at all times throughout the 6 second run. The MVDR approach is considered first. Figure 4 shows the real and imaginary components of the complex weights of the individual array elements as a function of time. Figure 4 focuses on the antenna array weights as a function of time. It is seen from Figure 4 that prior to the injection of the first interferer (J1), the magnitude of the array weights is minimal. Once interference is incident upon the elements of the adaptive antenna array, its presence is sensed by a variation in the structure of the covariance matrix 𝑅. In turn, the array weights are driven by the adaptive algorithm to minimize the incident interference by placing a null in the array pattern. This is evidenced by the step change in element weights soon after the introduction of CW interference (J1) at the 4 second time mark. A similar change to the adaptive array weights (not shown in Figure 4) is seen after introduction of J2 into the SOE at 4.3 seconds into the scenario. 56 Figure 5 depicts the input to the adaptive array at the reference element (in gray). The reference element is chosen to be antenna element 1 (AE1). Superimposed on it is the output signal (in black). Figure 6 depicts a zoomed in version of Figure 5. Both figures depict input to and output from the adaptive array after injection of both interference sources (J1 and J2). The figures represent a 1 millisecond snapshot in time after both interferers (J1 and J2) have been introduced into the scenario. Weights Real part of the adaptive array weights AE1 0.5 AE2 AE3 AE4 0 -0.5 3.97 3.99 4 4.01 4.02 Time in Seconds Imaginary part of the adaptive array weights 1 Weights 3.98 0 -1 3.97 3.98 3.99 4 4.01 Time in Seconds 4.02 Unnormalized amplitude Figure 4. Real and imaginary parts of the adaptive array weights for the MVDR algorithm. “AE” refers to antenna element. 4 x 10 4 2 0 -2 -4 0 0.5 1 Time in milliseconds Figure 5. Signal input to and output from the adaptive array. Input to the adaptive array is shown in gray. Output from the adaptive array is shown in black. The MVDR algorithm is implemented here. 57 The high level of interference cancellation is evidenced in Figure 5 and Figure 6. For the given signal level, level of noise floor and ISR, the actual cancellation of J1 by the adaptive processor is about 50 dB (CW ISR = 65 dB and the GPS signal is approximately 12 to 15 dB below the thermal noise floor). Given sufficient degrees of freedom, the SAP algorithm will mitigate interference to within 3dB of the ambient noise floor of the GPS signal (Compton, 1988). It should be noted that the phase imposed on the desired GPS signal at the array's output by the MVDR algorithm before and after the injection of continuous wave (J1) and broadband (J2) interferers is typically zero and is a Unnormalized amplitude byproduct of the distortion less constraint imposed by the MVDR algorithm. 400 200 0 -200 -400 0 0.5 1 Time in milliseconds Figure 6. Signal input to and output from the adaptive array. Zoomed in version of Figure 5. The next sets of figures correspond to that of the MV algorithm. Figure 7 displays the real and imaginary components of the complex weights of the individual array elements as computed by the adaptive array algorithm. Figure 8 depicts the signal input to the adaptive array at the reference element (in gray). The output of the adaptive array is superimposed on this (in black). Figure 9 shows a zoomed-in version of Figure 8. 58 Figure 8 and Figure 9 represent a 1 millisecond snapshot in time after both interferers (J1 and J2) have been injected into the scenario. Figure 10 depicts the phase imposed on the array's output by the MV adaptive algorithm before and after the injection of J1 and J2 at two different points in the scenario. Comparing Figure 7 and Figure 4, it can be seen that the non-directional constraint utilized by the MV algorithm translates to the array weights on the reference element (AE1) being unity whereas the directional constraint utilized by the MVDR algorithm does not translate into a constraint on a single antenna element. The directional constraint in equation 21 imposes a constraint on the inner product of the weights and the Weights steering vector towards the desired signal. 1 Real part of the adaptive array weights 0.5 AE1 0 AE2 AE3 AE4 4 4.005 Time in Seconds Imaginary part of the adaptive array weights Weights 3.995 0.1 0 -0.1 3.97 AE1 3.98 AE2 3.99 4 4.01 Time in Seconds AE3 AE4 4.02 Figure 7. Real and imaginary parts of the adaptive array weights for the MV algorithm. “AE” refers to antenna element. Unnormalized amplitude 59 4 x 10 4 2 0 -2 -4 0 0.5 Time in milliseconds 1 Unnormalized amplitude Figure 8. Signal input to and output from the adaptive array. Input to the adaptive array is shown in gray. Output from the adaptive array is shown in black. The MV algorithm is implemented here. 400 200 0 -200 -400 0 0.5 Time in milliseconds 1 Figure 9. Signal input to and output from the adaptive array. Zoomed in version of Figure 8. Upon close inspection of Figure 10, it is seen that there are two instances where the phase of the array in the look direction varies. The first instance is at the 4 second time mark when the CW interferer (J1) is introduced. The second variation occurs at the 4.3 second time mark when the BB interferer (J2) is introduced. Phase in radians 60 0 -0.5 -1 0 2 4 6 Time in Seconds Figure 10. Phase of the array in the look direction using the MV algorithm. If the platform attitude is invariant, the phase in the look direction varies purely as a function of the change in the complex weights of the adaptive array. In Figure 10, the net phase imposed by the array on the incoming GPS signal varies by 1.2 radians in a very short span of time for the MV algorithm. This rapid phase variation in the adaptive array weights mechanism may impose phase step/s onto the GPS signal that is/are beyond the lock range of the baseband GPS receiver‟s carrier phase tracking loop. The MVDR algorithm does not undergo such rapid transitions in phase due to the distortion-less constraint imposed by the algorithm. In this case, the baseband carrier phase tracking is not affected. The next section will explore an approach to implement a phase compensation for the MV algorithm. The phase compensation enables the user to achieve baseband carrier phase tracking comparable to the MVDR approach by maintaining phase lock on the GPS signal. 61 Phase Compensation as Applied to the Minimum Variance Algorithm As seen in the previous section, the minimum variance (MV) algorithm causes the phase of the adaptive array mechanism in the desired signal's look direction to change abruptly. This may cause the baseband GPS carrier phase tracking loop to break lock. The phase variation imposed on the signal in the look direction is deterministic. The central idea is to account for the upstream variations in the phase of the GPS signal at the downstream baseband processing level. One suitable candidate for this approach is to effect a change of phase in the locally generated copy of the carrier in the baseband GPS receiver. Upon close inspection of Figure 10, it is seen that there are two instances where the phase of the array in the look direction varies. The first instance is at the 4 second time mark when the CW interferer (J1) is introduced. The second variation occurs at the 4.3 second time mark when the BB interferer (J2) is introduced. Closer integration of the adaptive array processing mechanism along with the baseband software GPS receiver enables the user to achieve the above mentioned goal. The current implementation of the adaptive array processing mechanism is entirely in software. As illustrated in Figure 11, adaptive array processing is followed by conventional baseband signal processing. The box labeled "A&D" in Figure 11 implies “Accumulate and Dump” and refers to the integrator in the baseband tracking loop. The 62 phase compensation technique discussed here is targeted at the sequential GPS receiver realization shown in Figure 11 where NCO stands for Numerically Controlled Oscillator. Figure 11. Block diagram of the adaptive array and baseband GPS receiver depicting a control and data path between the adaptive algorithm section and the baseband PLL. However, it is also possible to achieve the same objective using a GPS receiver operating in the batch processing mode. In the batch processing mode, the GPS receiver does not implement classical tracking loops. These receivers continually estimate the code and carrier phase of the GPS signal using transform domain processing (Gunawardena, 2007). In such a case, the deterministic phase step imposed by the adaptive array may be compensated in the carrier phase measurement mechanism. The batch processing approach is only limited by the sampling rates of the ADC (Analog to 63 Digital Converter). In the sequential GPS receiver, the baseband carrier phase tracking loop is implemented as a third order PLL. When the SOE varies, the weights applied by the adaptive array onto the incoming GPS signal will vary as a result of weights adaptation. These result in phase variations imposed on the GPS signal. As discussed earlier, in the MV approach, the array processing mechanism may impose a phase step on the array‟s output. This is due to the rapid convergence of the adaptive array weights. This, in turn, affects the GPS signal fed to the baseband receiver. Such a phase step would impact the PLL discriminator‟s output. The GPS signal component entering the baseband receiver is given by: H z (t )  w * X (t ) (25) The change in SOE could be due to an interferer that was not transmitting at a prior point in time or a non stationary interference component. However, the rate of convergence of the adaptive array algorithm would typically determine how the array's phase variation manifests itself onto the GPS signal. When the algorithm converges rapidly, the phase variation could show up as a sizeable step change in the phase of the GPS signal which could exceed the tracking range of the baseband PLL. In the current implementation, the adaptive antenna array is composed of 4 antenna elements mounted on a common ground plane. These antenna elements are 64 mounted within a half wavelength of each other and are susceptible to the effects of mutual coupling. Each element would typically have a unique frequency response across azimuth and elevation. This may be denoted by: 𝐻𝑘 𝑓, 𝐴𝑧, 𝐸𝑙 𝐺𝑘 (𝑓) (26) where, 𝐴𝑧 is the Azimuth, 𝐸𝑙 is the elevation, 𝑓 is the frequency parameter and, 𝑘 is an index to represent an individual antenna element ( 𝑘 = 1 𝑡𝑜 𝑁; 𝑁 = 4) 𝐻𝑘 𝑓, 𝐴𝑧, 𝐸𝑙 is the frequency response across 𝑓, 𝐴𝑧, 𝐸𝑙 for element 𝑘. 𝐺𝑘 (𝑓) is the frequency response of the receiver array's individual RF path leading from the 𝑘'th antenna element to the baseband digital processing. This frequency response is assumed to account for the effects of mutual coupling between the elements of the antenna array. For a signal impinging on the antenna array from a given 𝐴𝑧 and 𝐸𝑙 there is a deterministic geometric signal delay (or advance) from the reference element to the other elements of the array depending on the direction of arrival of the signal and the chosen reference antenna element. This is captured by the information in the steering vector 𝑆𝑙 . As mentioned earlier, steering vector information is not necessary for the purposes of unconstrained adaptive array processing but is used here for the purposes of modeling the geometric wave fronts of the received signals. 65 The desired signal at the baseband digital intermediate frequency (IF) output of the individual antenna elements may be given by: 𝐻𝑘 𝑓, 𝐴𝑧, 𝐸𝑙 𝐺𝑘 𝑓 𝑆𝑙,𝑘 (27) with 𝑆𝑙,𝑘 being the geometric steering vector for desired source 𝑙 at the 𝑘'th antenna element. This can be viewed as the signal out of the first block in Figure 1 (Down conversion and A/D sampling) and is the input to the adaptive array processor block. When there is a variation in the SOE as seen at the antenna array, the complex weights in the adaptive array mechanism will vary to minimize the power (variance) of the received signal. This will result in an updated set of weights used in the adaptive array processing. For the sake of clarity, let the weights used by the adaptive mechanism prior to a variation in the SOE at time 𝑡 be 𝑤𝑘,𝑡 . Let, 𝑤𝑘,𝑡 = 𝛼𝑘,𝑡 + 𝑗𝛽𝑘,𝑡 where, 𝛼𝑘,𝑡 is the real part of the array weight as seen at the 𝑘'th array element at time 𝑡. 𝛽𝑘,𝑡 is the imaginary part of the array weight as seen at the 𝑘'th array element at time 𝑡. (28) 66 Following the notation, wk ,t 1 refers to the adaptive array weight at time 𝑡 + 1 for antenna element 𝑘. The array weights may also be represented in polar form as: 𝑕𝑘,𝑡 𝑒 𝑖𝜓 𝑘,𝑡 (29) with, 𝛽𝑘,𝑡 𝛼𝑘,𝑡 (30) 𝑕𝑘,𝑡 = 𝑎𝑏𝑠(𝑤𝑘,𝑡 ) (31) 𝜓𝑘,𝑡 = tan−1 and, The objective of the phase compensation technique as applied to the MV algorithm is to make the baseband carrier tracking loop's phase discriminator mostly transparent to any deterministic phase variations that may occur as a result of the adaptive array weight adaptation upstream. It is to be noted that equation 27 represents a desired signal 𝑙 from a given direction (𝐴𝑧, 𝐸𝑙) at a given antenna element 𝑘. For the purpose of characterizing the adaptive algorithm's functionality, it is assumed that the interference is injected into the scenario between time 𝑡 and 𝑡 + 1. The representation of the desired signal in equation 27 is valid prior to (time 𝑡) and after (time 𝑡 + 1) incidence of interference. Given the high rate of array weights adaptation (1 kHz) and the fact that the line of sight vector from the satellite to the user does not vary by an appreciable amount during this time, equation 27 67 is assumed to be constant. Furthermore, equation 27 may be represented in the time domain by a net effective magnitude and phase given by: 𝑚𝑘 𝑒 𝑖𝜃 𝑘 (32) 𝑚𝑘 = 𝑎𝑏𝑠(𝐻𝑘 𝑓, 𝐴𝑧, 𝐸𝑙 𝐺𝑘 𝑓 𝑆𝑙,𝑘 ) (33) 𝜃𝑘 = 𝑎𝑟𝑔(𝐻𝑘 𝑓, 𝐴𝑧, 𝐸𝑙 𝐺𝑘 𝑓 𝑆𝑙,𝑘 ) (34) where, and, for a given antenna element 𝑘. Pursuant to the above stated assumption, equation 32 is invariant between time 𝑡 and 𝑡 + 1. At this point, the signal is multiplied by its respective antenna weight that was computed by the adaptive array algorithm. At time 𝑡, this is given by: 𝑕𝑘,𝑡 𝑒 −𝑖𝜓 𝑘 ,𝑡 𝑚𝑘 𝑒 𝑖𝜃 𝑘 (35) When the interferer is incident on the antenna array, the adaptive array algorithm senses a change in the structure of the computed covariance and changes its weights between time 𝑡 and 𝑡 + 1 to minimize the power of the interferer. This may result in an abrupt change in the phase of the adaptive array weights. The signal at time 𝑡 + 1 may be given by: 68 𝑕𝑘,𝑡+1 𝑒 −𝑖𝜓 𝑘 ,𝑡+1 𝑚𝑘 𝑒 𝑖𝜃 𝑘 (36) From equations 29, 32, 35 and 36, it is seen that change to the desired signal at the output of the adaptive array for element 𝑘 between time 𝑡 and 𝑡 + 1 is given by: (𝑕𝑘,𝑡+1 𝑒 −𝑖𝜓 𝑘 ,𝑡+1 − 𝑕𝑘,𝑡 𝑒 −𝑖𝜓 𝑘 ,𝑡 )𝑚𝑘 𝑒 𝑖𝜃 𝑘 (37) The effective change in the signal as seen at the baseband GPS receiver will be given by the sum of the changes as imposed by the individual antenna array elements and can be given by: 𝑁 (𝑕𝑘,𝑡+1 𝑒 −𝑖𝜓 𝑘 ,𝑡+1 − 𝑕𝑘,𝑡 𝑒 −𝑖𝜓 𝑘,𝑡 )𝑚𝑘 𝑒 𝑖𝜃 𝑘 (38) 𝑘=1 which, in turn, becomes: 𝑁 𝑁 𝑕𝑘,𝑡+1 𝑒 −𝑖𝜓 𝑘 ,𝑡+1 𝑚𝑘 𝑒 𝑖𝜃 𝑘 − 𝑘=1 𝑕𝑘,𝑡 𝑒 −𝑖𝜓 𝑘 ,𝑡 𝑚𝑘 𝑒 𝑖𝜃 𝑘 (39) 𝑘=1 The impact of the adaptive array on the baseband carrier tracking loop is a function of the relative phase change of the desired signal between time 𝑡 and 𝑡 + 1. It is this change in phase that needs to be captured to perform phase compensation in the baseband receiver. Here we make an approximation using the fact that 𝑚𝑘 𝑒 𝑖𝜃 𝑘 does not change appreciably between time 𝑡 and 𝑡 + 1. As a result, the effective phase change imposed upon the signal will be given by a change in the phase (Δζ) of the weights used at time 𝑡 versus time 𝑡 + 1. This is given by: 69 𝑁 𝑁 𝑕𝑘,𝑡+1 𝑒 −𝑖𝜓 𝑘,𝑡+1 − 𝑎𝑟𝑔⁡ Δζ = 𝑎𝑟𝑔 𝑘=1 𝑕𝑘,𝑡 𝑒 −𝑖𝜓 𝑘 ,𝑡 (40) 𝑘=1 This information (Δ𝜁) is conveyed to the carrier loop NCO across the control/data path in Figure 11 (depicted by the arrow). In the baseband carrier tracking loop, the NCO generates a local copy of the carrier that the NCO "expects" to match with the incoming carrier signal. If there has been a change to the signal phase as a result of the adaptive array mechanism upstream, the locally generated signal will not match the carrier signal that arrives at the baseband receiver. This abrupt change in the phase may be severe enough to lose lock on the carrier phase of the signal. To address this issue, the NCO utilizes the information Δ𝜁 and tunes the carrier NCO to generate a local "matched" copy of the incoming signal that has been affected by a phase change imposed upon it by the adaptive processing mechanism. The size of the carrier NCO tuning is Δ𝜁 translated to units of the NCO step size based on the given sampling rate and number of bits of NCO resolution used in the baseband carrier tracking loop. This information may also be used to compensate for the accumulated delta range error induced as a result of the carrier NCO tuning to account for the phase change imposed on the signal by the adaptive array processing. However, the accuracy of the range domain compensation is yet to be evaluated. The control/data path in Figure 11 may also carry information from the baseband tracking loops to the adaptive array mechanism. Information sent over this path may be used to control the rate of adaptation in the adaptive array processor. This can be 70 beneficial when there is excessive jitter introduced onto the phase of the signal as a result of the rapid phase variations imposed by the adaptive array mechanism. This aspect of adaptive array control is not addressed in this effort. The feed forward phase compensation implementation discussed here is validated using simulated data whose key parameters have been described in Table 3. Implementation Results The results shown in this section pertain to the cases of CW (J1) and BB (J2) interferers injected into the scenario at two different points in time as described in Table 3. As observed in Table 3, the setup introduces the first source of interference into the scenario four seconds after the start of the simulation. Figure 3 depicts the locations of the interferers J1 (CW interferer) and J2 (Broadband interferer) with respect to the satellite of interest (SV PRN 12). The interference mitigation achieved by the MV and MVDR algorithms has been covered in earlier sections (Figure 5, Figure 6, Figure 8, Figure 9). This section focuses on the impact of adaptive array processing on the baseband GPS tracking loops and the phase compensation mechanism used to alleviate this impact for the case of the MV adaptive array algorithm. The plots in this section depict the performance of the code and carrier phase tracking loops along with the corresponding discriminator outputs (Ward, 71 2006), code and carrier lock indicators (Van Dierendonck, 1996), estimates of the carrier to noise ratio (Van Dierendonck, 1996) and carrier loop filter accelerations. These tracking loop parameters are excellent indicators of the impact of different adaptive array processing mechanisms on the baseband GPS signal. Without loss of generality, a perfect receiver clock is assumed in these simulations. The first set of plots depicts the baseband receiver performance when an MVDR approach to array signal processing is used. The following equation represents the computation of the code lock indicator (Van Dierendonck, 1996): 𝐶 1 (𝜇𝑁𝑃 − 1) = 10 ∗ 𝑙𝑜𝑔10 𝑁𝑜 𝑇 𝑀 − 𝜇𝑁𝑃 (41) where, 𝐶 𝑁𝑜 is the carrier to noise ratio, 𝜇𝑁𝑃 is the lock detector measurement, 𝑇 is the predetection integration interval for the wideband power and, 𝑀 is the number of coherent PDI's multiples of 𝑇 used to compute the narrowband power. The carrier lock detector is given by (Van Dierendonck, 1996): 𝐶2∅𝑘 = 𝑁𝐵𝐷𝐾 𝑁𝐵𝑃𝐾 where, 𝑁𝐵𝐷𝐾 is the difference between the narrow band powers as seen in the in-phase (42) 72 and quadri-phase components of the signal. 𝑁𝐵𝑃𝐾 is sum of the narrowband powers as seen in the in-phase and quadri-phase components of the signal and, 𝐶2∅𝑘 is the normalized estimate of the cosine of twice the carrier phase. Figure 12 and Figure 13 depict the carrier and code lock indicators for the baseband receiver processing. The carrier lock indicator ranges between -1 and 1 while the code lock indicator ranges between 0 and 20 (Van Dierendonck, 1996). At high CNR values, these indicators are close to or at their maximum limit implying a good lock on the respective components of the GPS signal. The code lock indicator is based on comparing the narrowband power to the wideband power where the wideband power is computed based on 1millisecond in-phase (I) and quadri-phase (Q) accumulations and the narrow band power is computed based on 20 millisecond I and Q accumulations. The carrier lock indicator is based on computing the difference in narrow band power between the I and Q accumulations. At the CNR ranges shown in this scenario, under nominal tracking conditions, the lock indicators would stay in the range of 18-20 for code lock and 0.9 - 1.0 for carrier lock. The vertical lines in the figures indicate the points in time when interferers (J1 and J2) were introduced into the scenario. The first vertical black line corresponds to the introduction of a CW interferer (J1) and the second (dotted vertical) black line corresponds to the introduction of the BB interferer (J2). The code lock indicator has a marginal drop in value after introduction of CW interference while the carrier lock indicator recaptures its peak value. The carrier lock indicator‟s 73 performance is attributed to the phase constraint imposed upon the signal in the look direction by the MVDR algorithm. Figure 14 and Figure 15 depict the carrier and code discriminator outputs respectively. As seen from these figures, the discriminator outputs are noisier after introduction of interference. This also results in a drop in C/No as shown Carrier Lock indicator in Figure 16. 1 0.5 0 -0.5 -1 0 Code Lock indicator 2 4 6 Time in seconds Figure 12. Baseband GPS receiver carrier lock indicator (Adaptive array algorithm – MVDR). 20 15 10 5 0 0 2 4 6 Time in seconds Figure 13. Baseband GPS receiver code lock indicator (Adaptive array algorithm – MVDR). Carrier Discriminator Output 74 0.2 0.1 0 -0.1 -0.2 0 2 4 Time in seconds 6 Code Discriminator Output Figure 14. Phase discriminator output in units of cycles (Adaptive array algorithm – MVDR). 0.2 0.1 0 -0.1 -0.2 0 Carrier to Noise ratio (dB-Hz) 2 4 6 Time in seconds Figure 15. Normalized code discriminator output (Adaptive array algorithm – MVDR). 60 40 20 0 0 2 4 Time in seconds 6 Figure 16. Carrier to noise ratio (C/No) indicator. (Adaptive array algorithm – MVDR). 75 The C/No indicator updates at a prescribed rate (50 Hz) and is averaged over 100 milliseconds. As a result, injection of interference into the scenario at the four second mark does not manifest as a lowered C/No estimate instantaneously. Figure 17 shows the early, prompt and late power levels from the three sets of complex correlators that are used to track the GPS code and carrier phase. A temporary change in the power levels can be observed when the interferers are injected into the scenario. However, soon after the adaptive array‟s weights converge, the GPS signal‟s power levels are back to approximately the same levels as they were prior to the injection of interference. This is unique to constrained adaptive processing and is ascribed to the unity constraint imposed upon the desired GPS signal arriving from the look direction of the antenna array. In addition, the computed C/No before and after the injection of interference is in contrast to what is seen Figure 17 where the early, prompt and late correlator powers hold at the same level prior to the injection on interference. This is a byproduct of the antenna array having to maintain the required phase and gain response in the direction of the desired signal while, at the same time, devote its remaining degrees of freedom to suppressing interferers J1 and J2 simultaneously. A four element antenna array that performs Spatial Adaptive Processing (SAP) has three degrees of freedom (Compton, 1988). All three array degrees of freedom are used by the MVDR algorithm in this scenario. Effectively, the array has used one degree of freedom towards the MVDR constraint while using the other two to point nulls towards the directions of the 76 interferers. As a result, the array is unable to form deep nulls towards the interferers and lets some of the interfering energy past the adaptive array processor. This is reflected in the computed C/No in Figure 16. It is also evidenced by the relative noisiness in the early, prompt and late correlator power levels before and after the injection of the CW and broadband interferers. It is to be noted that small mismatches between the actual direction of arrival of the GPS signal and the look direction of the array do not lead to significant errors in the GPS signal's carrier phase measurements (Fante & Vaccaro, 2000). The MVDR performance characterization shall serve as a baseline to evaluate the MV algorithm which does not apply any directional constraints. Early power 11 4 x 10 2 0 0 1 2 3 Prompt power 4 5 6 1 2 3 Late power 4 5 6 1 2 3 4 Time in seconds 5 6 12 2 x 10 1 0 0 11 10 x 10 5 0 0 Figure 17. Early, prompt and late power out of the correlators. (Adaptive array algorithm – MVDR). Figure 18 depicts the baseband carrier loop filter acceleration. The dynamics introduced in the scenario is on the order of -100 cycles/s/s. This translates to -2g‟s of LOS acceleration onto the incoming GPS signal. From the above plots, it is observed that 77 the injection of interference at the 4 and 4.3 second time marks in the scenario does not cause a problem with the phase tracking. The next set of plots shall demonstrate the results of the baseband carrier tracking when the MV algorithm is used and no phase Carrier Acceleration (in Hz/s) compensation is applied to the baseband GPS signal. 0 -100 -200 -300 0 2 4 Time in seconds 6 Figure 18. Carrier loop filter acceleration. (Adaptive array algorithm – MVDR). As seen in Figure 19 and Figure 20, the injection of CW interference at the 4 Carrier Lock indicator second mark causes the carrier and code lock indicators to drop. 1 0.5 0 0 2 4 6 Time in seconds Figure 19. Baseband GPS receiver carrier lock indicator (Adaptive array algorithm – MV). 78 The current baseband software receiver is configured to break out of the code tracking/carrier reacquisition mechanism when carrier phase lock is lost. This is done in order to better observe the loss of lock phenomenon associated with the variation in phase Code Lock indicator of the GPS signal imposed by the MV algorithm. 20 15 10 5 0 0 2 4 6 Time in seconds Figure 20. Baseband GPS receiver code lock indicator (Adaptive array algorithm – MV). As seen in Figure 21, the carrier phase discriminator registers an instantaneous spike in its output at the 4 second mark. This is in contrast to the previous case when the baseband receiver was tracking the output of the MVDR algorithm. Figure 22 depicts the output of the code phase discriminator. The baseband GPS receiver employs a carrier aided code tracking loop. Figure 23, Figure 24 and Figure 25 depict the C/No, correlator powers and carrier loop filter acceleration respectively. Carrier Discriminator Output 79 0.2 0.1 0 -0.1 -0.2 0 2 4 Time in seconds 6 Code Discriminator Output Figure 21. Phase discriminator output in units of cycles (Adaptive array algorithm – MV). 0.1 0 -0.1 -0.2 0 Carrier to Noise ratio (dB-Hz) 2 4 6 Time in seconds Figure 22. Normalized code discriminator output (Adaptive array algorithm – MV). 60 40 20 0 0 2 4 Time in seconds 6 Figure 23. Carrier to noise ratio (C/No) indicator. (Adaptive array algorithm – MV) 80 Late power Prompt power Early power 11 2 x 10 1 0 0 11 x 10 10 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 Time in seconds 4 5 6 5 0 0 10 x 10 10 5 0 0 Carrier Acceleration (in Hz/s) Figure 24. Early, prompt and late power out of the correlators. (Adaptive array algorithm – MV). 100 0 -100 -200 -300 0 2 4 Time in seconds 6 Figure 25. Carrier loop filter acceleration. (Adaptive array algorithm – MV). From Figure 20 through Figure 25, it is seen that the baseband tracking loop loses carrier phase lock on the GPS signal. Comparing Figure 17 and Figure 24 it is seen that the MV approach, unlike the MVDR approach, loses all signal power soon after CW interference (J1) is injected into the scenario. This injection of interference causes a rapid change in the weights of the array which in turn causes an abrupt change in the phase of the GPS signal. This change in phase causes the loss of lock condition in the PLL. 81 The next set of the figures are for the MV approach along with the phase compensation technique applied to the baseband GPS receiver processing. Figure 26 and Figure 27 depict the receiver carrier and code lock indicators respectively. Figure 28 and Figure 29 depict the carrier and code discriminator outputs of the phase compensated baseband receiver processing. Figure 30 to Figure 32 show the C/No measurements, early, prompt, late correlator powers and the carrier loop filter acceleration outputs of the phase compensated baseband GPS receiver processing. It is shown that the phase compensation mechanism aids the carrier loop and helps avert the loss of carrier phase Carrier Lock indicator lock. 1 0.5 0 0 2 4 6 Time in seconds Figure 26. Phase compensated baseband GPS receiver carrier lock indicator (Adaptive array algorithm – MV). Code Lock indicator 82 20 15 10 5 0 0 2 4 Time in seconds 6 Carrier Discriminator Output Figure 27. Phase compensated baseband GPS receiver code lock indicator (Adaptive array algorithm – MV). 0.2 0.1 0 -0.1 -0.2 0 2 4 Time in seconds 6 Code Discriminator Output Figure 28. Phase compensated carrier phase discriminator output in units of cycles (Adaptive array algorithm – MV). 0.2 0.1 0 -0.1 -0.2 0 2 4 6 Time in seconds Figure 29. Phase compensated normalized code discriminator output (Adaptive array algorithm – MV). Carrier to Noise ratio (dB-Hz) 83 60 40 20 0 0 2 4 Time in seconds 6 Late power Prompt power Early power Figure 30. Carrier to noise ratio (C/No) indicator (Adaptive array algorithm – MV). Phase compensation is applied. 2 x 10 11 1 0 0 1 11 x 10 10 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 5 0 0 1 11 x 10 2 1 0 0 1 Figure 31. Early, prompt and later power out of the correlators (Adaptive array algorithm – MV). Phase compensation is applied. Carrier Acceleration (in Hz/s) 84 0 -100 -200 -300 0 2 4 Time in seconds 6 Figure 32. Carrier loop filter acceleration (Adaptive array algorithm – MV). Phase compensation is applied. From these figures, it is observed that the phase compensation method limits the output of the carrier phase discriminator hence limiting loop excursions. From Figure 17 and Figure 31, it is seen that there is a reduction in the correlator signal powers for the MV phase compensated approach as against that of the MVDR approach. This is due to the absence of a constraint upon the direction of arrival of the desired GPS signal. After injecting J2 into this scenario the MV algorithm utilizes all of its degrees of freedom to maintain the unity constraint on AE1 and mitigate the impact of J1 and J2 on the desired GPS signal (SV PRN 12). From Figure 25 and Figure 32, it is observed that excursions in the carrier loop accelerations were limited by the phase compensated MV approach in comparison to the uncompensated MV approach. The phase compensation approach enables maintaining carrier phase lock on the desired GPS signal. Phase compensation can be applied to each satellite signal's baseband tracking loop mechanism based on the 85 knowledge of the adaptive array weight variations. However, when phase compensation is applied , the delta range measurements cannot be guaranteed to be representative of the true delta range. It is possible to compensate the delta range measurements for the equivalent change in the phase as a result of the phase compensation. It is to be noted that this range domain compensation is not necessarily a true representation of the actual delta range variation due to phase compensation. 86 CHAPTER 5: MULTIPATH CHARACTERIZATION GPS carrier-phase measurements are subject to a range of error sources and noise. Most of the systematic biases such as tropospheric and ionospheric delay can be removed by differential GPS (DGPS) techniques. Multipath error on the carrier-phase measurement, however, does not correlate between antenna locations. Carrier multipath is very specific to the antenna phase center location with respect to the surrounding environment and the satellite-receiver antenna phase center geometry. As a result, carrierphase multipath is a major impediment to high accuracy differential techniques (Kalyanaraman & Braasch, 2003). In the absence of multipath, the receiver is able to track the direct incoming signal phase. However, in the presence of multipath, the tracking loops in the GPS receiver follow the composite incoming signal (direct plus multipath). In addition, the input to the correlation process is the composite signal rather than the desired direct component. The receiver‟s tracking loops are unable to differentiate between the direct and composite signals. They employ the concept of null tracking which yields a non-zero error in the estimate of the direct signal time-of-arrival and carrier phase (Kalyanaraman, Kelly, Braasch, & Kacirek, 2004). Various methods have been used to characterize multipath. The interaction of the direct and reflected signals (specular reflections) can be analyzed in terms of correlation 87 functions. The ideal correlation function is scaled in amplitude and shifted in time to represent the multipath component of the signal. Care is taken to ensure that the phase of the multipath signal (with respect to the direct) is also preserved in this process (Kalyanaraman, Braasch, & Kelly, 2006) . It should be noted, for this analysis, the correlation function side lobes are assumed to be zero. The impact of non-zero side lobes has been described in (Braasch & DiBenedetto, 2001). This section details the work performed in the areas of carrier phase multipath validation using coherent and noncoherent discriminator architectures for code tracking. In addition, validation of code multipath error characteristics under conditions of fading multipath has been performed. The work presented in the following section has been published in (Kalyanaraman et al., 2006). Carrier-phase multipath can be studied by analyzing a simple phasor diagram as seen in Figure 33 (Braasch & Van Dierendonck, 1999). Without loss of generality, the direct signal is assumed to have zero-phase. The phase of the multipath with respect to the direct is given by the angle  m . This representation allows one to resolve the multipath phasor into its in-phase (along the direction of the direct signal) and quadrature components. 88 The magnitudes of the direct and multipath phasors are given by: 𝐷 = 𝑅(𝜏𝑐 ) (43) 𝑀 = 𝛼𝑅(𝜏𝑐 − 𝛿) (44) where, R(  c ) is the correlation function for the PRN code at time lag 𝜏𝑐 ,  is the delay of the multipath with respect to the direct and  is the ratio of multipath-to-direct signal (otherwise referred to as the multipath-to-direct ratio: M/D).  c is the code tracking error of the DLL (delay locked loop). In order to determine the mathematical model for the composite phase,  c , the multipath is decomposed into its in-phase component, M I = M cos m  , and quadrature component, M Q = M sin  m  . The correlation values used above are from the prompt correlator. Having resolved the multipath into its components (Figure 34), it is now possible to determine the phase relation between the composite vector (direct plus multipath) and the direct signal. This quantity (represented in radians or fractions of a wavelength) is the error due to carrier-phase multipath, assuming that the receiver‟s carrier tracking loop is perfectly locked onto θc. Utilizing the arc tan relation in order to obtain the composite phase (similar to the manner in which the phase of a complex quantity is estimated), carrier-phase multipath error is given by: 89 θc = tan−1 𝑀𝑄 𝑀𝐼 + 𝐷 Composite Signal (45) Multipath c m Direct Figure 33. Phasor diagram depicts the direct and reflected components along with the phase relation between the direct signal, the multipath θm and the phase tracking error θc of the composite signal (Braasch & Van Dierendonck, 1999). C MQ c D MI Figure 34. Decomposition of the Phasor diagram gives the angle θc subtended by the composite signal with respect to the direct. This is the carrier phase tracking error (Braasch & Van Dierendonck, 1999). Rewriting equation 45 using equations 43 and 44 (Braasch, 1996), 90 θc = tan−1 αR τc − δ sin θm R(τc) + αR τc − δ cos θm (46) Where, θm = phase of the multipath with respect to the direct, θc = composite phase tracking error, τc = code tracking error in chips, and δ = delay of the multipath with respect to the direct. For M/D≤ 0dB, the maximum possible phase-tracking error is 𝑝𝑖/2 radians or a quarter wavelength. At the GPS L1 frequency this is approximately 4.8 cm (Braasch & Van Dierendonck, 1999). From equation 46, it can be seen that the magnitude of this carrier phase multipath error is independent of the choice of frequency (L1/L2). It is a function of M/D ratio and phase of the multipath with respect to the direct signal. This expression for the phase error as derived earlier is a complete characterization of carrier-phase multipath. The following figures apply to the case of infinite signal bandwidth in the receiver, implying that R(.) is a perfect triangular autocorrelation function. For the purpose of model validation, the characteristic curves obtained while simulating the finite bandwidth case do not depart significantly from the infinite bandwidth case (Brodin, 1996). The term τc in equation 46 is determined by generating the multipath-distorted discriminator curve and finding the difference between 91 the zero crossing that would be tracked by the Delay locked loop (DLL) in a GPS receiver, and the zero crossing of the corresponding error-free discriminator curve. In Figure 35 through Figure 38 the carrier-phase error envelopes for the coherent and the non-coherent code tracking modes are characterized through simulation. It is to be noted that while the code tracking is performed in two different modes (coherent and non-coherent), phase lock is maintained on the composite signal by a carrier PLL in both modes. The equation used to generate the code loop detector output for the coherent DLL in the simulation is: Delay  I E  I L (47) Where, 𝐼𝐸 is the in phase value from the early correlator, and 𝐼𝐿 is the in phase value from the late correlator The normalization of the correlation function in the simulation rendered any normalization by the in-phase value of the prompt correlator superfluous. The code loop detector for the non-coherent DLL is: 𝐷𝑒𝑙𝑎𝑦 = 𝐸𝑎𝑟𝑙𝑦 𝑃𝑜𝑤𝑒𝑟 − 𝐿𝑎𝑡𝑒 𝑃𝑜𝑤𝑒𝑟 (48) 92 With, Early _ Power being the total power in the early component of the correlation function, Late _ Power being the total power in the late component of the correlation function. The computer based simulation implements a slightly different version of the code detector as compared to the real NovAtel OEM3 (Original Equipment Manufacturer) and OEM4 hardware used to obtain bench test data to help validate the theory. The OEM3 hardware implements a non-coherent dot product discriminator using the in-phase values and quad-phase values of the early, prompt and late correlators in the receiver hardware. The OEM4 hardware was used to implement a coherent discriminator using only the in-phase values of the Early, Prompt and Late correlators. However, the results obtained in this section are valid for other implementations of the coherent and non-coherent code loop detectors. In the plots to follow, a simple scheme is used to differentiate between the coherent (solid line) and non-coherent (dotted line) cases. The envelope with the solid lines (inner envelopes) represents the maximum and minimum carrier-phase multipath errors for the coherent code-tracking mode. Similarly, the dotted lines (outer envelopes) represent the maximum and minimum carrier-phase multipath errors for the non-coherent code-tracking mode. The term “envelope” implies that the upper curve represents the maximum error bound and the lower curve represents the minimum error bound. The standard correlator chip spacing (1 chip spacing) was modeled in Figure 35 and Figure 93 36. Figure 35 demonstrates the carrier-phase multipath error envelope obtained for an M/D of –3dB. Figure 36 displays the results for an M/D of –10dB. It should be noted that in both cases (i.e. coherent and non-coherent), envelopes of the carrier-phase multipath error do not decrease uniformly with multipath delay. The slope of the envelope changes at approximately 0.7 chip. In addition, this anomalous characteristic at medium delay multipath is clearly seen only over the range of M/D‟s where the strength of the multipath is a sizeable fraction of the direct (i.e. strong multipath environment). Figure 35. Carrier phase multipath error envelope versus relative multipath delay for the standard correlator spacing (1.0 chip) using equation 46, M/D = -3dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. Comparing Figure 35 and Figure 36, it is also apparent that the error envelope is wider (i.e. occupies a slightly larger span of multipath delay) for higher M/D ratios. 94 Figure 37 depicts the carrier-phase error envelope for the narrow correlator (0.1 chip spacing) receiver architecture. The M/D ratio in this case is –3dB. It can be observed that the medium delay characteristic of the carrier phase error envelope is markedly different from the case of the standard correlator (Kalyanaraman et al., 2004). Specifically, the error envelope decreases more or less uniformly with multipath delay. The result for an M/D ratio of –10dB is shown in Figure 38. Figure 36. Carrier phase multipath error envelope versus relative multipath delay for the standard correlator spacing (1.0 chip) using equation 46, M/D = -10dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. It is to be noted that the carrier multipath envelopes depart from each other as a function of M/D ratio and correlator chip spacing. This departure is greater for higher M/D ratios and larger correlator spacings. 95 Figure 37. Carrier phase multipath error envelope versus relative multipath delay for the narrow correlator spacing (0.1 chip) using equation 46, M/D = -3dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. From the simulation results in Figure 35, Figure 36, Figure 37 and Figure 38 it is clear that the maximum carrier-phase multipath error for the strong multipath case departs from that of the weak multipath case. This is especially the case for the wide correlator architecture (1 chip spacing) as seen in Figure 35 and Figure 36. The departure is less obvious in the case of the narrow correlator spacing (0.1 chip) architecture. The coherent and non-coherent cases have almost identical carrier-phase error envelopes across the range of M/D ratios. 96 Figure 38. Carrier phase multipath error envelope versus relative multipath delay for the narrow correlator spacing (0.1 chip) using equation 46, M/D = -10dB. The figure depicts the envelopes obtained for coherent (solid line) and non-coherent (dotted line) code tracking modes. Simplified Models Previous researchers have developed simplified models to characterize carrierphase multipath error. This section describes the limitations of these models. Equation 49 characterizes the maximum carrier-phase multipath error (Ray, 2000) as a function of relative amplitude of the multipath, code-tracking error and time delay of the multipath relative to the direct. This equation for the maximum carrier-phase multipath is: 𝜃𝑐 = sin−1 𝛼𝑅(𝜏𝑐 − 𝛿) 𝑅(𝜏𝑐 ) (49) This equation is valid when the multipath component of the signal is orthogonal to the composite signal. Simulating this equation involves estimation of the code tracking error and the phase of the multipath with respect to the direct that orients the multipath orthogonal to the composite. In order to find the code tracking error, knowledge of the 97 multipath parameters (amplitude, delay and phase) is essential. Of these parameters, the simulation sets the amplitude and multipath delay. At this point, the phase of the multipath that maximizes the composite phase error is needed in order to obtain the corresponding code tracking error. The requirement that the multipath should be orthogonal to the composite poses a restriction on the phase of the multipath with respect to the direct. However, this does not give the actual phase of the multipath (with respect to the direct) that meets the orthogonality requirement. In fact, the phase of the multipath with respect to the direct can be found only if the code tracking error is computed. The above argument sheds some light onto the implementation complexities involving the simulation of equation 49. Simplifications to equation 49 for maximum carrier phase multipath have been published (Brodin, 1996). In certain circumstances (i.e. narrow correlator) the code tracking error can be considered negligible. It is observed that the maximum carrier phase multipath error is obtained when this code tracking error goes to zero. As a result, 𝜏𝑐 in equation 49 becomes zero. Exploiting the symmetric nature of the correlation function, the maximum carrier-phase multipath error is given by equation 50. Implementing equation 50 in simulation yields envelopes for the maximum carrier-phase multipath error as seen in Figure 39. The M/D ratios are –3dB and -10dB. The multipath error envelope decreases uniformly with relative multipath delay. 98 𝜃𝑐 = sin−1 𝛼𝑅(𝛿) (50) Figure 39. Carrier phase multipath error envelope versus relative multipath delay using equation 50, M/D = -3dB and -10dB. A closer look at Figure 39 would reveal that it is monotonic in nature and does not reveal any undulations in its envelope as a function of the correlator spacing and M/D ratio. It does not exhibit the change in slope at medium delays, as predicted by equation 46 and seen in Figure 35 and Figure 36. This is due to the simplified nature of the model used in equation 50. Figure 40 depicts the theoretical comparison between the envelopes derived using the comprehensive multipath model and the simplified model for an M/D ratio of -3dB. There is a noticeable difference between the maximum multipath errors across these two 99 envelopes at medium multipath delays. The true nature of the carrier phase multipath error is validated using bench test data. Carrier-phase error in meters 0.03 0.02 Complete Model 0.01 0 Simplified Model -0.01 -0.02 -0.03 0 0.2 0.4 0.6 0.8 1 Delay in chips 1.2 1.4 Figure 40. Carrier phase multipath error envelope versus relative multipath delay using equations 46 and 50, M/D = -3dB. The dotted lines represent equation 46 while the solid line represents equation 50. Bench Test Setup The first set of tests was performed to validate carrier-phase multipath theory for the non-coherent code tracking architecture. Zeta Associates Inc. performed bench tests and the results were provided to Ohio University for validation of theoretical multipath (Shallberg, 2003). The next set of tests to validate carrier-phase multipath theory for the coherent code tracking architecture was performed by Ohio University at Honeywell Corporation‟s Aerospace and Electronic Systems (AES) division at Olathe, Kansas. The fact that both sets of tests were not performed under the exact same conditions resulted in some minor differences between them. The theoretical basis behind the setups was the same. The setup to validate carrier-phase multipath for the non-coherent code tracking 100 architecture will be explained first. This is followed by the setup to validate carrier-phase multipath for the coherent code tracking architecture. Non-Coherent Code Tracking Architecture A multi-channel GPS hardware simulator (Spirent/ Northern Telecom STR2760) was used to generate a single multipath ray with specific amplitude and delay. Data was collected using a NovAtel OEM3 GPS receiver and included the time of measurement, pseudorange and carrier-phase. A special firmware load allowed the OEM3 receiver to operate in narrow and wide correlator tracking modes. This receiver employs a noncoherent code-tracking loop of the dot product type. Zero Doppler signals were produced by using a satellite simulated in a circular, zero inclination (equatorial) orbit. In a benchtest environment, different sources of error as seen on the GPS signal can be turned off or on (except thermal noise). In this case, all sources of error except multipath were turned off. The laboratory setup for the multipath tests consisted of a multi-channel GPS signal simulator, preamplifier, receiver and a computer to log the signal strength, pseudorange, and phase data at a 1 Hz rate. One channel of the GPS signal simulator was configured to generate a direct path PRN-1 satellite signal at -130 dBm. A second channel generated a multipath signal with the same PRN code. This multipath signal was generated at a lower signal level compared to the direct and had a variable delay relative to the direct path signal (Cox, Shallberg, & Manz, 1999). 101 Both standard (1-chip) and narrow (0.1-chip) correlator receivers were subjected to the same multipath scenarios. The tests were run for different multipath to direct power ratios (-2, -4, -7 and –10dB). Multipath delay for each test was increased in fractional steps of a wavelength. The wavelength steps were λL1, λL1*1.125, λL1*1.25, λL1*1.375, λL1*1.5, λL1*28, λL1*28.125, λL1*28.25…,λL1*2674.375, λL1*2674.5. This scenario uses a delay step size that samples in-phase, out-of-phase and at three intermediate points. In order to allow receiver pseudorange processing to settle at each multipath delay, the delay value was kept constant for a period of 20 seconds before changing to the next value (30 second dwell after a major step). In addition to simulating a satellite with multipath (PRN-1), a second satellite was simulated with a range rate of 100m/sec (PRN-2). The carrier-phase data from this second satellite is used to remove simulator/receiver trends from the carrier-phase measurements on PRN-1 using the following expression 𝑃𝑅𝑁1𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = 𝑃𝑅𝑁1𝑐𝑎𝑟 − (𝑃𝑅𝑁2𝑐𝑎𝑟 + 𝑇 ∗ 100 ) 𝐿1𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡 𝑕 where, PRN1_corrected is the corrected carrier phase on PRN 1 in L1 cycles, PRN1_car is the uncorrected carrier phase on PRN1 in L1 cycles, (51) 102 PRN2_car is the uncorrected carrier phase on PRN2 in L1 cycles, T is the time interval between successive measurements, L1_wavelength is the GPS carrier wavelength at L1 frequency. The receiver and simulator were phase locked to the same frequency reference. Data logged by the NovAtel receiver was in a proprietary format. This was converted to ASCII using NovAtel‟s file conversion utility. Data with M/D ratios of –2dB and –10dB and different correlator spacing (standard and narrow) is used for the bench test comparison with the simulation results in order to capture the effects of strong and weak multipath. Coherent Code Tracking Architecture Bench tests were performed at the Aerospace Electronics Division of Honeywell at Olathe, Kansas in order to validate carrier-phase multipath theory for coherent code tracking receivers. A multi-channel GPS hardware simulator (Spirent STR4760) was used to generate a single multipath ray with specific amplitude and delay. Data was collected using a NovAtel OEM4 GPS receiver and included the time of measurement, pseudorange and accumulated carrier-phase. A special firmware load allowed the OEM4 receiver to operate in narrow and wide correlator tracking modes. This special firmware load employed a coherent code-tracking loop. In the previous bench setup an OEM3 receiver was used to collect data to validate the theory for the case of the non-coherent code tracking loop. In a bench-test environment, different sources of error as seen on the 103 GPS signal can be turned off or on (except inherent hardware error sources). Two PRN‟s with minimal cross correlation were used in the multipath testing procedure. The satellites transmitting these PRN‟s were co-located in a geo-stationary orbit with the static user located at zero latitude, zero longitude and zero height. As a result zero Doppler signals were produced. A second channel generated a multipath signal with the same PRN code. This multipath signal was generated at a lower signal level compared to the direct and had a variable delay relative to the direct path signal. Both standard (1-chip) and narrow (0.1-chip) correlator receivers were subjected to the same multipath scenarios. The tests were run for different multipath-to-direct ratios (-3 and –10dB). Multipath delay for each test was increased in fractional steps of a wavelength. The wavelength steps were λL1, λL1*1.125, λL1*1.25, λL1*1.375, λL1*1.5, λL1*40, λL1*40.125, λL1*40.25. This scenario uses a delay step size that samples in-phase, out-of-phase and at three intermediate points every 40 carrier cycles, corresponding to a resolution in the delay axis of approximately 7.6 meters. In order to allow receiver pseudorange processing to settle at each multipath delay, the delay value was kept constant for a period of 35 seconds before changing to the next value (60 second dwell after a major step). 104 A second satellite in the simulation was co-located in geo-stationary orbit with PRN-10 and transmitted PRN-20. It serves as a reference since it was not corrupted by multipath. The single difference of the receiver measurements made on these two satellites will remove any common error effects as seen on the GPS signal (i.e. atmospheric errors (if any) and common clock biases). The cross-correlation functions of GPS PRN codes have 3-levels (63, -1 and –65 relative to a peak autocorrelation value of 1023). However, there are combinations of PRN codes that have better cross correlation properties than others. For example, there are PRN combinations in which the crosscorrelation function near a relative offset (τ) of zero are dominated by –1‟s rather than 63‟s and –65‟s. PRN's 10 and 20 have very low cross correlation side lobes, (i.e., -1‟s) and were chosen for this reason. Binary data were logged using the NovAtel receiver in its proprietary format. This was converted to ASCII using NovAtel‟s file conversion utility. Data with M/D ratios of –3dB and –10dB and different correlator spacing (standard and narrow) are used for the bench test comparison with the simulation results in order to capture the effects of strong and weak multipath. Data Analysis and Validation Data collected through bench testing under controlled conditions is compared to the theoretical error envelopes depicted in Figure 35 through Figure 38. Such data 105 proves to be extremely useful in validating the theoretical characterization of GPS carrier-phase multipath. A single multipath ray was injected from the GPS hardware signal simulator and the receiver‟s pseudorange and carrier-phase measurements were logged to a file. Given zero Doppler in the configuration between GPS signal simulator and the receiver, there should not be any change to the measured pseudoranges and carrier-phase measurements over time. Additional simulator/receiver trends were removed based on the measurements made on a second PRN that was transmitted without multipath added to it. As a result, any variation in the GPS carrier-phase measurements that are seen over the period of data collection is purely due to noise and multipath. The noise on the carrier phase measurement is in the millimeter range while the multipath error is in the centimeter range. Non-Coherent Code Tracking Architecture The non-coherent code-tracking mode was shown to affect the maximum carrierphase multipath error envelopes quite significantly for the case of the standard correlator at high M/D ratios. This simulation result is verified with bench data. 106 The first series of tests were conducted using the standard correlator architecture with the M/D ratios set at –2dB and –10dB. The plot of carrier-phase multipath error as a function of relative multipath delay can be seen in Figure 41. The M/D ratio chosen for this test run was –2dB. It is immediately obvious from Figure 41, that envelope of the multipath error as inferred from the bench data is not a uniformly decreasing function of relative multipath delay. This confirms the results of the simulation seen in Figure 35. In Figure 41 through Figure 44, the theoretical envelopes derived from equation 46 have been overlaid with the bench test data. The solid line represents the theoretically obtained carrier-phase multipath error envelope. The thickness of the solid line is an approximation of the 1-sigma noise bounds on the theoretical carrier-phase multipath error envelope. When the M/D ratio is –10dB, the non-uniform trend of the multipath error envelope seen in the –2dB case is no longer clearly observable (Figure 42). The next series of tests were conducted using the narrow correlator architecture. All other operating conditions were similar to the previous test run. The carrier-phase data collected with this narrow correlator architecture (0.1 chip) based GPS receiver is analyzed to observe the effects of multipath at different delays relative to the direct. M/D ratio for Figure 43 and Figure 44 were –2dB and –10dB respectively. 107 Figure 41. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -2dB. Figure 42. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -10dB. 108 Figure 43. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -2dB. Figure 44. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing, non-coherent code tracking. Bench test results overlaid with the theoretical envelope (solid line). M/D = -10dB As observed in the figures, the error plots constitute a single-sided envelope. This is ascribed to the manner in which the multipath delay was incremented. The multipath delay ranges from n  to (n+0.5)  and then steps to (n+1)  (n being a whole number). 109 As a result, the envelope does not manifest errors for multipath delays between (n+0.51)𝜆 to (n+0.99)𝜆. These delays contribute to the other (positive) side of the multipath error envelope as seen in the simulations. Given the symmetric nature of carrier-phase multipath errors, it will suffice to analyze one-sided carrier-phase multipath error envelopes. Upon comparison of the simulation results based on equation 50 and the bench test data, it is observed that equation 50 does not predict the non-uniformity in the multipath error envelope at medium delays for the standard correlator. This simplified expression for carrier-phase multipath error assumes that the code multipath error is very small and can be neglected. However, the code multipath error affects the estimation of the carrier-phase multipath error. Thus, equation 50 does not completely characterize carrier-phase multipath under all conditions. Specifically, for medium delay multipath, where the standard correlator experiences the largest code error, this simplified model is noticeably in error. From Figure 41 through Figure 44 it is seen that the data collected through bench testing agrees quite well with the theoretical envelope overlaid upon the data (solid line). This theoretical envelope is a one-sided representation of the envelopes obtained from Figure 35, Figure 36, Figure 37 and Figure 38, which were generated based upon equation 46. The non-uniform characteristic in the carrier phase envelope as predicted by 110 equation 46 is also observed in the data collected through extensive bench tests. This is quite evident from Figure 41. The simplified equations, due to their inherent assumptions, do not truly characterize the nature of carrier-phase multipath across all ranges of relative signal strengths and delays. Coherent Code Tracking Architecture The coherent code-tracking mode was shown to affect the carrier-phase multipath error envelopes appreciably for the case of the standard correlator at high M/D ratios. This simulation result is verified with bench data. The first series of tests was conducted using the standard correlator architecture with the M/D ratios set at –3dB and –10dB. The plot of carrier-phase multipath error as a function of relative multipath delay can be seen in Figure 45. The M/D ratio chosen for this test run was –3dB. The envelope of the multipath error as inferred from the bench data is not a uniformly decreasing function of relative multipath delay. This confirms the results of the simulation seen in Figure 35. In Figure 45 through Figure 48, the simulation data has been overlaid with the bench test data. The solid line represents the simulation data. The raw bench data has been averaged over each multipath delay value in order to minimize the effects of noise. As mentioned earlier, each delay value in the multipath profile was kept constant for a given period of time. Carrier-phase measurements from the latter half of each delay offset were used for the averaging procedure. Measurements from the first half of each delay offset in the profile were 111 excluded from the averaging procedure, as they represent a transient response while stepping through the delay profile. Carrier-phase multipath error for each step in the delay profile is depicted by a single averaged value. Figure 45. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -3dB. When the M/D ratio is –10dB, the multipath error envelope is given by Figure 46. The next series of tests were conducted using the narrow correlator architecture. All other operating conditions were effectively identical to the previous test run. The carrier-phase data collected with this narrow correlator architecture (0.1 chip) based GPS receiver is analyzed to observe the effects of multipath at different delays relative to the direct. Figure 47 and Figure 48 depict the comparison between theory and data for the narrow correlator (0.1 chip) with the M/D power ratios set at –3 and –10 dB respectively. 112 Figure 46. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -10dB. Figure 47. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -3dB. 113 Figure 48. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Averaged bench test results overlaid with the simulation results (Solid line), M/D = -10dB. As discussed for the previous test run, single sided envelopes are provided. The excellent agreement between the theory and bench test results suggest that the more complete model presented as equation 46 is necessary for fully bounding carrier-phase multipath. High Fidelity Multipath Modeling The simulations used to derive the carrier-phase error envelope as shown in Figure 45 through Figure 48 were based on a theoretical model (equation 46) that analyzed the interaction between the direct and multipath components of the GPS signals taking the amplitude, delay and phase of the multipath into account. A high-fidelity receiver baseband processing model is used to complement the previous simulations. This high-fidelity model implements the receiver code and carrier tracking loops and computes code and carrier multipath based on a truth reference. This model provides a 114 holistic approach that incorporates the effects of noise and band limiting. The following table lists the settings used for the receiver simulation that was implemented to characterize carrier phase multipath. The high fidelity multipath model and data were provided by Joseph M. Kelly of the Avionics Engineering Center at Ohio University. Table 4:Attributes of receiver model. Front End BW (MHz) Carrier Loop BW (Hz) Carrier Aiding Code Loop BW (Hz) Code Loop DLL Feedback Rates PRN MDR (multipath-to-direct) C/No (dB-Hz) 17 (double-sided) 15 (2nd order PLL) yes 0.125 1 chip, dot product, normalized 5Hz code / 100Hz carrier 10 -3dB / -10dB 45 In order to compare with the bench data, the simulation was run with multipath delay values defined as: *(40n, 40n+0.125, 40n+0.25, 40n+0.375, 40n+0.5) for n = 0 to 58, such that maximum delay is approximately 1.5 C/A code chips. Each step within a set whole number value of “n” is given a dwell time of 35 seconds. A settling time of 60 seconds is allowed when transitioning to a new set. Relative phase of the multipath signal, in this case, is anchored to the time delay of the multipath relative to the direct signal. The results of this model are compared on a one-to-one basis with the bench test data in the following figures. It is to be kept in mind that these are raw phase measurements and have a higher noise level unlike the averaged values seen in the previous section. The trace in black represents the bench test data while the trace in gray 115 represents the results of the receiver simulation. Figure 49 depicts the wide correlator (1.0 chip spacing) coherent multipath scenario with M/D set at –3dB. The bench data and the simulation results compare to a very good extent given the limitations imposed by noise. It can be observed that the non-linearity in the carrier-phase error envelope seen in the bench test data is matched by the high fidelity multipath modeling. Figure 50 depicts a zoomed-in version of Figure 49. The black line represents the envelope of the bench test data in Figure 49, Figure 51, Figure 53 and Figure 55. The black dotted trace in Figure 50, Figure 52, Figure 54 and Figure 56 are used for the zoomed in version of the plots and represents the bench data while the trace in gray represents the simulation data. Figure 49. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -3dB. 116 Figure 50. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -3dB. Zoomed in version of Figure 49. Figure 51 depicts the wide correlator (1.0 chip spacing) coherent multipath scenario with M/D set at –10dB. Figure 52 depicts a zoomed in version of Figure 51. The black dotted trace is the bench data and the trace in gray represents the simulation data. The next tests were performed for the narrow correlator (0.1 chip spacing) case with the multipath set at -3 and –10 dB respectively. The results are shown in Figure 53 and Figure 55. Figure 54 and Figure 56 depict zoomed in versions of Figure 53 and Figure 55 respectively. The black dotted trace is the bench data and the trace in gray represents the simulation data. 117 Figure 51. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -10dB. From Figure 49, Figure 51, Figure 53 and Figure 55 it is seen that the carrierphase multipath error obtained from the high fidelity simulation compares favorably with the bench test results. Figure 52. Zoomed in version of Figure 51. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -10dB. 118 Figure 53. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -3dB. Figure 54. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -3dB. Zoomed in version of Figure 53. 119 Figure 55. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), coherent code tracking. Bench test results (black) overlaid with the simulation results (gray), M/D = -10dB Figure 56. Bench test results (dotted black) overlaid with the simulation results (gray), M/D = -10dB. Zoomed in version of Figure 55. Analysis of Phase Rate Effects Relative motion of the satellite and/ or receiver with respect to a reflecting surface will induce amplitude variation and non-zero phase rates on the multipath component of 120 the received signal (Kelly, 2001). Receivers generally will reduce the effect of the multipath through effective averaging performed by the tracking loops (Kelly, Braasch, & DiBenedetto, May 2003). One approach to characterize fading multipath is to sweep the multipath signal across the preset range of multipath delays at a given rate. In the scenario, this rate was set at an eighth of a wavelength per second. Such sweeping effectively causes a constant phase-rate resulting in multipath fading. Both wide and narrow correlator architectures were subject to the same conditions of fading and bench data was collected for the –3dB and –10 dB cases. The receiver implementation included a carrier-aided coherent code-tracking loop. Given the high degree of carrier aiding of the code loop and extremely low code loop bandwidth (approximately 0.05 Hz), it can be seen from Figure 57 and Figure 58 that much of the anticipated code multipath error is filtered out by the code loop due to the phase rate of the multipath which was set at 1/8 Hz. This rate was chosen in order to display the averaging effect of the pseudorange multipath due to a slower code loop filter as mentioned above. Although the shape of the code error envelope is more or less preserved, the magnitude of the code multipath error as a function of relative multipath delay is extremely low in comparison to the static scenario (Kalyanaraman et al., 2004). This will be discussed later in this section. However, it is observed that the carrier-phase errors for the multipath-fading scenario are very similar to the static multipath scenario. This similarity is attributed to 121 the fact that carrier loop bandwidth (approximately 15 Hz) is much higher than the multipath fading frequency set at 1/8 Hz. This allows the carrier loop to keep track of the combination of the direct signal and the multipath. In such a case, the phase multipath errors do not get filtered out. Figure 57 and Figure 58 pertain to the standard correlator coherent code-tracking mode. The multipath fading frequency is 1/8 Hz and the M/D ratio is –3dB. The next set of figures (Figure 59 and Figure 60) pertain to the narrow correlator coherent codetracking mode with the multipath fading frequency at 1/8 Hz and the M/D ratio set to – 3dB. Figure 57. Pseudorange multipath error envelope versus relative multipath delay for the standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the code tracking errors obtained for coherent code tracking mode. 122 Figure 58. Carrier-phase multipath error versus relative multipath delay for the standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the carrier-phase errors obtained for coherent code tracking mode. Figure 59. Pseudorange multipath error envelope versus relative multipath delay for the narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the code tracking errors obtained for coherent code tracking mode. 123 Figure 60. Carrier-phase multipath error versus relative multipath delay for the narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. The plot indicates the carrier-phase errors obtained for coherent code tracking mode It is to be noted that the data displayed in Figure 59 and Figure 60 are for multipath delays between 0.1 and 1.5 chips. The data logging software in the receiver failed to log data between multipath delays of 0 to 0.1 chips. From this data it is seen that the shape of the code range multipath error envelope is mostly preserved for both standard and narrow correlator cases. However, the peak value of these pseudorange multipath errors is much less compared to the static case for the given M/D ratios as seen in Figure 61 and Figure 62. 124 Figure 61. Plot of pseudorange multipath error versus relative multipath delay for the static and fading multipath scenarios. Standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. Figure 61 depicts the pseudorange errors for the wide correlator strong multipath scenario with the M/D ratio set at –3dB. The dotted envelopes depict the maximum multipath error for the static case as a function of time delay while the inner envelope shows the fading multipath error. The peak value for the standard correlator receiver architecture with the M/D ratio set at –3dB is about +/-110 meters in the static case. The peak error as seen from the fading multipath data is approximately +/- 2 meters. 125 Figure 62. Plot of pseudorange multipath error versus relative multipath delay for the static and fading multipath scenarios. Narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. Figure 62 depicts the pseudorange errors for the narrow correlator strong multipath scenario with the M/D ratio set at –3dB. The dotted envelopes depict the maximum multipath error for the static case as a function of time delay while the inner envelope shows the fading multipath error. The peak value of the narrow correlator receiver architecture with the M/D ratio set at –3dB is about +/-11 meters in the static case. The peak error as seen from the fading multipath data is approximately +/- 0.5 meters. The code DLL is unable to follow the dynamics of the multipath fading and essentially filters out the pseudorange multipath. This is due to the extremely narrow code loop bandwidth allowed when using a high degree of carrier aiding in the code loop. The carrier tracking loop bandwidth is wide enough to accommodate significant line of sight Doppler and Doppler rates. This design aspect of the receiver effectively enables the carrier-tracking loop to follow the 1/8 Hz fading rate. As a result, the carrier phase 126 tracking error is not mitigated through loop bandwidth filtering as in the pseudorange multipath case. These can be seen from Figure 63 and Figure 64 which compare the carrier phase multipath error obtained for the static case with the carrier phase error from the fading case. The dotted black lines depict the envelope of the carrier-phase multipath error for the static case. The noisy gray profile depicts the carrier phase multipath error for the fading scenario. It is easily seen that a fading bandwidth of 1/8 Hz is very low compared to the bandwidth of the carrier-tracking loop and does little to affect the carrier phase multipath error. A typical example of fading multipath would be an airborne receiver picking up ground reflection multipath. The following equation (Kelly et al., May 2003) is used to compute the frequency difference between the direct and the multipath signal (ground reflection) reaching the airborne user from the satellite vehicle f diff  d p  f nom  dt c (52) where 𝑓𝑑𝑖𝑓𝑓 is the difference frequency between the direct and the multipath signal, 𝑓𝑛𝑜𝑚 is the nominal frequency of the GPS signal, and 𝑐 is the speed of light in meters per second 𝑑 ∆𝑝 𝑑𝑡 is the time derivative of the path length difference between the direct and the reflected (multipath) signal from the satellite vehicle and, 127 ∆𝑝 is the path length difference, is given by 2  h  sin( ) (53) where, 𝑕 is the height of the GPS user above the ground, and 𝛼 is the angle of elevation of the Satellite vehicle at the user and the ground reflection point (assuming a plane wave incidence for the direct and multipath components of the signal). Figure 63. Plot of carrier-phase multipath error versus relative multipath delay for the static and fading multipath scenarios. Standard correlator spacing (1.0 chip), M/D = -3dB. Fading frequency is 1/8 Hz. 128 Figure 64. Plot of carrier-phase multipath error versus relative multipath delay for the static and fading multipath scenarios. Narrow correlator spacing (0.1 chip), M/D = -3dB. Fading frequency is 1/8 Hz. For a typical aircraft approach and landing scenario, the „sink rate‟ (i.e., derivative of aircraft height, 𝑕) is approximately 2.5 m/s. This yields, via equations 52 and 53, frequency differences in the range of 2 – 20 Hz for satellite elevation angles in the range of 5 to 90 deg (Kelly, 2001). 129 CHAPTER 6: CONCLUSION The carrier phase distortions caused by unconstrained adaptive array processing and GPS carrier phase multipath serve to highlight the nature of possible GPS signal distortions for high accuracy ranging applications and applications that need to maintain continuous carrier phase tracking. The adaptive array processing work highlights the use of baseband compensation using a feed forward signal processing mechanism that integrates the upstream adaptive array signal processing along with the baseband GPS tracking loop mechanism. This is one example of feed-forward compensation. The concept may be used for various viable baseband signal processing augmentations depending on the type of application and the optimization criterion. Previous research (Moelker, 1998) had shown that variations in GPS carrier phase may exceed the range of phase steps tracked by a Costas PLL. The adaptive array processing results reinforce this conclusion. The MVDR array signal processing approach places a unity constraint towards the direction of the desired GPS signal. Hence, the phase of a signal in the desired look direction (direction of arrival of the desired GPS signal) is not distorted by the adaptive array algorithm. The distortion less constraint used by the MVDR approach requires use of an attitude sensor such as an inertial reference unit. However, in case of the MV array signal processing approach which does not use an 130 attitude reference, the desired GPS signal may undergo a rapid phase variation which may result in loss of carrier phase lock. The phase compensation approach outlined in this work uses knowledge of the adaptive array weights to compensate for the sizeable step changes in the phase of the GPS signal and shows considerable promise in implementing an MV alternative to the MVDR approach. As a result, the user does not need to place a distortion-less constraint on the adaptive array algorithm. The phase compensation technique allows the baseband receiver to maintain continuous carrier phase lock on the GPS signal while implementing the MV adaptive array processing algorithm. The purpose of the phase compensation is to make transparent, to baseband tracking loop, a phase step imposed by the adaptive array on the GPS signal. Intelligent integration of the GPS receiver processing with a source of adaptive array weights information enables the user to apply the necessary phase compensation in the baseband tracking loops. The underlying assumptions in this work have been made from the perspective of evaluating the performance of phase compensation methods. In a constrained adaptive array application (e.g. constrain adaptive array phase variations and/or compensate for these variations), the user typically needs to have knowledge of platform attitude (coarse as it may be). Fine platform attitude information is not necessary for the purpose of beam steering or beamforming for a GNSS antenna array. It is to be noted that group delay 131 distortions that arise during unconstrained or uncompensated adaptive array processing impact the GPS code phase measurements. In addition, the theoretical carrier-phase multipath error envelopes obtained for the coherent and non-coherent code tracking GPS receivers have been validated through bench tests. The validity of two simplified carrier-phase multipath models from equations 49 and 50 have been investigated. Carrier-phase multipath theory agrees well with the bench test data for the complete theoretical model in equation 46. The existence of a non monotonic characteristic in the carrier phase multipath error envelope, as predicted by equation 46, has been verified experimentally for the coherent and the non-coherent code tracking architectures. This corroborates the model utilized for the theoretical characterization of carrier-phase multipath in equation 46. The effect of multipath fading on code and carrier multipath errors has been investigated for the case of coherent codetracking loops, implementing wide and narrow correlator spacing. In addition, the utilization of carrier-aided code loops to mitigate the impact of fading multipath on pseudorange accuracy has been demonstrated. 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