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THE DESIGN OF DIGITAL FILTERS by WILLIAM M. RILEY, B.S. in E.E. A THESIS IN ELECTRICAL ENGINEERING Submitted to the Graduate Faculty of Texas Technological College in Partial Fulfillment of the Requirements for the Deg:ree of 1-·fP. . STER OF SCIENCE IN ~LECTRICAL ENGINEERING Approved _-_::.__\_____~~-V-~! " ·--·~-.:::-· · ·-·----=:.........___ _ Chairman of the Committee _________ --·'?f·-·~=--··- - --_:. ..!.__ _ _ ~----- ·--·-··--- .. -'---~~-·--- ---.....!C-~.,.,__ _~r--··-"""--·-~·--=r-~--¥- Dean of.the Graduate August, 1968 S~1ool <;,.__ ACKNOWLEDGEMENT S I am deeply grateful to Dr. D. L. Vines for his encouragement and guidance and to the other members of my graduate committee. Dr. J. P. Craig and Dr. T. A. Atchison, for their helpful criticism. I would also like to express my appreciation to the Texas Technological College Computer Center and to Collins Radio Company whose support made this research possible. 11 CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES I. INTRODUCTION The General Filter Equation A Realization Configuration The Problem II. METHODS AND PROCEDURES Coefficient Computation The Bilinear Z Transform Filter Realization The Computer Program III. ly. INTERPRETATION OF FINDINGS Exam-ple One: Coefficient Determination Example Two: Higher Order Filters CONCLUSION LIST OF REFERENCES APPENDIX A. The Computer Program, FILRSP B. Input Data Card Formats C. Example Output Data 5 11 LIST OF TABLES 1. Partial Fraction Coefficients 2. Filter Coefficients IV LIST OF FIGURES 1. Block Representation of a Digital Filter 2. Realization Configuration for Equation 8 3. Representation of the Bilinear Z-Transform 4. (a) Forward Feed Loop (b) One Partial Fraction Terra 5. Parallel-Cascade Filter Combination 6. Program Flow Chart 7. Butterworth Digital Filter 8. Elliptic Digital Filter CHAPTER I INTRODUCTION The term "digital filter" is used to represent a two port device whose input and output signals are sets of numerical data. The input set of data or series of numbers, could represent a sampled analog signal or be successive amplitude modulated pulses of a radar signal which have been numerically coded as digital words. The digital filter accepts the input, modifies it according to the desired transfer function, and yields as an output the modified set of numbers. A digital filter may be represented in block form as sbovn in Figure 1. Thus, a digital filter is related to discrete data in an analogous manner to an analog filter's relationship with continuous data. Historically the term "diglcal filter" is relatively young, but the concept datos back to the fifteenth century v/hen the classical works on numerical analysis were being developed [1]. In its broadest scope, digital filter includes the most sophisticated digital computer as well as the simplest adding machine. The electrical enginr-eu's interest in digital filters has developed frora a need" to simulate analog circuits avid systems, using digital comput-ers to a desire for new techniques for manipulation of discrete information in the rapidly developing area of digital .systems Since digital filters manipulate di:-crete signals, the mathematics cf difference equations is used for the r.,^.them.atical model Input DIGITAL X(nT) Figure 1. FILTER Output Y(nT) BLOCK REPRESENTATION OF A DIGITAL FILTER of systems containing digital filters. Hereafter, the term "filter" will be used to mean "digital filter" except where additional emphasis is desired or misinterpretation is likely. designated. Analog filters will be so The general equation for the filter transfer function is now derived. The General Filter Equation An expression for a series of sampled data in the time domain may be written [2] X*(t) = X(0) + X(T)5(t - T) + X(2T)6(t - 2T) + ... Z X(nT)6(t - nT) n=0 (1) xvhere X(nT) = 0 for t < 0, and 6(t - nT) is a unit impulse function at time t = nT. The starred term indicates a discrete quantity. Also, some authors use X*(z) to indicate the z-transform but here. X(z) will be used since the z itself will indicate a sampled quantity. The z-i:ransform of Equation 1 is the Laplace Transform of X=^(t) with the Laplace Operator, s> replaced by s - - In X and thus [2] X(2) - where z I X(nT)z~" n-0 is the conventional unit delay in z-transform tlieory. (2) The relationship between the unit delay In the time domain and the z domain is expressed as .-^ = e-J'"T (3) The transfer function for the filter Is defined as H(z) = |i?j- (4) Both the input and output are series of discrete numbers, hence Equation 4 may be written as CO Z YdnDz"™ CO Z X(nT)z~'^ n=0 Equation 3 contains two polynomials in z-k which are both of infinite order. To design realizable filters"with finite delay times and a finite number of components, the order of the numerator and denominator polynomials of the transfer function must be restricted to finite orders. Dividing each term of the I'uwera'ror and denominator by the input at time t -- 0, the transfer fur.cticn is put in the form M ^ -m a z !I(z) - - - ^ - ~ 1 + (6) E b z~^ n n=i where a = y(mT)/X(0), m = 0 , 1, 2, ..., M; and b - X(nT)/X(0), m m n = 1, 2, .,., N« This is the general equation for a digital filter. Realization Configuration The realization configuration used to implem.ent a filter using digital hardware is perhaps the most important part of the filter design. An incorrect choice of configuration may lead to an unreal- izable result. There are several criteria which may be used to decide between configurations. These criteria are: the number of mathematical operations required, such as additions and multiplications per cycle of operation, configurations which will reduce the ;)Ossibility of numerical error, such as avoiding the differencing of similar numbers [3], and the reduction of the quantity of digital hardware required, for example, choosing a configuration which will allow a shorter word register. There are several methods to determine a possible filter coafiguration for a given difference equation. One is simply to interpret the z-transfer function as the input .md output signals multiplied by various delays. Using Equation 6 as an exa'Tiple, the equation may be written -1 . -2 , ^ -K a + a, z T a,,z + . .. + a-,z _ _JOL 1 ^ ^ 1 -I- b.,z + b„z + ... + h z 1 2 N X (-J) Rearranging Y = X[a Note that Yz -k + a z"""" + ... H- a^^z"^] - YCb^z"^ f ... + b z~^] is the output delayed k time units. (8) Thus, Equation 8 may be represented by the configuration shown in Figure 2. This 0 t> \ INPUT - ^ — I 1—>- ..^iM > J -1 -1 ' -1 ' ^z r z -1 ' 1 I I •Ami^tmmmam _ _ — ^ -te" \ I 1 • Lt>" Figure 2. 1 REALIZATION CONFIGURATION FOR EQUATION 8 configuration utilizes N + M f 1 multipliers, N f M delays, and N + M + 1 adders. The Problem Two problems now confront the design engineer: A. How to determine the coefficients a and b for a desired filter response; m n r f and B. What physical configuration may be used to realize the filter? The answer to the first question has been the essential part of the recent research into digital filters. Methods of coefficient i determination are catalogued by Kuo and Kaiser [1] dependi.ng en one i of two filter types: filter. the nonrecursive or the recursive digital The recursive filter is described by Equation 6, and the non- recursive filter is described by Equation 6 vjith all the b n coefficients equal to zero. In other words, the nonrecursive filter's output is not a function of previous outputs. The nonrecur- sive filter coefficients, a , may be found by the use of classical numerical formulae [4], by use of a Fourier series expansion of the desired filter characteristic curve [5,6], or by the use of a min-max or least squares m.etliod [7J. Other recent articles by Radcr and Gold [8J and by Constaatinides [9,10] give another direct design procedure using the squared-magnitude-function for analog filters. Detailed examples are given for the Butterworth, Ghebysuev, and Elliptic type filters. 8 The coefficients of recursive filters may also be found using the above mentioned meth9ds. An alternate approach is the use of analog transfer functions which are mathematically transformed into digital transfer functions [1]. This method possesses the advantage that the engineer may use the highly developed techniques of analog filter design, which is the general method followed in this research, Of the overall effort on the subject of digital filters, a smaller portion deals with the question of realization configurations. The basic methods [1,2,11] are direct, parallel, and cascade forms. Combinations of these forms may be used [8] for' filter realization. I The purpose here then is to present answers to both the problem of coefficient determination and of deriving a realization configuration. Hopefully the inform.ation introduced in this research will aid in the realization of digital filter theory to modern digital system applications. CHAPTER II METHODS AND PROCEDURES The m.ethod of digital filter realization to be derived in this chapter is, in a sense, the reverse of that which has been shov^^l for the general filter equation. The analog transfer function is first arranged in a form, which facilitates coefficient computation, then a realization scheme which minimizes the possibility of numerical error is derived from the new equation. Two considerations were decisive in the choice of the design technique to be developed. Primarily, the bilinear z-transform was picked because it eliminates the frequency folding inherent in the standard z-transform [12] . This allows use of the resulting filter over a widsr portion of the Nyquist interval without introducing appreciable distortion. The secondary consideration was the use of the partial fraction expansion of the analog transfer function. The transfer function is thus reduced to a sum of terras, each of which have identical formats and thus the sarrie z-transform except for a difference of constants. All filter coefficients may be calculated using one method, regardless of variations between the initial analog transfer functions. Co efficient C c ra put a t i oiis_ Beginning with the general filter transfer function, the partial fraction expansion may be vjritten 10 N K. H(s) = I — ^ - — i=l ^ " . ^ (9) where the K.s are the partial fraction coefficients and may be complex numbers. Applying the bilinear z-transform to Equation 9 H(z) = H(s)I ^ ^-1 s = 3j1 + z and N H(z) = K.(l + z"h >: - - ^ - — ~i=l (1 - z ) - s^(l + z ) (10) -1 Notice that the teuin (1 + z ) may be factored out of the numerator summation so that Equation 10 may be rearranged as N K. H(z) - (1 + z ^) Z -^" 17 i=l (1 - s.) - (1 + s.)z ' (11) Dividing by (1 - s^) K, 1 N 1 - s. H(z) - (1 + z ) z ^^''^ TT";~ 1 - 1 1 - s. z 1 For convenience let C^ = K^/(l - s^) and D^ •= (1 + 3^/(1 - s ^ . Nov; Equation 10 becomes N H(z) = (1 -4- z n S C. -^=—;T i=l 1 - D.z 1 (13) 1 I Equation 13 is the desired form from which the filter coefficients, C^ and D , are easily calculated once the denominator roots, s., and the partial fraction coefficients. K., are known. 1 The Bilinear Z-Transform The bilinear z-transform is of the Mobius type [13] which is a mapping between the entire left half plane of the s domain into the area outside of the unit circle of the z " domain, as illustrated in Figure 3. The transform is 1 -1 1 f Z 1 Unfortunately tliLs transform causes a shift or "warping** of the frequency scale as can be seen from Equation 14. 1 - £ Set s = ju), then -ju)T J^'^ A = T ^ -jwT 1 + e -^ or (15) ! jw^^lI v/here w 4- / ^ J 1 \ - tan(—) is the analog filter frequency and w is the digital filter frequency. Hence, a filter derived using the bilinear Z-transform will have a frequency response related to the parent analog filter frequency response by (J - " arc tan (o)) (16) 12 jw —c- s = a a z I- j w = a + jb -1 1 - /. -1 1 + z ,-1 ^ 1 " s ' 1 4- s Fiv^ure 3 . Rliri^SS^-lNTATION OF THE EILTNKAR Z-TR.VNSFORH 13 Fortunately, compensation for this inherent frequency warping caused by the bilinear z-transform is possible. This is acconiplishc;d by prewarping the critical frequencies of the desired response characteristic. Then, after calculation of the analog transfer function using the prewarped frequencies, application of the bilinear z-transform causes the critical frequencies of the digital filter to be shifted back to their desired values. For a low-pass filter, the cutoff frequency of the digital filter is thus related to the cutoff frequency of the parent analog j filter by rearranging Equation 16 as ; w = tan(-^) pu) 0) ^ s where w po) (17) is the prewarped analog critical frequency, OJ, is the il desired critical frequency of the digital filter, and oi ^ is the sampling frequency. Filter Realization The filter realization scheme represents the actual layout of the physical filter which may be i.mplemen?:ed using digital hordware. The computer program to be explained in the next section is, in part, a simulation of this physical filter, and as such, may be used to determine the characteristics of the filter without a costly engineering model. If: Realization is based on Equation 13, which consists of several separate elements. First, the factor (1 + z ) is realized as a forward-positive-feed system as shown in Figure 4(a). Secondly, the summation of partial fractions must be realized. Consider the configuration shown in Figure 4(b). Solving for the transfer function E - W + F but F = ED.z •"• 1 then E==-^ 1 - D.z ^ 1 the output is Y = EC. 1 or V7C. 1 - D.z 1 •'• thus the transfer function is Y " ^=1 1 - «.z-l 1 The complete filter is now realized by summing all partial fractions in a parallel arrangement in cascade with the forv;ard feed portion shovm in Figure 4(a). The result is a combination parallel-cascade filter and is illustrated in Figure 5. 15 W -1 (a) Transfer function ~ = 1 + z -i-®-- (b) Transfer function W Y "} L -1 1 - D.z 1 Figure 4. (a) FORWARD FEED LOOP; (b) ONE PARTIAL FRACTION: TERM 16 OUTPUT Transfer function -1, N H(z) = (1 + z ) T. i-1 1.0 - D.z-1 1 Figure 5. . PARALLEL-CASCADE FILTER COMBINATION 17 Computer Program A computer program for designing low-pass digital filters using the filter realization explained above was written in FORTRAI'J V for the UNIVAC 1108 computer. The program, FILRSP, is included in Appendix A. The program is designed to accomplish any ore of three functions: (1) calculation of the filter coefficients using the standard z-transform filter, (2) calculation of the filter coefficients using the bilinear z-transform and find the filter response for a particular sampling frequency, and (3) to find the response of the parent analog filter transfer function. The program flow chart is sliowi in Figure 6. The INPliT CONTROL SECTION reads the input data cards and also lists toe input data on the printout for convenience. Input data cards are prepared according to the formats given in Appendix B. The input analog filter transfer function must be in the form of the ratio of two polynomials, x^hich may be factored or unfactored; the unfactored form being preferred. The program generates only low-pass prototype filters which •T:ay be converted to high-pass, bandpass, or bandstop filters by use of the familiar filter frequency transformations [11]. Thus, only tl/.e cutoff frequency of the low-pass filtr^i: must be prewarped. If the standard z-transform or the analog r.^:.-ipc;'.;se is specified, the prewarping is bypassed. 18 C START ^ INPUT CONTROL SECTION TREWARP CUTOFF ,FREQUENCY UP.RCUTINE ROOTS CALL 1:0" \?ACTORlT!i i SUJJRGUTINE CALL PARFAO »i I'RACTION I EXPANSION zir:: FREQLENC^ ?)0 LOO? i CALCULATE -1 SUnKGuTINE 5VA.V.I1 CALL CALCULATE H Xix-z; (CALCULATE [.AfrpLITUDE AND PHASE SUBROUTINE CALL Tl'LOT INCREMENT FREQUENCY ITT L CALCULATE" AND STORE OUTRUT NO ^^REQUENC "\LIMI;JI^ I , . Yh PRINT OUTPUT Fi;:'\ra &• PRO(^RAM FLOW CHART A^ SIOP J 19 If unfactored, the SUBROUTINE ROOTS factors the transfer function denominator. Next, the SUBROUTINE PARFAO expands the transfer function into a partial fraction form, the coefficients K. of Equation 9 being calculated. As the frequency cycle begins, the value of z according to Equation 3. is evaluated Then the transfer function is calculated by SUBROUTINE EVALH according to the type of filter requested. The amplitude and phase of the transfer function are calculated by taking the real and imaginary parts of the evaluated complex number, HET. The output graph is prepared by the SUBROUTINE TPLOT and stored on one of the utility tape units. Tlie frequency cycle is repeated until the desired limit, WMAX, is reached. After the frequency cycle is completed, the output data is printed in tabular "form and then plotted in graphical form for quick interpretation. Limitations of the program must be recognized. The SUBROUTINE PARFAO wiLl allow no n>ultiple valued roots in the denominator and no poles at the origin. Also, WMAX is limited to 999.9999 radians per second unless a format statement is changed. The phase angle output must be corrected because the computer arctangent routine autom.atically reduces the cingle to + 180 degrees or less, thus, for angles greater than 130 degrees the output v/ill appear to be in error. 20 Three illustrative examples of coefficient determination and filter response of the derived filter realization are presented in the next chapter, and interpretations of findings discussed. CHAPTER III INTERPRETATION OF FINDINGS The power of the digital filter is manifest in its ability to approximate analog filters, its consistent and accurate operation, and the capability to attain transfer functions which are difficult or impossible to physically realize using analog techniques. This is not to mention the added benefit that the filter design may be completely simulated on a general purpose computer without the expense oc building an engineering model. ' I Use of the techniques presented here must be based on rhe judgement of the design engineer. In order to have some information upon which a decision laay be based, two examples are presented and discussed. Example One: Coefficient Determination The details of coefficient determination are pointed out in a step by step procedure. Consider a filter with the followir/, desired characteristics: 1. Cutoff frequency, w, = 1.0 RPS-^ 2. Minimum less, at w - 2.0 RFS of 18.0 decibels 3. Maximally flat in the passband 4. Sampling rate, fo = 6.0 RPS ^Radians Per Second 21 22 Design of the filter coefficients is as follows: Step 1. Select an appropriate analog filter. A third order Butterworth analog filter fits the above criteria. Its transfer function may be written [14] U(s) :. 1 1 + 2s + 2s Step 2. (18) + s The analog filter cutoff frequency must be prewarped using Equation 17. oj = tan( ) po) 0) s = tan (•—) o - 0.57735 (19) The transfer function is now prewarped using the low-pass to lew-pass frequency transformation [11] p(jj Equation 16 becomes H(s) = --^^^ 2 3 B(l) + B(^>^+ B(3)s -I- B(4)s- where A(l) = 1.0 B(l) = 1.0 B(2) = 1.1547005 ^^-^-^ 23 B(3) = 0.66666660 B(4) - 0.19245007 Step 3. The transfer function is now expressed in partial fraction form. H(s) = 3 K. Z - --^— s — s i-1 i (22) where K - 3.0 + jO.O K^ - -1.5 - j0.36602550 K^ --- -1.5 !- j0.86602550 s - "0.57735026 + jO.O s^ - -0.28867511 + j0.49999998 S3 -^ "0,28867,^11 - :J 0.49999998 Step 4. Each paitlal fraction is transformed as suggested by Equat ion 13, thus o n{z) C - (1 + z"^)S ^-~~ i-1 1 - D.z 1 where C - 1.9019239 + jO.O C^ - "0.7S305939 - j0.976r^2719 C3 - -0.78505939 "^ JO.97662719 D - 0.26794919 + jO.O D^ - 0.34891529 !- j0.52337239 O3 '' 0.34891529 " .10.52337289 (23) 24 The above coefficients were calculated by the computer program, FILRSP. This digital filter's response and its parent analog fil- ter's response are shown in Figure 7. A point of major importance which is immediately obvious from a comparison of the response curves is the periodic nature of the digital filter response. The reason for this repetitive response is easily seen by the use of Euler's relationship and Equation 3 which gives z = e ^ = cos(coT) - jsin(ojT) (24) The unit delay is the difference of two periodic functions with a period of wT. 27r But, T = — ; thus, the period repeats every w = no) , s n = 1, 2, ... 0°. The frequency, w /2, is called the Nyquist frequency [2] and Shannon has shown [15] that information contained in frequencies higher than the Nyquist rate cannot be recovered from the digital signal. This imposes an upper limit to the useful frequency response of digital filters at 0)^/2 for most applications. Kuo and Kaiser [1] appropriately designate digital filters that are useful over the majority of the Nyquist rate as "wide-band" filters. In regard to the application of digital filters, notice that the filter will pass frequencies higher than the Nyquist rate, especially those near multiples of w exist as shown in Figure 7. where the repetitive passbands Although these high frequencies cannot be converted into useful infoirmation, they do .cause distortion of 0 3 10 0) U) o ,-J r-H (U •H O G) n 20 Digital Analog 30 T V- V- 4 6 Frequency -- RPS Fi.-ure 7. EUTTKllViORTII DIGITAL FILTER u) 8 26 the information in the frequency range below the Nyquist rate; this type distortion is termed "aliasing" [12]. To prevent aliasing, the input to a digital filter will be bandlimited to frequencies below the Nyquist rate. The phase response of this digital filter deviates less than five degrees from that of the analog filter in the passband. Example Tw^o: Filter Configuration The advantage of the filter configuration derived in Chapter I is brought forth by comparing its response for a high order filter with that of the standard z-transform of the same filter. The standard z-transform is explained in References 1 and 2. Consider a filter with the following desired characteristics: 1. Cutoff Frequency, co, = 1.0 RPS 2. Equiripple in the passbands and stopbands with Maximum passband ripple ~ 0.5 db Minim.um stopband ripple = 75 db 3. Step 1. Sam-pling rate, w = 3 . 0 KPS Selecu an appropriate analog filter such as an elliptic type filter which will provide the above characteristics. The transfer function is [11] 10 I (s - AS^) H(s) = ~ ^ (s - S ) i=l (25) 27 where the roots of the two polynomials are Numerator Roots: AS(1) = 0.0 + jl.0695414 AS(2) = 0.0 - jl.0695414 AS(3) = 0.0 + j1.1009005 AS(4) = 0.0 - jl.1009005 AS(5) = 0.0 + jl.1946271 AS(6) = 0.0 - jl.1946271 AS(7) - 0.0 + j1.4652816 AS(8) - 0.0 - jl.4652816 AS(9) = 0.0 f j2.5031313 AS(IO) = 0.0 - j2.5031313 Denominator Roots: S(l) - -0.006913 + jl.0010752 S(2) = -0.006913 - jl.0010752 S(3) - -0.0257616 4- jO.9756431 S(4) -- -0.0257616 - jO.9756431 S(3) - -0.0615122 + jO.9063786 S(6) - -0.0615122 - jO.9063786 S(7) = -0.1269215 H jO.7504391 S(8) = -0.1269215 - jO.7504391 S(9) = -0.2142976 + jO.4483675 S(10) = -0.21429/6 - jO.4483675 S(ll) = -0.2611853 + jO.O 28 Step 2. Prewarp the analog transfer function using u) = 1.7321 pu) (26) and Equation 20. Step 3. Expand the transfer function into partial fractions. The partial fraction coefficients, K., are given in Table 1. Step 4. Calculate the filter coefficients, C. and D.; see Table 2 for the tabulated values. The filter frequency response is shown in Figure 8. For comparison, a standard z-transform derived filter is calculated and its response also graphed in Figure 8. For exact response values see the actual computer program output data for this filter which is included in Appendix C. Characteristics of the bilinear z-transformed digital filter are found to be Cutoff frequency = 1 . 0 RPS Maximum passband ripple = 0.50001413 db Minimum stopband loss = 76.5 db The slope of the edge of the passband was calculated to be -2235 db per octave. Comparing the two filters' responses, it is obvious that the bilinear z-transform derived filter is superior. The minimum stop- band loss for the standard z-transformed filter is only 57.0 db even though the passband response is satisfactory. 29 K(l) = 4.2566700 - j 5.2365968 K(2) = 4.2566700 + j 5.2365968 K(3) = -3.7206656 + j25.811197 K(4) = -3.7206656 - j25.811197 K(5) = -27.370689 - j61.069924 K(6) = -27.370689 + j61.069924 K(7) = 110.96046 + j89.354376 K(8) = 110.96046 - j89.354376 K(9) = -232.52271 - j72.419789 K(10) = -232.52271 + j72.419789 K(ll) = Table 1. 297.79389 - j 0.00000086 Partial Fraction Coefficients 30 C(l D(l 3.3214965 + j -0.49784774 + j 0.51640595 0.86038636 C(2 D(2 3.3214965 - j -0.49784774 - j 0.51640595 0.86038636 C(3 D(3 -12.035886 + j -0.47065882 + j 5.2384513 0.85630556 C(4 D(4 -12.035886 -0.47065882 5.2384513 0.85630556 C(5 D(5 17.778919 - j 29.966240 -0.40008587 + j 0.85112108 C(6 D(6 17.778919 -f j 29.966240 -0.40008587 - j 0.85112108 C(7 D(7 6.0459044 + j 79.693464 -0.23219827 + j 0.81813364 C(8 D(8 6.0459044 - j 79.693464 -0.23219827 - j 0.81813364 C(9 D(9 -105.74512 - jll2.70700 0.10435222 -I- j 0.62547459 C(10) D(10) -105.74512 + jll2.70700 0.10435222 - j 0.62547459 C(li) D(ll) 205.03768 - j 0.37704422 -f- j Table 2. j j 0.00000059 0,0 Filter Coefficients 31 High order filters with transmission zeros require no special handling when realized using the bilinear z-transforra filter presented here since the bilinear filter is superior to the standard z-transform filter. The phase response of the elliptic digital filter is v;ithin 28% of the parent analog filter's phase response in the passband. This phase response is shown as curve "2" in the graphical data portion of Appendix C. 32 0 (0 CO o ' - % • « 10 - 20 •• 30 •• 40 -- 50 - H4 . » < . . Bilinear Standard - 0 0.5 1.0 Frequency - RPS Figure 8. ELLIPTIC DIGITAL FILTER 1.5 OJ CHAPTER IV CONCLUSION The digital filter coefficient determination method and realization scheme presented are easily applied to a known analog filter transfer function. The resulting digital filter will, for low sampling rates, have better cutoff characteristics in the stopband and comparable passband response. The digital filter is constructed using r.oit delay elements and, therefore, may only have phase lag and no phase lead. It should be noted that digital filters may not have feedback loops without delay elements since this would require an estimate of future signals at the summation point. As applied to modern electrical engineering, the recent advent of readily available digital computers and low cost integrated circuitry are resulting in the rapidly expanding use of digital control systems in areas previously dominated by analog electronics. For example, di.'^^ital frequency synthesizers and airborne digital guidance devices now dominate a previously all analog area of electronics. The importance of signal filtering has long been recog- nized in electrical engineering, and the ability to filter the discrete information'inherent to digital systems is and will play an important role in the future rate of growth of total digital systems. The digital filter design and realization method presented herein may be immediately applied to simulation problems v/ithout 33 34 modification. Suggested further study in this area includes physically constructing such a filter to determine the problems associated v/ith actual realization, as well as collection of performance data to confirm or abrogate predicted theoretical results. LIST OF REFERENCES 1. Kuo, Franklin F., and Kaiser, James F., Systems Analysis by P,i8_ltal Computer, John Wiley and Sons, Inc., New York, 1966, Chapter 7. 2. Tou, Julius T., Digital and Sampled-data Control Systems, McGraw-Hill, New York, 1959. 3. Atchison, T. A., Class Notes Texas Technological College, Unpublished, 1968. 4. Hildebrandj F. B., Introduction to Numerical Analysis, McGrawUill, New York, 1956."' ""' 5. Anders, E. ii., et. al.. Digital Filters, NASA Contractor Report CR-1'^6, December 1964. 6. Mcirtin, M. A., Digital Filters for Data Processing, General Electric ML'sile and Space Division, Technical Information Series Nuu.l^er 62SD484, August 1965. 7. Treitel, S., and Robinson, E. A., ''The Design of HighResolution Digital Filters," Ir^EE Transactions on Geoscience ELlctr^-tl^:;-'J?-» ^^^-^' ^'^~^^» ^^-^ • ^5 PP • 25-3'3," June 1966. . 8. Rader, C- M., and Gold, B., "Digital Filter Design Techniques in the Frequency Domain," Proceedings of the IEEE, vol. 55, pp. 149-171, February 1967. 9. Constantinides, A. G., "Synthesis of Chebychev Digital Filters," In-J tJ tu_te_jj^ _Elec_tr 1 cal Engineer s-E.1 ec tronic Letters, -/o 1. 3, p pT'T?.4 -12 6, Hi rc h' 19 6 7 . 10. Ibid.: "Elliptic Digital Filters," Institute of Electrical ^^•^Jii'l£^S^s3^Il£Sj^-:D.^F-J^'-^^ vol. 3, pp. 255-255, June 1967. 11. Golden, R. M., and Kaiser, J. F., "Design of V/ideband Sampleddata Filters," Tlji;^ Bell Sy.'item Teclmical JournaJ , July 1964, p. 1543. 12. Blackman, R. B., iuid Tukey, J. W., The Measurement of Power Spectra, Denver Publications, Inc., New York, 1959. 35 36 LIST OF REFERENCES (CONTINUED) 13. Silverman, R. A., Introductory Complex Analysis, Chapter 3, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. 14.' Collins Radio Company, Analog Filter Data, Unpublished, 1967. 15. Bennett, W. R., and Davey, J. R., Data Transmission, Chapter 5, McGraw-Hill, New York, 1965. APPENDIX A. The Computer Program, FILRSP B. Input Data Card Formats C. Example Output Data 37 APPENDIX A: THE COMPUTER PROGRAM, FILRSP The following pages contain the FORTRAN V digital computer program, FILRSP. The program was designed for use with the UNIVAC 1108 computer. 38 39 COMPLEX HET COMPLEX AS,PF,C»0»S,AA IN=8 INTEGER OUT 0UT=7 Clr'ENSION PF(iO),C(20),0(20),S(16) DIMENSION A(31)»B(10)»8R(31) DIMENSION AS(20) DIMENSION AA(20) DIMENSION PM(129),F(5),F0(5).SF(5) DIMENSION CARD(13) DIMENSION SAVEA(200)»SAVEB(200»129) 1 2 3 4 5 6 7 8 F0RMAT(13A6) FORMAT(IH ,3bH THE INpUT DATA FOR THIS PROBLEM IS) F0RMAT(12X,3('+X»I3)»3(7X,I1)) FOR:-AT (lOXI <* (6X»Fa .3)) FORMAT (22X#£J13.7) F0RMAT(22X,Eir>,8,2X»El5.6) FORf'.AT(lHl) FORMAT ( I H ./2X,'+H0ATA/3H W »7X»3H WT»7X»5H H M A G , 1 2 X » S H HMDB» A 1 2 X » 5 H H A N G , 1 2 X , S H R E A L » 1 2 X » 1 0 H IMAGINARY) 9 F0RMAT(mX»3{bX»E13.7)) 10 FORMATdH , 1 X » 3 5 H T H E DIGITAL FILTER COEFFICIENTS ARE) 21 FORMAT{26X.13) 8b FORMATdH .5X,3HC( » 1 2 , 5 H ) = , E 1 5 , 8 , 5 X y E 1 5 , 8 , / , A 6X,3H0( » I 2 , 5 H ) = , e i 5 . 6 f 5 X » E 1 5 . 8 ) 87 FORMATdH >t3H THE DC RESPONSE BEfC'AZ NORMALIZATION IS »E15,e/) 88 rOKMATdH »Fa,2» 2X»F8.'4f 2X, 5 (E15*8>2X)) 89 FORMATdH »F10.'*i2X» 102A1) 90 FORMATdH .13A6) PROGRAM CONSTANTS PI = 3,li*159265 TV;0PI::6,2831853 RA-3C>0./TV/CPI COi>-> 0 (0 , 3 ) M, ^^»IP»i-' T Y P E, r: F 0 R M, D F 0 R M GO TO 11 1 J F1 REAO(0»5)A{I) IFd.Lr.LM) KEY=K£Y-1 •ee—T^ i i 17 1=1+1 18 r.'EAO(0,6)AS(I) I F d . L T . M ) KEY=KEY-1 GO TO 11' IF(CFCRM.E0.2) GO TO 19 J'.:J•^1 LN^N-5-l READ(0,5)8(J) I F { J . L T A N ) KEY=KEY-I IF(J.EQ.LN) GO TO 20 GO TO 11 19 J=J+1 REAO(0,6)S(J) IF(J.LT.N) KEY=KEY-1 IF(J.Ev3.N) GO TO 20 GO TO 11 20 CCM'TINUE CALCULATED CONSTANTS T-TWOPI/WS WCAL= (WMAX-WyilN) /FLOAT (IP~1) ARG=(PI*WCO)/WS WPW=SlN{AF/i)/COS(ARG) CLEAR REGISTERS GO TO (79»80),OFORM 79 NSV=N PREWARP COEFFICIENTS I F { N T Y P E » N £ , 2 ) GO TO 59 DO 5 7 II.:;1»N B (ll + i} - B (11 >.l) / i' V;P'.^;<*. 1 1 ) V 57 CONTINUE IF(M.EQ,0) GO TO 59 UO 58 n = l/M 5d A ( 11 +1) -A (I i-t-l) / (WPW* + I I) 59 C O N T I N U E C NORMALUE COEFFiCtENTS 41 \^ ^ C 'y DO 70 I=1,LN tJd)rb(I)/B(LN) 70 COWTIfjUE KLVEK5E ORDER OF COEFFICIENTS FOR FACTOR SUBROUTINE no 78 1=1fLN NI=LN+1-I BKd)='i{Nl) 78 CONTINUE > CALL R00Tb(l.N,8R»S) N=NSV 8 0 CONTINUE IF(fJTYPE.NE.2) GO TO 8(t IF(NFu;iM,Lvi.l) GO TO 82 00 81 I=liM 81 AS(I)=ASd)*WPrf o2 IFCJFORM.EO.l) GO TO .»A»S, L»AA, AS»NFORM) WRITE(OUT»10) CALCULATION OF DIGITAL FILTER COEFFICIENTS GO TO (t»0,'+l»'+2) fNTYPL 40 00 43 IJ=1,N HF(IJ)=EXP(S(IJ)*T) C(iJ) = (T*AA{;iJ) )/PF(IJ) 40 OdJ)=1.0/PFdJ) GO Tu 46 '•l Ut' 44 lJ=i,N C(IJ)=AAdJ)/d.O-S(iJ)) 44 U(IJ) = (1.U + S(IJ) )/d.O-S(lJ)) GO TO 46 42 UO 45 IJrlfN 45 CdJ)=AA(lJ) odj)=OMPLX(o.o,n.o) 4 6 COMTIrUJE UO 115 IPf- = l,N WRI rE(i)UT,b6) IPF,C(IPJ- ) » IPFrD(iPF) lis CONTINUE Mr;Dt3=.o,o C FRE'^UErJCY DO LOOP 0 0 l u O 1 = 1.Il-* I f - d . G T . l ) GO TO 2 0 3 GO TO 204 203 •,y=FL0AT(l-l)+.VCAL4WMIN ;;04ft'T= W*l ^ CAuL L7AL! I ( S, W » T, H,- fVTYpE»C . 0 > A-^ »HET) HN.AG=CABS(HEr) IF(HMAvJ,LL,0,0) (.0 TO bO HlMG=AIMAG(Hc.T) HREAL=ReAL(HET) H M N G = A FAN?. (HIM5»HREAL) HANG=HANG*KA 42 CALCULATE NORMALIZING FACTOR IFCI.GT.l) GO TO 205 C0NST=i,/HMA6 WRITE(0UT,87)HMAG WRITE(0UT»7) WRITE(0UT,8) 205 CONTINUE HMA6=HMAG*C0NST GO TO 65 60 HANG r 0,0 65 CONTINUE IF(HMAG,LE.O.O) GO TO 66 OUTPUT CONTROL SECTION HMDB=20.*ALOGIO(HMAG) 66 CONTINUE Fd)=90,+HMDB F(2)=HANS F(3)=HMA© L=I e CALL TPL0T(L»3,SF»F»F0,.TRUE.».TRUE.iPM) WRITE(0UT»88)W,WT»HMAG,HMOB.HANG»HREAL»HIMG SAyEA(I)=W DO 110 K=15fll6 SAVEa(I,K)=PM(K) 110 CONTINUE 100 CONTINUE WRITE(0UT»7) 00 1010 I0UT=1,IP WRlTE(0UT.89)SAVEA(I0UT)i(SAVEB(I0UT»K)»K=15»ll6) lOlU CONTINUE 99 CONTINUE END 43 // 1 2 3 5 6 1000 3000 SUBROUriNE ROOTS(NPOLY,N,A,ROOT) INTEGER OUT 0UT=7 IN=8 COMPLEX ROOT DIMENSION A(31),B(30),C(30) DIMENSION AG(3l),AGN(3l) DIMENSION R00T(l6),Hd6) F0RMAT(E15,8) FORMAT(!<•) FORMATdH ,7H ROOT= »El5,8,3X,2H+I ,E15.8) FORMATdH ,7H ROOT= »El5.8»3X»2H~I ,E15,8) FORMATdH , 7H R00T= »E15.8) ERR =0,0 FORMATd**) NPR0B=1 NCOUNT = 0 DO 3000 LL= 1»16 H(LL) = 0.0 ROOT(LL) = (O.OrO.O) CONTINUE LL = 0 WHITE(OUT#510)NPR08 510 FORMATdH »9H0OATA S £ T l 3 ) WRITE(OUT»507)N 507 FORMATdH »19H POLYNOMIAL M=N+1 DEGREE=I3//) WRIT-: (OUT»1002)M 1002 FORMATdH »5n THE ,I4,13H COEFFICIENTS ARE WRITE(OUT»1001)iA(I)»I=l»M) 1001 FORMATdH »E15.3) FIRST=Ad) DO 500 1=1,N 500 ••OOO 2060 1006 «HITt(0UT,6)R00T2 LL = LL + 1 ROOT(LL) = CMPLX(ROOT2,0,0) H(LL) = RHR N=N-i GOTO 590 359 DO 3bti |T=1,N 3oU AdT)=iJdT) GOTO 401 J20 R00T2=a(N-l)/B(M-2) 41J WKlTE(0UT,6)K00T2 wRirL(JUT,t>)R00T4 LL = LL + I ROOT(LL) = CfiPLX(ROOT2,0.0) H(LL) = RRR LL = LL + 1 ROOT(LL) = CMPLX(ROOT4,0.0) H(LL) = RRR G0T037J 390 P=.J(N-1) y=.:)(N) iJG0T0=2 GO TO 204 370 WRITE(OUT,506) bUo FORMATdH //) 509 CONTINUE Li- (NCOUNT) 3001,3001, 3002 3U02 CALL NROOT(ROOT»LL,H) 30J1 CONTINUE IUU7 CO.ITINUE RL TURN EfiJ tl7,8) 49 / V SUBROUTINE NROOT(X,N»R) COMPLEX XtZP REAL IMP,MM»M,IMZ INTEGER RI INTEGER OUT 0UT=7 IN=8 DIMENSION Xd6),Rd6),ARG(100),RZdOO),IMZdOO) B = 6,2831853 WRITE(0UTr3) 3X»18H THE CONSTANT l5»/) 3 FORMATdH »13H0THE POWER IS DO 21 I = 1»N IR = R(I) WRITE(0UT,4)IR,X(I)1 ZERO =0,00 ALIM = O.OOOOOOl 4 FORMATdH0#5X,14, 3X,2E14,7) 21 CONTINUE DO 5 J = l.N RJ = R(J) ZP = X(J) RR = 1.0/R(J) RP = REAL(ZP) IMP = AIMAG(ZP) MM = RP**2 + IMP**2 M = SQRT(MM) RHO = ATAN2(IMP,RP) ROOT = M**(1./RJ) ANGR = RHO/R(J) RI = R(J) DO 10 JJ = l»RI C = JJ -I ARG(JJ) = ANGR - K B + O / R C J ) 10 CONTINUE DO 11 JK = 1»RI AA = ARGCJK) RZ(JK) = ROOT*COS(AA) IM2(JK) = R00T*SIN(AA) ARZ = RZ(.JK) AIMZ = IMZ(JK) ARZ = ABS(ARZ) AIMZ = ABS(AIMZ) IF(ARZ-ALIM)101d01»100 101 RZ(JK) = ZERO 100 I F ( A ! M Z - A L I M ) 1 0 2 , 1 0 2 » 1 0 3 102 IMZ(JK) = ZERO 103 CONTINUE 11 CONTINUE WRITE(0UT#25) 25 FORMATdH » / / / ) WRITE(0UT,12)RI TEXAS TECHNOLOGICAL COLLEGE LUBBOCK. TEXAS tIRPAOV 50 12 FORMATdH ,5H0THE rI2,10H ROOTS ARE,//. IIH ,3X,10H REAL PART ,5X, lOH IMAG PART DO 22 I = If RI WRITE(0UT,13)RZ(I),IMZ(I) 13 FORMATdH ,E14.7, 2X. E 1 4 . 7 ) 22 CONTINUE 5 CONTINUE RETURN END y ./) SUBROUTINE PARFAO(N,A,S»L»C»R.NFORM) COMPLEX S,C,0,PN COMPLEX R DIMENSION R{20) DIMENSION A(31),5(16)iC<20),PN(20),D(20) INTEGER OUT 0UT=7 IN=8 w:aTE(0UTfl2) 12 FORMATdH ,4aH THE PARTIAL FRACTION EXPANSION COEFFICIENTS ARE/) DO 100 1=1,N 200 DO 210 K=1»N IF(I.EQ.K) GO TO 209 D(K)-S(I)-S(K) GO TO 208 209 O(K)=1.0 208 IF(K,EG,1) GO TO 210 D(K)nD(K)*0(K-l) 210 CONTINUE COMMENT* CALCULATION OF fiUMERATOR GO TO (215,315)»NFORM ^15 DO 220 K=1»L IF(K.EQ,L) GO TO 219 PN(K)--A(K)*{S(n**(K--l))+PN(K-l) GO TO 220 219 PN(K)= A(K) 220 CONTINUE 60 TO 330 315 00 320 K=1»L IF(!<,EQ.l) GO TO 319 P;N(K) = {S(I)-R(K-1))*PN(K-1) GO TO 320 319 PN(K)=1.0 320 CONTINUE 330 v:(I)=PN(L)/0(N) WRITE(0UT»11>I»C(I) „ ^v ou T irtc fly* 11 FORMATdH ,1.0X,2HK(.I2.'!H) - ,E15.8,2X,2H I.E15.8/) 100 CONTINUE RETURN END 51 ^ SUBROUTINE EV'ALH(S, W, T ,N,NTYPE»C »D, AA ,H) COMPLEX C . D , A A , H , S J W , Z E T , S DIMENSION Sd6),C(16),D(16), A A d 6 ) H=CMPLX(0.0,0,0) SUW=CMPLX(0,0,W) WT=W*T CO=COS(WT) SI=SIN(WT) ZET:::CHPLX(CO,-SI) GO TO (19,21,23),NTYPE 19 DO 20 /.rl.N 20 H=C(I)/(D(I)-ZET)+H GO TO 30 21 00 22 I:;1,N 22 Hr.C(I)/(l,0-O(I)*ZET)+H Hr;H«{1.0+2ET) GO TO 30 23 DO 24 1=1,N 24 H=AA(l)/(SjW-!-{I>)+H 30 CONTINUE RETURN EMQ y c C C C C C C C C C C C SUBROUTIME TPLCT (T f NM, ?.F, F, FO; SYM, PC, PM) OtflNlTION OF TPtOr ARGUMfNTS T = TJME IN pf^INTER ROWS NM = MAXIMUM N'JM'3Z:'i Of" CURVES C:::iil3 PLOTTEO, NM N Q T GREATER rH^rJ 9 SF = NAME OF f-CAtE FACTOR ffATfriX F - NAME OF T:;:; " U N C T T Q N M A T R I X ti£!NG PLOTTED FO = OfUGIN MA'r'raX ill PRINTER COLUMNS SYM = T IF NM SEPARATE PRINT SYMBOLS DESIRED. PC = F IF TPl.OT PLOrS, T IF *MAIN* PLOTS INTEGER OUT 0UT=7 IN-8 INTEGER WIDTH,T WIDTH-129 DIMENSION 8(20)»PM(129) 0 i MENS I ON F (; !M) »SF ( • ' 'M) , FC (i\,M) LOGICAL SYMfPC DATA COT,BLANK,AST»0H,FUUS/1H,,IH »lH*,lHOdH+/ DATA B d ) ,B(2) ,8(3) »B(tn ,G (^3)/IHl, 1H2 ; 1H3,1H4, 1 H 5 / DATA aJ6) ,8(7) ,8.(8) ,8 (9)/i'ri6 j .i:i7i IHS . 1H9/ IK(SYM) GO TO 16 DO 17 J=1,NM 17 6(J)nAST 52 C C INITIALIZE PRINT MATRIX TO BLANKS C 16 DO 3 J=l,WIDTH 3 PM(J)=BLANK C C DO 2 Nrl,NM K = F(N)*SF(N) + FO(N) IF(.NOT.((K.LE.WIDTH).AND.(K.GE.l))) GO TO 2 PM(K)=8(N) 2 CONTINUE 111 CONTINUE C C FILL IN T AXIS COORDINATE IN PRINT MATRIX C DO 21 J=l,WIDTH FJ=J DO 21 N=1,NM IF(IFIX(FJ).NE.IFIX(F0(N)))G0 TO 21 IF( ; ( T / 1 0 ) * 1 0 - T).EO.0) GO TO 22 PM(J)=DOT GO TO 21 PM(.J)=PLUS 21 CONTINUE ^ . • > C C FILL IN FUNCTION AXIS EVERY lOO TIME UNITS IF(T.EQ,1) GO TO 112 I F ( ( T / 1 0 0 ) • 100 - T)llll»112,llll 112 CONTINUE JP=5 DO 11 J-1,WIDTH IF(J.EQ.JP) GO TO 14 PM(VI)=DOT GO TO 11 14 PM(J)=PLUS JP=JP + 10 11 CONTINUE 1111 CONTINUE IF (PC) GO TO 20 WRITE(0UT.15) (PM(J) ;J=:l r WIDTH) 15 FORMATdH ,129Al) 20 CONTINUE RETURN END APPENDIX B: INPUT DATA FORMATS The following pages illustrates the exact formats for all necessary input data to run a filter program using FILRSP. 53 54 -Q- II ^ I.. '"' I)l u. II 2 y. -4 l» l-H uJ i J cJ cvl > t-i e- o ft. LL 01 H Q H C3 ll II 0- ;z M 3 J 11 ^ ll < il I w OJ -€>: (0 V v^' ^ :z Jl Jl ^- "5 < u. u. tO in J A. lU o ^^ o Q. UJ 75" (V Q 4 o II 1" II 1 ^ }-' o D < J '--0-1 1.L tU •p 4_ ifj J Jr z JtL 1.'- .J Q lU a 55 V/l)e re M - Order of numorator (13) N = Order of Denominator (13) IP = Number of p o i n t s t o be p l o t t e d (13) NTYPE = Desired program action (II) Type - Action 1 Standard Z-Transform 2 Bilinecir Z-Transform 3 Analog Response only NFORM = 2 1 DFORM = 2 1 I f numerator i s f a c t o r e d (II) I f nurfsrator i s not f a c t o r e d I f denominator i s factored (II) I f denominator i s not f a c t o r e d WS = Sampling frequency in RPS (F8o3) WMIN = Minimum frequency t o be p l o t t e d (F8,3) WMAX = Maximum frequency t o be p l o t t e d f F8c3) WCO - Desir-ed cutoff frequency f o r d i g i t a l f i l t e r (F8.3) Coefficient formats are E13o7 Root Formats aix; E 1 5 . 8 , 2X, E15,8 APPENDIX C: EXAMPLE OUTPUT DATA A complete output data listing is enclosed on the following pages. 56 57 THE INPUT DATA Iron THI S PPOBLEM IS ELLIPTIC FILTER - bIL INE AR Z~TRAN5F0RM CONTROL DATA M= 10 N= 11 I P = n i NTYPE =2 NF0P>^ = ? OFORMz? CONTROL DATA WS= 3, 0 AMnj= >/MA X= 0.0 2.0 WCC= 1,0 PLOTTER DATA F0( 1)=1 5. F0(2)=(65. F0(3)=5,0 PLOTTER DATA SF( 1)=1 .0 SF(2)= 0. 1 SF(3)=100. NUMERATOH ROOT A S(l) = 0.0 I 1. 0695414 AS(2) = 0.0 I-- 1 . 0695414 i>(3) I 1. 1009005 = 0.n A S(4) I = 0.0 - 1 . 1009005 A S(5) I I. 1946271 = 0.0 A S{6) I 1 . 1946271 = 0.0 A S(7) I 4652816 I. = 0.0 A !• S(8) 4652S16 1 . = 0.0 A I S(9) 5031313 2. = 0.0 A I. 5031313 (10) ? . = 0.0 AS I 0010752 b(l) 1. = -0 .006913 DENOMINATOR ROOT I- 1 . 0010752 S(2) ----0 .006913 I 0. 9756431 b(3) =-0 .0257616 !•- 0 . 9756431 S(4) = -0 ,0257616 I 0. 9063786 b(5) -"0 .0615122 !•-n.9063786 S(r,)--a .0615122 I 7504391 S(7) = -0 I 0. 7504391 s(a)= -0 .1269215 ,12(^9215 I- 0 . 4483675 i)(4)=-a .2142976 I- 0. 4403675 (in) --0 .?142976 I- 0 . 0 S (11) =-0 .2611853 p. THE PARTIAL FKAC rioM EX PArjSION COEFFICIEN TS ARE K( 1) = .42S6b700+0l -.52365968+01 K( 2) = ,42566700+01 ,52365970+01 K( 3) = -.37206656+01 .25811197+02 K( 4) = ",37206654+01 -.25811197+02 K( 5) = -.27370689+02 -.610699:^4 + 02 K( 6) = -.27370689+02 ,61069924+02 K( 7) = ,ii096nf-6+n3 ,89354376+02 K( 8) = .11090045+03 -,893S4373+02 K( 9) = -.23252^.71+n3 -,72419789+02 K(10) = ",23252270+03 ,72419789+02 K(ll) = .29779309+03 -,86149362-06 58 THE DIGITAL FILTER COEFFICIENTS ARE C( 1 = ,33214965+01 ,51640595-00 0( 1 = -.49784774-00 ,86030636-00 C{ 2 = ,33214965+01 ,51640590-00 D( 2 = -.49784774-00 ,8603^^636-00 C( 3 = -,12035866+02 ,52304513+01 0( 3 = -,47065882-00 ,85630556-00 C( 4 = -,12035886+02 ,52384514+01 D( 4 = -,47065882-00 ,85630556-00 C( 5 = ,17778919+02 ,2^^966240 + 02 D( 5 = -.40008587-00 ,85112108-00 C( t> = .17778919+02 ,29966240+02 U( 6 .85112100-00 = -.40008567-00 C( 7 .79693464+02 = ,60459044+01 0( 7 .81813364-00 = -.23219827-00 C( d .79693461+02 = .604^:9044+01 D( a ,81813364-00 = -.23219827-00 C( 9 .11270700+03 = -.l057'4512 + 03 D( 9 .62547459-00 = .10435222+00 C( 10 .11270700+03 = -.10574511+03 0( 10 ,62547459-00 = .10435222+00 C( .593157U0-06 = .20503758+03 11 Li( .00000000 = .37704422-00 11 ,52203764+03 THE DC RESPONSE BEl'ORE NORWALIZATION IS 59 \r c^i c<, ; * 1a' ct I f . 1CO CM »^ 1 1fO 1f . r o ro 1 r^ 1»0 1ro 1 1 ro O lO l O ro K; K-; I r o f O r^ r^ C 1CD O 1 1O io o o o O o o c: •r At 1in ^c I T At O vO ro ro (T CC e vO . * CsJ vO ^-4 • H CO , - • ro o ro xC ro i n ro o CM 0^ r-l CM tVJ r-t (c^ i n CVJ CVI ( O —« : t 1cr 'w-^ . - t vO CO r- r ' At At O 0 ^ > - ^ - o l n ( ^ ^ o ^ - ^ - . * ( M •M '>o CM cr (T .-t i n CM v-* i n CO ,"• l O . - 1 At c r f ^ A t O vO O A t CC - • At 'C vO ro oc K> o • t f O r j o v D f O O ^ f O O vD m CVJ vJD tr o> r~- -d- 00 o O CC vD CO o CM A t i n r>- CO CO Z i n C3> CO r-» i n CO CM i n tor-4 , , - 1 'cviCMrvjrOfOrorOAt :*• ^ A t i T At ::i O 1ro r- + + + + + + + + XMAGI 1 :'^ t o 1r^ rr r^. r^ o o o o o o o o oo O fvJ h+- + + + + + + + + + + + + ++ + + + CD flC in O ro »H O CC CM xO (Vi ,-t ec : t o ^ :* ^-^ »-» o o 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lO CVI <-^ •-« c M m r - A t o v O r o c c t C A T A t f ^ ( T At ro tc xO «L cr ro ro *-« oc CJ CO 0^ CM cr CM ro in 1-i ^-i O r» < CO «o r- At o 0 0 OJ ( T At o , - t cr o 00 ro ro rc vC o - O At vC n .-t CM i n ro \0 in CD ^ ^ xD i n xO cr r o r - C M O r O x O r O i t C M l T ) cr cr o^ c^ CO xO At .-• I*>- ro r^ •-• At • ^ O O CO vO r O r-« At At At A t A t At 1 1 1 1 1 At r o » o Cij CJ * 1 1 1 1 1 1 1 1 •^ xO OJ , - « . - • CM r O 1 1 KlK)rOrOfOiOrOrtK7fOK>fOiOfOfOrOK)iOiO{Vl(\l<\iCVJrorO(OK)iOrOtOrOfOi^rOK)f'')»OiOK) o o o O O o o c o o O o o o o c o o o o o o o o o o o o o o o o c C O o o o o + + rO + JO + O+ + 1+^ »-• + i+n CC + CO + —< + + l+^ h +- + + + + + + + + + + + + + + ++ + + + + + d+ .+- t vO •H At r - c M x O L O i n r o c M i n r - ^ rO CM A t r*cr o •f d - CVJ f Vi CVJ o ;* CVJ CO »J3 CM r o vO xO : t cr i n r^ cr w-\ cr r H vC in r"> r- tf) »-4 in in >o »-« r~ i nCO i CVJ 0 0 i n CM in \0 ro vo COO U") CM ro CM f *C O o in CO CVI vO i n CM £) cu vO t o CM r O O^ • - • a ; t r- vO ^ r - CO r - o xD h - At r-J CVJ o in vo i n o r o f O r^ ro ro ro o r- CM .O (T o ro O A t ro ^ -^ CVJ ro ro O OJ vO c CM O At i n »-t ^ ( T CO A t »-H 0 - Hm o 1 + + + + ++ 1 1 1 1 1 1 1 1 1 1 i CM CM CM CVi CV CM CM ro ro r o roro C o o o o o c o o o o o n ro to »o ro ro lO ro rO lO ro o O O CD O o o o o o o + + + + + + + + + ++ + + + + + + + + + + + + ^ ^ .-( CO i n CM vo CO cr \0 r- x\j -o in i - t o h- cr »-• l O CM CM t ^ O xO r-4 ro r- c CC c - <\.€ - 4 . H »-« .-4 O O o c: O O o c o o c o c c o c c o oo o o a o o o o o O O c o o o 1 1 1 1 1 1 + i f i 1 1 1 1 1 1 1 1 i 1 (j^ i n i_n. \ti — o in ::t - in cr ^ ;3- vO cr CJ (T m ^ cr OV C\J ^ m r ^ i n c o v D o c v o v o r - CM CO CM At r - c r r-< _ Qx m ( O i n CVJ _ i T > c^ o At - cr o o vD h - in o CM »JD h - vJ3 O^ f ^ f O i n r*- .h- cr- cr At l O r - r^; vC At .-< o d- vO vC u" At CT' : t CM At OPi CVJ •-< f^ i f CO : i " ("J -C UO J(T r- o (*j ^ ^ Ki c- invX3 ( T . n • ' ) ' \ ! ::*• c r CT' -•• f ^ CC 0O N O ' - ^ ' - ' O O — I O < - t ov cr- CO ^0 to c^ vn --< vO.-» J»c^c^'0^o^cocoJD^• 1 1 1 1 1 1 1 1 1 1 1 #—X i n vD O xO :i oc CM - - ' O CM rO i n ro cr C^ CJ CC At O d C3 AT Al- O At cr i nw-4 •-• *.* a ro r o At r^ in vC' CC " - I c ^i n •-I xO in cr ro o v-4 K5 r - xo r o t o c r x c ^ r o c M x c c r r » m x o c r r~ ; t o in cr ro r ~ . - « o ^ i n x o r ^ ^ r ^ r ^ r - i n c r At At At t o CM CM • - ^ ^ x D C M — t i n r O O ^ ' H C M r O A t ^ • • • 1 1 1 1 1 o o o 0 o o 1 1 1 t o o CC J": CM1 • - • ( ro in cr CM CM 1 r*At m cC -t o S\ Lf) _^ c C^ cr • 1 • • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o oo o o o o o o O O O o o o o o o o o o 0 o o o 1 1 i 1 1 1 1 11 1 1 1 1 u^ cr cr r^ xD r^ i n CM (T K^ I T r - - • 1 C At m xD in cr I f xc I f1 c r uO CVJ c r At O in vO oij c cr CC At xO —• O CM o o CM i n »-» ro C1Vi n ^ o xO 1 •-* c' o m cr t""; r^ c IP • n xO CC CM ' CC ( ^ CO : t 1 o >• -o r^ cr cr in CC cr CC • c' cr J* cr 0^ ( r cc 1*- 0 n : t t a' c cr cr cr cr cr cr (T' cr O' cr (7" .96013776-00 .96634033-00 .97314536-00 1 1 i o o c ^ o o o o o 1 1 1 1 1 1 i 1 1 v-t OJ 0 0 w-< ^ O O O O o ^ + en r- r~ o •-4 vc r- c ro ro CM CC in At o CM (O cr r^ r- ' D r - i n A t CO 0 ^ At i n CM o r^ o ic r- m 0 0 At I f ' CM a A t I f ) xO xC At - ^ CJ c r .96220032-00 ,9577180'+-00 .95365901-00 .95015889-00 .94734563-00 .94533751-00 .94424061-00 .94414591-00 -94512480-00 1 • 0 < I o3>x)r>-.OAtooa-4o(raor-'.nto^'Hc:>crcOvDindrro;M'4oa|r-^jit'OAi^o7vr. oil-Jln^•CT»•-•^Oln^-cocoJt-ocoooJAtu^f^cr-<'Olnr^(r'-^cMAt^o^orM^/Ooo!^•-• g5^x^l^binSSi^25S?Q3cvjr--ina;roi-cMxoo^Oior^^ina^^^^xOojhO^ o o o - r - ^ c o r x i o j . ^ r O A t :t>;-)-ninxOxor>f-r- c o x i c r c r o o o - ^ ' - 4 f M C M ^ " ] " ^ = ^ ^ ' ^ - ^ ^ ^ ^ ^ ^ ^ ^ . . - < ^ ^ ^ ^ - 4 ^ - « orMAt.£»:0O0jAtnX!orv!*naDor>jAtx0cCOCMAtx0c0OCMA^xC<0O(M-tx000OfM^_0 o o o o o -4 ^ . 1 -4 - i CJ ,M CJ -J CVJ -o ro ro .-o " ! d- At -t ^ At -n -n .-^ J-) n ^ 0 X3 o £) r- r^ f- r• • , , , • . . . . « • • • • • • • < 60 »o c o oo o o o o o o o o o o c o o o o o O O O in ro ro o At CC h . CO At . - I .-« CM r o r o CO CM r o vO o At i n + + »Oror^ 1 1 1 1 1 1 1 1 1 1 1 ^ f ^ c v j r ^ > x ) » - i A t r o r * - ' - < r o c c x O x O h - ( r t n O (T O C! (T. i n At I T CJ O \C xO c v i c r c M r o o c r » - i o c ^ r o — t . - i d - ( r c r c r o c D ^ r O d - c v K i A t i f . AtcrrOcc • - t c M A t < y > i n r o r > - i n : t « H i n a r o : t — t c M o c j ^ i o i ^ A t o f o r ^ i v C i t crA:r i n » - i o r o c r r ^ o ; t » - « » H ' £ i n . - « r - r - . H r o AtcjAtr-Atrof>-.tf^Atca-»-ir»-CUin.-«rO O o < 3 ^ c O ' - « A t i n a A t r o r ~ c r ^ O A t (•^ - ^ A t nxoeocor>-.* oxDo^ooocrxocoAtin xO in Pi OJ vi) r j ;0 u3 vo f^ rI .-«^^.-(^<\JOJ^^^^^^ o o o o o o o o o o o o o < i t 1 1 1 1 1 1 1 1 1 1 in^;^- cMinmrocM^Hcor-o,-! r»oiocoAtocjrO(Oin»- - t in ct .*•") ^ CM CO »-t CM CM ••^ •-» •-< xCroa3'-^cciOxo.H,-»..4 I I I I i I ro ro o c o o o o c o o o o c o o o o o o o o o c c o o o o o o o o o o o o o c o o o o o + + +- + + + + + + + + + + + + + + + + + + + + + + + + + + + + -^ cr + o+ C+M ^oxCro^.- - o « O x o r ^ c r c r i n x o ^ A t r ^ e c L n c r c c CM c ; x O C J C C O x O C M r - O d - ' D r O O ^ U 3 0 » O C r v O If c M o i n o ^ i n c c a e c f O ^ o o o c M A t r ^ i n c r A t c r CV « - r ~ c r e o i n ^ c r r ^ ' - < r - r o A t r ~ r - i n ! o c - i r , ro r-i •-*CT'f^inrocr»-«m«-i—•••^•H^^.-i.-iCxicMf').-". I I c OO—l.:tOOOO o O O O O O O O O i 1 t 1 1 + 1 1 1 xD ro o CVJ ro xD •-• CM ^ r- —< c^i oc cr cc< —» c .^ ro r o c o i n A t r - i n o ^ t At r - i n i n c o r o c r c r o x ; CC i f o — t r o o ^ c u i cr oo CVJ o o -< CM ro -^ to c r A t c o i n r H r - c r ! ^ At r O O j r ^ x C ^ i O A t ' - i I 1 I o o o o I in i If CV If) h in cr xO At O —< ^ X L'> if) CM At i n r- cr cr cr I I I I t i l l I I I I I I I I I I 1 1 l++-«- + + + + + + + + + + + + + + + + + + + + + + + + + + + + r-f^tocroccAtr-crcMr^ocxccr coro(nr~or-f^crtOAtAtcr«-toxOAtxorocjoc O x O « - * i n ^ A t r O C J C J C C » ^ — 4 C M 0 J At AtAt o x o n r - c r r o o x o r o i n i ^ ) i o r o i n o - ^ i n x O i n A t A t r o c M i n o — i c r » - * . : t r ^ in o c r c M L n i ^ . - « h - c r « - i r - < : : f i n c r t o c r v O i n c j , - » c c x o c O ' - ' A t i n — i r ~ - . - » i n c c c r " - l o j r o •-icrcr-^incotrr^.•Oh-•<-»»<•) » - l o . r ^ t o . ^ r ^ • o ^ tC\OOCCO^xO»-'»-tcO>Cro«-»—ixO c . c j a c M ' - ' c o a c M c r - h - ^ — r r o r ^ f o — i r - r r ^ c r c M h-AtocrxCcrroinocjror~in--^ mAtCcClfinoifxOLOrOrOOtCcrOrOinxCir c v j o o o o c 7 » - ^ v O o o c M r ^ x o r > - crAtci^»-ta3r~xOxor-cocj^rr^xOxO —iroinincrf^cor-co-^'or-r^r- f^cp.-i n — < f ^ A t « - f r - c r r - ^ o i n c ; -•• o o xD cu c- At o i r l O r O C M C V J C M ^ - t A t r - O f O l f / C O o r j If 1 r^ c r- If. 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