Transcript
BioMedSigProcAna
Signals: Acquisition, Conversions, and Types Josef Goette Bern University of Applied Sciences, Biel/Bienne Institute of Human Centered Engineering - microLab
[email protected]
February 7, 2017
Contents 1 Signal Acquisition
1
2 Analog-to-Digital Conversion 2.1 Sampling and the Sampling Theorem . . . . . . . 2.2 Compressive Sampling . . . . . . . . . . . . . . . 2.3 Quantization and Adc Architectures . . . . . . .
5 7 16 24
3 Signal Types
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4 System Types
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Notation and Symbols
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References
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Josef Goette, 2007–2017 All rights reserved. This work may not be translated or copied in whole or in part without the written permission by the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software is forbidden.
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Signal Acquisition
You might also want to consult [EBB05, Section 10.4]. Biomedical Signals and Interferences • biomedical signals – are often very small signals (weak signals) – typically contain unwanted interferences or noise • noise – extraneous ∗ thermal noise in sensors ∗ 50 Hz interference ∗ ... – intrinsic ∗ from adjacent tissues or organs ∗ example: electro-cardiogram (Ecg) signals may be affected by the bio-electric activity from adjacent muscles ∗ ...
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Signal-Acquisition Chain condition
sense
store
process
quantity to be observed
acquire
sense: sensing the reality by sensors condition: analog signal conditioning acquire: data acquisition system including analog-to-digital (Ad) conversion store: data storage and display process: digital signal processing to suppress noise and to extract specific information
In more detail, the blocks in the above diagram may be characterized as follows: 1. The sensors convert the signals to be observed—which might be of electrical, magnetic, chemical, mechanical, acoustical, optical, or another nature—into an electric analog signal. These sensors are the interfaces between the biological system and the electrical recording instruments.
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BioMedSigProcAna As an example consider Ecg signals: They are measured with electrodes that have a silver/silver-chloride (Ag/AgCl) interface attached to the body; these electrodes detect the movement of ions. It is important to make sure that the sensor used to detect a biological signal of interest does not adversely affect the properties and characteristics of the signals it is measuring. 2. Analog signal conditioning usually means that the signals are amplified and filtered by analog means. Operational amplifiers are electronic circuits that are used primarily to increase the amplitude or size of biological signals, which are often weak; a thousand-fold boosting is not unusual. Analog filters remove noise, compensate for distortions caused by the sensor, and limit the frequency band to frequencies smaller than a certain frequency fB . Frequencyband limitation becomes important in the process of sampling of analog (continuous-time) signals to generate discrete-time signals; see our discussion later in the present document. 3. The data acquisition system employs analog-to-digital (Ad) converters; they convert the analog signal at their input to a calibrated digital signal. The Ad conversion process might be thought of a two step process: First, there is a time-discretization (sampling), and second, there is an amplitude-discretization (quantization). Whereas the sampling process is a linear operation and does not introduce distortions if it is carried-out correctly, the quantization process is non-linear and does introduce distortions. We below discuss in more detail discretization and quantization. 4. Digital signal processing is applied to reduce noise and 2 SigAcqui
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BioMedSigProcAna other interferences in the stored digital signal, but also to extract useful information from the signal to improve the understanding of the physiological meaning of the originally measured quantity.
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Analog-to-Digital Conversion
The analog-to-digital (Ad) conversion process might be interpreted to consist of the two processes of sampling and of quantization. We briefly discuss below in Subsection 2.1 the classical sampling process, that is, the essentials of the Shannon/Nyquist sampling theory, and in Subsection 2.3 the quantization process. Concerning literature on standard data acquisition, we refer for classical Ad- and Da-converters to [Raz95];1 but we note that you might also want to study oversampling Ad-conversion techniques2 in, for example, [NST97] or [ST05]. The article [LRRB05] discusses commercially available analog-to-digital converters3 and their key performance parameters. Over the last few years, an alternative sampling/sensing theory has evolved, which seems to contradict the Shannon sampling theorem, in that it enables faithful signal recovery from data generated with highly sub-Nyquist-rate sampling. This new theory is named compressed sensing or compressive sampling and is based on the observation that the Nyquist rate is a sufficient but not a necessary condition. We given in Subsection 2.2 a short account of these emerging new ideas. As a starting literature, you might want to read [Bar07] and/or [CW08].
1 Such
classical converters are often also called Nyquist-rate converters. Ad converters are usually called Σ∆ converters; other authors name them also ∆Σ converters. 3 We note that the article is from 2005. Therefore, it will not describe the most recent devices. 2 Oversampling
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BioMedSigProcAna Yet another family of alternative sampling methods that seem to violate the standard (the simplest form of the) sampling theorem is what is often called “event-based” (or Lebesgue) sampling. To get a quick overview, you might want to read the magazine paper [GAG+ 13]; we mention that the esophageal longterm electrocardiography project at our HuCE-microLab implements in its analog-to-digital converters such a Lebesgue sampling, see [MNH+ 12b], [MNH+ 12a], [MNH+ 15], and [MNH+ 17].
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2.1
Sampling and the Sampling Theorem
The sampling process is often also called the continuous-todiscrete (C2d) conversion. Sampling: Involved Signals x(t)
x[n] = ˆ x (t = n · Ts )
C2d Ts = 1/fs
C2d: ideal continuous-to-discrete converter x(t): continuous-time signal Ts = 1/fs : parameter specifying the uniform sampling of the continuous-time signal x(t) every Ts seconds; fs = ˆ sampling frequency x[n]: sequence = ˆ discrete-time signal (ordered sequence of (real) numbers)
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Sampling: Time-Domain Appearance • ideal continuous-to-discrete conversion • “Dirac-sampling” • Ts = 1/fs sampling interval x(t)
x[n]
........... ....... ............ .............. ..... ..... ... ... .....1.0 . . . . . . . . . . . . .... ................... ... ... · · · .... .... . . . −Ts
0
Ts
2Ts
3Ts
•
t
- C2d
1.0
−1
• 0
•
•
2
··· n 3 -
• 1
Note our notation: Whereas a continuous-time (analog) signal x(·) uses parentheses for its time argument, the discrete-time signal x[·] uses brackets; this notation is most often used in the signal processing literature. The discrete-time signal x[n] is a sequence of numbers whose individual values are the samples of the analog signal. The integer index n indicates the order of the values (samples) in the sequence. Note that this index n—the discrete time—is dimensionless: once the samples are taken from the continuous-time signal x(t), the time scale information is lost. The discrete-time signal is just the sequence of numbers carrying no information about the sampling period, which is the information needed to reconstruct the time scale.
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Sampling: Time-Domain Appearance (Matlab) plot for continuous−time signal 2
1
fs=200 Hz
amplitude
Ts=5 msec 0
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−2 −0.01
−0.005
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time t [sec]
plot for discrete-time signal
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amplitude
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-1
-2 -2
-1
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time index n
The plot in the second panel of the above figure is often called a “lolly-pop” or “tinker-toy” plot; Matlab’s command stem() generates such plots.
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Sampling: A Most Simple Example • continuous-time signal x(t) = A cos (2πf t + φ) • sampling by rate fs = 1/Ts • discrete-time signal x[n] = x t = n · Ts
= A cos 2π f · Ts n + φ = A cos 2π fˆ n + φ |{z} | {z } = ˆ ω ˆ = ˆ fˆ = A cos ω ˆn + φ
fˆ = ˆ f Ts = f /fs = ˆ normalized frequency ω ˆ= ˆ ωTs = ˆ normalized radian frequency
Note that the normalized frequency fˆ—the discrete-time frequency—is dimensionless: [fˆ] = [f · Ts ] = Hz · sec = 1. Correspondingly, the normalize radian frequency ω ˆ —the discrete-time radian frequency—has just units of rad, that is, it is also a dimensionless quantity. These observations are consistent with the discrete-time—the index n—being dimensionless. So again, after sampling the time-scale information is lost.
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Sampling: Frequency-Domain Appearance • band-limited continuous-time signal (limited to maximum frequency fB ) • correct sampling • fB is smaller than half of sampling frequency fs spectrum of Ct signal
....... ...... .... ....... ....... ..... .... ............ ....
··· −2fs
−fs
0
···
fB
fs
2fs
f
spectrum of sampled signal
....... .... ... .... .... ... · · · ... ...... ...... .... .. ...... .. −2fs
....... .... ... .... .... ... .... ....... ....... .... . .... .. −fs
. .......... ...... .... ...... ....... ..... ... ......... ....
. .......... ...... .... ...... ....... ..... . . . ... ....... ...
−fs /2
0
fs /2
fs
. .......... ...... .... ...... ....... ..... · · · ... ......... .... 2fs
f
Note that we abbreviate continuous-time signals as Ct signals; sampled signals are called discrete-time (Dt) signals or sequences (ordered sets of real numbers).
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Sampling: Frequency-Domain Appearance (2) • band-limited continuous-time signal (limited to maximum frequency fB ) • incorrect sampling ; aliasing • fB is larger than half of sampling frequency fs spectrum of Ct signal
........... ......... .... ............ ....... ..... . . . . . ....... ....... ... .... ........ .
··· −2fs
−fs
0
fB
··· fs
2fs
f
spectrum of sampled signal
.................. ........................... .......................................................................................... .......................................................................................... ....... . ................ ...... ........... ............................................................................. ........... ...... ................................................................................. ........ ......... . . . . . . . . . . . . · · · ........... ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ......... ............. · · · . . . . . . . . . . . . . . . . . . . . . . . ............ .. ... ................. ... .................................................................... ................ .................................................................... ... ................ .... . ...... −2fs
−fs
−fs /2
0
fs /2
fs
2fs
f
Sometimes it is distinguished between aliasing and folding: In aliasing, a higher frequency appears as a lower frequency with correct phase, in folding a higher frequency also appears as a lower frequency but with a phase shifted by 180 degrees. You might think on the wheel of a stage coach in a western motion picture: If the wheel is turning sufficiently slow, you correctly see the movement of the spokes; if the wheel turns faster, you see the spokes and with them the wheel moving in the opposite direction, which is folding; if the wheel turns even faster, the spokes again move in the correct direction but you see the wheel turning in a too slow velocity, which is aliasing.
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Sampling: Frequency-Domain Appearance (3) • broadband continuous-time signal (not limited to a maximum frequency) • continuous-time anti-aliasing filter • practically correct sampling • filtered-signal bandwidth is smaller than half of sampling frequency fs spectrum of Ct signal
................................. f.....B . ..... ... .. .............. .. ..... ....... .... ....... ............................ . . . ··· ··· . . . . . . . . . . . . . . . . ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... . . f anti-aliasing filter ..... ........ ................ ................ . ..................... ... .. ...
−fs
−fs /2
0
fs /2
fs
spectrum of sampled signal
. . ....... ......... ......... ...... ....... ....... ... .... .... .. .... ....... ....... .... · · · .... ....... ....... .... · · · ... ...... ...... .... ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... −fs
−fs /2
0
fs /2
fs
f
We mean, in the above slide, by “practically correct sampling” that the analog signal conditioned by the anti-aliasing filter is practically band-limited, that is, its frequency components above half of the sampling frequency are sufficiently attenuated. In the sketch above we have represented the useful signal spectrum by the butterfly shape, and the noise by the low dotted spectrum which is very broadband. Because the sampling device—the C2d block—sees that total signal, useful signal plus noise, we understand that the anti-aliasing filter is absolutely 2 SigAcqui
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BioMedSigProcAna needed to prevent stacking of the broadband noise in the baseband of the discrete-time signal; note that the useful signal itself would not need that anti-aliasing filter, but (most often broadband) noise is always present. To additionally appreciate the need of an anti-aliasing filter, consider the following very practical situation: A system with analog and digital parts uses a mother clock with frequency fm ; the digital part generates out of the mother clock a sampling clock fs = fm /M with M and integer. Without using an anti-aliasing filter, the mother clock signal with frequency fm is folded down to the base-band and acts here as an interference. (To the vicinity of which frequency in the base-band is the mother clock frequency folded down?)
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Sampling: Shannon Sampling Theorem given: • continuous-time signal x(t) • frequencies f in x(t): f ≤ fmax then: reconstruction of continuous-time signal x(t) from discrete-time signal x[n] = ˆ x(n · Ts ) (sequence of samples) if: fs = ˆ
1 > 2 · fmax Ts
We note that the sampling theorem gives the limit for a possible reconstruction, but it does not give an algorithm for doing the reconstruction. We also note that the theorem addresses the worst case, and that Nyquist-rate sampling is, in the absence of extra information,4 essentially optimal for band-limited signals. Concerning notions, we have the Nyquist rate, which is the minimum sampling rate that is necessary to sample a given band-limited signal; the maximally allowed signal frequency for a given sampling frequency fs —which of course is fs /2—is called the Nyquist frequency. 4 Section 2.2 introduces situations of extra information and how this extra information might be exploited by newer approaches called compressive sampling.
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2.2
Compressive Sampling
Basic Observations • many signals are naturally compressible • examples – natural images are well compressed by the Jpeg standard – natural images are even better compressed by the Jpeg2000 standard • exploit compressibility to devise novel methods for data acquisition – which generate far lesser samples – which, therefore, need lesser storage or less time for data transmission – which need far lesser time for data collection – but which require far more processing effort for signal reconstruction than traditional approaches
Note that the arguments given above just state that certain signals do not contain too much information and are, therefore, compressible. But also note that compression approaches such as the algorithms in Jpeg or in Jpeg2000 are procedures that 2 SigAcqui
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Big Question, Note the Provocation • is it possible to obtain the performance of adaptive compression from a fixed set of measurements? – surprising answer: Yes – surprising fact: measurements benefit form randomness • the previous statements seem to contradict 1. mountains of serious modern engineering work (Shannon, Nyquist, . . . ) 2. the fundamental theorem of linear algebra which states that “we need as many equations as we have unknowns” • ; “periodically reconsider the impossible”
We cite above Doug Williams, the editor of the Special Section in the Signal Processing Magazine form march 2008. We are well aware that we might have in a certain problem more equations than unknowns—resulting, for example, from more measurements than are degrees of freedom—giving us the freedom to look for least squares solution and enhancing robustness of our procedures against perturbations; but having less equations than unknowns . . . ? Are there certain hidden “equations” (additional information)?
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Sparse Signals • sparsity measured by number of “non-zero” elements • example: time-domain sparse signal • example: frequency-domain sparse signal • example: . . . • general: signal x[n] ≡ x sparse in a convenient basis Ψ x = Ψs , s= ˆ vector of coefficients, sparse ˆ matrix of basis vectors (columns) where Ψ = x= ˆ signal, in general non-sparse vector
We indicate above that sparsity is measured by the number of “non-zero” elements: If the signal vector x has N elements, then the basis matrix Ψ has N (column) basis-vectors, but the coefficient vector s has only k < N non-zero coefficients, and the signal is termed k-sparse. For practically relevant cases, we have k ≪ N . Also in practice there will not be a coefficient vector with all but k elements exactly zero, but, instead, these coefficients are so small that they must be considered to be below the noise floor.
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Example: Time-Sparse Signal signal in time domain 0.5
x[n]
0 −0.5 −1
50
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...and its spectrum (abs) 3.5 3
|X[k]|
2.5 2 1.5 1 0.5 0
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Example: Frequency-Sparse Signal signal in time domain 30 20
x[n]
10 0 −10 −20 −30 50
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...and its sinus−spectrum 20 15
X[k]
10 5 0 −5 −10 −15 50
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Example: Wavelet-Sparse Signal signal in time domain 2
x[n]
1 0 −1 −2
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...and its spectrum (abs) 30
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We note that the signal in the above plots is neither sparse in the time- nor in the frequency domain. However, it is sparse in a basis that we have built with Daubechies-wavelet filters of length 8.5
5 We have worked with the public domain Matlab toolbox sparco-1.2 from the University of British Columbia, Ca.
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Compressive Sampling • the k-sparse signals can be “sampled” (measured ) using m > const · k samples • there are m ≪ N equations for a signal having N elements • reconstructions by optimization procedures ; is a kind of non-linear filtering • minimum ℓ0 norm reconstruction ; is feasible but not practical • minimum ℓ1 norm reconstruction ; is feasible and practical • minimum ℓ2 norm reconstruction ; does not work
The ℓ0 norm of a vector is just the number of its non-zero elements; the ℓ1 norm is the sum of its absolute values (the “Manhattan distance”); and the ℓ2 norm is the usual Euclidean distance measure. Minimum ℓ0 -norm reconstruction is not practical, because the needed algorithms are combinatorial algorithms; in contrast to that, minimum ℓ1 -norm reconstruction leads just to a linear program, which is well suited for practical numerical computations.
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Quantization and Adc Architectures
2.3
Appearance of Quantization • at input: continuous amplitudes • at output: discrete amplitudes (finite number of states) • most often: the number of states (of quantization steps) is a power-of-two • quantization errors = ˆ non-linear distortions xq [n]
x[n] •
• •
x[n]
• •
- Quantizer
•
xq [n] -
• •
•
•
−1 0
1
n
n −1 0
1
2
3
2
3
The acronym Adc in the above title stands for analog-todigital converter. Note that the quantization errors are, as stated, non-linear distortions, but that most analysis approaches use a linear model describing the quantization effect as an additive white noise process which is uncorrelated with the quantized signal; for a more detailed discussion we refer to our later document [Goe17c]. 2 SigAcqui
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Adc Architectures • many (most) converters also contain the sample-and-hold circuit • successive-approximation architecture • dual-slope architecture • flash architecture • more special architectures and variants • Σ∆ architecture • non-conventional: event-driven Adcs
Most sample-and-hold circuits come in form of track-andhold circuits, [Raz95]: In one operation phase, the tracking phase, a switch connects the input signal to a storage element, and the storage element reproduces, after a certain transient time, the input signal. In the second operation phase, the hold phase, the switch is opened and the signal on the storage element remains constant and allows the subsequent analog-todigital converter to do its work. Successive approximation converters are also called algorithmic converters. There working principle is as follows: They create their n-bits number output in n operation steps. In the first step they decide on the most significant bit by comparing the input signal to a half-scale threshold and set this bit to one if the signal is above the threshold or let it on zero otherwise; so the output word of the converter after the first step would be either 100 ...0 or 000 ...0. In the second step, a digitalto-analog converter (Dac) converts this preliminary result back 2 SigAcqui
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BioMedSigProcAna to an analog signal, which in turn is subtracted from the input signal giving a rest; with this rest, the next bit is created and the process repeats itself until all n bits are produced. We see that these converter architectures keep-on splitting the voltage range into halves to finally find out the level of the input signal. Dual slope analog-to-digital converters base their decision on the n bits on time measurements—which can be made quite precise: A first phase of operation pre-charges a capacitor from zero to a voltage Vin by a current proportional to the input signal. In the second phase, a known constant current dis-charges the capacitor to zero; a counter measures the time needed for dis-charging, and this time (the counter value) is a digital output proportional to Vin . Because the technique is very precise and cancels most component variations, dual-slope converters are high-resolution Adcs; they are, however, relatively slow. Flash architectures are parallel processing structures and realize the fastest Adcs. They simultaneously compare the input voltage to be converted to a set of reference voltages to very quickly find the quantization interval the input voltage is in. The reference voltages might be realized by a ladder of resistors, which must be sufficiently accurate for the targeted resolution. We mention that the reference voltages might likewise by realized by capacitive voltage division. There are additional and more special architectures and variants and enhancements of the above architectures; we refer, as a starting point for self studies, to [Raz95], and to [MW98], and to the literature referenced therein. Whereas the architectures discussed so far are useful for Adcs that might operate at the Nyquist sampling rate,6 Σ∆ architectures are different in that they initially generate a very fast (much oversampled) sequence of low-resolution samples, which in turn is digitally post-processed with the goal to exchange 6 Note that when operating at or only slightly above Nyquist rate, the converters need an anti-aliasing filter with sufficiently high performance, that is, a filter that has a steep roll-off.
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BioMedSigProcAna speed (high sampling rate) for resolution (lower sampling rate for processed sequence that has higher resolution); the following pages 25 to 39 discuss these Σ∆-converter issues in more detail. On pages 40 to 44 we finally give a short account on event-driven analog-to-digital converters.
Σ∆: Oversampling • simplifies the design of the (analog) anti-aliasing filter spectrum of Ct signal
....... .... .... ...... ....... ..... · · · .. .... ... .. .. ...... ... 0
···
fB
f
fs
spectrum of oversampled signal
.............................................................................. ....................... ....... ....... ... ....................... .... ...................... .. ..... ........ . .... ...... ....... ..... . ....................... . . . . . . . ...................... · · · .. .... ... .. ....................... ... ............. ..... ....................... .. ...... ... . ....................
0
fB
fs
··· f
The notion “oversampling” means that the sampling frequency fs is much higher than the minimally required frequency 2fB , recall the sampling theorem stated on page 15. Note that the analog anti-aliasing filter has much relaxed requirements in that the required filter roll-off is less steep than in previous cases with Nyquist sampling, compare the graphics on page 13; such a flatter roll-off directly translates into a filter of lower order which is simpler to design, and needs less and lesser precise analog electrical components.
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Σ∆: Quantization Noise Power • assume equidistant quantization • total quantization noise σε2 depends only on number of bits • quantization noise is evenly spread over spectrum • in signal band [−fB , fB ] is higher or lower noise density quantization noise power for low fs ...................................................................................................................................................................... σε2 /fs
..................................................................................................................................................................................................................................................... .............................................................................................................................................................................. ...............................................................................
··· −fs
−fs /2
0
fs /2
··· f
fs
quantization noise power for higher f˜s
.............................................................................................................................................................................................................................................................................................................................................. σε2 /f˜s ... ............................................. ......................................................................... ... . . .
··· −f˜s
−f˜s /2
−fB
0
fB
f˜s /2
··· f˜s
f
The analog-to-digital conversion process introduces quantization noise ε[n] = ˆ x[n] − xq [n]; x[n] is the analog input signal, and xq [n] is its quantized version, respectively. The usually adopted model for this noise process is a white noise that is also uncorrelated with the converted signal. White noise means that the total noise power comes in a flat (white) spectrum Sn (f ) = const, and, starting from the noise spectrum, the total noise is its integral over the Nyquist band: total noise =
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28
const · df = const · fs = σε2 ;
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BioMedSigProcAna thus the constant level of the noise spectrum is σε2 /fs . We treat Adc noise in our document [Goe17c]. The figure on page 28 illustrates two cases: The upper panel is for the situation of Nyquist sampling fs = 2fB —the signal band extends to half of the sampling frequency. The lower panel shows an example for the same signal band but with a oversampling by 2 (f˜s = 4fB ). For the example in the lower panel we see that the quantization noise level is half of that in the upper panel.
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Σ∆: Classical Delta Modulation • a coded modulation system • a simplified modulation scheme in the realm of Pcm • a French Patent from 1946 (Us Patent from 1953)
pulse generator
···
···
modulator
x(t)
···
··· to channel
comparator
x e(t)
integrating system
The acronym Pcm stands for pulse-code modulation. Pcm is a scheme to represent a sequence of uniformly sampled analog values by a sequence of digital words; the two characterizing parameters are sampling rate and number of bits. Pcm has been invented in 1937 by Alec Harley Reeves, a British scientist and engineer. Pcm is omnipresent in that many well known systems use it: Compact Disc (Cd), Digital Versatile Disc (Dvd), Bluray Disc (Bd), but also digital audio and video in computers and digital telephony, to name just a few. In our discussion of delta modulation and delta de-modulation as a starting idea, which 2 SigAcqui
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BioMedSigProcAna finally leads to our topic, the Σ∆ oversampling analog-to-digital converter, we base on [Pan65]; by using that older book as our starting point, we pay tribute to the earlier engineers working in the field. The modulator in the diagram on page 30 just gives a sequence of two-level pulses with the meaning that a positive pulse indicates that the comparator found that x(t) is larger than the feed-back signal x e(t), and a negative pulse indicates the reverse. As the diagram indicates, the feed-back signal is something like an integral of the pulse train sent to the channel; it is roughly a piecewise constant approximation of the signal x(t): Theoretically, if the pulse train were a train of Dirac impulses, and the integrating system were a true integrator, we would obtain an exact staircase signal as x e(t); in the systems of the time of the invention of delta modulators, the integrating systems have been realized as Rc circuits with large time constants, see, for example, [Pan65, Chapter 22]. We also note that we show only the most basic situation here. There are various details improving the basic structure including modulators using not only a 1-unit code but an n-unit code, and feed-back systems having double (or even multiple) integration.
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Σ∆: . . . and its De-Modulation • on the receiver side an integrating system mimics the integrating system in the feedback of the modulator: b x ˜(t) ≈ x e(t)
• recall: the integrating system in the feedback of the modulator is an estimator if the incoming signal: x e(t) ≈ x(t)
• the present additional low-pass filter smooths the staircaseb˜(t) leaving the integrating system like signal x ··· from channel
··· integrating system
b˜(t) x
Lp filter
x b(t)
If we have an ideal channel without distortions or filtering effects, the integrating systems in the modulator and that in the de-modulator both “see” the same bipolar impulse train. If additionally these two integrating systems were exactly the b ˜(t) = x e(t). same,7 then their output signals would be equal: x 7 In earlier days the integrating system has been, as we have already noted, realized as Rc circuits. Therefore, the two systems will not be exactly the same due to element tolerances.
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Σ∆: Towards Sigma-Delta Modulation • to get the idea: we assume linear systems an sub-systems • then: we might move the integrating system in the demodulator from the output to the input of the channel • then: we next might move the integrating system from the output to the input of the modulating system, that is, to x(t) • then: we finally might combine the effects of the two integrating systems, one from the input to the comparator and the other from the feedback to the comparator, into one integrating system following the comparator • note: the comparator is essentially a ± adder
We note that, of course, not all systems of the combination delta-modulator/delta-demodulator are linear. However, to find a result, we might well work as if they were linear; if the obtained results can be proven to be correct, by other means of course, the result is true independent of how we have obtained it. A note on notions is appropriate: Many authors also name the considered structure a delta-sigma (∆Σ) modulator, because the first operation is building the difference (the “delta,” ∆), and the next operations is integrating (building the “sum,” Σ). The ∆Σ modulator has first been described by Inose, Yasuda, and Murakami in 1962. But it took some time with further advances in Vlsi technology to render possible the manufacturing of cost effective Σ∆ oversampling analog-to-digital converters.
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Σ∆: The Oversampled Ad Converter • sigma-delta modulator (here only 1-bit code) • no channel • digital low-pass filter • down-sampling x(t)
+ −
analog integrator
1-bit Adc
1 M th band b / / Lp filter
↓M
xq [n]
1-bit Dac ←−−−− sigma-delta modulator −−−−→←−−−−−−− decimator −−−−−−−→
The block containing the symbol “↓ M ” carries from the samples of its input sequence only every M th sample over to its output sequence. This operation is called “down-sampling by M .” The symbols “ 1/ ” and “ b/ ” indicate that the corresponding (digital) signal lines have width of 1 bit and b bits, respectively. The low-pass filtering together with the subsequent downsampling is called “decimation.” The “analog integrator” in the above system is most often realized as a switched-capacitor circuit, [Mit98], [ST05], but it might also be implemented as an active Rc system as discussed in [ST05], or even as a passive Rc system.8 8 Whereas switched-capacitor circuits are discrete-time (Dt) systems, active or passive Rc circuits are continuous-time systems. The qualification “analog” concerns the amplitudes of the signals, which are continuous in analog systems as opposed to quantized in digital systems.
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BioMedSigProcAna Applications that apply Σ∆ oversampling analog-to-digital converters are typical low-frequency applications, because the oversampling ratio is usually quit large in practice rendering the sigma-delta modulator a much faster system than the application itself. Areas of application include, but are not restricted to, measurement, instrumentation, and sensors; digital spectrum analyzers; digital telephony; and digital audio, [Mit98].9 As an example we might mention the compact disk encoding in digital audio, where the (audio) output sampling rate is 44.1 kHz, but the employed sigma-delta converter typically runs at a rate of 3′ 175.2 kHz.10
9 Data acquisition in biomedical systems is, of course, also a favored application, because we often have slow signals. 10 The oversampling ratio is M = 72.
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Σ∆: Noise Shaping of Modulator to simplify rough analysis: linear approximation • integrator ; accumulator in Dt • 1-bit Adc: linearized ; quantization noise ε[n] • 1-bit Dac: just a delay ε[n] x[n]
+ −
1 1 − z −1
+
y[n]
z −1
Y (z) = 1 X(z) zero noise Y (z) = 1 − z −1 E(z) zero input Let the input to the integrator, which becomes an accumulator in the Dt world, be named u[n], and its output be named w[n], respectively. Then the accumulators operation is given by w[n] = w[n − 1] + u[n] . In the z domain we thus describe this accumulator by W (z) 1 . = U (z) 1 − z −1 2 SigAcqui
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BioMedSigProcAna We model the effect of the analog-to-digital converter by a source ε[n] of white noise; we give a slightly more detailed discussion on this topic in our later document [Goe17c]. And, as mentioned, the digital-to-analog converter in the feedback path is just a delay z −1 . Because of linearity, we can compute the useful transfer function as well as the noise transfer function of the system in the above slide using superposition; the rule to compute in the given simple situation is “direct / (1 + loop).” We thus obtain the useful transfer function as 1 Y (z) 1 1−z −1 =1. = = −1 −1 z X(z) zero noise (1 − z ) + z −1 1 + 1−z−1 The noise transfer function describing the noise shaping is11 1 − z −1 1 Y (z) = 1 − z −1 . = = −1 z E(z) zero input (1 − z −1 ) + z −1 1 + 1−z−1
We see that this noise transfer-function is the transfer function of a first-difference system: In the time domain, the corresponding difference equation reads y[n] = ε[n] − ε[n − 1]; it sets Dc to zero, it suppresses low frequencies, and it lets pass high frequencies. Thus it shifts noise out of low-frequency bands, where the useful signal resides—compare the spectral representation on page 27—, into high-frequency bands. We may verify the described behavior by computing the frequency response corresponding to the noise-shaping transω ) = 1 − e−j ωˆ . We fer function Hns (z) = ˆ 1 − z −1 :12 Hns (ˆ then obtain the modulation noise spectral density as Sy,n (ˆ ω) = 2 |Hns (ˆ ω )| Sε (ˆ ω ) with |Hns (ˆ ω )| obtained as 2|sin(ˆ ω/2)| and Sε (ˆ ω) being white noise with (constant) spectral density σε2 , also compare page 28, but note that there we show the quantization 11 We 12 See
have the noise ε[n] described in the z domain by E(z) •—• ε[n]. more details in our document [Goe17b].
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BioMedSigProcAna noise power spectral density as a function of natural frequency f , whereas here we have it as a function of normalized radian 2 frequency ω ˆ .13 Obviously, |Hns (ˆ ω)| = 4 sin2 (ˆ ω /2) has the characteristic already described: It is zero at ω ˆ = 0, that is, at Dc, it is small for low frequencies, and it has the highest value at ω ˆ = π/2, which is the Nyquist frequency. The true complete signal y[n] that leaves the modulator on page 36—useful signal plus noise—is a binary sequence that has then in its low frequency contents mainly the useful signal and in its high frequency contents merely noise. With a subsequent low-pass filter, a Σ∆ oversampling analog-to-digital converter tries to separate the useful signal from the noise, see the diagram on page 34. Because the resulting filter output-signal has low-frequency contents only, it is additionally down-sampled to compact the signal. Also note that due to the computations in the low-pass filter, the samples at the output are b-bit samples in contrast to the binary samples at the input of the filter. The final sequence xq [n] resulting after decimation—low-pass filtering and subsequent down-sampling—has thus traded resolution for rate: The fast input sequence with binary samples (low resolution) has been converted into a slower output sequence with samples of b-bits resolution. Most often the discussed low-pass filter is implemented as an Fir filter, which has the advantage that it can realize a linear phase, but this advantage comes for the price of a quite high filter order. The simplest Fir low-pass filter is a moving average filter, or a cascade of such moving-average filters. Often, these moving average filters are realized as recursive running sum filters, which are also known as cascaded integrator comb (Cic) filters, see our discussion in [Goe17a].14 13 On page 28 we use the description with natural frequency f to make clear the relation of the frequencies of the useful (initially analog) signal to the frequencies and power of noise. 14 We note already here that the discussed filters needed for Σ∆ oversampling analog-to-digital converters are fixed-point filters, see our discussion
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BioMedSigProcAna The decimation by M , that is, the M th band low-pass filtering with subsequent down-sampling by M , might also be realized in a cascade of lower-order decimation operations. For example, a decimation by M = 8 could be realized by a cascade of 3 decimation by 2 operations, in which case each section of the cascade would use a half-band low-pass filter; the direct decimation by M = 8 would need an 8th band low-pass filter. Also Cic structures are usually implemented as cascaded decimation operations of lower order. After the present short introduction, there is much more to be said about Σ∆ oversampling analog-to-digital converters: There are not only first-order modulators as discussed, but also 2nd-order modulators and even higher-order modulators. These modulators come with various different structures. Additionally, there are not only low-pass but also band-pass modulators and corresponding decimation filters. Finally, there are many details to be considered for the implementation of Σ∆ oversampling analog-to-digital converters. As starting points for a more advanced study you might consider reading [NST97] and [ST05]; note that [ST05] comes with a free Matlab toolbox named The Delta-Sigma Toolbox.
in [Goe17c]; realizing moving-average filters as Cic filters is only feasible for fixed-point filters.
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Event-Driven Adcs • goal: reduction of power consumption • keywords: irregular sampling & asynchronous designs • members of event-driven Adc family – level-crossing sampling – continuous-time quantization – others: event occurs if some error exceeds a predefined limit
For the family of event-driven analog-to-digital converters we refer to the tutorial paper [Tsi10]. Reduction of power consumption is important in mobile applications, and, in our realm, very important for implants. “Irregular sampling” refers to discrete time signal who’s samples are no longer equidistantly distributed along the time axis; an alternative notion is “non-uniform sampling.” The notion “asynchronous designs” refers to the electronic implementations of these converter structures—the digital parts of the circuits work without clocks. There are various realizations of event-driven Adcs with our focus being the level-crossing samplers, see below. If you are interested in continuous-time quantizers and “others,” you read [Tsi10].
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Level-Crossing Sampling .... •.. ... .•.... .•.... . .. •.. •.. ... .•.... . . . . . • . . . . .. .. .. ... ..... • ... •..... ... . ..•. . ............ . . . • •... .....•. ............ .•. •..... ..... .... .•....... . •................... ..•.. •.. •.............................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...............•.. . . . . •.. .•.. .. •... ..... ... .... •.. .. . ... •... • ... .... •.. .. .... t x(t)
ti−1
ti
ti+1
• predefined levels ; times ti where signal crosses levels • signal is represented by ∆i = ˆ ti − ti−1 values • few samples for flat (inactive) signal & many samples for steep (active) signal
Note that we do not have two samples for the same level in sequence—see for example the third and the forth sample, where na¨ıvely we would place the forth sample earlier at the time instant where the signal passes again the level of the third sample. We insist to the given rule in order to realize a hysteresis that avoids the generation of too many samples merely due to noise that may make the signal to jitter at a level. We note that in time segments with active signal there are many samples, but the time-difference values ∆i are small. Because we measure the ∆i values by a counter, we see that during active phases the corresponding counter values have only a few non-zero bits in positions left adjacent to the least significant
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BioMedSigProcAna bit. We may take advantage of this observation when coding the samples for compact storage. We finally note that the level-crossing sampling is sometimes also called Lebesgue sampling.
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Level-Crossing Sampling & Additional Compression • from graphics on page 41: a horizontal signal section does not generate samples • the above case is a straight line which is horizontal • . . . and it stays on its lane—a horizontal band • idea to allow additional compression: not only horizontal straight line segments constitute inactive signal segments • . . . but straight line segments of other slopes, too • amount of signal curvature is responsible for signal exiting a presently active lane
Concerning a horizontal signal section, we have to add the following argument: If the signal of the section stays between two crossing levels, it does obviously generate no samples. If the signal of the section stays exactly on a crossing level, however, we can be sure that in practice this signal is corrupted by a (possibly very small) noise which activates the generation of one sample; because a following sample cannot be one from the same level, it is clear that also this case does generate no additional samples. The idea for approaches to additional compression can also be understood from the point of view of the reconstruction of a signal that is sparsely and non-uniformly sampled: Basically, we try to reconstruct the signal by piecewise polynomials—splines, and we ask what samples must be generated during the compression process to yield, on one hand, as few as possible samples 2 SigAcqui
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BioMedSigProcAna and to reconstruct the signal, on the other hand, with as low as possible distortion. Usually, the piecewise polynomials are of low orders: Whereas orders zero an one lead to piecewise constant and to piecewise linear reconstructions, we prefer orders two and three to yield quadratic and cubic splines. For one approach worked out in detail, we refer to our paper [MNH+ 15].
Level-Crossing Sampling With Additional Compression
; signal and its sample instances: typical appearance ..... .•.. ... ..... .•.... . ... ... ... ..... ......... . . . . . . ... ... .•.. ..... .•.. ... ... ..... . .•..................... . . . .... ... . ...... ...•...... ... . . . . . . . . . . . . . • • • • . . . . . . . . . . . . . . . . . . . . . . . . ......................................................... . . . .................. .. ... ........... ................... ... .•... •...... . . •... ... ... ... ... ..• •... .... .... x(t)
ti−1
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3
Signal Types
Time- and Amplitude Quantization time values continuous
discrete discrete-time signal x[n]
continuous discrete
amplitude values
analog signal x(t)
................... .... .... . . . .. ... ... ........................ .... . . . . . . ...... . . . . . . .................. ....
• •
•
•
•
t
n −1
0
1
2
3
digital signal xq [n]
discrete-amplitude signal x(t) •
• • • •
t
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4
System Types
Analog, Discrete-Time, and Digital Systems u(t)
............. ..... ..... ................ ..
t
•
u[n] •
• •
continuous-time (Ct) (analog) system u(t) -
··· n
u[n] -
uq [n]
• • • ··· •
n
−1 0 1 2 3
y(t)
Dt system
y[n]
- digital system
y[n]
• •
• • •
··· n
- −1 0 1 2 3
yq [n] • •
digital system ud [n]
t
-
discrete-time (Dt) system
•
−1 0 1 2 3
•
Ct system
y(t)
..........................................
• • •
yd [n]
··· n
- −1 0 1 2 3
We note that combinations of the shown basic system-types exist. For example, the continuous-to-discrete (C2d) converter, see page 7, takes a continuous-time signal at its signal input15 and produces a discrete-time signal at its output. Likewise, the quantizer, see page 24, takes a discrete-time signal (continuous amplitude values) at its input and outputs a digital signal. 15 Note that the C2d converter has also a parameter input which specifies the used sampling rate.
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BioMedSigProcAna Note that these two examples, continuous-to-discrete converter as well as quantizer, are static (non-dynamic) systems; they do not contain memory elements, hence the present output solely depends on the present input.
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Notation and Symbols b
Often stands for the number of bits used to quantize the (real) sample of a signal.
f
A (natural) frequency in Hz.
fB
A (natural) band-limiting frequency in Hz.
fmax
Maximum (natural) frequency in the realm of the sampling theorem; see page 15.
fs
Sampling frequency in Hz; see page 7.
fˆ
Discrete-time, normalized, frequency: fˆ = ˆ f /fs = ˆ f · Ts ; see page 10. Note that f has no physical units.
Hns (z)
Noise-shaping transfer function; see below page 36.
k
An integer for counting; sometimes also used to denote discrete-time time.
m
An integer for counting; sometimes also used to denote discrete-time time.
M
An integer, usually to denote a down-sampling ratio.
n
Discrete-time time variable (an integer number); see page 7.
N
An integer, to denote a number of elements.
s
Vector of coefficients in sparse signal representations; see page 19.
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BioMedSigProcAna Sn (·)
Noise spectrum; see page 28. Note that the noise spectrum might be the spectrum of a Ct (continuoustime) or of a Dt (discrete-time) noise signal; furthermore, it might be expressed as a function of natural frequency or as a function of radian frequency. The context makes clear for what we use the symbol.
t
Continuous-time time variable (a real number); see page 7.
Ts
Sampling interval, Ts = 1/fs ; see page 7.
Vin
Input voltage (into an Adc).
x(t)
Continuous-time (analog) signal; see page 7.
x b(t)
We denote the estimation of a signal x(t) by x b(t); see page 32.
x[n]
Discrete-time signal; see page 7.
xq [n]
Quantized discrete-time signal; see page 24.
x
Vector of discrete-time signal components; see page 19.
X(z)
If we have a discrete-time signal x[n], X(z) •—• x[n] denotes its z-transform; see page 36.
ε[n]
Quantization noise, ε[n] = ˆ x[n] − xq [n], where x[n] is the continuous-amplitude input signal, and xq [n] is its quantized version, see page 28.
φ
Denotes a phase shift; see page 10.
ω
Continuous-time radian frequency, ω = 2πf , with f the natural frequency in Hz; see page 10.
ω ˆ
Discrete-time radian frequency, ω ˆ = ωTs , with ω the continuous-time radian frequency in rad/sec, and Ts the sampling interval; see page 10.
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BioMedSigProcAna Ψ
Matrix of basis (column) vectors for signal representation in the realm of sparse signals; see page 19.
σε2
Total quantization-noise power; see page 28.
Hns (ˆ ω)
Frequency response of noise-shaping transfer function Hns (z): Hns (ˆ ω ) = H(z)|z=ejωˆ ; see below page 36.
ℓ0
Denotes the ℓ0 vector norm, which is the number of non-zero elements of a vector; see page 23.
ℓ1
Denotes the norm ℓ1 of a vector, which is the sum of the absolute values of the components of the vector; see page 23.
ℓ2
Denotes the Euclidean norm ℓ2 of a vector, which is the square root of the sum of the squares of the components of the vector; see page 23.
↓M
Denotes down-sampling by M : From a discrete-time signal (a sequence of samples), only every M th sample is retained; see page 34.
b
/
2 SigAcqui
If we indicate in a block diagram a signal line by this symbol, we mean that the (quantized) signal is b bits wide; see page 34.
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BioMedSigProcAna We next list (most of) the used abbreviations. Ad
Analog-to-digital.
Adc
Analog-to-digital converter.
Bd
Blu-ray disc.
Ct
Continuous-time (signal or system).
Cd
Compact disc.
Cic
Cascaded integrator comb (filter); see our discussions following page 36.
C2d
Continuous-to-discrete (converter, conversion); see page 7.
Da
Digital-to-analog.
Dac
Digital-to-analog converter.
Dc
Direct current; a signal is a Dc signal, if it is constant (contains only the frequency zero).
Dt
Discrete-time (signal or system).
Dvd
Digital versatile disc.
Pcm
Pulse-code modulation; see page 30.
Rc
Resistor-capacitor (circuit).
Sc
Switched-capacitor (circuit).
Vlsi
Very large scale integration; see page 33.
Σ∆
Denotes a special architecture for analog-to-digital conversion; see page 25 and the following.
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References [Bar07]
Richard G. Baraniuk. Lecture notes: Compressive sensing. IEEE Signal Processing Magazine, 24(4):118–124, July 2007.
[CW08]
Emmanuel J. Cand`es and Michael B. Wakin. An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2):21–30, March 2008.
[EBB05]
John D. Enderle, Susan M. Blanchard, and Joseph D. Bronzino. Introduction to Biomedical Engineering. Elsevier Academic Press, Amsterdam, second edition, 2005. Bfh-ti Biel/Bienne Library 615.47 ENDER.
[GAG+ 13]
Graham C. Goodwin, Juan Carlos Ag¨ uero, Mauricio E. Cea Garrido, Mario E. Salgado, and Juan I. Yuz. Sampling and sampled-data models. IEEE Control Systems Magazine, 33(5):34–53, October 2013.
[Goe17a]
Josef Goette. Biomedical Signal Processing and Analysis—Event Detection: Qrs-Complexes in Ecg Signals. Bern University of Applied Sciences, Script at the Bfh-ti Biel/Bienne, HuCEmicroLab, February 2017.
[Goe17b]
Josef Goette. Biomedical Signal Processing and Analysis—Filtering for Removing Artifacts. Bern University of Applied Sciences, Script at the Bfhti Biel/Bienne, HuCE-microLab, February 2017.
[Goe17c]
Josef Goette. Biomedical Signal Processing and Analysis—On Fixed-Point Filter Realizations.
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BioMedSigProcAna Bern University of Applied Sciences, Script at the Bfh-ti Biel/Bienne, HuCE-microLab, February 2017. [LRRB05]
Bin Le, Thomas W. Rondeau, Jeffrey H. Reed, and Charles W. Bostian. Analog-to-digital converters. IEEE Signal Processing Magazine, 22(6):69– 77, November 2005.
[Mit98]
Sanjit K. Mitra. Digital Signal Processing: A Computer Based Approach. Mc Graw Hill, New York, 1998. Bfh-ti Biel/Bienne Library 621.391 MITRA.
[MNH+ 12a] Thanks Marisa, Thomas Niederhauser, Andreas Haeberlin, Josef Goette, Marcel Jacomet, and Rolf Vogel. Asynchronous Ecg time sampling: Saving bits with Golomb-Rice encoding. Computing in Cardiology, Krakow, Poland, 9–12 September 2012. [MNH+ 12b] Thanks Marisa, Thomas Niederhauser, Andreas Haeberlin, Josef Goette, Marcel Jacomet, and Rolf Vogel. Asynchronous time encoding: An approach to sub-Nyquist rate sampling. Biomedizinische Technik/Biomedical Engineering, 57:176, August 2012. [MNH+ 15]
Thanks Marisa, Thomas Niederhauser, Andreas Haeberlin, Reto A. Wildhaber, Rolf Vogel, Marcel Jacomet, and Josef Goette. Bufferless compression of asynchronously sampled Ecg signals in cubic Hermitian vector space. IEEE Trans. Biomed. Eng., 63(12):2878–2887, December 2015.
[MNH+ 17]
Thanks Marisa, Thomas Niederhauser, Andreas Haeberlin, Reto A. Wildhaber, Rolf Vogel, Josef Goette, and Marcel Jacomet. Pseudo asynchronous
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BioMedSigProcAna Adc for Ecg signal acquisition. Biomed. Circuits Syst., 11, 2017.
IEEE Trans.
[MW98]
Vijay K. Madisetti and Douglas B. Williams. The Digital Signal Processing Handbook. Ieee Press, Crc Press Llc, Boca Raton, Florida, 1998. Bfhti Biel/Bienne Library 621.391 MADIS.
[NST97]
Steven R. Norsworthy, Richard Schreier, and Gabor C. Temes. Delta-Sigma Data Converters. John Wiley & Sons, New York, 1997.
[Pan65]
Philip F. Panter. Modulation, Noise, and Spectral Analysis—Applied to Information Transmission. Mc Graw Hill, New York, 1965.
[Raz95]
Behzad Razavi. Principles of Data Conversion System Design. Ieee Press, New York, 1995.
[ST05]
Richard Schreier and Gabor C. Temes. Understanding Delta-Sigma Data Converters. Ieee Press, Wiley Interscience, Hoboken, N.J., 2005.
[Tsi10]
Yannis Tsividis. Event-driven data acquisition and digital signal processing—a tutorial. IEEE Trans. Circuits Syst. II, 57(8):577–581, August 2010.
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