Digital Signal Processing Course. Nathalie Thomas Master SATCOM

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1 Digital Signal Processing Course Nathalie Thomas Master SATCOM

2 Chapitre 1 Introduction 1.1 Signal digitization : sampling and quantization Signal sampling Let's call x(t) the original deterministic signal, T e the sampling period and {x(kt e )}, k Z the sampled signal (periodic sampling). It is possible to associate the following continuous model to {x(kt e )}, k Z : x e (t) = x(t) T e (t), where T e represents the Dirac comb of width T e. If X(f) is the Fourier trnsform of x(t), then : {x(kt e )} k Z F T F e X(f nf e ) (1.1) where F e = 1 T e is the sampling frequency. The Fourier transform of x(t) thus becomes periodical due to the sampling process. To keep the same information in the sampled signal, compared to the original one, F e must be chosen in order to fulll the Shannon condition : n F e > 2F max, (1.2) if F max denotes the maximum frequency of X(f). 1

3 1.1.2 Quantization A quantized signal is a signal whose amplitudes can only take a nite number of values. The number of possible values is given by the number of quantization bits : with nb bits il will be possible to code 2 nb levels on the signal dynamic D. A well done quantization (no clipping, a quantization step q = D 2 nb thin enough) is equivalent to add a noise, n Q (t), to the original (unquantized) signal x(t) : x Q (t) = x(t) + n Q (t), (1.3) where x Q (t) denotes the quantized signal. This noise is uniform on [ q 2, q 2] and the quantization signal to noise ratio can be written as : SNR Q (db) = 10 log 10 ( Px P nq ) where P nq = E [ q n 2 ] Q = n 2 2 Qp nq dn Q = n 2 R q Q 1 q dn Q = q which leads to SNR Q (db) = 6 nb + constante, where the constant depends on the considered signal. The operation of quantization is a non linear irreversible operation. However, if it is done with good conditions (no clipping), she can be transparent with the number of quantization bits today available on processors Examples on an image Figure 1.1 presents an image composed of pixels, coded on nb = 8 bits and a sampled version with an undersampling factor of 4 ( pixels). This image (Barbara) is often used in image processing. It allows to see the Moire phenomenon : some structures, dierent from the ones contained in the original image, appear due to undersampling. Figure 1.2 presents Barbara coded on nb = 4 bits and on nb = 2 bits. 2

4 Figure 1.1 Original image (quantized on 8 bits, pixels), Undersampled image (factor 4) For example in the last case, only 2 nb = 2 2 = 4 gray levels can be observed (instead of 2 8 = 256 from black to white on the original image). Figure 1.2 Image quantized on nb = 4 bits, Image quantized on nb = 2 bits 3

5 1.2 Advantages and drawbacks of digital processing Work with digital signals oers several advantages : Robustness against noise : BER = f(snr), Stableness and reproducibility of the devices, New functions, such as adaptive ltering or channel coding. The main drawback is an increased bandwidth for the digital signal. But, in order to reduce it, source coding can be used. 1.3 Digital implementation for the signal processing tools Several signal processing tools are currently used to extract some information from the signal (needed bandwidth for transmission, defect detection...), to construct or modify some signals (modulation, noise ltering...) : Fourier transform, Cross and self correlation functions, Power spectral density, Filters, All this tools will have to be implemented digitally. For that, several approximations, estimations will have to be done in order to construct digitally implementable tools from the theoretical ones. The objective of this course is to present and explain these modications and their impacts in order to be able to correctly use the digital tools and correctly analyze the obtained results. 1.4 Processing time - real time processing A processing is accomplished in real time when a signal sample is given at the output of the processing for each input sample (i. e. each sampling period T e ). A basic digital signal processing operation is a MAC = Multiplication Accumulation (+/ operation). Processing time will be given as the number of MAC per second. 4

6 Chapitre 2 Digital Fourier Transform (DFT) For a given signal x(t), the Fourier Transform X(f) and its inverse are given by : X(f) = 2.1 From FT to DFT R x(t)e j2πft dt x(t) = R X(f)e +j2πft df Approximations have to be done, leading to some eects that must be known in order to be able to correctly carry on a spectral analysis using the digital Fourier transform Sampling Signal sampling : x(t) {x(kt e )} k Z (2.1) leads to a periodization of its Fourier transform, that is approximated by the sum of the rectangle areas under the curve, forgetting T e multiplicative factor : X(f) X 1 ( f) = + k= j2πk f x(k)e (2.2) where f = f F e is the normalized frequency. Note that x(kt e ) is denoted x(k) as the k th element of the vector x representing the digital sampled signal. 5

7 X 1 ( f) is periodic with period 1 (X 1 (f) is periodic with period F e ). Be careful, it is then important to respect the Shannon sampling condition to avoid aliasing. Be also careful to the lecture of the spectrum. Indeed, if the digital Fourier transform is observed on a period between 0 and F e, the positive part will be observed between 0 and F e /2, while the negative part will be between F e /2 and F e Limited signal duration to N points {x(kt e )} k Z {x(kt e )} k=0,...,n 1 This signal knowledge on a limited number of points (dimension of the array representing the digital signal) leads to a distortion on the Fourier transform : where w(k) = Giving : with X 1 ( f) X 2 ( f) = { N 1 k=0 1, k=0,..., N-1 ; 0, elsewhere. x(k)e j2πk f = + k= X 2 ( f) = X 1 ( f) W 1 ( f), W 1 ( f) = + k= w(k)e j2πk f. j2πk f x(k)w(k)e This distortion of the Fourier transform implies a limited separating power for the digital spectral analysis (possibility to separate 2 spectral patterns with very close frequencies) and a given ripple rate (some ripples appear around abrupt transitions on the spectrum). These parameters (separating power and ripple rate) are given by the shape of W 1 ( f) and, more precisely, by the width of its central lobe and the amplitude of its secondary lobes (see gure 2.1). It is possible to obtain dierent separating powers and ripple rates for the same spectral analysis by limiting the studied signal with dierent "truncation (or weighted, or apodization) windows" w(k) (not necessarily a rectangular one). It is possible to obtain that way several shapes for W 1 ( f) and so several versions for the digital Fourier transform of the same signal. 6

8 Figure 2.2 presents some classical truncation windows, while gure 2.3 presents their Fourier transforms. We can observe that the central lobes are narrower or wider, leading to a greater or lesser separating power for the digital spectral analysis. We can also notice that the secondary lobes are more or less attenuated, leading to a greater or lesser ripple rate for the digital spectral analysis. When the truncation window is not a rectangular one, we speak about weighted Fourier tranform. A digital spectral analysis must be carried out using several truncation windows. Each used window will allow to extract dierent informations. Figure 2.4 presents an example for which each used window allows to highlight dierent spectral components. Could you nd what is the corresponding signal? Figure 2.1 Fourier transform of the rectangular window : Dirichlet Core. 7

9 Figure 2.2 Some examples of truncation windows. Figure 2.3 Fourier transform of some truncation windows. 8

10 Figure 2.4 Several versions for the Fourier transform of the same signal. 9

11 2.1.3 N points are computed for the Fourier transform As the digital signal cannot be continuous in time and unlimited, we are only able to compute a nite number of points for the digital Fourier transform : X 2 ( f) { ( n )} X 2 N n=0,...,n 1 (2.3) The fact that we are able to compute a limited number of points for the Fourier transform has an impact on the spectral analysis resolution. Indeed the spectral analysis resolution depends on the number of points computed on a period of the DFT : computation step equal to Fe N (or 1 N for normalized frequencies). In order to increase the spectral resolution, it is possible to use the zero padding interpolation method. From x(k), given on N points, a new signal is dened, by adding some zeros at the end of the vector : y(k) = The DFT of this new signal : Y (n) = N 1 k=0 { x(k) k=0,..., N-1 0 k=n,..., MN-1. kn j2π x(k)e MN, n = 0,..., MN 1 has a thinner computation step on the same scale : MN points spaced from F e MN (or 1 MN with normalized frequencies) are computed between 0 and F e (a DFT period, between 0 and 1 with normalized frequencies), instead of N points spaced from Fe N are computed between 0 and F e (or between 0 and 1 with normalized frequencies). So the spectral resolution is higher. Figures 2.5 and 2.6 propose dierent plots for the modulus of the digital Fourier transform of a cosine function whose normalized frequency is 0.2, whose number of points is N = 1000 and for dierent values of parameter MN. Another impact of the frequency sampling is a time periodization. Indeed, we must consider, when working with digital signals, that they are periodical in the time domain, as well as their Fourier transform (frequency domain). This point has an important impact : it is not possible to keep the important property of the theoretical Fourier transform consisting in converting a product to a convolution product and inversely. Indeed, as we will see 10

12 Figure 2.5 Fourier transform for a cosine function of normalized frequency 0.2 and for several values of the zero padding parameter. it in the following (section??), the Digital Fourier Transform converts a product into a circular convolution product, i.e. a convolution product between periodical signals. However, it is possible to make the circular convolution product being the same as the linear (classical) convolution product by adding at least N zeros at the end of the signals (assuming signals are given on N points) DFT and idft expressions All the approximations previously studied (sampling in the time domain, limited signal duration, sampling in the frequency domain) lead to the following tool to compute the digital Fourier transform : X(n) = N 1 k=0 kn j2π x(k)e N, n = 0,..., N 1 11

13 Figure 2.6 Fourier transform for a cosine function of normalized frequency 0.2 and for several values of the zero padding parameter : zoom on the peaks in gure

14 By doing the same way, it is possible to obtain the inverse digital Fourier transform : x(k) = 1 N N 1 n=0 2.2 DFT Properties Linearity kn +j2π X(n)e N, k = 0,..., N 1 where λ is a scalar. DF T [x 1 (k) + λx 2 (k)] = DF T [x 1 (k)] + λdf T [x 2 (k)], Time translation => phase rotation Hermitian symmetry X (n). DF T [x(k k 0 )] = X(n)e j2π k 0 n N Denoting X(n) the DFT of a real signal x(k) : X (N n) = X( n) = Circular convolution Be careful : the digital Fourier transform converts a product into a circular convolution product : N 1 X 1 (n)x 2 (n)df T 1 x 1(k) x 2 (k) = x 1 (p)x 2 ([k p] modulo N ) if X 1 (n) denotes the DFT of x 1 (k) and X 2 (n) the DFT of x 2 (k) Parseval equality p=0 N 1 k=0 x(k) 2 = 1 N N 1 n=0 X(n) 2 13

15 2.2.6 Fast Fourier Transform (FFT) DFT is able to be computed using a fast algorithm called FFT algorithm. The principle (see the following section) is to break the DFT into smaller DFTs. The condition (for the basic Cooley Tuckey algorithm) is that the number of samples N has to be a power of Fast Fourier Transform : Cooley Tuckey basic Algorithm Principle The Fourier transform X(n) computed for the N points of the digital signal x(k) is given by : X(n) = N 1 k=0 kn j2π x(k)e N N 1 = k=0 x(k)w kn N, n = 0,..., N 1 (2.4) denoting W N = e j 2π N. We talk about a N-order DFT. Its computation time can be evaluated to N 2 +/ operations. Let's assume that N is a power of 2 : N = 2 p. We can decompose the signal into interleaved sequences. First decomposition : where : X 1 (n) = N/2 1 i=0 X(n) = X 1 (n) + W n N X 2(n), n = 0,..., N 1, (2.5) N/2 1 x(2i)w in N/2, X 2(n) = i=0 x(2i+1)w in N/2, i = 0,..., N/2 1 (2.6) The computation time can be evaluated, after this rst decomposition, to 2 ( ) N N +/ operations, which is already N 2, especially if N is high. 14

16 Second decomposition : X 1 (n) = X 11 (n)+w n N/2 X 12(n), X 2 (n) = X 21 (n)+w n N/2 X 22(n), n = 0,..., N/2 1, (2.7) where : X 11 (n) = X 12 (n) = N/4 1 i=0 N/4 1 i=0 N/4 1 x 1 (2i)W in N/4, X 12(n) = i=0 N/4 1 x 2 (2i)W in N/4, X 22(n) = with x 1 (p) = x(2i) et x 2 (p) = x(2i + 1), i, p = 0,..., N/2 1. i=0 x 1 (2i+1)W in N/4, i = 0,..., N/4 1 (2.8) x 2 (2i+1)W in N/4, i = 0,..., N/4 1 (2.9)... Last decomposition : We go on like this till arriving to the smallest possible Fourier transform relating to two points, which is called the "buttery" of the Fourier transform (gure 2.7). Then, N p 2-order Fourier transform (or butteries) will have been computed, giving a computation time equal to Nlog 2 (N) N 2 +/ operations. Figure 2.7 Buttery of the FFT (radix-2 algorithm ) : 2-order Fourier transform = 2 +/ operations Example for N = 2 3 = 8 First decomposition : 15

17 X(n) = X 1 (n) + W n N X 2(n), n = 0,..., 7, where X 1 (n) is on x(0), x(2), x(4), x(6) and X 2 (n) is on x(1), x(3), x(5), x(7). Second and last decomposition : X 1 (n) = X 11 (n) + W n N/2 X 12(n), n = 0,..., 3, where X 11 (n) is on x(0), x(4) and X 12 (n) is on x(2), x(6). X 2 (n) = X 21 (n) + W n N/2 X 22(n), n = 0,..., 3, where X 21 (n) is on x(1), x(5) and X 22 (n) is on x(3), x(7). Corresponding graph We obtain p = 3 columns of N 2 = 4 butteries, that is p ( ) N 2 2 = pn = log 2 (N) N = 3 8 +/ operations. Note that the signal samples are not given in the natural order to the algorithm input. A "bit reversal" algorithm can be used to give them in the right order. Figure 2.8 FFT graph for N =

18 Chapitre 3 Estimators for cross and auto correlation functions The correlation function can be estimated in the time or frequency domain. 3.1 Time domain estimation Biased estimator Assuming ergodic signals, x(k) and y(k) on N points, and using the law of large numbers, the positive part of the cross correlation function between x(k) and y(k) can be estimated by : R xy (k) = 1 N N 1 n=0 x(n)y (n k) 0 k N 1 (3.1) Actually, as the sum concerns only N k samples of the product x(n)y (n k), this estimator is biased : [ ] E Rxy (k) = N k N R xy(k). (3.2) The bias is multiplicative and triangular (see gure 3.1 in examples) 17

19 3.1.2 Unbiased estimator An unbiased estimator can be dened as follows : R xy (k) = 1 N 1 x(n)y (n k) 0 k N 1 N k n=k Note that for k N, the estimator variance increases dramatically. Indeed, when k N a few points are used to compute the estimation. So it will vary a lot between several signal realizations (see gure 3.2 in examples) Processing time evaluation Estimation in the time domain requires about N + (N 1) = N(N + 1) 2 N 2 2 +/ operations. This computation time only evaluates the computation of the positive part of the cross correlation function, knowing that the negative part can be obtained by using the hermitian symmetry property : R xy ( k) = R xy(k), which is still true in digital Examples : cosine function, line of a SAR (Synthese Aperture Radar) image Figure 3.1 plots a biased estimation of a cosine autocorrelation function, obtained using the Matlab function xcorr.m. We can observe the multiplicative triangular bias. Figure 3.2 plots an unbiased estimation of a cosine autocorrelation function, obtained using the Matlab function xcorr.m with parameter 'unbiased'. We can observe the estimation variance on the sides of the analysis window. Figure 3.3 compares biased and unbiased estimations for the covariance function (autocorrelation mean 2 ) of a real signal (line of a SAR image) to the theoretical covariance. For real signals, the biased estimation is often closer to the theory. 18

20 Figure 3.1 Biased estimation of a cosine autocorrelation function Figure 3.2 Unbiased estimation of a cosine autocorrelation function 19

21 Figure 3.3 Estimation of the covariance computed for a line of a SAR image 3.2 Frequency domain estimation Denition For large data sets, a frequency domain estimation should be preferred (see below). For that, we can rst note that the correlation function between x(k) and y(k) can be written as a convolution : R xy (k) (x z) (k) with z(k) = y ( k) Then, it can be estimated in the frequency domain like this : R xy (k) = IDF T (X(n)Y (n)) where X(n) = DF T (x(k)) and Y (n) = DF T (y(k)) and IFT stands for Inverse Digital Fourier Transform. This allows to reduce the processing time to about 3Nlog 2 (N) + N << N 2 2 +/ operations. Note that this solution is based on the fact that it is possible to replace N 1 n=0 by + n= in the cross correlation expression and also on the property of the Fourier transform (or inverse Fourier transform) to convert a product into a linear convolution 20

22 product. But, we saw previously that this is not true : digital signal must be considered as periodical and the digital Fourier transform converts a product into a circular convolution product (convolution between periodical signals). However, it is possible, using zero padding, to make the circular convolution product and the linear convolution product being exactly the same. For that, the number of added zeros must be, at least, the same as the number of signal samples. 3.3 Some properties Hermitian symmetry R xy ( k) = R xy(k) For real signals : Upper boundaries R xy ( k) = R xy (k) R x (k) R x (0) = P x, P x = R x (0) denoting the power of signal x. R xy (k) 1 ( Rx (0) + 2 R ) y (0) 21

23 Chapitre 4 Power Spectral Density (PSD) estimation Power Spectral Density reects the contribution of each frequency to the signal average power. It is given by the Fourier transform of the signal autocorrelation : S x (f) = F T [R x (τ)] It can be digitally estimated with two basic estimators and variants. Variants are dened in order to reduce the estimation variance. 4.1 Basic estimators Periodogram The periodogram is based on the PSD expression for deterministic signals with nite energy (which is nally the case for all the digital signals) : Ŝ x (n) = 1 DF T [x(k)] 2 N Correlogram The correlogram is based on the denition of the PSD : [ Ŝ x (n) = DF T Rx (k)] 22

24 The correlogram can be biased or unbiased, depending on used R x (k). Note that both basic estimators are based on the digital Fourier transform. Note also that PSD estimations using the periodogram or the biased correlogram will give the same result, as you will have to observe and explain it during the laboratories Main issues with the basic extimators Periodogramme and biased correlogram are non consistent estimators : They are biased : and thus [ Ŝ x (n) = DF T Rx (k)] ] E [Ŝx (n) = DF T [ N k N ] S x (n) [ ] ( ) 2 DF T N k N = N 2 sin(πfn) sin(πf) is called the Fejer kernel. It can be shown, for a white Gaussien noise, that the variance of Ŝx(n) tends to S 2 x(n) when the number of signal samples N tends to innity. It remains more or less true for other signals. The basic estimators variance does not tend to 0 when N +. The biased correlogram can, in turn, lead to negative values for the PSD. 4.2 Some examples of variants Cumulative (or Bartlett) Periodogram To implement the cumulative periodogram the N point signal x(k) is divided into M parts : x i (k), i = 1,..., M of L = N M points. Then an averaged periodogram is done : Ŝ x (n) = 1 LM M DF T [x i (k)] 2, i=1 leading to a reduced variance for the estimated PSD. The main drawback for this estimator is the bias enhancement due to the widening of the Fejer kernel central lobe (L times wider). Moreover, the obtained PSD resolution is decreased (L computed points on F e, instead of 23

25 N) Modied periodogram Another way to reduce the basic estimator variance consists in ltering : [ Ŝ xfiltered (n) = DF T Rx (k) w(k)] = Ŝx(n) W (n) (4.1) We can use here several truncation windows on the cross correlation function (see the chapter on Fourier transform) Welch Periodogram The principle is the same as for Bartlett periodogram but the dierent parts x i can overlap, leading to a lower estimation variance for the same PSD resolution compared to the cumulative periodogram. Typically the overlapping length is half the window length. 4.3 Example on a line of a SAR image Figure 4.1 plots the theoretical PSD for a line of a SAR image and compares it to several estimations : periodogram, biased correlogram and unbiased correlogram. We can note that, actually, periodogram and biased correlogram estimations are the same and closer to the theoretical PSD compared to the unbiased correlogram estimation. Figure 4.2 plots the theoretical PSD for a line of a SAR image and its estimation using a cumulative periodogram. It shows that the estimation variance has been reduced thanks to the average on several signal realizations (or several parts of the signal). 24

26 Figure 4.1 PSD of a line of a SAR image Figure 4.2 PSD of a line of a SAR image using a cumulative periodogram estimation 25

27 Chapitre 5 Digital ltering lters. We focus in this chapter on rational and time invariant linear digital 5.1 A tool : z transform As Laplace transform allows us to study analog time invariant linear lters, z transform will allow the study of time invariant linear digital lters Denition Convergence X(z) = ZT [x(n)] = + n= x(n)z n, z C The convergence region is the set of complex numbers such as X(z) converges. We will use Cauchy criterium : lim n n v(n) < 1 v(n) n=0 converges to get a sucient convergence condition : X(z) = + x(n)z n + + n=0 n=1 x( n)z n converges for 0 R x z < R + x < 26

28 with : R x = lim n + x(n) 1/n, R + x = 1 lim n + x( n) 1/n Example : X(z) = + z n n=0 converges for z > 1 : R x = 1 and R + x = Properties Linearity ZT [ax(n) + by(n)] = azt [x(n)] + bzt [y(n)] Convergence : if R + = min(r + x, R + y ) and R = max(r x, R y ) the domain of convergence includes ]R, R + [. Time shift ZT [x(n n 0 )] = z n 0 ZT [x(n)] Same domain of convergence as X(z) = ZT [x(n)]. Scaling ( z ZT [a n x(n)] = X a) Convergence : ar x z < ar + x Derivability z transform denes a Laurent series which is indenitely derivable term by term in its domain of convergence. We deduct : ZT [nx(n)] = z dx(z) dz Same domain of convergence as X(z) = ZT [x(n)]. 27

29 5.1.4 Convolution product The convolution product between x(n) and y(n) is dened by : x(n) y(n) = + k= x(k)y(n k) Then : ZT [x(n) y(n)] = X(z)Y (z) and its domain of convergence can be larger than the intersection between the domains of convergence of X(z) and Y (z) Inverse transform Denition z inverse transform is dened by : x(n) = 1 X(z)z n 1 dz j2π C + where C + is a closed curve included in the convergence ring. Proof The expression of the z inverse transform comes directly from the computation of : J(n, k) = z n k 1 dz C + With the residue theorem we can show : J(n, k) = Then : 1 X(z)z n 1 dz = 1 j2π C + j2π C + ( + k= { ( = 1 + ) x(k) J(n, k) = x(n) j2π k= 0, n k ; j2π, n = k. x(k)z k ) z n 1 dz Note : tables exist for z transforms and inverse z transforms. 28

30 Reminder : If z i is a single pole of g(z) : If z i is a pole of order α : Residue [g(z)] z=zi = 5.2 Denitions Linearity Residue [g(z)] z=zi = lim z zi (z z i )g(z) [ ] 1 α 1 (α 1)! z α 1 (g(z)(z z i) α ) z=z i If y 1 (n) and y 2 (n) are the outputs corresponding to the inputs x 1 (n) and x 2 (n), the lter is linear if it gives an output y 1 (n) + y 2 (n) for an input x 1 (n) + x 2 (n) Time invariance If y(n) is the output corresponding to the input x(n), the lter is time invariant if it gives the output y(n n 0 ) for the input x(n n 0 ), n 0 representing a delay of n 0 samples Impulse response and transfer function We can write the output y(n) of a digital lter as a function of the input x(n) and the lter impulse response h(n) : y(n) = + k= h(k)x(n k) = + k= h(n k)x(k) = x(n) h(n) The z transform of h(n), H(z), is the transfer function of the lter. If Y (z) and X(z) are respectively the z transform of y(n) and x(n), then : H(z) = Y (z) X(z) 29

31 5.2.4 Frequency response and Group Propagation Time (GPT) The frequency response and GPT associated to the lter are given by : GP T ( f) = 1 2π 5.3 Realisability H( f) = [H(z)] z=e j2π f dϕ H ( f) d f A digital lter is realizable if : It is causal : h(n) = 0 for n < 0, It is stable : + n= h(n) <, Its impulse response h(n) is real., with ϕ H ( f) [ = Arg H( f) ] 5.4 Rationnal digital lters The rational digital lters are dened by a rational transfer function : H(z) = which gives in the time domain (by xing a 0 = 1) : N 1 k=0 b kz k M 1 k=0 a kz k (5.1) M 1 N 1 y(n) = a k y(n k) + b k x(n k). (5.2) k=1 k= Innite Impulsionnal Response Filters (IIR) Equations (5.1) and (5.2) dene lters called Innite Impulsionnal Response Filters (IIR). Each new value for the impulse response can be obtained using the previous ones : M 1 h(n) = a k h(n k), for n N (5.3) k=1 Thus it is possible to obtain an innite number of points (or coecients) for the impulse response. 30

32 5.4.2 Finite Impulsionnal Response Filters (FIR) and : They are dened as follows : H(z) = y(n) = N 1 k=0 N 1 k=0 b k z k b k x(n k). The coecients (points) of the FIR lter impulse response are directly given by the b k coecients : h(n) = N 1 k=0 b k δ(n k) = b n (5.4) because δ(n k) = { 1 pour n = k 0 pour n k. (5.5) The main interest of FIR lters is that they do not have stability problems. Another interest is that their GPT can be easily constant : if the lter impulse response is odd or even (see later) Stability Assuming N < M and doing a simple element decomposition, we get : H(z) = N 1 k=0 b kz k M 1 M 1 k=0 a kz = A k k 1 p k z 1 k=0 Then, doing an inverse z transform (causal solution) : h(n) = M 1 k=0 A k p n k u(n), where u(n) stands for the Heaviside function. + n= h(n) will be bounded, and so the system will be stable, if p k < 1 k. A digital IIR lter is stable if all the poles of its transfer function H(z) are of modulus lower than 1. 31

33 5.4.4 Synthesis Introduction Digital lter synthesis corresponds to the procedure allowing to obtain the coecients dening the lter which will comply with the given specications. To implement a real time digital lter we have to implement it in the time domain. FIR synthesis FIR lters, also called non recursive lters, can be synthesized by a Fourier series development of the ideal response lter H ideal ( f) : H ideal ( f) = + k= j2π fk h ideal (k) e where the Fourier series coecients h ideal (k) represent the elements of the impulse response, or lter "coecients". In practise their number must be limited to N, called lter order. This limitation can be modeled by the use of a weighted window w(n) of length N : h real (n) = h ideal (n) w(n) leading to an approximated frequency response : H real ( f) = H ideal ( f) W ( f) where W ( f) is the Fourier transform of w(n). FIR synthesis method will then consists in determining the order N and the weighted window w allowing H real ( f) to cope with the desired specications. Be careful, in order to be able to implement the lter, its impulse response has to be causal. If not it must be shifted, then introducing a delay : ( ) h realcausal = h real n N 2 if N is even or hrealcausal = h ( ) real n N 1 2 if N is odd. This shift does not modify the modulus of the frequency response but adds a linear phase. Assuming N is even, we obtain : H realcausal ( f) = H real ( f)e jπ fn, 32

34 Giving : f) Hrealcausal ( = H real ( f) [ Arg H realcausal ( f) ] [ = Arg H real ( f) ] π fn [ If Arg H real ( f) ] = constant, which is true if the impulse response h real (n) is even or odd, then the group propagation time of the obtained lter is constant and equal to N/2 : GP T ( f) = 1 dϕ H ( f) 2π d f = 1 2π ( πn) = N 2 Note that optimization methods can also be used to improve the synthesis of digital lters. For example one consists in minimizing the mean square error between the specications H( f) and the frequency response of the lter obtained with the previous method. Another one, using an iterative algorithm called Remez, allows to reach the best approximation of the specied transfer function with constant amplitude oscillations. IIR synthesis IIR lters are recursive lters thanks to a feedback loop. Their synthesis is based on analog lter libraries. From the specications on H( f), we deduct those on H(f), choose an analog model and x its parameters in order to obtain a transfer function H(p) satisfying the specications, and, then, apply the bilinear transform to obtain H(z) : H (z) = [H(p)] p= 2 1 z 1. Te 1+z 1 The bilinear transform allows to pass from H(p) to H(z) keeping the frequency response and the stability of the lter. However it introduces a distortion on the frequency axis that may be compensated when passing from H( f) to H(f) by doing f = 1 πt e tan(π f), where T e is the sampling frequency. The innite impulse response allows a more selective ltering compared to FIR lters for a lower computation time. However the group delay of IIR lters is not constant but can be approached as so in a limited bandwidth. 33

35 5.4.5 Implantation Direct structure Figure 5.1 Direct structure This implantation needs two delayed lines and M + N + 1 multiplication/addition operations. Canonic structure With an intermediate variable : W (z) = X(z) M 1 k=0 a kz k we can obtained a simplied structure, called canonic. Figure 5.2 Canonic structure This implantation only needs one delayed line and M + N + 1 multiplication/addition operations. 34

36 Decomposed structure Most of the time a digital lter is implemented broken down into rst and second order cells : H(z) = b 0 + b 1 z a 1 z 1 H(z) = b 0 + b 1 z 1 + b 2 z a 1 z 1 + a 2 z 2 This can be done with series or parallel cells. Series decomposed structure (cascade) H(z) = M 1 i=0 H i (z), with H i (z) being rst and second order cells. Figure 5.3 Series structure Parallel decomposed structure H(z) = M 1 i=0 H i (z), with H i (z) being rst and second order cells. Figure 5.4 Parallel structure Non recursive structure This is the structure of FIR lters. 35

37 Figure 5.5 Non recursive structure 36

38 Chapitre 6 References Signal and Systems, by Simon Haykin and Barry Von Veen, Wiley Digital Signal Processing, by Alan V. Oppenheim, Ronald W. Schafer, Prentice-Hall. Documents on complex variable, Laplace transform, z tranform : http ://dobigeon.perso.enseeiht.fr/teaching/complexe.html 37