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Digital Signal Processing Introduction Course R.WEBER SP2 ESI module Traitement Numérique

Digital Signal Processing Introduction Course R.WEBER SP2 ESI module Traitement Numérique. A] Time and Frequency description of a digital signal B] Digital Filtering : Analysis and Design Tools. PART A Time and Frequency description of a digital signal. Temporal Description. ?.

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Digital Signal Processing Introduction Course R.WEBER SP2 ESI module Traitement Numérique

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  1. Digital Signal Processing Introduction Course R.WEBERSP2 ESI module Traitement Numérique A] Time and Frequency description of a digital signal B] Digital Filtering : Analysis and Design Tools

  2. PART ATime and Frequency description of a digital signal

  3. Temporal Description ? What really happens here ? x=[ 2.1 -0.5 1.3 0.5 0 1.7 -0.4 0.3 -1.9 -0.8 ]; -1 0 1 2 3 4 5 6 7 8 9 10 n (time index or relative time) Dr. R.Weber / Digital Signal Processing

  4. Theoretical spectral description !  Remarks : ! X(f+1)=X(f) • periodicity of the spectrum • f can be limited to [0, 0.5] for real signal (spectrum symmetry) • f can be limited to [- 0.5 , 0.5] or [0 , 1] for complex signal ?  The problem : Yes, with the Fourier Transform  The answer : Dr. R.Weber / Digital Signal Processing

  5. Theoretical spectrum of basic signals  impulse x = [ 1 zeros(1,1023)]; f 0 1.5 -1.5 1 -0.5 -1 0.5  Complex exponential x = A.*exp(2*pi*j*(0:1023)*fo); f fO 0 1.5 -1.5 1 -0.5 -1 0.5 x = A.*cos(2*pi*(0:1023)*fo);  sine wave f fO 0 1.5 -1.5 1 -0.5 -1 0.5 Dr. R.Weber / Digital Signal Processing

  6. Theoretical spectrum of basic signals  White Gaussian Noise x = .*randn(1, 1024); f 0 1.5 -1.5 1 -0.5 -1 0.5  General case : time and frequency are related f ? • How will appear • a pure sine wave, • a pure impulse ? n specgramdemo(x,Fs); Dr. R.Weber / Digital Signal Processing

  7. Basic spectral properties (1)  Pure delay :  Modulation or frequency shift: Dr. R.Weber / Digital Signal Processing

  8. Basic spectral properties (2)  Product and Convolution:  Parceval relation : Dr. R.Weber / Digital Signal Processing

  9. Practical spectral description (1) ! No digital implementation possible X=fft(x); FFT(x) = DFT(x) with M a power of 2  Theoretical Formulation :  Approximations : The Discrete Fourier Transform • n[0, M-1] • Only f / f=k/M Dr. R.Weber / Digital Signal Processing

  10. Practical spectral description (2) Theoretical spectrum (n]-,+ [) |X(f)| M Windowed spectrum (n]0,M]) h(n) H(f) n f 0 0.5 Practical spectrum (n]0,M], f k/M) |X(k)| n n k/M 0 0.5 1/M 2/M Possibility to reduce distortion by using different kinds of windows, h(n) Dr. R.Weber / Digital Signal Processing

  11. Windowing Hamming Rectangular Blackmann-Harris Hann WINtool sous Maltab Dr. R.Weber / Digital Signal Processing

  12. DFT properties  Inverse DFT : x=ifft(X);  Parceval relation : Power spectral density cov(x) Dr. R.Weber / Digital Signal Processing

  13. Power Spectral density(1) with ? What is the measured power spectral density of a sinus wave when using a classical DFT ?  Spectral line case :a complex exponential or s(n) is sine wave DFT so then When s=sin(2*pi*(0:1023)*20/1024); plot((abs(fft(s)).^2)/1024) Dr. R.Weber / Digital Signal Processing

  14. Power Spectral density (2) ! Problem : each slice of M samples will give a different power spectral density estimation  Solution : averaging of all the local power spectral density L samples M samples 50% overlap  Impact of the windowing psd(s,M,Fs,[h(0) h(1) … h(M-1)]) =  Random noise case : f=k/M Dr. R.Weber / Digital Signal Processing

  15. Signal Power ? What is the mean power ? psd psd f 0 f 1 0.5 - 0.5 0.5 0 ? What is the mean power of the sum of signals ? Dr. R.Weber / Digital Signal Processing

  16. Signal Processing goals • s’(n)  s(n) • Power(b’(n)) 0  Goal : y(n)=s’(n)+b’(n) x(n)=s(n)+b(n) processing Signal to noise ratio :  Example : Spectral point of view Temporal point of view s(n) x(n)=s(n)+b(n) Dr. R.Weber / Digital Signal Processing

  17. PART BLinear Digital Filtering :Analysis and Design Tools

  18. Linear Digital filtering y = filter(b,a,x); b = [ b0 b1 … bN]; a = [ 1 a1 … aD]; 1. Filter if x1(n) y1(n) Filter thenx1(n)+x2(n)  y1(n)+y2(n) Filter if x2(n) y2(n) Filter y(n-T) then x(n-T) Filter if x(n) y(n)  Formulations : temporal view x(n) Filter y(n) non recursive part recursive part  Basic properties : Linearity Temporal reproducibility Dr. R.Weber / Digital Signal Processing

  19. Impulse response The impulse response (n) h(n) =(n) =h(n) x = [ 1 zeros(1,29)]; h = filter(b,a,x);  Formulation : y(n) x(n) Filter  Consequences : Non recursive Finite impulse response (FIR) Recursive Infinite impulse response (IIR) Dr. R.Weber / Digital Signal Processing

  20. Application to stability Always stable h(n) c=0.985 c=0.95 c=0.995 Example : c=0.999 c=1 c=1.001  Formulation :  Consequences : Finite impulse response (FIR) Infinite impulse response (IIR) Risk of instability Example : x = [ 1 zeros(1,400)]; c=0.985; b= 1; a=[1 -2*c*cos(2*pi*0.01) c.^2]; h = filter(b,a,x); plot(h) Dr. R.Weber / Digital Signal Processing

  21. Convolution Filter with impulse response h(n) x(n) x(0)(n) + x(1)(n-1) + x(2)(n-2)  Remarks : + which is coherent ! x(3)(n-3) +  Formulation : = -- A recursive filter can be approximated by a non recursive one h(n) Dr. R.Weber / Digital Signal Processing

  22. The Z-transform  Application to the filtering: ZT ZT  Formulation :  useful (and simple) properties: • Given X(z)=ZT(x) and Y(z)=ZT(y) • linearity : ZT(x(n) + y(n)) = X(z) + Y(z) • convolution : Zt[(x@y)(n)] = X(z) Y(z) • ZT(x(n+1)) = z X(z) ; ZT(x(n+k)) = zk X(z) • ZT(x(n-1)) = z-1 X(z) ; ZT(x(n-k)) = z-k X(z) These make the ZT very interesting ! Dr. R.Weber / Digital Signal Processing

  23. Stability rule o zeros x poles Im(z) Instable filter Stable filter Re(z)  Formulation : o 1 x Filter stability if |zp,i|<1 o x ? x Verify this rule with the previous example 1 x o x o zplane(b,a); Dr. R.Weber / Digital Signal Processing

  24. Frequency response (1) Filter with impulse response h(n)  Formulation : x = [ 1 zeros(1,1023)]; c=0.985; b= 1; a=[1 -2*c*cos(2*pi*0.2) c.^2]; h = filter(b,a,x); H= fft(h); f=(0:1023)/1024; plot(f,10*log10(abs(H).^2)); Dr. R.Weber / Digital Signal Processing

  25. Simple filter design method h(n) happrox(n) n n L values • Draw the frequency response, H(f), you are looking for. • Compute the impulse response, h(n), by applying an inverse Fourier Transform • By truncating the length of this response, you obtain a sequence of values which are also the non-recursive coefficients of an approximate FIR version of your filter. H(f) FT-1 rectangular windowing f Checking : Ripple can not be removed even with L large Solution : other type of windowing (hamming, hanning, blackmannharris…) Dr. R.Weber / Digital Signal Processing

  26. Frequency response (2)  Another formulation : ? c=0.985; b= 1; a=[1 -2*c*cos(2*pi*0.2) c.^2]; %zero padding to increase frequency resolution b=[b zeros(1,1023)]; a=[a zeros(1,1020)]; H= fft(b)./fft(a); f=(0:1023)/1024; plot(f,10*log10(abs(H).^2)); [H,f]=freqz(b,a,1024,1); Dr. R.Weber / Digital Signal Processing

  27. Frequency response (3) zplane(b,a);  Another formulation : f=0.25 X O f=0.5 f=0 O X Dr. R.Weber / Digital Signal Processing

  28. Frequency response (4) Dr. R.Weber / Digital Signal Processing

  29. Phase response (1) x(n) Filter  pure delay : ? ? ? ?  general case : Time responses for the given frequencies grpdelay(b,a,1024,1); freqz(b,a,1024,1); Dr. R.Weber / Digital Signal Processing

  30. Phase response (2)  linear phase filter : Yes, if FIR filters with symmetric or antisymmetric coefficients No, if IIR Dr. R.Weber / Digital Signal Processing

  31. Filtered Power Spectral density ? Problem : What is the psd after filtering ? x(n) Filter y(n) |H(f)| 2 1 0 f 0.5 f0 Dr. R.Weber / Digital Signal Processing

  32. Filter Specifications(1) Passband Stopband Ripple Attenuation Transition band 2 1 3 4 5 Dr. R.Weber / Digital Signal Processing

  33. Filter Specifications (2)   Order or complexity Attenuation or • If  Ripple or  Transition Band  General rules : Ripple Attenuation Transition band then • For a given specification, IIR is always less complex than FIR But be careful to the stability (and the phase linearity) ! Dr. R.Weber / Digital Signal Processing

  34. Basic Design Methods Bandpass :  Basic FIR methods : ordre 63  64 coefficients, linear phase Least-square Equiripple order 12  26 coefficients  Basic IIR methods : order 24  50 coefficients Cauer or Elliptic Butterworth Dr. R.Weber / Digital Signal Processing

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