Digital Signal Processing

Transcription

1 Digital Signal Processing Master SATCOM Objectives: 1- Being able to use the following tools in order to conduct a digital signal spectral analysis: Digital Fourier Transform (DFT) Power Spectral Density (PSD) Auto and cross correlations 2- Being able to synthesize a digital filter in order to create or modify signals. Provided documents ( 1- The slides 2- A detailed course and exercises with solutions 3- The texts for the practical work 4- Some elements for Matlab programming. A written exam including the analog part Nathalie Thomas

2 Chapter 1 Digital Signal Processing Introduction Master SpaceCOM Signal digitization - Main digital processing tools - Processing time / real time processing Nathalie Thomas

3 Analog signal: Defined at all times by real values Analog signal: Defined at all times by real values Example : Digitization: x(t) = sin (2pf t) Sampling quantization Digital signal: time Defined at discrete times and taking amplitudes in a finite set of values. T = 1/f

4 Sampled signal: Defined at discrete times by real values Is it possible to keep all the signal information in the collected samples? Example : k = sample index No, it s impossible x(kt e ) = sin (2pf kt e ) T e : sampling period time Note : here is a periodical sampling Of course it is! On one condition : F e = 1 T e > 2 F max

5 Signal de parole : Amplitude Example 1 When the sampling frequency is reduced, the periodizations come closer and closer and at a moment they appear in the signal bandwidth, generating some aliasing : F e > 2F max time temps F e = 2F max -F e - F max F max F e frequency Obtained spectrum using the Fourier transform : Energy distribution -2F e -F e - F max F max F e 2F e F e < 2F max - F max F max frequency Fourier transform of the well sampled signal: -4F e -3F e -2F e -F e F e 2F e 3F e 4F e - F max F max Fourier transform of the sampled signal with F e = F max / 12 : -F e - F max F max F e fréquences frequency

6 Original image: 512*512 pixels, quantized on 8 bits Example 2 Under sampled image with a factor of 2: 256*256 pixels, quantized on 8 bits 5 image sous échantillonné d"un facteur 2 image d"origine Under sampled image with a factor of 4 : 128*128 pixels, quantized on 8 bits 2 4 image sous échantillonné d"un facteur Moiré effects

7 Signal Dynamic Quantized signal: Defined at all times by a finite number of values Example (uniform quantifization by truncating on nb = 2 bits) : x(t) = sin (2pf t) {V1, V2, V3, V4} x(t) = sin (2pf t) Original signal Example : x(t) = sin (2pf t) V1() V2(1) Signal quantized on 2 bits time V3(11) q time V4(1) The number of possible values for the signal amplitudes after quantization is given by the number of quantization bits. With nb bits it is possible to code 2 nb levels on the signal dynamic. The signal dynamic is divided into 2 nb = 2 2 = 4 levels with the same size (uniform quantifization) and all the amplitudes belonging to the same level are assigned the same value coded on nb=2 bits. The level width gives what is called the quantization step q : q = Signal dynamic 2 nb

8 Is it possible to keep the same information in the quantized signal? But the quantized and unquantized signals can be very close Yesss!! A well done quantization adds a noise, called quantization noise, to the original signal (unquantized). It can be shown that the quantization signal to noise ratio (SNR) is given by: Tough luck but no SNR Q (db) = 1 log P unquantized signal P quantization noise = 6nbconstant Quantization is an irreversible operation. The quantized signal will never be exactly The constant depends on the considered signal, but, watever is the signal, we can note that, when the number of quantization bits nb increases, the signal to noise ratio also increases quickly leading to a transparent operation. the same as the unquantized one.

9 Example 1 Quantized signal on 16 bits => 2 16 = levels: Original image: 512*512, Quantized on 8 bits image d"origine Example 2 Amplitude temps time Quantized signal on nb = 4 bits => 2 4 = 16 levels: Amplitude Quantized image on 4 bits Quantification : 4 bits 5 5 Quantized image on 2 bits Quantification : 2 bits time

10 512 pixels Digital signal: Example 2 : Defined at discrete times by a finite 45 Quantification : 8 bits number of values = sampling and quantized signal Original image: 512*512, Quantized on 8 bits Zoom Example 1 : 75 V1() x(kt e ) = sin (2pf kt e ) {V1, V2, V3, V4} image d"origine gray level among 256 (2 8 ) V2(1) V3(11) temps V4(1) T e : sampling period Finaly the binary information associated to the sinusoid, or digital sinusoid, will be given by: pixels

11 France Inter RTL2 Le Mouv' France Culture Nostalgie Skyrock NRJ Virgin Radio RTL RMC France Info Fun Radio BFM Digital signal processing Why? Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering To suppress/reduce unwanted components : noise, some frequencies Fourier Amplitude Transform Energy distribution Needed bandwidth (for transmission on carrier frequency) Example 1 time temps frequency What tools? Example of shared bandwidth: FM band in Toulouse Digital Fourier transform (DFT) Auto and cross correlation functions Power spectral density (PSD) Digital Filters 87.5 MHz FM band 18 MHz Transmission channel Bandpass Reserved for France Info frequency

12 Example 2 Digital signal processing Why? Different V Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering To suppress/reduce unwanted components : noise, some frequencies filters Several Signals carrying the same binary information -V V -V V time time time What tools? -V Digital Fourier transform (DFT) Auto and cross correlation functions Fourier Transform Several Occupied bandwidth Power spectral density (PSD) Digital Filters fréquences frequency

13 Digital signal processing Why? 1 Transmitted signal and transmitted signal spectrum signal émis Example 3 Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering To suppress/reduce unwanted components : noise, some frequencies What tools? Digital Fourier transform (DFT) Auto and cross correlation functions Power spectral density (PSD) Digital Filters Power/frequency (db/rad/sample) spectre du signal émis Normalized Frequency ( p rad/sample) Power/frequency (db/rad/sample) Received signal and received signal spectrum After the receiver filter signal reçu après filtrage spectre du signal reçu après filtrage Normalized Frequency ( p rad/sample) BER = Power/frequency (db/rad/sample) signal reçu Received signal and received signal spectrum Before the receiver filter spectre du signal reçu Normalized Frequency ( p rad/sample) BER = 7.8e-4 Filtering SNR = 1 db

14 Digital signal processing 1 Transmitted signal and transmitted signal spectrum signal émis Example 4 Why? -1 Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering To suppress/reduce unwanted components : noise, some frequencies What tools? Digital Fourier transform (DFT) Auto and cross correlation functions Power spectral density (PSD) Digital Filters Power/frequency (db/rad/sample) Power/frequency (db/rad/sample) spectre du signal émis Normalized Frequency ( p rad/sample) signal reçu après filtrage Received signal and received signal spectrum After the receiver filter spectre du signal reçu après filtrage Normalized Frequency ( p rad/sample) BER = 7.53e-4 Power/frequency (db/rad/sample) Received signal and received signal spectrum Before the receiver filter signal reçu spectre du signal reçu Normalized Frequency ( p rad/sample) BER =,179 Filtering SNR = db

15 Transmitted Image Image émise 1 Example 5 Digital signal processing 2 Why? Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering To suppress/reduce unwanted components : noise, some frequencies What tools? Image émise Image reçue Received image After the receiver filter Image reçue Received Image Before the receiver filter Image reçue BER = 7.86e-4 Digital Fourier transform (DFT) Auto and cross correlation functions Power spectral density (PSD) Digital Filters BER = Filtering SNR = 1 db

16 Transmitted Image Image émise 1 Example 6 Digital signal processing Why? Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering Image émise Image reçue Received Image Before the receiver filter Image reçue To suppress/reduce unwanted components : noise, some frequencies What tools? Digital Fourier transform (DFT) Received image 5 After the Image receiver reçue filter BER =.1581 Filtering Auto and cross correlation functions 4 Power spectral density (PSD) Digital Filters BER = e-4 SNR = db

17 Digital signal processing Example 7 Why? Extract some information: Needed bandwidth to transmit the signal? Defects detection Construct/Modify signals: To adapt them to a transmission channel: modulation, filtering To suppress/reduce unwanted components : noise, some frequencies What tools? Digital Fourier transform (DFT) Auto and cross correlation functions Power spectral density (PSD) Digital Filters t (secondes) DSP signal réel 1 T(X) Signal réel filtré Signal réel filtré deux fois Fréquences (Hz) Temps (s) Statistique de test, seuil considéré Signal réel Indices période de signal T(X) seuil Zoom Signal réel t (secondes) DSP signal réel 1 After highpass filtering to suppress the 5 Hz frequency: Fréquences (Hz) Detection of higher frequency components, sign of potential occuring Problems (arcs)

18 Example 8 Digital signal processing Why? Amplitude Extract some information: Needed bandwidth to transmit the signal? time Signal detection in noise Construct/Modify signals: To adapt them to a transmission channel: Noise? modulation, filtering To suppress/reduce unwanted components : noise, some frequencies Autocorrelation function R x (t) Not only What tools? Some noise Digital Fourier transform (DFT) Auto and cross correlation functions Power spectral density (PSD) Digital Filters and some signal t

19 Digital signals Digital tools for processing Processing time What differences between the theoritical and the digitally implemented tools? Digital processing Processing time? Real time processing? Examples : Biased estimation for the auto autocorrelation function: Basic digital processing Digital Fourier transform (DFT): Fourier Transform, Digital filtering (Finite Impulse Response filter): Auto and cross correlation functions, Power spectral density, Processing time = Number of Multiplication/Accumulation (MAC) Filtering. Real Time processing : processing time < T e

20 Chapter 2 Digital Signal Processing Master SpaceCOM Digital Fourier Transform (DFT) Nathalie Thomas

21 Fourier Transform (FT) Digital Fourier Transform (DFT) Example FT of the signal: Modulus of the signal FT: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) With Matlab : %Computation of the signal DFT X=fft(x); %visualization of the DFT modulus figure; plot(abs(x))?? Visualization of the signal DFT modulus: ? 92? Two cosines? With frequencies 11 and 92 Hz?

22 Fourier Transform (FT)?? Digital Fourier Transform (DFT) 1- Time sampling 2- Signal of limited duration 12 => Sampled and time limited signal 3- Frequency sampling What impacts??

23 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Shannon Condition!! Example F e > 2F max F e < 2F max Aliasing

24 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! How to read the spectrum!! Two diracs but only one cosine Visualization on [, F e ] : - F e 2

25 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! How to read the spectrum!! Example Matlab simulation: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) %Computation of the signal DFT X=fft(x); Visualization of the DFT modulus: - Two cosine? No only one! %Visualization of the DFT modulus figure; plot(abs(x))

26 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Scales (time and frequencies)!! Example Matlab simulation: Signal : %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples DFT modulus: - %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) %Computation of the signal DFT X=fft(x); %Visualization of the DFT modulus figure; plot(abs(x)) One period = 1 seconds?? frequencies f =11 Hz and F e - f =92 Hz??

27 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Scales (time and frequencies)!! Example Matlab simulation: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) Signal: 1 period = 1 signal samples %Computation of the signal DFT X=fft(x); %Visualization of the DFT modulus figure; plot(abs(x)) Size of the vector = 11 samples Sample index Not time in seconds!!

28 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Scales (time and frequencies)!! Example Matlab simulation: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples Signal: %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot([:te:n*te],x) %Computation of the signal DFT X=fft(x); %Visualization of the DFT modulus figure; plot(abs(x)) One period =,1 seconds 1 period =,1 seconds Time (seconds)

29 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Scales (time and frequencies)!! Example Matlab simulation: DFT modulus: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples Two cosine? No only one! %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) %Computation of the signal DFT X=fft(x); %Visualization of the DFT modulus figure; plot(abs(x)) th 92 th 11 Size of the vector = 11 samples f =1 DFT samples Sample index Not frequency in Hz!!

30 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Scales (time and frequencies)!! Example Matlab simulation: DFT modulus: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples Two cosine? No only one! %Signal generation x=cos(2*pi*f*[:te:n*te]); 3 25 f = 1 Hz %Signal plotting figure; plot(x) 2 15 %Computation of the signal DFT X=fft(x); %Visualization of the DFT modulus figure; plot(linspace(,fe,length(x)),abs(x)) f F e -f F e Frequency (Hz)

31 Fourier Transform (FT) 1- FT of the sampled signal 1- Time sampling => spectrum periodization!! Scales (time and frequencies)!! Example %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) Matlab simulation: %Computation of the signal DFT X=fft(x); 5 %Visualization of the DFT modulus figure; plot(linspace(-fe/2,fe/2,length(x)),fftshift(abs(x))) F e /2 - f f F e /2 The spectrum is centered around Frequency (Hz)

32 Fourier Transform (FT) 2- FT of the time limited signal 2- Digital of limited duration [, L] Example Signal of finite duration Signal of infinite duration x Truncation window

33 Fourier Transform (FT) 2- FT of the time limited signal 2- Signal of limited duration [, L]!! Spectrum (DFT) distortion!! Example FT of the time unlimited signal: FT of the time limited signal * FT of the truncation window:

34 Fourier Transform (FT) 2- FT of the time limited signal 2- Signal of limited duration [,L]!! Spectrum (DFT) distortion!! Arising problems: Example 1 Example 2 Sum of two cosines with close frequencies Sum of two cosine with very different powers => Only one cosine!! Limited separating power => Only one cosine!! Ripples around transitions

35 Fourier Transform (FT) 2- FT of the time limited signal 2- Signal of limited duration [,L]!! Spectrum (DFT) distortion!! Example Sinus Signal of limited duration cardinal Unlimited Signal x Truncation window of length T : w(t) FT Ripple amplitudes Separating power

36 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 12- Sampled and time limited signal!! Spectrum (DFT) distortion!! Example Sinus cardinal Dirichlet Core Signal of limited duration -.2 Unlimited Signal x Truncation window on N samples: w(k) FT Ripple amplitudes xxx x Separating power

37 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 12- Sampled and time limited signal!! Spectrum (DFT) distortion!! Example 1.8 Signal of limited duration Use other truncation windows (weight windows)? Unlimited signal x RECTANGULAR truncation window rectangulaire Hamming Blackman

38 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 12- Sampled and time limited signal!! Spectrum (DFT) distortion!! Use other truncation windows (weight windows)? Examples Tracés Digital des fenêtres weight windows de troncature Tracés FT des of TF digital des fenêtres weight windows de troncature Several amplitudes for the associated ripples Several separating power Rectangular rectangulaire Hamming Blackman Rectangular rectangulaire Hamming Blackman

39 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 2- Sampled and time limited signal!! Spectrum (DFT) distortion!! Use other truncation windows (weight windows)? Example 1 Sum of two cosines with close frequencies %Example1 %Parameters f1=2; %frequency for cosine 1 f2=27; % frequency for cosine 2 Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x1=cos(2*pi*f1*[:te:n*te]); x2=cos(2*pi*f2*[:te:n*te]); %Rectangular window x=x1x2; X_V1=fft(x,496); %Hamming window w=window(@hamming,length(x)); x=(x1x2).*w.'; X_V2=fft(x,496); %Plots figure subplot(2,1,1) plot(linspace(,1,496),log1(abs(x_v1))); title( Rectangular window'); subplot(2,1,2) plot(linspace(,1,496),log1(abs(x_v2))); title('hamming window'); 1 1 Rectangular fenetre naturelle window Hamming fenetre de Hamming window 1 1-2

40 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 2- Sampled and time limited signal!! Spectrum (DFT) distortion!! Use other truncation windows (weight windows)? %Example 2 %Paramètres f1=2; %frequency for cosine 1 f2=32; % frequency for cosine 2 Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x1=cos(2*pi*f1*[:te:n*te]); x2=.5*cos(2*pi*f2*[:te:n*te]); x=x1x2; %Rectangular window X=fft(x,496); figure subplot(3,1,1) semilogy(linspace(,1,496),abs(x)); axis([ 1 1^-5 1^5]) title( Rectangular window'); %Hamming window w=window(@hamming,length(x)); x=(x1x2).*w.'; X=fft(x,496); subplot(3,1,2) semilogy(linspace(,1,496),abs(x)); title('hamming window'); %Blackman window w=window(@blackman,length(x)); %w=blackman(n); x=(x1x2).*w.'; X=fft(x,496); subplot(3,1,3) semilogy(linspace(,1,496),abs(x)); title('blackman window'); Example 2 Sum of two cosine with very different powers Rectangular fenetre naturelle window Hamming fenetre de Hamming window Blackman window fenetre de Blackman

41 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 2- Sampled and time limited signal!! Spectrum (DFT) distortion!! Example 3 Rectangular fenetre naturelle window X(f) X(f) Fréquences Normalizednormalisées frequency fenetre Hamming de Hamming window Signal??? X(f) Fréquences Normalized normalisées frequency fenetre Blackman de Blackman window Fréquences Normalized normalisées frequency

42 Fourier Transform (FT) 2-TF FT signal of the échantillonné sampled and time et de limited durée limitée signal 2- Sampled and time limited signal!! Spectrum (DFT) distortion!! Example 3 - Rectangular fenetre naturelle window X(f) 1 3 cosines with the same power 4 cosines: 3 with the same power 1 with a lower power X(f) Fréquences Normalizednormalisées frequency fenetre de Hamming 1 2 Hamming window Fréquences Normalized normalisées frequency fenetre Blackman de Blackman window 2 cosines with the same power but something is strange X(f) 1 3 cosines with different powers Fréquences Normalized normalisées frequency Proposed solution: Use SEVERAL truncation (or weight) windows for one analysis

43 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling Example Matlab simulation: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) Plot of the DFT modulus: Two cosine? No only one! But where are the Dirichlet cores??? %Computation of the signal DFT X=fft(x); %Visualization of the DFT modulus figure; plot(linspace(,fe,length(x)),abs(x)) f F e -f F e Frequency (Hz)

44 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling!! Bad spectrum visualization!! Example DFT modulus DFT modulus f F e -f f (Hz) f F e -f f (Hz) F e /N F e /N

45 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling!! Bad spectrum visualization!! Proposed solution: interpolation with Zero Padding Mdulus of X(n) Example Modulus of Y(n) x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xx x xx xxx xx x x x x x x x x x x x f F e -f F e /N f (Hz) x x x x x x x x x x x xx x xx xxx xx x x x x x x x x x x x f F e -f F e /N F e /MN f (Hz) N points spaced from F e /N between and F e MN points spaced from F /MN between and F

46 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling!! Bad spectrum visualization!! Proposed solution: interpolation with Zero Padding Exemple Matlab simulation: %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot(x) %Computation of the signal DFT X=fft(x,128); %Visualization of the DFT modulus figure; plot(linspace(,fe,length(x)),abs(x)) DFT modulus: f F e -f F e Two cosine? No only one! Here are the Dirichlet cores! Frequency (Hz)

47 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling!! Bad spectrum visualization!! Matlab simulation: Proposed solution: interpolation with Zero Padding Example Plots of the DFT modulus For several value of the Zero Padding parameter: 6 sans no ZP 6 ZP : 128 %Parameters f=1; %cosine frequency Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]); %Signal plotting figure; plot([:te:n*te],x) %Computation of the DFT X1=fft(x); X2=fft(x,128); X3=fft(x,256); X4=fft(x,512); Zero Padding parameter % Plots of the DFT modulus figure; subplot(2,2,1) plot(linspace(,fe,length(x1)),abs(x1)) xlabel( Frequency (Hz) ) subplot(2,2,2) plot(linspace(,fe,length(x2)),abs(x2)) xlabel( Frequency (Hz) ) subplot(2,2,3) plot(linspace(,fe,length(x3)),abs(x3)) xlabel( Frequency (Hz) ) subplot(2,2,4) plot(linspace(,fe,length(x4)),abs(x4)) xlabel( Frequency (Hz) ) TFD(x) TFD(x) Fréquences Frequency (Hz) ZP : Fréquences Frequency (Hz) TFD(x) TFD(x) Fréquences Frequency (Hz) ZP : Fréquences Frequency (Hz)

48 Fourier Transform (FT) Digital Fourier Transform (DFT) DFT DFT -1 Inverse Fourier transform (FT -1 ) Digital inverse Fourier transform(dft -1 ) 3- Frequency sampling => signal periodization x(k) is periodical of period N!! A product is changed into a circular convolution product by DFT or DFT -1!! Linear («classical») convolution: Circular convolution:

49 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling => signal periodization!! Circular convolution = linear convolution!! Example (N=3) x(k) is periodical of period N Linear convolution: Circular convolution: x 1 (p) : x 2 (p) : x 1 (p modulo N ) : x 2 (p modulo N ) : x 2 (k-p) : p= p= p= p= p= p= p= p= p= p= p= p= p= p=

50 Fourier Transform (FT) Digital Fourier Transform (DFT) 3- Frequency sampling => signal periodization Example (N=3) x(k) is periodical of period N By adding N zeros Linear convolution = circular convolution

51 Inverse Digital Fourier Transform (DFT -1 ) Digital Fourier Transform (DFT) 4- DFT properties: Linearity Translation => phase rotation Hermitic symetry Circular convolution Parseval equality Fast Fourier Transform Algorithm (FFT) for a lower computation time: Nlog 2 (N)<<N 2 for a DFT (or DFT -1 ) on N points

52 Inverse Digital Fourier Transform (DFT -1 ) Digital Fourier Transform (DFT) 5- An algorithm to reduce the computation time : Cooley Tuckey FFT (Fast Fourier Transform) DFT of order N = 2 p : N 2 (/x) operations First decomposition: N2(N/2) 2 << N 2 (/x) operations N (/x) operations Even indexes (DFT of order N/2) Odd indexes DFT of order N/2 Second decomposition: 2(N/2)4(N/4) 2 (/x) operations Even indexes Odd indexes Even indexes Odd indexes 4 DFTs of order N/4 Np DFT of order 2 = N log 2 (N) (/x) operations

53 Inverse Digital Fourier Transform (DFT -1 ) Digital Fourier Transform (DFT) Time sampling => spectrum periodization - Respect Shannon condition - Pay attention to the way you read the spectrum and the scales (in time and frequency) Signal of limited duration => spectrum distortion - Use several weight windows for one signal analysis => several separating powers and amplitudes for the ripples appearing in the frequency domain Frequency sampling => signal periodization - Pay attention to the lecture of the spectrum => use the zero padding interpolation technique to have a good visualization - DFT and DFT -1 convert a product into a circular convolution product => if a linear convolution is required: linear convolution = circular convolution thanks to zero padding FFT = Fast Fourier Transform algorithm - Condition N=2 p => decomposition of the DFT into interlaced subseries - Processing time: Nlog 2 (N) << N 2

54 Chapter 3 Digital Signal Processing Master SpaceCOM Digital auto and cross correlation functions Nathalie Thomas

55 (Hermitic symetry) Example Matlab simulation:?? 6 %Parameters f=1; %frequency of the cosine Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples %Signal generation x=cos(2*pi*f*[:te:n*te]rand*2*pi); % Computation and plotting of the biased % autocorrelation Rx=xcorr(x); figure; plot([-n*te:te:n*te],rx); xlabel('time (s)'); ylabel('r_x') R x Temps (s) Time (s)

56 (Hermitic symetry) BIAISED estimator Example Matlab simulation: %Parameters f=1; %frequency of the cosine Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples Multiplicative Triangular bias %Signal generation x=cos(2*pi*f*[:te:n*te]rand*2*pi); % Computation and plotting of the biased % autocorrelation Rx=xcorr(x); figure; plot([-n*te:te:n*te],rx); xlabel('time (s)'); ylabel('r_x') R x Temps (s) Time (s)

57 (Hermitic symetry) UNBIASED estimator Example Matlab simulation: %Parameters f=1; %frequency of the cosine Fe=1; %sampling frequency Te=1/Fe; %sampling period N=1; %number of generated samples % Computation and plotting of the unbiased % autocorrelation for several signal realizations x1=cos(2*pi*f*[:te:n*te]rand*2*pi); Rx1=xcorr(x1,'unbiased'); x2=cos(2*pi*f*[:te:n*te]rand*2*pi); Rx2=xcorr(x2,'unbiased'); x3=cos(2*pi*f*[:te:n*te]rand*2*pi); Rx3=xcorr(x3,'unbiased'); figure plot(rx1); hold on; plot(rx2,'r'); plot(rx3,'g') legend('realization 1','realization 2','realization3') xlabel('time (s)'); ylabel('r_x') R x réalisation 1 réalisation 2 réalisation Temps (s) Time (s) R x zooms réalisation realization 1 1 réalisation realization 2 2 réalisation3 realization 3 R x High variance on the sides réalisation 1 réalisation 2 réalisation Temps (s) Time (s) Temps (s) Time (s)

58 Processing time for the computation of a digital cross correlation function 1 Time domain estimation (biased or unbiased) Processing time ~ N 2 (/x) operations 2- Frequency domain estimation Linear convolution: If: then: And: DFT Processing time: N3Nlog 2 (N) (/x) operations The frequency domain estimation allows to reduce the pocessing time: N3Nlog 2 (N) << N 2 (/x) operations If: - we can put an infinite sum in the expression of R xy (k) instead of,, N-1 - the DFT converts a linear convolution product into a product BUT!! Signals are periodical!!

59 Chapter 4 Digital Signal Processing Master SpaceCOM Power Spectral Density (PSD) Nathalie Thomas

60 Two basic estimators Periodogram Note: périodogram biased correlogram Correlogram 14 Drawback: inconsistent estimators Convolutive bias: Fejer core : The variance does not depend on the signal observation length: Frequences normalisées Normalized frequency

61 PSD PSD Two basic estimators Examples: signal samples Simulated DSP simulée PSD Theoretical DSP théorique PSD DSP x(t) Temps Fréquences Normalized normalisées frequency 1 signal samples Simulated DSP simulée PSD Theoretical DSP théorique PSD DSP 2.5 x(t) Temps Fréquences Normalized normalisées frequency

62 Modified versions Objective: reduce the estimation variance when the signal observation length increases Cumulative periodogram (Bartlett) : x(t) Temps L N=ML

63 PSD PSD PSD Modified versions Cumulative periodogram (Bartlett) : Examples : N= 1 samples, 1 window DSP Simulated simulée PSD DSP Theoretical théorique PSD N= 1 samples, M=1 windows of L= 1 samples DSP Simulated simulée PSD DSP Theoretical théorique PSD N= 1 samples, M=1 windows of L=1 samples 1.4 Simulated DSP simulée PSD Theoretical DSP théorique PSD DSP 3 DSP 1.8 DSP Fréquences normalisées Normalized frequency Fréquences normalisées Drawback: for a given signal observation length, the bias increases when the variance is reduced Example : (N=1, M=1) => (N=1, M=1, L=1) Normalized frequency Fréquences normalisées Normalized frequency 1 samples 1 samples Fejer core: Frequences normalisées Normalized frequency Frequences normalisées Normalized frequency 2/N

64 Modified periodogram: PSD PSD Modified versions Modified correlogram: Weigth window Drawback: abrupt variations are smoothed, the separating power is reduced Welch periodogram = cumumative and modified periodogram - Sliding window => M > M windows of length L allowed to overlap - Modified periodogram on each signal section - Example: M=1 windows of L=1 samples M=1 windows of L=1 samples Overlapping = L/2, Hamming windowing Simulated PSD Theoretical PSD DSP simulée DSP théorique Simulated PSD Theoretical PSD DSP simulée DSP théorique N= 1 samples DSP.6 DSP Fréquences Normalized normalisées frequency Normalized Fréquences normalisées frequency

65 PSD PSD PSD PSD PSD Modified versions Examples of cumulative periodograms: 7 6 K= 1 samples, 1 window Simulated DSP simulée PSD Theoretical DSP théorique PSD K= 1 samples, M=1 windows of L= 1 samples Simulated PSD DSP simulée Theoretical PSD DSP théorique K= 1 samples, M=1 windows of L=1 samples Simulated DSP simulée PSD Theoretical DSP théorique PSD DSP 4 3 DSP DSP Fréquences Normalized normalisées frequency Fréquences Normalized normalisées frequency 2 Simulated DSP simulée PSD 1.8 Theoretical DSP théorique PSD Fréquences Normalized normalisées frequency Simulated DSP simulée PSD Theoretical DSP théorique PSD Welch periodogram: (overlapping = L/2) DSP Fréquences Normalized frequency normalisées DSP Fréquences Normalized normalisées frequency

66 Chapter 5 Digital Signal Processing Master SpaceCOM Digital Filtering Nathalie Thomas

67 -9 Analog linear time invariant filters A TOOL FOR THEIR STUDY: THE LAPLACE TRANSFORM Definition : Analysis of ANALOG linear time invariant systems Reminders Transfer function o Time analysis (step response, ramp response): stability (poles of H(p) with negative real parts), rapidity (rise time, settling time), precision (static error, following error) o Frequency analysis (response to a sinusoidal input) : Bode diagrams=> cutoff frequency, bandpass, stopband attenuation, resonnance o Example : Frequency response Step response Bode diagram

68 Digital linear time invariant filters A TOOL FOR THEIR STUDY: THE Z-TRANSFORM Definition: Analysis of DIGITAL linear time invariant systems o Main properties: Linearity: Time shift: Convolution product: o Existence: o o inversion: Note: tables of ZT exist

69 Digital linear time invariant filters DEFINITIONS Linearity: Filter Filter Filter Time invariance: Filter Filter Impulse response and transfer function: Impulse response Digital impulse (Kronecker symbol) Filter Transfer fonction Frequency response and group propagation time (GPT) Frequency response Note: GPT

70 Digital linear time invariant filters REALISABILITY o Causality: o Stability: Three conditions on the impulse response: o Real impulse response:

71 Digital linear time invariant filters RATIONNAL FILTERS By analogy with analog filters: Example : i(t) R Differential equation in the time domain x(t) C y(t) Rationnal transfer function in p-domain Digital filters definition: %filtering with Matlab y=filter(b,a,x); Filtered signal Definition of the wanted filter (two sets of coefficients) Signal to filter RATIONNAL transfer function in z-domain Recurrence equation in the time domain (Note: ) Closed-loop system IIR filters = INFINITE IMPULSE RESPONSE filters:

72 Filtres numériques Digital linear linéaires time invariants filters dans le temps STABILITE DES FILTRES IIR STABILITY RATIONNELS DE TYPE RII Condition on the poles of the transfer function: Stability condition for the digital rationnal filters (hypothesis : N<M) (causal solution (1) ) Condition on the coefficients : o 1 st order filters Stability triangle for 2 nd order IIR filters 1 Stability condition for the 1 st order IIRs o 2 nd order filters Stability condition for the 2 nd order IIRs - 1 Stable filter Unstable filter (1) ) The inverse ZT is not unique. It will be different depending on the chosen closed curve to compute it: see exercises (

73 Digital linear time invariant filters FINITE IMPULSE RESPONSE FILTERS (FIR) Defined by: Non-recursive filters o Always stables o Only defined by one set of coefficients b k, k=,, N-1 %Filtering with Matlab y=filter(b,1,x); Filtered signal Definition of the wanted FIR filter (only one set of coefficients) Signal to filter

74 Digital linear time invariant filters RATIONAL FILTERS: SYNTHESIS AND IMPLEMENTATION Specifications to follow: TEMPLATE SYNTHESIS COEFFICIENTS defining a filter compliant with the given specifications IMPLANTATION on Matlab : IMPLANTATION in real time: Filtered signal Sets of coefficients defining the filter to be used y=filter(b,a,x); Signal to filter IIR filter FIR filter Note: A=[1] for FIR filters To be computed in T e seconds (sampling period)

75 Ideal target frequency responses: Digital linear time invariant filters THE TEMPLATE Low-pass filter High-pass filter Band-pass filter Notch filter Template to follow: o On the modulus of the frequency response Examples : LOW-PASS filter HIGH-PASS filter Cutted bandwidth Bandpass Cutted bandwidth Bandpass Cutted bandwidth Bandpass o Transition area Transition area On the phase of the frequency response: phase or all-pass filter. Transition area Transition area

76 Ideal target frequency responses: Digital linear time invariant filters THE TEMPLATE Low-pass filter High-pass filter Band-pass filter Notch filter Template to follow: o On the modulus of the frequency response Examples : Low-pass filter LINEAR SCALE Low-pass filter LOGARITHM SCALE (dbs) Cutted bandwidth Bandpass Cutted bandwidth Bandpass Cutted bandwidth o Transition area Transition area On the phase of the frequency response: phase or all-pass filter. Transition area

77 Digital linear time invariant filters RATIONAL FILTERS: SYNTHESIS AND IMPLEMENTATION Specifications to follow: TEMPLATE There is not only one filter following the template SYNTHESIS COEFFICIENTS FIR synthesis IIR synthesis Two different methods OF A FILTER following the given specifications IMPLANTATION on Matlab : IMPLANTATION in real time: Filtered signal Sets of coefficients defining the filter to be used y=filter(b,a,x); Signal to filter IIR filter FIR filter Note: A=[1] for FIR filters To be computed in T e seconds (sampling period)

78 Digital linear time invariant filters FIR SYNTHESIS Example: DIGITAL low-pass filter Specifications: TEMPLATE SYNTHESIS COEFFICIENTS Sampling Signal limitation to N coefficients FILTER ORDER Frequency periodisation Distortions : - Transitions are smoothed - Ripples appeararound

79 Digital linear time invariant filters FIR SYNTHESIS Example: DIGITAL low-pass filter Specifications: TEMPLATE Non causal filter: SYNTHESIS COEFFICIENTS Causal filter: (Hyp : N is odd) Introduction of a delay

80 Digital linear time invariant filters FIR SYNTHESIS Example: CAUSAL DIGITAL low-pass filter Introduction of a delay in the impulse response: Introduction of a delay between the input and the output signals: signal filtré Filtered signal (Hyp: N is odd) (Hyp: N is odd) (Hyp : odd order) Power/frequency (db/rad/sample) spectre du signal filtré signal filtré Filtered signal Normalized Frequency ( p rad/sample) (order = 81) le) spectre du signal filtré (order = 21)

81 Digital linear time invariant filters FIR SYNTHESIS Example: CAUSAL DIGITAL low-pass filter (Hyp : N is odd) A constant value is added to the GPT: Constant if h(n) odd or even The GPT for FIR filters is constant If h(n) is odd or even

82 Digital linear time invariant filters FIR SYNTHESIS Parameters allowing to follow the template: ORDER and WEIGTH WINDOW Order influence (rectangular window) Order Ordre = 51 = 51 Order Ordre = 11 = 11 Order Ordre = 21 = Truncation window influence (order = 21) Rectangular Fenêtre rectangulaire window Hamming Fenêtre de window Hamming Blackman Fenêtre de window Blackman H(f).6 H(f) Fréquences Frequency (Hz) Transitions are more or less attenuated Fréquences Frequency (Hz) (Hz) The ripples amplitudes are more or less important

83 Digital linear time invariant filters IIR SYNTHESIS Specifications: TEMPLATE SYNTHESIS COEFFICIENTS Specifications to follow: ~ H ( f )? Specifications to follow: H ( f ) f = ሚfF e Analog patterns library: Butterworth, Tchebycheff, Cauer, Bessel f = 1 πt e tan(π ሚf) Predistorsion Transfer function: H ( p )? Transfer function: H ( z ) BILINEAR TRANSFORM: H z = H(p) p= 2 T e 1 z 1 1z 1 The stability and the frequency response are kept BUT the frequency axis is distorded: ሚf = 1 π arctan(πft e) An example of IIR synthesis is given in the exercises document (

84 Digital linear time invariant filters RATIONAL FILTERS IMPLANTATION Direct structure Canonic structure MN1 (/x) operations, 2 delayed lines MN1 (/x) operations, 1 delayed line

85 Digital linear time invariant filters RATIONAL FILTERS IMPLANTATION Decomposed structures Series (or cascade) : Parallel : H i (z) : first or second order cells Non recursive structure (FIR)

86 To go further Some references: Digital Signal Processing: Course and exercices with solutions, by Nathalie Thomas, ENSEEIHT documentation : 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.html