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Practical Signal Processing Concepts and Algorithms using MATLAB

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Practical Signal Processing Concepts and Algorithms using MATLAB. Spectral Analysis of Signals. Section Outline. Signal statistics Discrete Fourier transform Power spectral density estimation Time-varying spectra. Statistical Signal Processing. time. time. ensemble. System. time.

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Practical Signal Processing Concepts and Algorithms using MATLAB

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  1. Practical Signal Processing Concepts and Algorithms using MATLAB Spectral Analysis of Signals

  2. Section Outline • Signal statistics • Discrete Fourier transform • Power spectral density estimation • Time-varying spectra

  3. Statistical Signal Processing time time ensemble System time

  4. Example: Crosscorrelation y x >> [c,lags] = xcorr(x,y) >> stem(lags,c)

  5. Correlation and Covariance The functions xcorr and xcov estimate the cross-correlation and cross-covariance sequences of random processes. The cross-correlation sequence is a statistical quantity defined as where xn and yn are stationary random processes, , and E{·} is the expected value operator which measures the similarity between the two waveforms.The covariance sequence is the mean-removed cross-correlation sequence                           ,

  6. or, in terms of the cross-correlation, • Example on xcorr • >> y = x; • >> x = [1 1 1 1 1]'; • >> xyc = xcorr(x,y) • >> stem(xyc) • Example on xcov >>ww = randn(1000,1); % Generate uniform noise with %mean = 1/2. >> [cov_ww,lags] = xcov(ww,10,'coeff'); >> stem(lags,cov_ww)

  7. Discrete Fourier Transform (DFT) Discrete Fourier Transform: Complex nth roots of unity Signal y (top) and transform Y (bottom)

  8. Fast Fourier Transform (FFT) FFT y Pad / Chop Efficient DFT Y Buffer Y = fft(y,n) Window length length(y) FFT length n fast

  9. Fast Fourier Transform (FFT) • The discrete Fourier transform, or DFT, is the primary tool of digital signal processing. • The foundation of the Signal Processing Toolbox is the fast Fourier transform (FFT), a method for computing the DFT with reduced execution time. • Many of the toolbox functions (including z-domain frequency response, spectrum and cepstrum analysis, and some filter design and implementation functions) incorporate the FFT.

  10. Cont. t = (0:0.001:1); %0.001 is sampling x = sin(2*pi*50*t) + 2*sin(2*pi*120*t); y = fft(x); %Compute DFT of x m = abs(y); %magnitude f = (0:length(y)/2-1)*1000/length(y); %Frequency vector plot(f,m(1:1:(length(m)-1)/2)) %before filter grid [b,a] = butter(9,100/500,'high'); %design filter c=filter(b,a,x); %implement filter figure(2) y = fft(c); %Compute DFT of c(filtered x) m = abs(y); % Magnitude f = (0:length(y)/2-1)*1000/length(y); %Frequency vector plot(f,m(1:1:(length(m)-1)/2)) %after filter Grid

  11. Spectral Analysis with the FFT ySampled signal FsSamples/unit time n = length(y)Number of samples t = (0:n-1)/Fs Principal range dt = 1/FsTime increment Y = fft(y)DFT of sampled signal abs(Y)Amplitude of DFT abs(Y).^2/length(Y)Power of DFT f = (0:n-1)*(Fs/n)Frequency (cycles/unit time) (n/2)*(Fs/n) = Fs/2Nyquist frequency p = 1./fPeriod (unit time/cycle)

  12. FFT Example >> Fs = 100; >> t = 0:1/Fs:10-1/Fs; >> y = sin(2*pi*15*t) + sin(2*pi*30*t); >> Y = fft(y,512); >> f = (0:length(Y)-1)*(Fs-1)/length(Y); >> Power = Y.*conj(Y)/length(Y); >> plot(f,Power) >> title('Periodogram') Symmetric about Fs/2 = 50 Use fftshift to center at 0

  13. FFT Demos >> sigdemo1 >> playshow fftdemo >> phone >> playshow sunspots

  14. Aliasing Revisited Original signal Spectral copy 5 Hz sine wave sampled at 15 Hz 5 Hz sine wave sampled at 7.5 Hz +Fs +Fs -Fs -Fs Principal range Principal range No overlap/aliasing Overlap/aliasing

  15. Power Spectral Density (PSD) Ryy( f ) Total signal power of analog signal y = Ryy( f ) is the DFT of the autocorrelation functionryy(t) Estimate PSD from a finite sample Nonparametric Parametric Subspace Welch pwelch Multitaper pmtm Burg pburg Yule-Walker pyulear EV peig MUSIC pmusic

  16. Periodogram >> [Pxx,w] = periodogram(x) Welch >> [Pxx,w] = pwelch(x) Multitaper >> [Pxx,w] = pmtm(x,nw) Nonparametric Methods

  17. Parametric Methods • Yule-Walker AR Method • >> [Pxx,f] = pyulear(x,p,nfft,fs) • Burg Method • >> [Pxx,f] = pburg(x,p,nfft,fs) • Covariance and Modified Covariance Methods • >> [Pxx,f] = pcov(x,p,nfft,fs) • >> [Pxx,f] = pmcov(x,p,nfft,fs) Order of AR model

  18. Parametric Methods (Continued)

  19. Subspace Methods • Eigenvector Method • >> [S,f] = peig(x,p,nfft,fs) • Multiple Signal Classification (MUSIC) Method • >> [S,f] = pmusic(x,p,nfft,fs)

  20. Spectrum Viewer in SPTool 1. Select signal in Signal Viewer in SPTool. 2. Select Create Spectra Spectrum Viewer. Analysis method Display window

  21. Time-Varying Spectra >> [B,f,t] = specgram(x,nfft,fs,window,numoverlap)

  22. Spectrogram Demos >> specgramdemo >> xpsound

  23. Example: Reduced Sampling Rate >> HAL9000

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