Practical Signal Processing Concepts and Algorithms using MATLAB

Spectral Analysis of Signals

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Practical Signal Processing Concepts and Algorithms using MATLAB

Spectral Analysis of Signals_________________________________________________________________________________________________________

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Section Outline

• Signal statistics• Discrete Fourier transform• Power spectral density estimation• Time-varying spectra


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Statistical Signal Processing

System

time

time

time

ens

em

ble


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Example: Crosscorrelation

x

y

)()( dnxny

>> [c,lags] = xcorr(x,y)>> stem(lags,c)


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Correlation and CovarianceThe 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

,


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or, in terms of the cross-correlation,

Example on xcorrExample on xcorr

>> y = x; >> x = [1 1 1 1 1]'; >> xyc = xcorr(x,y)

>> stem(xyc)

Example on xcovExample on xcov

>>ww = randn(1000,1); % Generate uniform noise with

%mean = 1/2.

>> [cov_ww,lags] = xcov(ww,10,'coeff');

>> stem(lags,cov_ww)


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Discrete Fourier Transform (DFT)

Discrete Fourier Transform:

1

011

n

jj

jkk yY

nie /2

Complex nth roots of unity Signal y (top) and transform Y (bottom)


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Fast Fourier Transform (FFT)

EfficientDFT

Pad /Chop

y Y

Y = fft(y,n)Window length length(y)

FFT length n

fast

Buffer

FFT


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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.


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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


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Spectral Analysis with the FFT

y Sampled signal

Fs Samples/unit time

n = length(y) Number of samples

t = (0:n-1)/Fs Principal range

dt = 1/Fs Time 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/2 Nyquist frequency

p = 1./f Period (unit time/cycle)


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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


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


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Aliasing Revisited

5 Hz sine wave sampled at 15 Hz 5 Hz sine wave sampled at 7.5 Hz

Original signal

Spectral copy

Principal range Principal range

No overlap/aliasing Overlap/aliasing

+Fs

-Fs -Fs

+Fs


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Power Spectral Density (PSD)

Nonparametric Parametric

Welchpwelch

Multitaperpmtm

Yule-Walker

pyulearMUSICpmusic

Estimate PSD from a finite sample

Subspace

Burg pburg

EVpeig

Ryy( f ) is the DFT of the autocorrelation function ryy(t)

Ryy( f )

Total signal power of analog signal y = y

yy dffR )(


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Nonparametric Methods

• Periodogram

>> [Pxx,w] = periodogram(x)

• Welch

>> [Pxx,w] = pwelch(x)

• Multitaper

>> [Pxx,w] = pmtm(x,nw)


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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


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Parametric Methods (Continued)

Burg Covariance Modified Covariance Yule-Walker

Characteristics No windowing No windowing No windowing Applies window to data

Minimizes forward and backward prediction errors

Minimizes forward prediction error

Minimizes forward and backward prediction errors

Minimizes forward prediction error

Advantages High resolution for short data sets

Better resolution than Y-W for short data sets

High resolution for short data sets

As good as other methods for large data sets

Model always stable Able to extract frequencies from data consisting of p or more pure sinusoids

Able to extract frequencies from data consisting of p or more pure sinusoids

Model always stable

No spectral line-splitting

Disadvantages Peak locations sensitive to initial phase

Model can be unstable Model can be unstable Performs relatively poorly for short data sets

Spectral line-splitting for sinusoids in noise, or very large p

Frequency bias for sinusoids in noise

Peak locations slightly dependent on initial phase



Minor frequency bias for sinusoids in noise

Conditions for Nonsingularity

p must be ≤ 1/2 the input frame size

p must be ≤ 2/3 the input frame size

Autocorrelation matrix always nonsingular


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Subspace Methods

• Eigenvector Method

>> [S,f] = peig(x,p,nfft,fs)

• Multiple Signal Classification (MUSIC) Method

>> [S,f] = pmusic(x,p,nfft,fs)


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Spectrum Viewer in SPTool

1. Select signal in Signal Viewer in SPTool.2. Select Create Spectra Spectrum Viewer.

Display window

Analysis method


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Time-Varying Spectra

>> [B,f,t] = specgram(x,nfft,fs,window,numoverlap)


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Spectrogram Demos

>> specgramdemo>> xpsound


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Example: Reduced Sampling Rate

>> HAL9000

Practical Signal Processing Concepts and Algorithms using MATLAB

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