Estimation of spectra
• Fourier Transform• Periodogram• Spectral leakage & resolution & bias• Tapered Periodogram• Welch‘s method• Confidence Intervals
Further reading on spectral analysis:Bendat & Piersol: Random data, Wiley 2000Percival & Walden: Spectral analysis for physical applications, Cambridge 1993
Spectral leakage & resolution
randn('state',0)fs = 1000; % Sampling frequency 1000 Hzt = (0:fs/10)/fs; % 100 samplesA = [1 2]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 0.1*randn(size(t));Hs = spectrum.periodogram;psd(Hs,xn,'Fs',fs,'NFFT',1024)
randn('state',0)fs = 1000; % Sampling frequency 1000 Hzt = (0:fs/15)/fs; % 67 samplesA = [1 2]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 0.1*randn(size(t));Hs = spectrum.periodogram;psd(Hs,xn,'Fs',fs,'NFFT',1024)
L = 100 samples L = 67 samples
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Periodogram Power Spectral Density Estimate
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Periodogram Power Spectral Density Estimate
Leakage introducesAuto regressive process AR(4)
θθ −= }ˆ{Ebias
4,4,44,34,24,14, 9238.06535.28106.37607.2 tttttt XXXXX ε+−+−= −−−−
)1(,,1,0,1}ˆ{ )( −±±=⎟⎟⎠
⎞⎜⎜⎝
⎛−= NlagnsobservatioNs
NsE p Kτ
τττ
Noisy data
randn('state',0)fs = 1000; % Sampling frequency 1000 Hzt = (0:fs/10)/fs; % 100 samplesA = [1 2]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 2*randn(size(t));Hs = spectrum.periodogram;psd(Hs,xn,'Fs',fs,'NFFT',1024)
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Periodogram Power Spectral Density Estimate
Tapered Periodogram I
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Time domain
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Normalized Frequency (×π rad/sample)
Mag
nitu
de (d
B)
Frequency domain
Rectangular
Hamming
Hanning
fs = 1;t0 = 25;t = [-t0:1/fs:t0];L = numel(t);wvtool(rectwin(L),hamming(L),hanning(L))
Tapered Periodogram II
randn('state',0)fs = 1000; % Sampling frequency 1000 Hzt = (0:fs/10)/fs; % 100 samplesA = [1 2]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 0.1*randn(size(t));Hs = spectrum.periodogram('rectangular');psd(Hs,xn,'Fs',fs,'NFFT',1024)
randn('state',0)fs = 1000; % Sampling frequency 1000 Hzt = (0:fs/10)/fs; % 100 samplesA = [1 2]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 0.1*randn(size(t));Hs = spectrum.periodogram('Hamming');psd(Hs,xn,'Fs',fs,'NFFT',1024)
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Periodogram Power Spectral Density Estimate
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Periodogram Power Spectral Density Estimate
Welch‘s methodThe signal is divided into overlapping segments
randn('state',1)fs = 1000; % Sampling frequency 1000 Hzt = (0:0.3*fs)/fs; % 301 samplesA = [1 2]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 5*randn(size(t));Hs = spectrum.periodogram('rectangular');psd(Hs,xn,'Fs',fs,'NFFT',1024)
randn('state',1)fs = 1000; % Sampling frequencyt = (0:0.3*fs)./fs; % 301 samplesA = [2 8]; % Sinusoid amplitudes (row vector)f = [150;140]; % Sinusoid frequencies (column vector)xn = A*sin(2*pi*f*t) + 5*randn(size(t));Hs = spectrum.welch('rectangular',150,50);psd(Hs,xn,'Fs',fs,'NFFT',512);
nseg = 1 nseg = 3
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Welch Power Spectral Density Estimate
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Periodogram Power Spectral Density Estimate
Chi2 Confidence interval
{ }( ){ }
nsegPP
PPE
afP
xx
xx
xx
xx
xx
/1,21
1ˆ
211
ˆvar
ˆ2
)(ˆ
2
2
≈−
≤≤+
=
=
εεε
υ
χυ
34.00209249354920432534
46.0035514998383138543432
37.2043119238544651424
meanBootstrap samples
384142924665313425
2045535515434969
51544298744454493281
Original sample
Bootstrap Confidence intervalhttp:\\people.revoledu.com\kardi\tutorial\bootstrap\
