This example shows how to measure signal similarities. It will help you answer questions such as: How do I compare signals with different lengths or different sampling rates? How do I find if there is a signal or just noise in a measurement? Are two signals related? How to measure a delay between two signals (and how do I align them)? How do I compare the frequency content of two signals? Similarities can also be found in different sections of a signal to determine if a signal is periodic.
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Consider a database of audio signals and a pattern matching application where you need to identify a song as it is playing. Data is commonly stored at a low sampling rate to occupy less memory.
% Load data load relatedsig.mat; figure ax(1) = subplot(311); plot((0:numel(T1)-1)/Fs1,T1,'k'); ylabel('Template 1'); grid on ax(2) = subplot(312); plot((0:numel(T2)-1)/Fs2,T2,'r'); ylabel('Template 2'); grid on ax(3) = subplot(313); plot((0:numel(S)-1)/Fs,S); ylabel('Signal'); grid on xlabel('Time (secs)'); linkaxes(ax(1:3),'x') axis([0 1.61 -4 4])
The first and the second subplot show the template signals from the database. The third subplot shows the signal which we want to search for in our database. Just by looking at the time series, the signal does not seem to match to any of the two templates. A closer inspection reveals that the signals actually have different lengths and sampling rates.
[Fs1 Fs2 Fs]
ans = 4096 4096 8192
Different lengths prevent you from calculating the difference between two signals but this can easily be remedied by extracting the common part of signals. Furthermore, it is not always necessary to equalize lengths. Cross-correlation can be performed between signals with different lengths, but it is essential to ensure that they have identical sampling rates. The safest way to do this is to resample the signal with a lower sampling rate. The resample function applies an anti-aliasing(low-pass) FIR filter to the signal during the resampling process.
[P1,Q1] = rat(Fs/Fs1); % Rational fraction approximation [P2,Q2] = rat(Fs/Fs2); % Rational fraction approximation T1 = resample(T1,P1,Q1); % Change sampling rate by rational factor T2 = resample(T2,P2,Q2); % Change sampling rate by rational factor
We can now cross-correlate signal S to templates T1 and T2 with the xcorr function to determine if there is a match.
[C1,lag1] = xcorr(T1,S); [C2,lag2] = xcorr(T2,S); figure ax(1) = subplot(211); plot(lag1/Fs,C1,'k'); ylabel('Amplitude'); grid on title('Cross-correlation between Template 1 and Signal') ax(2) = subplot(212); plot(lag2/Fs,C2,'r'); ylabel('Amplitude'); grid on title('Cross-correlation between Template 2 and Signal') xlabel('Time(secs)'); axis(ax(1:2),[-1.5 1.5 -700 700 ])
The first subplot indicates that the signal and template 1 are less correlated while the high peak in the second subplot indicates that signal is present in the second template.
[~,I] = max(abs(C2)); timeDiff = lag2(I)/Fs
timeDiff = 0.0609
The peak of the cross correlation implies that the signal is present in template T2 starting after 61 ms.
Consider a situation where you are collecting data from different sensors, recording vibrations caused by cars on both sides of a bridge. When you analyze the signals, you may need to align them. Assume you have 3 sensors working at same sampling rates and they are measuring signals caused by the same event.
figure, ax(1) = subplot(311); plot(s1); ylabel('s1'); grid on ax(2) = subplot(312); plot(s2,'k'); ylabel('s2'); grid on ax(3) = subplot(313); plot(s3,'r'); ylabel('s3'); grid on xlabel('Samples') linkaxes(ax,'xy')
The maximum value of the cross-correlations between s1 and s2 and s1 and s3 indicate time leads/lags.
[C21,lag1] = xcorr(s2,s1); [C31,lag2] = xcorr(s3,s1); figure subplot(211); plot(lag1,C21/max(C21)); ylabel('C21'); grid on title('Cross-Correlations') subplot(212); plot(lag2,C31/max(C31)); ylabel('C31'); grid on xlabel('Samples') [~,I1] = max(abs(C21)); % Find the index of the highest peak [~,I2] = max(abs(C31)); % Find the index of the highest peak t21 = lag1(I1) % Time difference between the signals s2,s1 t31 = lag2(I2) % Time difference between the signals s3,s1
t21 = -350 t31 = 150
t21 indicates that s2 lags s1 by 350 samples, and t31 indicates that s3 leads s1 by 150 samples. This information can now used to align the 3 signals.
s2 = [zeros(abs(t21),1);s2]; s3 = s3(t31:end); figure ax(1) = subplot(311); plot(s1); grid on; title('s1'); axis tight ax(2) = subplot(312); plot(s2); grid on; title('s2'); axis tight ax(3) = subplot(313); plot(s3); grid on; title('s3'); axis tight linkaxes(ax,'xy')
A power spectrum displays the power present in each frequency. Spectral coherence identifies frequency-domain correlation between signals. Coherence values tending towards 0 indicate that the corresponding frequency components are uncorrelated while values tending towards 1 indicate that the corresponding frequency components are correlated. Consider two signals and their respective power spectra.
Fs = FsSig; % Sampling Rate [P1,f1] = periodogram(sig1,,,Fs,'power'); [P2,f2] = periodogram(sig2,,,Fs,'power'); figure t = (0:numel(sig1)-1)/Fs; subplot(221); plot(t,sig1,'k'); ylabel('s1'); grid on title('Time Series') subplot(223); plot(t,sig2); ylabel('s2'); grid on xlabel('Time (secs)') subplot(222); plot(f1,P1,'k'); ylabel('P1'); grid on; axis tight title('Power Spectrum') subplot(224); plot(f2,P2); ylabel('P2'); grid on; axis tight xlabel('Frequency (Hz)')
The mscohere function calculates the spectral coherence between the two signals. It confirms that sig1 and sig2 have two correlated components around 35 Hz and 165 Hz. In frequencies where spectral coherence is high, the relative phase between the correlated components can be estimated with the cross spectrum phase.
[Cxy,f] = mscohere(sig1,sig2,,,,Fs); Pxy = cpsd(sig1,sig2,,,,Fs); phase = -angle(Pxy)/pi*180; [pks,locs] = findpeaks(Cxy,'MinPeakHeight',0.75); figure subplot(211); plot(f,Cxy); title('Coherence Estimate'); grid on; hgca = gca; hgca.XTick = f(locs); hgca.YTick = .75; axis([0 200 0 1]) subplot(212); plot(f,phase); title('Cross Spectrum Phase (deg)'); grid on; hgca = gca; hgca.XTick = f(locs); hgca.YTick = round(phase(locs)); xlabel('Frequency (Hz)'); axis([0 200 -180 180])
The phase lag between the 35 Hz components is close to -90 degrees, and the phase lag between the 165 Hz components is close to -60 degrees.
Consider a set of temperature measurements in an office building during the winter season. Measurements were taken every 30 minutes for about 16.5 weeks.
load officetemp.mat % Load Temperature Data Fs = 1/(60*30); % Sample rate is 1 sample every 30 minutes days = (0:length(temp)-1)/(Fs*60*60*24); figure plot(days,temp) title('Temperature Data') xlabel('Time (days)'); ylabel('Temperature (Fahrenheit)') grid on
With the temperatures in the low 70s, you need to remove the mean to analyze small fluctuations in the signal. The xcov function removes the mean of the signal before computing the cross-correlation. It returns the cross-covariance. Limit the maximum lag to 50% of the signal to get a good estimate of the cross-covariance.
maxlags = numel(temp)*0.5; [xc,lag] = xcov(temp,maxlags); [~,df] = findpeaks(xc,'MinPeakDistance',5*2*24); [~,mf] = findpeaks(xc); figure plot(lag/(2*24),xc,'k',... lag(df)/(2*24),xc(df),'kv','MarkerFaceColor','r') grid on xlim([-15 15]); xlabel('Time (days)') title('Auto-covariance')
Observe dominant and minor fluctuations in the auto-covariance. Dominant and minor peaks appear equidistant. To verify if they are, compute and plot the difference between the locations of subsequent peaks.
cycle1 = diff(df)/(2*24); cycle2 = diff(mf)/(2*24); subplot(211); plot(cycle1); ylabel('Days'); grid on title('Dominant peak distance') subplot(212); plot(cycle2,'r'); ylabel('Days'); grid on title('Minor peak distance') mean(cycle1) mean(cycle2)
ans = 7 ans = 1.0000
The minor peaks indicate 7 cycles/week and the dominant peaks indicate 1 cycle per week. This makes sense given that the data comes from a temperature-controlled building on a 7 day calendar. The first 7-day cycle indicates that there is a weekly cyclic behavior of the building temperature where temperatures lower during the weekends and go back to normal during the week days. The 1-day cycle behavior indicates that there is also a daily cyclic behavior - temperatures lower during the night and increase during the day.