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Signal Proces sing Toolbox Signal Processing Toolbox Examples

Measuring Signal Similarities

This example s hows how to meas ure signal sim ilarities. It will help you answer questions s uch 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 meas ure 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.

Comparing Signals w ith Different Sampling Rates

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


plot((0:numel(T2)-1)/Fs2,T2,'r'); ylabel('Template 2'); grid on


plot((0:numel(S)-1)/Fs,S,'b'); 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 databas e. Just by looking at the time se ries, the signal does not seem to m atch 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 d ifference between two signals but this can easi ly be remedied by extracting the comm on

part of signals . Furthermore, it is not always necess ary to equalize lengths. Cross-correlation can be pe rformed between signals wi th

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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 resamplefunction 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

Finding a Signal in a Measurement

We can now cross-correlate signal S to templates T1 and T2 with the xcorrfunction to determine if there is a match.

[C1,lag1] = xcorr(T1,S);

[C2,lag2] = xcorr(T2,S);

figure


plot(lag1/Fs,C1,'k'); ylabel('Amplitude'); grid on

title('Cross-correlation between Template 1 and Signal')


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.

Measuring Delay Between Signals and Aligning Them

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 signa ls, you may need to align them . Assume you have 3 sensors working at same s ampling rates and they are

measuring s ignals caused by the sam e event.

figure,

ax(1) = subplot(311); plot(s1,'b'); 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')


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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


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linkaxes(ax,'xy')

Comparing the Frequency Content of Signals

A power spectrum disp lays the power pres ent in each frequency. Spectral coherence identi fies frequency-dom ain corre lation between

signals. Coherence values tending towards 0 indicate that the corresponding frequency components are uncorrelated while values

tending towards 1 i ndicate that the corresponding frequency componen ts 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 mscoherefunction 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 phas e between the correlated

components can be es timated with the cross spectrum phas e.

[Cxy,f] = mscohere(sig1,sig2,[],[],[],Fs);

Pxy = cpsd(sig1,sig2,[],[],[],Fs);

phase = -angle(Pxy)/pi*180;


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[pks,locs] = findpeaks(Cxy,'MinPeakHeight',0.75);

figure

subplot(211);

plot(f,Cxy); title('Coherence Estimate');grid on;

set(gca,'xtick',f(locs),'ytick',.75);

axis([0 200 0 1])

subplot(212);

plot(f,phase); title('Cross Spectrum Phase (deg)');grid on;

set(gca,'xtick',f(locs),'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.

Finding Periodicities in a Signal

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 sm all fluctuations in the signal. The xcovfunction 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 estim ate of the cross -covariance.


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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

set(gca,'Xlim',[-15 15])

xlabel('Time (days)')

title('Auto-covariance')

Observe dominant and minor fluctuations i n 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


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The minor peaks indicate 7 cycle/week and the dominant peaks indicate 1 cycles per week. This m akes s ense gi ven that the data comes

from a tem perature-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 l ower 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 l ower during the night and increase during the day.

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