Magnitude-squared coherence
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Syntax
cxy = mscohere(x,y)
cxy = mscohere(x,y,window)
cxy = mscohere(x,y,window,noverlap)
cxy = mscohere(x,y,window,noverlap,nfft)
cxy = mscohere(___,'mimo')
[cxy,w] = mscohere(___)
[cxy,f] = mscohere(___,fs)
[cxy,w] = mscohere(x,y,window,noverlap,w)
[cxy,f] = mscohere(x,y,window,noverlap,f,fs)
[___] = mscohere(x,y,___,freqrange)
mscohere(___)
Description
cxy = mscohere(x,y)
findsthe magnitude-squared coherence estimate, cxy
,of the input signals, x
and y
.
If
x
andy
areboth vectors, they must have the same length.If one of the signals is a matrix and the other isa vector, then the length of the vector must equal the number of rowsin the matrix. The function expands the vector and returns a matrixof column-by-column magnitude-squared coherence estimates.
If
x
andy
arematrices with the same number of rows but different numbers of columns,thenmscohere
returns a multiple coherence matrix.The mth column ofcxy
containsan estimate of the degree of correlation between all the input signalsand the mth output signal. See Magnitude-Squared Coherence for more information.If
x
andy
arematrices of equal size, thenmscohere
operatescolumn-wise:cxy(:,n) = mscohere(x(:,n),y(:,n))
.To obtain a multiple coherence matrix, append'mimo'
tothe argument list.
cxy = mscohere(x,y,window)
uses window
todivide x
and y
into segmentsand perform windowing. You must use at least two segments. Otherwise,the magnitude-squared coherence is 1 for all frequencies. In the MIMOcase, the number of segments must be greater than the number of inputchannels.
cxy = mscohere(x,y,window,noverlap)
uses noverlap
samplesof overlap between adjoining segments.
example
cxy = mscohere(x,y,window,noverlap,nfft)
uses nfft
samplingpoints to calculate the discrete Fourier transform.
example
cxy = mscohere(___,'mimo')
computesa multiple coherence matrix for matrix inputs. This syntax can includeany combination of input arguments from previous syntaxes.
[cxy,w] = mscohere(___)
returnsa vector of normalized frequencies, w
, at whichthe magnitude-squared coherence is estimated.
example
[cxy,f] = mscohere(___,fs)
returnsa vector of frequencies, f
, expressed in termsof the sample rate, fs
, at which the magnitude-squaredcoherence is estimated. fs
must be the sixthnumeric input to mscohere
. To input a samplerate and still use the default values of the preceding optional arguments,specify these arguments as empty, []
.
[cxy,w] = mscohere(x,y,window,noverlap,w)
returnsthe magnitude-squared coherence estimate at the normalized frequenciesspecified in w
.
[cxy,f] = mscohere(x,y,window,noverlap,f,fs)
returnsthe magnitude-squared coherence estimate at the frequencies specifiedin f
.
[___] = mscohere(x,y,___,freqrange)
returnsthe magnitude-squared coherence estimate over the frequency rangespecified by freqrange
. Valid options for freqrange
are 'onesided'
, 'twosided'
,and 'centered'
.
example
mscohere(___)
withno output arguments plots the magnitude-squared coherence estimatein the current figure window.
Examples
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Coherence Estimate of Two Sequences
Open Live Script
Compute and plot the coherence estimate between two colored noise sequences.
Generate a signal consisting of white Gaussian noise.
r = randn(16384,1);
To create the first sequence, bandpass filter the signal. Design a 16th-order filter that passes normalized frequencies between 0.2π and 0.4π rad/sample. Specify a stopband attenuation of 60 dB. Filter the original signal.
dx = designfilt('bandpassiir','FilterOrder',16, ... 'StopbandFrequency1',0.2,'StopbandFrequency2',0.4, ... 'StopbandAttenuation',60);x = filter(dx,r);
To create the second sequence, design a 16th-order filter that stops normalized frequencies between 0.6π and 0.8π rad/sample. Specify a passband ripple of 0.1 dB. Filter the original signal.
dy = designfilt('bandstopiir','FilterOrder',16, ... 'PassbandFrequency1',0.6,'PassbandFrequency2',0.8, ... 'PassbandRipple',0.1);y = filter(dy,r);
Estimate the magnitude-squared coherence of x
and y
. Use a 512-sample Hamming window. Specify 500 samples of overlap between adjoining segments and 2048 DFT points.
[cxy,fc] = mscohere(x,y,hamming(512),500,2048);
Plot the coherence function and overlay the frequency responses of the filters.
[qx,f] = freqz(dx);qy = freqz(dy);plot(fc/pi,cxy)hold onplot(f/pi,abs(qx),f/pi,abs(qy))hold off
Multiple Coherence and Ordinary Coherence
Open Live Script
Generate a random two-channel signal, x
. Generate another signal, y
, by lowpass filtering the two channels and adding them together. Specify a 30th-order FIR filter with a cutoff frequency of 0.3π and designed using a rectangular window.
h = fir1(30,0.3,rectwin(31));x = randn(16384,2);y = sum(filter(h,1,x),2);
Compute the multiple-coherence estimate of x
and y
. Window the signals with a 1024-sample Hann window. Specify 512 samples of overlap between adjoining segments and 1024 DFT points. Plot the estimate.
noverlap = 512;nfft = 1024;mscohere(x,y,hann(nfft),noverlap,nfft,'mimo')
Compare the coherence estimate to the frequency response of the filter. The drops in coherence correspond to the zeros of the frequency response.
[H,f] = freqz(h);hold onyyaxis rightplot(f/pi,20*log10(abs(H)))hold off
Compute and plot the ordinary magnitude-squared coherence estimate of x
and y
. The estimate does not reach 1 for any of the channels.
figuremscohere(x,y,hann(nfft),noverlap,nfft)
Coherence of MIMO System
Open Live Script
Generate two multichannel signals, each sampled at 1 kHz for 2 seconds. The first signal, the input, consists of three sinusoids with frequencies of 120 Hz, 360 Hz, and 480 Hz. The second signal, the output, is composed of two sinusoids with frequencies of 120 Hz and 360 Hz. One of the sinusoids lags the first signal by π/2. The other sinusoid has a lag of π/4. Both signals are embedded in white Gaussian noise.
fs = 1000;f = 120;t = (0:1/fs:2-1/fs)';inpt = sin(2*pi*f*[1 3 4].*t);inpt = inpt+randn(size(inpt));oupt = sin(2*pi*f*[1 3].*t-[pi/2 pi/4]);oupt = oupt+randn(size(oupt));
Estimate the degree of correlation between all the input signals and each of the output channels. Use a Hamming window of length 100 to window the data. mscohere
returns one coherence function for each output channel. The coherence functions reach maxima at the frequencies shared by the input and the output.
[Cxy,f] = mscohere(inpt,oupt,hamming(100),[],[],fs,'mimo');for k = 1:size(oupt,2) subplot(size(oupt,2),1,k) plot(f,Cxy(:,k)) title(['Output ' int2str(k) ', All Inputs'])end
Switch the input and output signals and compute the multiple coherence function. Use the same Hamming window. There is no correlation between input and output at 480 Hz. Thus there are no peaks in the third correlation function.
[Cxy,f] = mscohere(oupt,inpt,hamming(100),[],[],fs,'mimo');for k = 1:size(inpt,2) subplot(size(inpt,2),1,k) plot(f,Cxy(:,k)) title(['Input ' int2str(k) ', All Outputs'])end
Repeat the computation, using the plotting functionality of mscohere
.
clfmscohere(oupt,inpt,hamming(100),[],[],fs,'mimo')
Compute the ordinary coherence function of the second signal and the first two channels of the first signal. The off-peak values differ from the multiple coherence function.
[Cxy,f] = mscohere(oupt,inpt(:,[1 2]),hamming(100),[],[],fs);plot(f,Cxy)
Find the phase differences by computing the angle of the cross-spectrum at the points of maximum coherence.
Pxy = cpsd(oupt,inpt(:,[1 2]),hamming(100),[],[],fs);[~,mxx] = max(Cxy);for k = 1:2 fprintf('Phase lag %d = %5.2f*pi\n',k,angle(Pxy(mxx(k),k))/pi)end
Phase lag 1 = -0.51*piPhase lag 2 = -0.22*pi
Modify Magnitude-Squared Coherence Plot
Open Live Script
Generate two sinusoidal signals sampled for 1second each at 1kHz. Each sinusoid has a frequency of 250Hz. One of the signals lags the other in phase by π/3radians. Embed both signals in white Gaussian noise of unit variance.
fs = 1000;f = 250;t = 0:1/fs:1-1/fs;um = sin(2*pi*f*t)+rand(size(t));un = sin(2*pi*f*t-pi/3)+rand(size(t));
Use mscohere
to compute and plot the magnitude-squared coherence of the signals.
mscohere(um,un,[],[],[],fs)
Modify the title of the plot, the label of the x-axis, and the limits of the y-axis.
title('Magnitude-Squared Coherence')xlabel('f (Hz)')ylim([0 1.1])
Use gca
to obtain a handle to the current axes. Change the locations of the tick marks. Remove the label of the y-axis.
ax = gca;ax.XTick = 0:250:500;ax.YTick = 0:0.25:1;ax.YLabel.String = [];
Call the Children
property of the handle to change the color and width of the plotted line.
ln = ax.Children;ln.Color = [0.8 0 0];ln.LineWidth = 1.5;
Alternatively, use set
and get
to modify the line properties.
set(get(gca,'Children'),'Color',[0 0.4 0],'LineStyle','--','LineWidth',1)
Input Arguments
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x
, y
— Input signals
vectors | matrices
Input signals, specified as vectors or matrices.
Example: cos(pi/4*(0:159))+randn(1,160)
specifiesa sinusoid embedded in white Gaussian noise.
Data Types: single
| double
Complex Number Support: Yes
window
— Window
integer | vector | []
Window, specified as an integer or as a row or column vector.Use window
to divide the signal into segments:
If
window
is an integer, thenmscohere
divides x and y intosegments of lengthwindow
and windows each segmentwith a Hamming window of that length.If
window
is a vector, thenmscohere
dividesx
andy
intosegments of the same length as the vector and windows each segmentusingwindow
.
If the length of x
and y
cannotbe divided exactly into an integer number of segments with noverlap overlappingsamples, then the signals are truncated accordingly.
If you specify window
as empty, then mscohere
usesa Hamming window such that x
and y
aredivided into eight segments with noverlap
overlappingsamples.
For a list of available windows, see Windows.
Example: hann(N+1)
and (1-cos(2*pi*(0:N)'/N))/2
bothspecify a Hann window of length N
+1.
Data Types: single
| double
noverlap
— Number of overlapped samples
positive integer | []
Number of overlapped samples, specified as a positive integer.
If window is scalar, then
noverlap
mustbe smaller thanwindow
.If
window
is a vector, thennoverlap
mustbe smaller than the length ofwindow
.
If you specify noverlap
as empty,then mscohere
uses a number that produces 50%overlap between segments. If the segment length is unspecified, thefunction sets noverlap
to ⌊N/4.5⌋,where N is the length of the input and output signals.
Data Types: double
| single
nfft
— Number of DFT points
positive integer | []
Number of DFT points, specified as a positive integer. If youspecify nfft
as empty, then mscohere
setsthis argument to max(256,2p),where p=⌈log2N⌉ forinput signals of length N.
Data Types: single
| double
freqrange
— Frequency range for magnitude-squared coherence estimate
'onesided'
| 'twosided'
| 'centered'
Frequency range for the magnitude-squared coherence estimate,specified as 'onesided'
, 'twosided'
,or 'centered'
. The default is 'onesided'
forreal-valued signals and 'twosided'
for complex-valuedsignals.
'onesided'
— Returns theone-sided estimate of the magnitude-squared coherence estimate betweentwo real-valued input signals, x and y.If nfft is even, cxy hasnfft
/2+1 rows and is computed over the interval [0,π] rad/sample.Ifnfft
is odd,cxy
has(nfft
+1)/2rows and the interval is [0,π) rad/sample.If you specify fs, the corresponding intervalsare [0,fs
/2] cycles/unit time for evennfft
and[0,fs
/2) cycles/unit time for oddnfft
.'twosided'
— Returns thetwo-sided estimate of the magnitude-squared coherence estimate betweentwo real-valued or complex-valued input signals,x
andy
.In this case,cxy
hasnfft
rowsand is computed over the interval [0,2π) rad/sample.If you specifyfs
, the interval is [0,fs
)cycles/unit time.'centered'
— Returns thecentered two-sided estimate of the magnitude-squared coherence estimatebetween two real-valued or complex-valued input signals,x
andy
.In this case,cxy
hasnfft
rowsand is computed over the interval (–π,π] rad/samplefor evennfft
and (–π,π) rad/samplefor oddnfft
. If you specifyfs
,the corresponding intervals are (–fs
/2,fs
/2]cycles/unit time for evennfft
and (–fs
/2,fs
/2)cycles/unit time for oddnfft
.
Output Arguments
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cxy
— Magnitude-squared coherence estimate
vector | matrix | three-dimensional array
Magnitude-squared coherence estimate, returned as a vector,matrix, or three-dimensional array.
w
— Normalized frequencies
vector
Normalized frequencies, returned as a real-valued column vector.
f
— Frequencies
vector
Frequencies, returned as a real-valued column vector.
More About
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Magnitude-Squared Coherence
The magnitude-squared coherence estimate isa function of frequency with values between 0 and 1. These valuesindicate how well x
corresponds to y
ateach frequency. The magnitude-squared coherence is a function of thepower spectral densities, Pxx(f) and Pyy(f),and the cross power spectral density, Pxy(f),of x
and y
:
For multi-input/multi-output systems, the multiple-coherencefunction becomes
for the ithoutput signal, where:
X corresponds to the array of m inputs.
PXyi isthe m-dimensional vector of cross power spectraldensities between the inputs and yi.
PXX is the m-by-m matrixof power spectral densities and cross power spectral densities ofthe inputs.
Pyiyi isthe power spectral density of the output.
The dagger (†) stands for the complex conjugatetranspose.
Algorithms
mscohere
estimates the magnitude-squaredcoherence function [2] using Welch’soverlapped averaged periodogram method [3], [5].
References
[1] Gómez González, A., J. Rodríguez, X. Sagartzazu,A. Schumacher, and I. Isasa. “Multiple Coherence Method inTime Domain for the Analysis of the Transmission Paths of Noise andVibrations with Non-Stationary Signals.” Proceedingsof the 2010 International Conference of Noise and Vibration Engineering,ISMA2010-USD2010. pp.3927–3941.
[2] Kay, Steven M. Modern SpectralEstimation. Englewood Cliffs, NJ: Prentice-Hall, 1988.
[3] Rabiner, Lawrence R., and Bernard Gold. Theoryand Application of Digital Signal Processing. EnglewoodCliffs, NJ: Prentice-Hall, 1975.
[4] Stoica, Petre, and Randolph Moses. SpectralAnalysis of Signals. Upper Saddle River, NJ: PrenticeHall, 2005.
[5] Welch, Peter D. “The Use of FastFourier Transform for the Estimation of Power Spectra: A Method Basedon Time Averaging Over Short, Modified Periodograms.” IEEE® Transactionson Audio and Electroacoustics. Vol. AU-15, 1967, pp. 70–73.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced before R2006a
expand all
R2024a: Code generation support for single-precision variable-size window inputs
The mscohere
function supports single-precision variable-size window inputs for code generation.
R2023b: Use single-precision data
The mscohere
function supports single-precision inputs.
See Also
cpsd | periodogram | pwelch | tfestimate
Topics
- Cross Spectrum and Magnitude-Squared Coherence
- Compare the Frequency Content of Two Signals
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