both vectors, they must have the same length.
If one of the signals is a matrix and the other is a vector, then the length of the vector must equal the number of rows in the matrix. The function expands the vector and returns a matrix of column-by-column magnitude-squared coherence estimates.
matrices with the same number of rows but different numbers of columns,
mscohere returns a multiple coherence matrix.
The mth column of
an estimate of the degree of correlation between all the input signals
and the mth output signal. See Magnitude-Squared Coherence for more information.
matrices of equal size, then
cxy(:,n) = mscohere(x(:,n),y(:,n)).
To obtain a multiple coherence matrix, append
the argument list.
a vector of frequencies,
f] = mscohere(___,
f, expressed in terms
of the sample rate,
fs, at which the magnitude-squared
coherence is estimated.
fs must be the sixth
numeric input to
mscohere. To input a sample
rate and still use the default values of the preceding optional arguments,
specify these arguments as empty,
no output arguments plots the magnitude-squared coherence estimate
in the current figure window.
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
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 on plot(f/pi,abs(qx),f/pi,abs(qy)) hold off
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
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 on yyaxis right plot(f/pi,20*log10(abs(H))) hold off
Compute and plot the ordinary magnitude-squared coherence estimate of
y. The estimate does not reach 1 for any of the channels.
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
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*pi Phase lag 2 = -0.22*pi
Generate two sinusoidal signals sampled for 1 second each at 1 kHz. Each sinusoid has a frequency of 250 Hz. One of the signals lags the other in phase by π/3 radians. 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));
mscohere to compute and plot the magnitude-squared coherence of the signals.
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])
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 = ;
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;
get to modify the line properties.
set(get(gca,'Children'),'Color',[0 0.4 0],'LineStyle','--','LineWidth',1)
y— Input signals
Input signals, specified as vectors or matrices.
a sinusoid embedded in white Gaussian noise.
Complex Number Support: Yes
Window, specified as an integer or as a row or column vector.
window to divide the signal into segments:
If the length of
be divided exactly into an integer number of segments with
samples, then the signals are truncated accordingly.
If you specify
window as empty, then
a Hamming window such that
divided into eight segments with
For a list of available windows, see Windows.
specify a Hann window of length
N + 1.
noverlap— Number of overlapped samples
Number of overlapped samples, specified as a positive integer.
window is scalar, then
be smaller than
window is a vector, then
be smaller than the length of
If you specify
noverlap as empty,
mscohere uses a number that produces 50%
overlap between segments. If the segment length is unspecified, the
noverlap to ⌊N/4.5⌋,
where N is the length of the input and output signals.
nfft— Number of DFT points
Number of DFT points, specified as a positive integer. If you
nfft as empty, then
this argument to max(256,2p),
where p = ⌈log2 N⌉ for
input signals of length N.
freqrange— Frequency range for magnitude-squared coherence estimate
Frequency range for the magnitude-squared coherence estimate,
'centered'. The default is
real-valued signals and
'twosided' for complex-valued
'onesided' — Returns the
one-sided estimate of the magnitude-squared coherence estimate between
two real-valued input signals,
nfft is even,
nfft/2 + 1 rows and is computed over the interval [0,π] rad/sample.
nfft is odd,
nfft + 1)/2
rows and the interval is [0,π) rad/sample.
If you specify
fs, the corresponding intervals
fs/2] cycles/unit time for even
fs/2) cycles/unit time for odd
'twosided' — Returns the
two-sided estimate of the magnitude-squared coherence estimate between
two real-valued or complex-valued input signals,
In this case,
and is computed over the interval [0,2π) rad/sample.
If you specify
fs, the interval is [0,
'centered' — Returns the
centered two-sided estimate of the magnitude-squared coherence estimate
between two real-valued or complex-valued input signals,
In this case,
and is computed over the interval (–π,π] rad/sample
nfft and (–π,π) rad/sample
nfft. If you specify
the corresponding intervals are (–
cycles/unit time for even
nfft and (–
cycles/unit time for odd
cxy— Magnitude-squared coherence estimate
Magnitude-squared coherence estimate, returned as a vector, matrix, or three-dimensional array.
w— Normalized frequencies
Normalized frequencies, returned as a real-valued column vector.
Frequencies, returned as a real-valued column vector.
The magnitude-squared coherence estimate is
a function of frequency with values between 0 and 1. These values
indicate how well
x corresponds to
each frequency. The magnitude-squared coherence is a function of the
power spectral densities, Pxx(f) and Pyy(f),
and the cross power spectral density, Pxy(f),
For multi-input/multi-output systems, the multiple-coherence function becomes
for the ith output signal, where:
X corresponds to the array of m inputs.
PXyi is the m-dimensional vector of cross power spectral densities between the inputs and yi.
PXX is the m-by-m matrix of power spectral densities and cross power spectral densities of the inputs.
Pyiyi is the power spectral density of the output.
The dagger (†) stands for the complex conjugate transpose.
 Gómez González, A., J. Rodríguez, X. Sagartzazu, A. Schumacher, and I. Isasa. “Multiple Coherence Method in Time Domain for the Analysis of the Transmission Paths of Noise and Vibrations with Non-Stationary Signals.” Proceedings of the 2010 International Conference of Noise and Vibration Engineering, ISMA2010-USD2010. pp. 3927–3941.
 Kay, Steven M. Modern Spectral Estimation. Englewood Cliffs, NJ: Prentice-Hall, 1988.
 Rabiner, Lawrence R., and Bernard Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
 Stoica, Petre, and Randolph Moses. Spectral Analysis of Signals. Upper Saddle River, NJ: Prentice Hall, 2005.
 Welch, Peter D. “The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms.” IEEE® Transactions on Audio and Electroacoustics. Vol. AU-15, 1967, pp. 70–73.