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 and y 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 and y arematrices with the same number of rows but different numbers of columns,then mscohere returns a multiple coherence matrix.The mth column of cxy 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 and y arematrices of equal size, then mscohere 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.

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, then mscohere divides x and y intosegments of length window and windows each segmentwith a Hamming window of that length.
If window is a vector, then mscohere divides x and y intosegments of the same length as the vector and windows each segmentusing window.

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 than window.
If window is a vector, then noverlap mustbe smaller than the length of window.

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(256,2^p),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 has nfft/2+1 rows and is computed over the interval [0,π] rad/sample.If nfft 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 even nfft and[0,fs/2) cycles/unit time for odd nfft.
'twosided' — Returns thetwo-sided estimate of the magnitude-squared coherence estimate betweentwo real-valued or complex-valued input signals, x and y.In this case, cxy has nfft rowsand is computed over the interval [0,2π) rad/sample.If you specify fs, 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 and y.In this case, cxy has nfft rowsand is computed over the interval (–π,π] rad/samplefor even nfft and (–π,π) rad/samplefor odd nfft. If you specify fs,the corresponding intervals are (–fs/2, fs/2]cycles/unit time for even nfft and (–fs/2, fs/2)cycles/unit time for odd nfft.

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, P_xx(f) and P_yy(f),and the cross power spectral density, P_xy(f),of x and y:

$C_{x y} (f) = \frac{{| P_{x y} (f) |}^{2}}{P_{x x} (f) P_{y y} (f)} .$

For multi-input/multi-output systems, the multiple-coherencefunction becomes

$\begin{matrix} C_{X y_{i}} (f) = \frac{P_{X y_{i}}^{†} (f) P_{X X}^{- 1} (f) P_{X y_{i}} (f)}{P_{y_{i} y_{i}} (f)} \\ = [\begin{matrix} P_{x_{1} y_{i}}^{*} (f) & \dots & P_{x_{m} y_{i}}^{*} (f) \end{matrix}] {[\begin{matrix} P_{x_{1} x_{1}} (f) & P_{x_{1} x_{2}} (f) & \dots & P_{x_{1} x_{m}} (f) \\ P_{x_{2} x_{1}} (f) & P_{x_{2} x_{2}} (f) & \dots & P_{x_{2} x_{m}} (f) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ P_{x_{m} x_{1}} (f) & P_{x_{m} x_{2}} (f) & \dots & P_{x_{m} x_{m}} (f) \end{matrix}]}^{- 1} [\begin{matrix} P_{x_{1} y_{i}} (f) \\ ⋮ \\ P_{x_{m} y_{i}} (f) \end{matrix}] \frac{1}{P_{y_{i} y_{i}} (f)} \end{matrix}$

for the ithoutput signal, where:

X corresponds to the array of m inputs.
P_{Xy_i} isthe m-dimensional vector of cross power spectraldensities between the inputs and y_i.
P_XX is the m-by-m matrixof power spectral densities and cross power spectral densities ofthe inputs.
P_{y_i}_{y_i} 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.

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Magnitude-squared coherence - MATLAB mscohere (2024)

Syntax

Description

Examples

Coherence Estimate of Two Sequences

Multiple Coherence and Ordinary Coherence

Coherence of MIMO System

Modify Magnitude-Squared Coherence Plot

Input Arguments

`x`, `y` — Input signals
vectors | matrices

`window` — Window
integer | vector | `[]`

`noverlap` — Number of overlapped samples
positive integer | `[]`

`nfft` — Number of DFT points
positive integer | `[]`

`freqrange` — Frequency range for magnitude-squared coherence estimate
`'onesided'` | `'twosided'` | `'centered'`

Output Arguments

`cxy` — Magnitude-squared coherence estimate
vector | matrix | three-dimensional array

`w` — Normalized frequencies
vector

`f` — Frequencies
vector

More About

Magnitude-Squared Coherence

Algorithms

References

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™.

Version History

R2024a: Code generation support for single-precision variable-size window inputs

R2023b: Use single-precision data

See Also

Topics

MATLAB Command

Americas

Europe

Asia Pacific

Magnitude-squared coherence - MATLAB mscohere (2024)

Syntax

Description

Examples

Coherence Estimate of Two Sequences

Multiple Coherence and Ordinary Coherence

Coherence of MIMO System

Modify Magnitude-Squared Coherence Plot

Input Arguments

x, y — Input signals vectors | matrices

window — Window integer | vector | []

noverlap — Number of overlapped samples positive integer | []

nfft — Number of DFT points positive integer | []

freqrange — Frequency range for magnitude-squared coherence estimate 'onesided' | 'twosided' | 'centered'

Output Arguments

cxy — Magnitude-squared coherence estimate vector | matrix | three-dimensional array

w — Normalized frequencies vector

f — Frequencies vector

More About

Magnitude-Squared Coherence

Algorithms

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

GPU ArraysAccelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Version History

R2024a: Code generation support for single-precision variable-size window inputs

R2023b: Use single-precision data

See Also

Topics

MATLAB Command

Americas

Europe

Asia Pacific

`x`, `y` — Input signals
vectors | matrices

`window` — Window
integer | vector | `[]`

`noverlap` — Number of overlapped samples
positive integer | `[]`

`nfft` — Number of DFT points
positive integer | `[]`

`freqrange` — Frequency range for magnitude-squared coherence estimate
`'onesided'` | `'twosided'` | `'centered'`

`cxy` — Magnitude-squared coherence estimate
vector | matrix | three-dimensional array

`w` — Normalized frequencies
vector

`f` — Frequencies
vector

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™.