Cross power spectral density


Pxy = cpsd(x,y)
Pxy = cpsd(x,y,window)
Pxy = cpsd(x,y,window,noverlap)
[Pxy,W] = cpsd(x,y,window,noverlap,nfft)
[Pxy,F] = cpsd(x,y,window,noverlap,nfft,fs)
[...] = cpsd(...,'twosided')


Pxy = cpsd(x,y) estimates the cross power spectral density, Pxy, of two discrete-time signals, x and y, using Welch's averaged, modified periodogram method of spectral estimation.

The input signals may be either vectors or two-dimensional matrices. If both are vectors, they must have the same length. If both are matrices, they must have the same size, and cpsd operates columnwise: Pxy(:,n) = cpsd(x(:,n),y(:,n)). If one is a matrix and the other is a vector, then the vector is converted to a column vector and internally expanded so both inputs have the same number of columns.

For real x and y, cpsd returns a one-sided CPSD and for complex x or y, it returns a two-sided CPSD.

cpsd uses the following default values:



Default Value


FFT length which determines the frequencies at which the PSD is estimated

For real x and y, the length of Pxy is (nfft/2+1) if nfft is even or (nfft+1)/2 if nfft is odd. For complex x or y, the length of Pxy is nfft.

If nfft is greater than the signal length, the data is zero-padded. If nfft is less than the signal length, the segment is wrapped so that the length is equal to nfft.

Maximum of 256 or the next power of 2 greater than the length of each section of x or y


Sampling frequency



Windowing function and number of samples to use for each section

Periodic Hamming window of length to obtain eight equal sections of x and y


Number of samples by which the sections overlap

Value to obtain 50% overlap

    Note   You can use the empty matrix, [], to specify the default value for any input argument except x or y. For example, Pxy = cpsd(x,y,[],[],128) uses a Hamming window, default noverlap to obtain 50% overlap, and the specified 128 nfft.

Pxy = cpsd(x,y,window) specifies a windowing function, divides x and y into overlapping sections of the specified window length, and windows each section using the specified window function. If you supply a scalar for window, Pxy uses a Hamming window of that length. x and y are divided into eight equal sections of that length. If the signal cannot be sectioned evenly with 50% overlap, it is truncated.

Pxy = cpsd(x,y,window,noverlap) overlaps the sections of x by noverlap samples. noverlap must be an integer smaller than the length of window.

[Pxy,W] = cpsd(x,y,window,noverlap,nfft) uses the specified FFT length nfft in estimating the CPSD. It also returns W, which is the vector of normalized frequencies (in rad/sample) at which the CPSD is estimated. For real signals, the range of W is [0, π] when nfft is even and [0, π) when nfft is odd. For complex signals, the range of W is [0, 2π).

[Pxy,F] = cpsd(x,y,window,noverlap,nfft,fs) returns Pxy as a function of frequency and a vector F of frequencies at which the CPSD is estimated. fs is the sampling frequency in Hz. For real signals, the range of F is [0, fs/2] when nfft is even and [0, fs/2) when nfft is odd. For complex signals, the range of F is [0, fs).

[...] = cpsd(...,'twosided') returns the two-sided CPSD of real signals x and y. The length of the resulting Pxy is nfft and its range is [0, 2π) if you do not specify fs. If you specify fs, the range is [0, fs). Entering 'onesided' for a real signal produces the default. You can place the 'onesided' or 'twosided' string in any position after the noverlap parameter.

cpsd(...) plots the CPSD versus frequency in the current figure window.


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CPSD of Colored Noise Signals

Generate two colored noise signals and plot their CPSD. Specify a length-1024 FFT and a 500-point triangular window with no overlap.

rng default
h = fir1(30,0.2,rectwin(31));
h1 = ones(1,10)/sqrt(10);
r = randn(16384,1);
x = filter(h1,1,r);
y = filter(h,1,x);

Cross Spectrum Phase of Lagged Sinusoids

Generate two 100 Hz sinusoidal signals sampled at 1 kHz for 296 ms. One of the sinusoids lags the other by 2.5 ms, equivalent to a phase lag of π/2. Both signals are embedded in white Gaussian noise of variance 1/42. Reset the random number generator for reproducible results.


Fs = 1000;
t = 0:1/Fs:0.296;

x = cos(2*pi*t*100)+0.25*randn(size(t));
tau = 1/400;
y = cos(2*pi*100*(t-tau))+0.25*randn(size(t));

Compute and plot the magnitude of the cross power spectral density. Use the default settings for cpsd. The magnitude peaks at the frequency where there is significant coherence between the signals.


Plot the phase of the cross spectrum. The ordinate at the high-coherence frequency corresponds to the phase lag between the sinusoids.

[Pxy,F] = cpsd(x,y,[],[],[],Fs);


hold on
hold off

title('Cross Spectrum Phase')

More About

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Cross Power Spectral Density

The cross power spectral density is the distribution of power per unit frequency and is defined as


The cross-correlation sequence is defined as


where xn and yn are jointly stationary random processes, <n<, and E {· } is the expected value operator.


cpsd uses Welch's averaged periodogram method. See the references listed below.


[1] Rabiner, Lawrence R., and B. Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975, pp. 414–419.

[2] Welch, Peter D. "The Use of the 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, June 1967, pp. 70–73.

[3] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.

Introduced before R2006a

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