Cross-Spectrum Explained (with MATLAB)

Cross-Spectrum: Concept and Mathematics

The cross-spectrum measures the frequency-domain correlation between two signals, showing how the signals relate in magnitude and phase at each frequency.

For two discrete-time signals x[n] and y[n], the cross-spectral density is defined as:

S_xy(f) = X(f) · Y^*(f)

• X(f) = FFT{x[n]} → Fourier transform of x[n]
• Y(f) = FFT{y[n]} → Fourier transform of y[n]
• Y*(f) → Complex conjugate of Y(f)
• |Sxy(f)| → Magnitude shows correlation strength at each frequency
• ∠Sxy(f) → Phase shows relative phase difference

Cross-spectrum is widely used in:

• Beamforming
• Coherence analysis
• Direction-of-arrival estimation (DoA)
• Identifying frequency-dependent relationships between signals

MATLAB Code Example

This MATLAB example demonstrates the cross-spectrum between two sinusoidal signals with a phase difference:

%% Cross-Spectrum Demonstration
clc; clear; close all;

%% PARAMETERS
Fs = 1000;           % Sampling frequency [Hz]
T = 1/Fs;
t = 0:T:1-T;         % 1 second duration
f_sig = 50;          % Signal frequency [Hz]

%% SIMULATE TWO SIGNALS
x = sin(2*pi*f_sig*t);           % Reference signal
y = 0.8*sin(2*pi*f_sig*t + pi/4); % Signal with 45° phase shift

%% COMPUTE CROSS-SPECTRUM
X = fft(x);
Y = fft(y);
S_xy = X .* conj(Y); % Cross-spectrum

%% FREQUENCY VECTOR
N = length(t);
f = Fs*(0:N-1)/N;

%% PLOT
figure('Color','w');

% Magnitude of cross-spectrum
subplot(2,1,1)
plot(f(1:N/2), abs(S_xy(1:N/2))/N, 'LineWidth',1.5)
grid on
xlabel('Frequency [Hz]')
ylabel('|S_{xy}(f)|')
title('Cross-Spectrum Magnitude')

% Phase of cross-spectrum
subplot(2,1,2)
plot(f(1:N/2), angle(S_xy(1:N/2))*180/pi, 'LineWidth',1.5)
grid on
xlabel('Frequency [Hz]')
ylabel('Phase [deg]')
title('Cross-Spectrum Phase')

Explanation:

• S_xy = X .* conj(Y) computes the cross-spectrum.
• The magnitude plot shows which frequencies are strongly correlated.
• The phase plot shows the phase difference at each frequency.
• This is a simple demonstration for understanding frequency-domain correlation.