Correlogram in MATLAB

 

Steps to compute correlogram of an input signal

1. Compute the autocorrelation function of narrowband_signal

2. Computes the Fast Fourier Transform (FFT) of the autocorrelation function (acf), resulting in corr_spectrum.

3. freq = (0:N-1)*(fs/N); Constructs a frequency vector (freq) corresponding to the FFT results, spanning from 0 Hz to just under the Nyquist frequency (fs/2). Where, N = Number of Samples in the Input Signal

4. Plots the magnitude of the FFT (abs(corr_spectrum)) against the frequency vector (freq), showing the correlogram of the narrowband signal.

 

Output 

Copy the MATLAB Code from here

% The code is written by SalimWireless.Com % The code is written by SalimWireless.Com clc; clear; close all; % Parameters fs = 1000; t = 0:1/fs:1-1/fs; N = length(t); % Generate narrowband signal f0 = 50; x = sin(2*pi*f0*t) + 0.5*randn(size(t)); % Autocorrelation (biased, unnormalized) acf = xcorr(x, 'biased'); % Keep non-negative lags acf = acf(N:end); % Correlogram (FFT of ACF) Sxx = real(fft(acf)); % One-sided UNNORMALIZED spectrum powerSpec = Sxx(1:floor(N/2)+1); % One-sided scaling (except DC & Nyquist) powerSpec(2:end-1) = 2 * powerSpec(2:end-1); % Frequency axis f = (0:floor(N/2)) * (fs/N); % Plot unnormalized spectrum in dB figure; plot(f, 10*log10(powerSpec)); xlabel('Frequency (Hz)'); ylabel('Power (dB)'); title('Unnormalized Correlogram – Narrowband Signal'); grid on;

Try Interactive Online Simulator

We have developed a web-based simulator for the Correlogram, Bartlett, and other spectral estimation methods to make these techniques easier to understand and help users learn complex signal processing concepts through interactive simulations.

Try our simulators: Periodogram, Correlogram, Bartlett, Blackman–Tukey, and Welch methods.

Other Spectral Estimation Techniques

The Windowed Periodogram Approach

To estimate the Power Spectral Density (PSD) of discrete signals, researchers often turn to the windowed periodogram. By applying a specific window function to the raw data, this method minimizes "spectral leakage," a common error where energy from one frequency spills into adjacent ones. This step is vital for high-fidelity frequency analysis.

Standard Periodogram Foundations

The traditional periodogram is a direct estimation technique derived from the Discrete-Time Fourier Transform (DTFT):

P_x(f) = (1/N) | ∑_n=0^N-1 x[n] e^{-j 2 π f n} |²

In this formula:

• x[n]: The sampled input signal.
• N: The total count of samples.

Because it involves an abrupt cutoff of the signal, the standard periodogram is prone to significant spectral leakage.

Using Windowing to Enhance Accuracy

By multiplying the signal by a window function w[n] before transforming it, we can smooth out the edges:

P_x(f) = (1 / (N · U)) | ∑_n=0^N-1 x[n] w[n] e^{-j 2 π f n} |²

Definitions:

• w[n]: The selected window weights.
• U = (1/N) ∑_n=0^N-1 |w[n]|²: A constant used to normalize the signal's power.

Standard Window Variations

• Rectangular: Basic truncation without smoothing. All values are 1 for 0 ≤ n ≤ N-1.
• Hamming: Designed to lower the peaks of sidelobes using: 0.54 - 0.46 cos(2 π n / (N-1)).
• Hann: Provides a gentle fade-in and fade-out at the signal boundaries: 0.5 [1 - cos(2 π n / (N-1))].
• Blackman: Offers even lower sidelobes by adding a second cosine term, though it widens the main spectral peak.

Methodology

1. Divide the data into blocks of length N.
2. Multiply each block by the chosen window function.
3. Run an FFT or DTFT on these windowed segments.
4. Average the results to stabilize the estimate.

The Correlogram Technique

This method calculates the PSD by taking the Fourier transform of the signal’s estimated autocorrelation sequence.

P_x(f) = ∑_k=-(N-1)^N-1 R_x[k] e^{-j 2 π f k}

Here, R_x[k] represents the autocorrelation at lag k. To ensure the PSD never drops below zero, a biased estimate is typically used (dividing by N). While an unbiased estimate (dividing by N-k) exists, it can sometimes produce mathematically impossible negative PSD values.

Bartlett’s Method

Bartlett’s technique aims to reduce the "noise" (variance) of the periodogram by splitting the signal into M distinct, non-overlapping parts and averaging their individual periodograms.

P_x(f) = (1 / (M · N)) ∑_m=0^M-1 | ∑_n=0^N-1 x_m[n] e^{-j 2 π f n} |²

Pros: It reduces variance by a factor of M.
Cons: It reduces the detail (resolution) of the frequency map because each segment is shorter than the original signal.

Blackman-Tukey Method

This approach focuses on windowing the autocorrelation function itself rather than the raw signal.

P_x(f) = ∑_k=-K^K R_x[k] w[k] e^{-j 2 π f k}

By smoothing the autocorrelation sequence with w[k], the resulting PSD is less jagged. This is highly effective in radar, sonar, and speech analysis, though it requires more processing power for long datasets.

Welch’s Method

An evolution of Bartlett’s method, Welch’s technique allows segments to overlap (usually by 50%) and applies a window to each segment before averaging.

P_x(f) = (1 / (K · L · U)) ∑_k=0^K-1 | ∑_n=0^L-1 x_k[n] w[n] e^{-j 2 π f n} |²

Welch's method is the industry standard for many applications—from analyzing EEG brainwaves to assessing wireless communication spectra—because it offers the best balance between reducing noise and preventing spectral leakage.

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