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Periodogram in MATLAB


Step 1: Signal Representation

Let the signal be x[n], where:

  • n = 0, 1, ..., N-1 (discrete-time indices),
  • N is the total number of samples.

Step 2: Compute the Discrete-Time Fourier Transform (DTFT)

The DTFT of x[n] is:

X(f) = ∑ x[n] e-j2Ï€fn

For practical computation, the Discrete Fourier Transform (DFT) is used:

X[k] = ∑ x[n] e-j(2Ï€/N)kn, k = 0, 1, ..., N-1

Here:

  • k represents discrete frequency bins,
  • f_k = k/N * f_s, where f_s is the sampling frequency.

Step 3: Compute Power Spectral Density (PSD)

The periodogram estimates the PSD as:

S_x(f_k) = (1/N) |X[k]|²

Where:

  • S_x(f_k) represents the power of the signal at frequency f_k.
  • The factor 1/N normalizes the power by the signal length.

Step 4: Convert to Decibels (Optional)

For visualization, convert PSD to decibels (dB):

S_xdB(f_k) = 10 log₁₀(S_x(f_k))

Practical Notes

  • Frequency Resolution: Depends on the signal duration T = N / f_s. Higher N gives finer frequency resolution.
  • Windowing (Optional): Use a window function (e.g., Hamming, Hann) to reduce spectral leakage: x'[n] = x[n] * w[n].
  • Frequency Range: PSD spans frequencies:
    • f_k = k/N * f_s for positive frequencies.
    • f_k = -(N-k)/N * f_s for negative frequencies.

     

    MATLAB Code

    clc;
    clear;
    close all;

    % Define signal parameters
    N = 256; % Number of samples
    fs = 1000; % Sampling frequency in Hz
    t = (0:N-1)/fs; % Time vector

    % Generate a sample signal (sum of two sinusoids)
    f1 = 50; % Frequency of first sinusoid in Hz
    f2 = 120; % Frequency of second sinusoid in Hz
    x = sin(2*pi*f1*t) + 0.5*sin(2*pi*f2*t) + randn(1, N)*0.1; % Signal with noise

    % Compute DFT of the signal
    X = fft(x, N);

    % Calculate the periodogram
    Pxx = (1/N) * abs(X).^2;

    % Generate frequency vector
    f = (0:N-1)*(fs/N);

    % Keep only the positive frequencies (up to Nyquist frequency)
    Pxx = Pxx(1:N/2+1);
    f = f(1:N/2+1);

    % Plot the periodogram
    figure;
    plot(f, 10*log10(Pxx), 'LineWidth', 1.5);
    xlabel('Frequency (Hz)');
    ylabel('Power/Frequency (dB/Hz)');
    title('Periodogram');
    grid on;

    Output 

     


     

     

     

     

    Copy the MATLAB Code above from here

     

    Further Reading

    1. Periodogram and Windowed Periododgram in details
    2. Correlogram in MATLAB
    3. Bartlett Method in MATLAB
    4. Blackman-Tukey Method in MATLAB
    5. Welch's Method in MATLAB

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  Website: www.salimwireless.com
  Interests: Signal Processing, Telecommunication, 5G Technology, Present & Future Wireless Technologies, Digital Signal Processing, Computer Networks, Millimeter Wave Band Channel, Web Development
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