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How Windowing Affects Your Periodogram


The windowed periodogram is a widely used technique for estimating the Power Spectral Density (PSD) of a signal. It enhances the classical periodogram by mitigating spectral leakage through the application of a windowing function. This technique is essential in signal processing for accurate frequency-domain analysis.

 

Power Spectral Density (PSD)

The PSD characterizes how the power of a signal is distributed across different frequency components. For a discrete-time signal, the PSD is defined as the Fourier Transform of the signal’s autocorrelation function:

Sx(f) = FT{Rx(Ï„)}

Here, Rx(Ï„)}is the autocorrelation function.

FT : Fourier Transform

 

Classical Periodogram

The periodogram is a non-parametric PSD estimation method based on the Discrete Fourier Transform (DFT):

Px(f) = \(\frac{1}{N}\) X(f)2

Here:

  • X(f): DFT of the signal x(n)

  • N: Signal length

However, the classical periodogram suffers from spectral leakage due to abrupt truncation of the signal.

 

Windowing to Mitigate Spectral Leakage

Spectral leakage can be minimized by applying a window function to the signal before computing the DFT. The resulting PSD estimate is called the windowed periodogram:

Pw(f) = \(\frac{1}{NW}\) Xw(f)2

Here:

  • w(n): Window function

  • W: Window normalization factor

Common Window Functions

  • Rectangular Window: Equivalent to the classical periodogram.

w[n]=1, 0≤n≤N−1

w[n]=0, otherwise

Where, N is the window length

  • Hamming Window: Reduces sidelobe amplitudes, improving frequency resolution.

w[n]=0.5(1−cos(\(\frac{\ 2\pi n}{N - 1}\ \))), 0≤n≤N−1

Where, N is the window length

  • Hanning Window: Similar to Hamming but with less sidelobe attenuation.

w[n]=0.54 – 0.46cos(\(\frac{\ 2\pi n}{N - 1}\ \)), 0≤n≤N−1

Where, N is the window length

  • Blackman Window: Offers even greater sidelobe suppression but at the cost of wider main lobes.

w[n]=0.42 – 0.5(cos(\(\frac{\ 2\pi n}{N - 1}\ \)) + 0.08(cos(\(\frac{\ 4\pi n}{N - 1}\ \)), 0≤n≤N−1

Where, N is the window length

 

Implementation Steps

  1. Segment the Signal: Divide the signal into overlapping or non-overlapping segments of length N.

  2. Apply a Window Function: Multiply each segment by a window function w(n).

  3. Compute the DFT: Calculate the DFT of the windowed segments.

  4. Average the Periodograms: For overlapping segments, average the periodograms to reduce variance.

     

Properties of the Windowed Periodogram

  • Bias: Windowing introduces bias in the PSD estimate as the window modifies the signal spectrum.

  • Variance: Averaging periodograms (Welch method) reduces variance but decreases frequency resolution.

  • Trade-Off: The choice of window affects the trade-off between spectral resolution and leakage suppression.

     

    MATLAB Code

    clc;
    clear;
    close all;

    fs = 48000;
    t = 0:1/fs:0.02;
    f_ping = 12000;

    % Base sine wave
    sine_wave = sin(2*pi*f_ping*t)';

    % Apply windows
    w_rect = ones(size(sine_wave));
    w_hann = hann(length(sine_wave));
    w_hamming = hamming(length(sine_wave));
    w_blackman = blackman(length(sine_wave));

    % Windowed signals
    s_rect = sine_wave .* w_rect;
    s_hann = sine_wave .* w_hann;
    s_hamming = sine_wave .* w_hamming;
    s_blackman = sine_wave .* w_blackman;

    % FFT
    Nfft = 4096;
    f = fs*(0:Nfft/2-1)/Nfft;

    % Function to compute and normalize spectrum
    get_norm_fft = @(sig) abs(fft(sig, Nfft))/max(abs(fft(sig, Nfft)));

    S_rect = get_norm_fft(s_rect);
    S_hann = get_norm_fft(s_hann);
    S_hamming = get_norm_fft(s_hamming);
    S_blackman = get_norm_fft(s_blackman);

    % Mainlobe power (±2 bins around peak)
    mainlobe_bins = 2;

    % Function to compute power ratio
    compute_power_ratio = @(S) ...
    deal( ...
    sum(S.^2), ... % Total power
    max(1, find(S == max(S), 1)), ... % Peak bin
    @(peak_bin) sum(S(max(1,peak_bin-mainlobe_bins):min(Nfft,peak_bin+mainlobe_bins)).^2), ...
    @(total, main) 10*log10((total-main)/main) ... % dB sidelobe/mainlobe ratio
    );

    % Calculate ratios
    [total_r, peak_r, get_main_r, get_slr_r] = compute_power_ratio(S_rect);
    main_r = get_main_r(peak_r); slr_r = get_slr_r(total_r, main_r);

    [total_h, peak_h, get_main_h, get_slr_h] = compute_power_ratio(S_hann);
    main_h = get_main_h(peak_h); slr_h = get_slr_h(total_h, main_h);

    [total_ham, peak_ham, get_main_ham, get_slr_ham] = compute_power_ratio(S_hamming);
    main_ham = get_main_ham(peak_ham); slr_ham = get_slr_ham(total_ham, main_ham);

    [total_b, peak_b, get_main_b, get_slr_b] = compute_power_ratio(S_blackman);
    main_b = get_main_b(peak_b); slr_b = get_slr_b(total_b, main_b);

    % Display Results
    fprintf('Window | Mainlobe Power | Sidelobe Power | Sidelobe/Main (dB)\n');
    fprintf('------------|----------------|----------------|--------------------\n');
    fprintf('Rectangular | %14.4f | %14.4f | %18.2f\n', main_r, total_r - main_r, slr_r);
    fprintf('Hann | %14.4f | %14.4f | %18.2f\n', main_h, total_h - main_h, slr_h);
    fprintf('Hamming | %14.4f | %14.4f | %18.2f\n', main_ham, total_ham - main_ham, slr_ham);
    fprintf('Blackman | %14.4f | %14.4f | %18.2f\n', main_b, total_b - main_b, slr_b);

    % Plot
    figure;
    plot(f, 20*log10(S_rect(1:Nfft/2)), 'k'); hold on;
    plot(f, 20*log10(S_hann(1:Nfft/2)), 'r');
    plot(f, 20*log10(S_hamming(1:Nfft/2)), 'g');
    plot(f, 20*log10(S_blackman(1:Nfft/2)), 'b');
    legend('Rectangular','Hann','Hamming','Blackman');
    xlim([f_ping-3000 f_ping+3000]); ylim([-100 5]);
    xlabel('Frequency (Hz)'); ylabel('Magnitude (dB)');
    title('Windowing Effects on Spectrum');
    grid on;

    Output 

    Window      | Mainlobe Power | Sidelobe Power | Sidelobe/Main (dB)
    ------------|----------------|----------------|--------------------
    Rectangular |         3.5771 |         4.9562 |               1.42
    Hann        |         4.3630 |         8.4370 |               2.86
    Hamming     |         4.2367 |         7.3928 |               2.42
    Blackman    |         4.4940 |        10.2410 |               3.58

     

     








Applications

  • Signal Processing: Analyzing frequency content of time-varying signals.

  • Communications: Evaluating spectrum occupancy in wireless systems.

  • Bioinformatics: Investigating periodicities in biological signals (e.g., EEG, ECG).

  • Seismology: Characterizing seismic wave frequencies.

     

    Further Reading

    1. Periodogram in MATLAB

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