Skip to main content

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

Try Interactive Online Simulator to see the Effect

📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

MIMO Channel Matrix | Rank and Condition Number

MIMO / Massive MIMO MIMO Channel Matrix | Rank and Condition...   The channel matrix in wireless communication is a matrix that describes the impact of the channel on the transmitted signal. The channel matrix can be used to model the effects of the atmospheric or underwater environment on the signal, such as the absorption, reflection or scattering of the signal by surrounding objects. When addressing multi-antenna communication, the term "channel matrix" is used. Let's assume that only one TX and one RX are in communication and there's no surrounding object. Here, in our case, we can apply the proper threshold condition to a received signal and get the original transmitted signal at the RX side. However, in real-world situations, we see signal path blockage, reflections, etc.,  (NLOS paths [↗]) more frequently. The obstruction is typically caused by building walls, etc. Multi-antenna communication was introduced to address this issue. It makes diversity app...
Read more

BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)

Bit Error Rate (BER) & SNR Guide Analyze communication system performance with our interactive simulators and MATLAB tools. 📘 Theory 🧮 Simulators 💻 MATLAB Code 📚 Resources BER Definition SNR Formula BER Calculator MATLAB Comparison 📂 Explore M-ary QAM, PSK, and QPSK Topics ▼ 🧮 Constellation Simulator: M-ary QAM 🧮 Constellation Simulator: M-ary PSK 🧮 BER calculation for ASK, FSK, and PSK 🧮 Approaches to BER vs SNR Calculation What is Bit Error Rate (BER)? The BER indicates how many corrupted bits are received compared to the total number of bits sent. It is the primary figur...
Read more

UGC NET Electronic Science Previous Year Question Papers with Solutions

Home / Engineering & Other Exams / UGC NET 2026 PYQ ⬇️ Download Papers and Solutions 📋 Exam Pattern 💡 Preparation Tips ❓ FAQs 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2025 - Question Paper Download PDF June 2025 - Solved Paper + Explanation ...
Read more

How to Mount Google Drive in Google Colab

How to Mount Google Drive in Google Colab Google Colab provides temporary storage during a session. Any files stored in the /content directory will be deleted when the runtime disconnects. To store datasets, trained models, and results permanently, it is recommended to mount your Google Drive in Colab. Mounting Google Drive allows your notebook to access files directly from your Drive and save outputs there so they remain available even after the Colab session ends. Step 1: Import the Drive Module First import the Google Colab drive module. from google.colab import drive Step 2: Mount Google Drive Run the following command to mount your Google Drive. from google.colab import drive drive.mount('/content/drive') After running the command: A link will appear in the output. Click the link and log in to your Google account. Copy the authentication code provided. Paste the code back into the notebook. Or, a Google authentication page will a...
Read more

Wiener Filter in MATLAB

  MATLAB Code  % Wiener Filter Based on Wiener-Hopf Equation % This script demonstrates how to apply the Wiener filter to recover % a reference signal from a noisy signal using the Wiener-Hopf equation. % The filter minimizes the mean squared error between the noisy signal and the reference signal. clear; close all; clc; % Signal Parameters fs = 4000; % Sampling frequency (Hz) T = 1; % Total recording time (seconds) L = T * fs; % Signal length (samples) tt = (0:L-1) / fs; % Time vector ff = (0:L-1) * fs / L; % Frequency vector % Generate Reference Signal (a sinusoid) y = sin(2 * pi * 120 * tt); % Reference sinusoidal signal y = y(:); % Ensure column vector % Create Noisy Signal by Adding Gaussian Noise x = 0.50 * randn(L, 1) + y; % Noisy signal x = x(:); % Ensure column vector % Define Filter Order (Number of Coefficients) N = 200; % Apply Wiener Filter using custom function [xest, b, MSE] = wienerFilt(x, y, N); % Plot Results figure; subplot(411); plot(tt, x, 'k'), hold on, p...
Read more

Overmodulation & Distortion in AM

Overmodulation in AM and How It Causes Distortion 1. AM Signal Equation s(t) = A c [1 + μ m(t)] cos(2Ï€ f c t) A c = carrier amplitude m(t) = normalized modulating signal (|m(t)| ≤ 1) μ = modulation index 2. Modulation Index μ = A m / A c - Normal AM: 0 < μ ≤ 1 → no distortion - Overmodulation: μ > 1 → distortion occurs 3. Envelope and Overmodulation A(t) = A c [1 + μ m(t)] - For undistorted AM: 1 + μ m(t) ≥ 0 at all times - If μ > 1: 1 + μ m(t) < 0 at negative peaks → carrier flips Example: Let m(t) = cos(2Ï€ f m t), A c = 1 V, μ = 1.2 Minimum envelope: A min = A c [1 - 1.2] = -0.2 V Negative amplitude → envelope crosses zero → 180° phase flip 4. Mathematical Consequence -A c cos(θ) = A c cos(θ + Ï€) This phase reversal is what causes distortion in the demodulated signal. 5. Instantaneous AM Signal s...
Read more

Phase Modulation (PM) & Demodulation

Advanced Analysis of Phase Modulation Phase Modulation (PM): Theoretical Foundations and Spectral Dynamics 1. Analytical Characterization Phase Modulation (PM) is a subset of Angle Modulation , where the information residing in the message signal \( m(t) \) is mapped linearly onto the instantaneous phase of a high-frequency carrier. Unlike Amplitude Modulation (AM), PM is a non-linear modulation process, resulting in an expansion of the signal bandwidth into an infinite dimensional Hilbert space. \[ S_{PM}(t) = A_c \cos\left[ 2\pi f_c t + \phi(t) \right] = A_c \cos\left[ 2\pi f_c t + K_p m(t) \right] \] The instantaneous frequency \( f_i(t) \) is defined as the time derivative of the total angle: \[ f_i(t) = \frac{1}{2\pi} \frac{d\theta_i(t)}{dt} = f_c + \frac{K_p}{2\pi} \frac{dm(t)}{dt} \] ...
Read more

QPSK Online Simulator (Signal Generation)

Simulator for QPSK Modulation Quadrature (4-PSK) Bitstream (Even length) Carrier Freq (Hz) Samples Per Symbol Run QPSK Simulation The Math Behind QPSK Quadrature Phase Shift Keying (QPSK) is a form of digital modulation that transmits two bits per symbol by changing the phase of a carrier wave. s(t) = A cos(2Ï€f c t + θ n ) Phase (θ n ): Each pair of bits (dibit) corresponds to a specific phase shift. In Gray coding, we use: "00" → Ï€/4 (45°) "01" → 3Ï€/4 (135°) "11" → 5Ï€/4 (225°) "10" → 7Ï€/4 (315°) Efficiency: Since 4 phases are used, QPSK carries double the data of BPSK in the same bandwidth. ...
Read more