Windowed Periodogram The windowed periodogram is a widely used technique for estimating the Power Spectral Density (PSD) of a discrete-time signal. It improves the classical periodogram by mitigating spectral leakage through the application of a window function. This is essential for accurate frequency-domain analysis. Classical Periodogram The periodogram is a non-parametric PSD estimation method based on the Discrete-Time Fourier Transform (DTFT): \[ P_x(f) = \frac{1}{N} \left| \sum_{n=0}^{N-1} x[n] e^{-j 2 \pi f n} \right|^2 \] Where: \(x[n]\) : Discrete-time signal \(N\) : Signal length The classical periodogram suffers from spectral leakage due to abrupt truncation of the signal. Windowing to Mitigate Spectral Leakage Apply a window function \(w[n]\) to the signal before computing the DTFT: \[ P_x(f) = \frac{1}{N \cdot U} \left| \sum_{n=0}^{N-1} x[n] w[n] e^{-j 2 \pi f n} \right|^2 \] Where: ...

MATLAB Code   % MATLAB script to compare different Spectral Estimation Methods using Built-in Functions % Sample signal: Sine wave with noise fs = 1000; % Sampling frequency (Hz) T = 1; % Signal duration (seconds) t = 0:1/fs:T-1/fs; % Time vector f_signal = 50; % Signal frequency (Hz) signal = sin(2*pi*f_signal*t) + 0.5*randn(size(t)); % Sine wave + noise % Method 1: Periodogram (Using built-in function) [pxx_periodogram, f_periodogram] = periodogram(signal, [], [], fs); % Method 2: Bartlett Method (Using built-in pwelch with default settings) [pxx_bartlett, f_bartlett] = pwelch(signal, [], [], [], fs); % Method 3: Welch Method (Using built-in pwelch with segment overlap) segmentLength = 256; % Length of each segment overlap = 128; % Overlap between segments [pxx_welch, f_welch] = pwelch(signal, segmentLength, overlap, [], fs); % Method 4: Blackman-Tukey Method (Using built-in cpsd function with autocorrelation) % Compute the Cross Power Spectral Density (CPSD) using autocorrelatio...

   MATLAB Code % The code is developed by SalimWireless.com clc; clear; close all; % Parameters Nbits = 2000; % Number of bits Rb = 1000; % Bit rate (bps) Fs = 10*Rb; % Sampling frequency (Hz) Ts = 1/Fs; % Sampling interval T = 1/Rb; % Bit duration SamplesPerSymbol = Fs/Rb; % Number of samples per bit % Generate random bit sequence data = randi([0 1], 1, Nbits); % Create the baseband message signal (upsampled bits) message_signal = zeros(1, Nbits * SamplesPerSymbol); for i = 1:Nbits message_signal((i-1)*SamplesPerSymbol + 1 : i*SamplesPerSymbol) = data(i); end t_message = (0:length(message_signal)-1)*Ts; % MSK Modulation mskMod = comm.MSKModulator('BitInput', true, 'SamplesPerSymbol', SamplesPerSymbol); mskSig = step(mskMod, data.'); % GMSK Modulation BT = 0.3; % Gaussian BT product gmskMod = comm.GMSKModulator('BitInput', true, 'BandwidthTimeProduct', BT, 'Sa...

  This plot shows the power distribution of an OFDM signal in frequency domain. The central flat region corresponds to the 64 subcarriers each spaced 15 kHz apart, giving a total bandwidth of 64 X 15 = 960 kHz . The PSD decreases outside this band due to the spectral shape of each subcarrier.  MATLAB Code  % The code is developed by SalimWireless.com clear; close all; clc; % PARAMETERS N = 64; % subcarriers Delta_f = 15e3; % subcarrier spacing Fs_base = 1e6; % base sampling rate M = 16; % 16-QAM fc = 2e6; % carrier BW = N * Delta_f; % approximate occupied BW desired_Fs = 10e6; % high sampling rate for oversampling % OVERSAMPLING L = ceil(desired_Fs / Fs_base); Fs_high = L * Fs_base; fprintf('Fs_base = %.1f kHz, Fs_high = %.1f kHz, L=%d\n', Fs_base/1e3, Fs_high/1e3, L); % GENERATE 16-QAM SYMBOLS data = randi([0 M-1], N, 1); qamSymbols = qammod(data, M, 'UnitAveragePower', true); % MAP TO IFFT B...

  Implementation Steps 1. FFT Computes the Frequency Content of a Signal FFT converts a time-domain signal to the frequency domain. If: The signal is sampled at rate $f_s$ You compute an $N_{\text{FFT}}$-point FFT Then each FFT bin corresponds to a frequency resolution of: $$\Delta f = \frac{f_s}{N_{\text{FFT}}}$$ So the FFT gives you accurate frequency content, assuming the signal is stationary and adequately sampled (Nyquist criterion met).  2. Magnitude Squared Gives Power (Not Amplitude) $$P[k] = |X[k]|^2$$ This gives power at each frequency bin, not just amplitude. It represents how much energy is present at each frequency. It's a key step for PSD.  3. Normalization Makes the PSD Physically Meaningful The equation: $$\text{PSD}[k] = \frac{|X[k]|^2}{N_{\text{FFT}} \cdot f_s \cdot U}$$ is derived from first principles and ensures that the u...

Comparison of Periodogram Techniques Overview of spectral estimation methods and their characteristics. Periodogram Fast but high variance. Take the raw time-domain signal x[n]x[n] of length NN. Optionally apply a window (e.g., Hann, Hamming) to reduce spectral leakage. Compute the DFT or FFT of the (optionally windowed) signal. Estimate the power spectral density (PSD) from the squared magnitude of the FFT. Welch’s Method Averaged periodogram with reduced variance. Divide the signal into overlapping segments. Apply a window function to each segment. Compute the FFT and PSD of each windowed segment. Average the PSDs across all segments. Correlogram ...

  In general, we compute the power spectral density (PSD) of a noisy periodic signal. However, in this article, you will learn how to add noise directly to the PSD of a signal. This process is approximately equivalent to adding noise to a clean signal and then computing its PSD. Here, I will discuss both the theoretical background and the MATLAB implementation. Steps 1. First, compute the Fast Fourier Transform (FFT) of the clean signal. Then, calculate the Power Spectral Density (PSD) from the FFT. 2. In our case, ensure that the PSD is in the linear scale . Next, compute the noise power from the given Signal-to-Noise Ratio (SNR) using: noise_power = signal power / linear SNR 3. Then, generate Additive White Gaussian Noise (AWGN) using the formula: AWGN noise = sqrt(noise_power) * randn       where randn generates a Gaussian-distributed signal with a mean of 0 and a variance of 1 .   MATLAB Co...

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: S x (f) = FT{R x (τ)} Here, R x (τ)}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): P x (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 ...

Welch’s Method An approach for estimating the power spectral density by averaging periodograms to lower variance. The signal is split into overlapping sections. A window function is applied to each section. The FFT is computed for each windowed section to obtain its power spectrum. The power spectra are then averaged across all sections. Copy the MATLAB Code from here % The code is developed by SalimWireless.com % The code is developed by SalimWireless.com clc; clear; close all; % Input Signal and Parameters Fs = 1000; % Sampling frequency t = 0:1/Fs:0.3; % Time vector x = cos(2*pi*200*t) + randn(size(t)); % Signal: 200 Hz cosine + noise % Welch's Method Parameters segmentLength = 256; % Length of each segment overlapFraction = 0.5; % Fractional overlap (50%) overlapSamples = floor(segmentLength * overlapFraction); % Overlap in samples step = segmentLength - overlapSamples; % Step size N = length(x); % Length of the input ...

Power Spectral Density Estimation Using the Periodogram 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 -j2πfn For practical computation, the Discrete Fourier Transform (DFT) is used: X[k] = ∑ x[n] e -j(2π/N)kn , k = 0, 1, ..., N-1 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 S_x(f_k) ...

Estimate PSD by segmenting the signal into \(K\) non-overlapping segments, computing periodograms, and averaging: \[ P_x(f) = \frac{1}{K \cdot M} \sum_{m=0}^{K-1} \left| \sum_{n=0}^{M-1} x_k[n] e^{-j 2 \pi f n} \right|^2 \] Where \(x_k[n]\) is the m-th segment, \(K\) is number of segments, and \(M\) is the segment length. Steps to calculate Spectral power density using Bartlett Method 'M' is the length of each segment for the Bartlett method, set to 100 samples. 'K' is the number of segments obtained by dividing the total number of samples N by the segment length 'M'. psd_bartlett_broadband is initialized to store the accumulated periodogram. For each segment k, x_k extracts the k-th segment of the broadband signal. P_k computes the periodogram of the k-th segment using the FFT. The periodograms are accumulated and averaged over all segments. The PSD is plotted in ...

  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 ...