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 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_x dB (f_k) = 10 lo

  Main Properties The mean and autocorrelation do not change over time. A wide-sense stationary (WSS) process has a constant mean, constant variance, and an autocorrelation function that depends only on the time difference (lag), not the absolute time. For a WSS input to an LTI system, you are expected to study the output's statistical properties (such as mean, variance, and autocorrelation). You will find that the output signal is also a WSS signal. If your input signal has zero mean and unit variance, then the LTI output will have the same nature as the input signal, but: The mean of the output is scaled by the DC gain of the LTI system. The variance of the output is scaled by the total power gain of the system. MATLAB Code %The code is developed by SalimWireless.com clc; clear; close all; % Generate a wide-sense stationary (WSS) signal with 0 mean and unit variance N = 1000; % Length of the signal X = randn(1, N); % WSS signal % Define the time indices t1 and t2 t1 = 0; % Time i

  We all know that for Fourier spectral analysis, you will get an impulse of significant magnitude at the frequency values of sine or cosine waves. Here, we will discuss the spectral analysis or frequency components when we add, subtract, multiply, convolve, integrate or differentiate signals, etc. For sinusoidal signals Addition When we add two sinusoidal signals, the frequency components of the combined signal will be the frequencies of the individual signals.  Subtraction Again, for subtraction, the frequency components of the combined signal will be the frequencies of the individual signals. But if the amplitudes and periods for both signals are the same, then the combined signal will be null. Multiplication The frequency domain components for the time domain multiplication of two sinusoidal signals will be f1 ± f2, where f1 is the frequency of the first sinusoidal signal and f2 is for the other. Convolution The frequency domain components for the time-domain convolution of two sin

  MATLAB Script % The code is developed by SalimWireless.Com clc; clear; close all; % Parameters numSymbols = 1000; % Number of symbols to simulate symbolIndices = randi([0 1], numSymbols, 1); % Random binary symbols (0 or 1) % ASK Modulation (BASK) askAmplitude = [0, 1]; % Amplitudes for binary ASK askSymbols = askAmplitude(symbolIndices + 1); % Modulated BASK symbols % FSK Modulation (Modified BFSK with 90-degree offset) fs = 100; % Sampling frequency symbolDuration = 1; % Symbol duration in seconds t = linspace(0, symbolDuration, fs*symbolDuration); fBase = 1; % Base frequency frequencies = [fBase, fBase]; % Same frequency for both % Generate FSK symbols with 90° phase offset fskSymbols = arrayfun(@(idx) ...     cos(2*pi*frequencies(1)*t) * (1-idx) + ...     1j * cos(2*pi*frequencies(2)*t) * idx, ...     symbolIndices, 'UniformOutput', false); % Extract last points (constellation points) fskConstellation = cellfun(@(x) x(end), fskSymbols); % PSK Modulation (BPSK) pskSymbols

Add Image: Rows: Columns: Image Combining Instructions 1. Upload Images Click on the Add Image input field to upload an image. Repeat this step for additional images. Make sure the total number of images matches or exceeds the grid size you want (based on the Rows and Columns inputs). 2. Set the Grid Size Adjust the Rows and Columns fields to define the grid layout for the combined image. For example, setting Rows = 2 and Columns = 2 will create a 2x2 grid for your images. 3. Combine Images Once you have selected your images and set the grid size, click the Combine! button to merge the images into one canvas. 4. Download the Combined Image After the images are combined, a preview of the merged image will appear. To save the merged image, click the Download Combined Image button, and it will download as a JPEG file named combined-image.jpg . Additional Notes

When two signals are convolved in the time domain, what frequency components will be present in the frequency domain? Is it similar to (frequency 1 + frequency 2) \text{(frequency 1 + frequency 2)} (frequency 1 + frequency 2) and (frequency 1 - frequency 2) \text{(frequency 1 - frequency 2)} (frequency 1 - frequency 2) ? No, it isn’t. That formula applies to the time-domain multiplication of two sinusoidal signals. According to the Discrete Convolution Theorem, convolution of two discrete signals in the time domain is equivalent to multiplication of their DFTs in the frequency domain: F{x[n] ∗ h[n]} = X[k] ⋅ H[k] where  X[k]  and  H[k]  are the DFTs of  x[n]  and  h[n] , respectively. Thus, the convolution  y[n]  in the time domain can be computed by taking the inverse DFT of the product: y[n] = F -1 {X[k] ⋅ H[k]} In general, the frequency components present in X[k]⋅H[k] correspond to the frequencies where both X[k] and H[k] have significant values. Frequencies in X[k] that align with

Power spectral density (PSD) tells us how the power of a signal is distributed across different frequency components, whereas Fourier Magnitude gives you the amplitude (or strength) of each frequency component in the signal. Steps to calculate the PSD of a signal Firstly, calculate the first Fourier transform (FFT) of a signal Then, calculate the Fourier magnitude of the signal The power spectrum is the square of the Fourier magnitude To calculate power spectrum density (PSD), divide the power spectrum by the total number of samples and the frequency resolution. {Frequency resolution = (sampling frequency / total number of samples)} MATLAB Script % The code is written by SalimWireless.com clear close all clc fs = 40000; % sampling frequency T = 1; % total recording time L = T .* fs; % signal length tt = (0:L-1)/fs; % time vector ff = (0:L-1)*fs/L; y = sin(2*pi*100 .* tt) .* sin(2*pi*1000 .* tt); y = y(:); % reference sinusoid % Allow user to input SNR in dB snr_db = input('Enter th

  Suppose, you have two arrays A = [1,2,3,4], and B = [10,11,12]. Then the convolution result, C will be [10,31,64,97,80,80]. The length of C is length of A + length of B -1 . Convolution is a very useful concept in signal processing. We've often taken consideration that transmitted signal is convolved with channel impulse response (CIR) in the time domain. And in the frequency domain they are in multiplication form. Lets discuss how mathematically find the convolution of two arrays or signals. For the above example, the formula for convolution is: \[ C[n] = (A * B)[n] = \sum_{m= 0}^{m} A[m] B[n - m] \]  Where, m is the length of A      and, n varies from 0 to m + p -1   where, p is the length of B   Convolution Formula The convolution of two continuous-time signals \( x(t) \) and \( h(t) \) is given by: \[ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) \, d\tau \] For discrete-time signals