Impulse Response h[n] of an ARMA System Step-by-Step Solution 1. Start with the Transfer Function: Given: H(z) = (1 + 0.3z⁻¹) / (1 - 0.75z⁻¹ + 0.5z⁻²) This is an ARMA(2,1) system where: MA (numerator) coefficients: [1, 0.3] AR (denominator) coefficients: [1, -0.75, 0.5] 2. Find the Impulse Response h[n]: We want the inverse Z-transform of H(z). Instead of doing partial fraction decomposition, we use the system's difference equation. 3. Recursive Computation Using the Difference Equation: From the system equation: y[n] + 0.75y[n−1] − 0.5y[n−2] = x[n] + 0.3x[n−1] Assume x[n] = δ[n] (unit impulse): x[0] = 1 , others are 0. Then y[n] = h[n] 4. Compute h[n] values: n = 0: y[0] + 0 = 1 → h[0] = 1 n = 1: y[1] + 0.75*1 = 0.3 → h[1] = -0.45 n = 2: y[2] - 0.3375 - 0.5 = 0 → h[2] = 0.8375 n = 3: y[3] + 0.628125 + 0.225 = 0 → h[3] = -0.853125 5. Table of Fi...

  MATLAB Code clc; clear all; close all; % Number of samples to generate n = 100000; % Generate Uniform distribution between 0 and 1 r = rand(1, n);  % rand generates numbers in the range [0, 1] % Transform to the range [-1, 1] a = -1; b = 1; uniform_values = a + (b - a) * r; % Plot the histogram of the generated uniform distribution figure; histogram(uniform_values, 30, 'Normalization', 'pdf');  % Normalized to show probability density title('Uniform Distribution between -1 and 1'); xlabel('Value'); ylabel('Probability Density'); % Generate Gaussian distribution (Standard Normal Distribution) gaussian_values = randn(1, n);  % Standard normal distribution (mean = 0, std = 1) % Plotting the Gaussian distribution figure; histogram(gaussian_values, 30, 'Normalization', 'pdf');  % Normalized to show probability density title('Gaussian Distribution (Standard Normal)'); xlabel('Value'); ylabel('Probability Density...

Singular Value Decomposition (SVD) SVD can be performed on any rectangular or square matrix. In SVD, U and V are unitary matrices (orthogonal if the matrix is real), satisfying the conditions UU H = I and VV H = I. Computing the condition number is often important—it is defined as the ratio of the largest singular value to the smallest non-zero singular value in the diagonal matrix of singular values. A high condition number indicates a nearly singular or ill-conditioned matrix.   For a Matrix,   Step 1: W e normalize each column We get,  H = We divided the elements of the first column by √(2² + 3²) = √13 , and proceeded similarly for the other columns. Here singular values are not in decreasing order. Step 2: Now we arrange the singular values in decreasing order H =   That implies, H =  U Σ V H Again assume, the first matrix is  U  (unitary matrix), the middle one is  Σ  (eigenmatrix) ,...

  MATLAB Code clc; clear; close all; % Parameters Rb = 1e6; % Bit rate (1 Mbps) SNR_dB = 0:2:20; % SNR range in dB beta_values = [0, 0.2, 0.5, 0.8, 1.0]; % Different roll-off factors numBits = 1e5; % Number of bits disp("For different roll-off (β) factors and a symbol rate of 1 MHz:"); % Simulation BER = zeros(length(beta_values), length(SNR_dB)); for b = 1:length(beta_values) beta = beta_values(b); bandwidth = (1 + beta) * (Rb / 2); % Bandwidth calculation timeBandwidthProduct = (1 + beta) / 2; % Time-bandwidth product calculation fprintf('Beta = %.1f, Bandwidth = %.2f MHz, Time-Bandwidth Product = %.2f\n', beta, bandwidth / 1e6, timeBandwidthProduct); for s = 1:length(SNR_dB) snr = 10^(SNR_dB(s) / 10); % Convert dB to linear EbN0 = snr * Rb / bandwidth; % Adjust for bandwidth noiseVar = 1 / (2 * EbN0); % Transmit random BPSK symbols bits = randi([0, 1], numBits, 1); symbols = 2 * bits - 1; noise = sqrt(noiseVar) * randn(numBits, 1); received = symbols...

  MATLAB Code clc; clear; close all; % Parameters fs = 1000; % Sampling frequency msgLength = 100; % Length of the message pnLength = 50; % Length of PN sequence silenceLength = 20; % Length of silence before and after lagAmount = 50; % Amount of lag (can be negative for lead) threshold = 0.5; % Threshold for correlation peak detection % Generate Unique PN Sequences pnPrefix = 2 * (randi([0, 1], 1, pnLength) - 0.5); pnPostfix = 2 * (randi([0, 1], 1, pnLength) - 0.5); % Generate Message originalMessage = (randi([0, 1], 1, msgLength)); message = 2*originalMessage - 1; % Construct Dataframe dataframe = [pnPrefix, message, pnPostfix]; % Introduce Lag or Lead if lagAmount > 0 %laggedFrame = [zeros(1, lagAmount), dataframe(1:end - lagAmount)]; laggedFrame = [zeros(1, lagAmount), dataframe]; else laggedFrame = [dataframe(-lagAmount + 1:end), zeros(1, -lagAmount)]; end % Correlation with PN Sequences corrPrefix = xcorr(laggedFrame, pnPrefix); corrPostfix = xcorr(laggedFrame, pnPostfi...

 Copy the MATLAB Code from here % The code is developed by SalimWireless.com clc; clear; close all; % Define a bit sequence bitSeq = [0, 1, 0, 0, 1, 1, 1, 0, 0, 1]; % Perform MSK modulation [modSignal, timeVec] = modulateMSK(bitSeq, 10, 10, 10000); % Plot the modulated signal subplot(2,1,1); samples = 1:numel(bitSeq); stem(samples, bitSeq); title('Original message signal'); xlabel('Time (s)'); ylabel('Amplitude'); % Plot the modulated signal subplot(2,1,2); samples = 1:10000; plot(samples / 10000, modSignal(1:10000)); title('MSK modulated signal'); xlabel('Time (s)'); ylabel('Amplitude'); % Perform MSK demodulation demodBits = demodMSK(modSignal, 10, 10, 10000); % Function to perform MSK modulation function [signal, timeVec] = modulateMSK(bits, carrierFreq, baudRate, sampleFreq) % Converts a binary bit sequence into an MSK-modulated signal % Inputs: % bits - Binary input sequence % carrierFreq - Carri...

  MATLAB Code % Set parameters fs = 1000; % Sampling rate in Hz t = 0:1/fs:1; % Time vector spanning 1 second analogSignal = (sin(2 * pi * 2 * t) + 1) / 2; % Generate a 2 Hz sine wave, scaled to [0,1] samplesPerPulse = 100; % Define number of samples per pulse ppmWidthFraction = 0.1; % Fraction of samplesPerPulse used for pulse width % Perform PPM modulation [ppmSignal, ppmPulseTrain] = ppmModulate(analogSignal, samplesPerPulse, ppmWidthFraction); % Perform PPM demodulation demodulatedSignal = ppmDemodulate(ppmSignal, samplesPerPulse); % Display the demodulated signal values disp('Demodulated Signal:'); disp(demodulatedSignal); % Plot results figure; % Plot original analog signal subplot(3, 1, 1); plot(t, analogSignal, 'b'); title('Original 2 Hz Sine Wave'); xlabel('Time (s)'); ylabel('Amplitude'); % Plot modulated PPM signal subplot(3, 1, 2); stem(ppmSignal(1:1000), 'filled'); title('Pulse Position Modulated (PPM) Signal'); xlabe...

  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...