LU Decomposition using Doolittle Factorization We can write an m X n matrix A as a product of two matrices, L and U . And A = L*U L = $\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ l21 & 1 & 0 & 0 \\ l31 & l32 & 1 & 0 \\ l41 & l42 & l43 & 1 \end{bmatrix}$ ; U = $\begin{bmatrix} u11 & u12 & u13 & u14 \\ 0 & u22 & u23 & u24 \\ 0 & 0 & u33 & u34 \\ 0 & 0 & 0 & u44 \end{bmatrix}$ L= lower triangular matrix; U= upper triangular matrix Procedure- Choose a matrix ( m X n) (e.g., 3X 3, 3 X 4, 4 X 4, etc.,) Initialize the L and U matrices. For L matrix, take a matrix with all diagonal elements assigned to 1, and the matrix elements above the diagonal are zeroes. L matrix size will be ( m X m ). The values of matrix elements below the main diagonal can be assigned to l 21 , l 31 , etc., and so on. $$\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ l21 & 1 & 0 & 0 \\ l31 & l32 ...

LU Decomposition using Crout’s Method Procedure- Choose a matrix ( m X n) (e.g., 3X 3, 3 X 4, 4 X 4, etc.,) The Crout’s matrix decomposition algorithm differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout’s method returns a lower triangular matrix and a unit upper triangular matrix. Initialize the L and U matrices. L matrix size will be ( m X m ). The values of matrix elements below the main diagonal can be assigned to l 21 , l 31 , etc., and so on. And the matrix elements above the diagonal are zeroes. $$\ \begin{bmatrix} l11 & 0 & 0 & 0 \\ l21 & l22 & 0 & 0 \\ l31 & l32 & l33 & 0 \\ l41 & l42 & l43 & l44 \end{bmatrix} $$ l 21 , l 31 , etc. are unknown For matrix U , take a matrix with all diagonal elements assigned to 1, and the matrix elements below the diagonal are zeroes. Size of matrix U will be as same as matrix A (...

Discriminator Receiver Where is a Discriminator Receiver Used? It is used in receivers for: Frequency Modulation (FM) signals FM broadcast radio systems Two-way radios (walkie-talkies) TV sound transmission (analog TV systems) What Does a Discriminator Do? In FM: Information is carried by frequency changes of the carrier. The amplitude remains constant . A discriminator: Converts frequency variations into voltage variations . ...

Frequency Shift Keying (FSK) FSK is a digital modulation technique where each bit is represented by a different carrier frequency: 0 → f 0 ,   1 → f 1 The transmitted FSK signal can be represented as: s(t) = A cos(2π f i t),   i ∈ {0,1} When transmitted over a noisy channel, Additive White Gaussian Noise (AWGN) spreads the received signal. This simulation shows the effect of noise on the FSK constellation by plotting each symbol based on its frequency and amplitude. Number of Random Bits: Signal-to-Noise Ratio (SNR) in dB:

Binary Phase Shift Keying (BPSK) Simulation BPSK is a type of digital modulation where each bit is represented by one of two possible phases of a carrier signal: 0 → -1,   1 → +1 The transmitted BPSK signal can be written as: s(t) = √(2E_b) * cos(2πf_c t + πb),   b ∈ {0,1} When transmitted over a noisy channel, Additive White Gaussian Noise (AWGN) affects the signal. At the receiver, the demodulated signal is compared with a threshold (0) to detect the bit. This simulation demonstrates the effect of noise on BPSK constellation points and allows you to calculate the Bit Error Rate (BER). Number of Bits: SNR (dB):

QPSK Constellation with AWGN Visualizing modulation symbols under different SNR conditions Input Parameters Number of Random Bits: Signal-to-Noise Ratio (SNR) in dB: Theory In 4-Phase Shift Keying (4-PSK), each symbol encodes 2 bits using one of four possible phases: 0, π/2, π, 3π/2 . The transmitted complex baseband signal is: x[n] = A · exp(jθ_n) where A is the symbol amplitude, and θ_n ∈ {0, π/2, π, 3π/2} is the phase corresponding to the transmitted bits. When the signal passes through an additive white Gaussian noise (AWGN) channel, the received signal is: r[n] = x[n] + w[n] where w[n] ~ N(0, σ²) is the complex AWGN. The signal-to-noise ratio (SNR) in decibels is defined as: SNR (...

  MATLAB Code for Zak Transform (Time-Frequency Transform)  %% The code is developed by www.salimwireless.com clc; clear; close all ; %% Step 1: Parameters N = 16; % Total number of symbols M = 4; % Lattice rows (time slots) SNR_dB = 20; % Noise level modOrder = 4; % QPSK %% Step 2: Generate random QPSK symbols data = randi([0 modOrder-1], N, 1); txSymbols = pskmod(data, modOrder, pi/4); disp( 'Transmitted symbols:' ); disp(txSymbols); %% Step 3: Discrete Zak Transform (map to 2D lattice) lattice = zeros(M, N/M); for n = 0:N-1 m = mod(n, M) + 1; % row (time slot) k = floor(n / M) + 1; % column (frequency slot) lattice(m,k) = lattice(m,k) + txSymbols(n+1)*exp(-1j*2*pi*(k-1)*n/N); end % Optional: Apply simple phase rotation per column to illustrate Zak Transform %for k = 1:N/M % lattice(:,k) = lattice(:,k) .* exp(-1j*2*pi*(k-1)*(0:M-1)'/N); %end disp( 'Zak lattice:' ); disp(lattice); %% Step 4: Serialize lattice for transmissio...

Zak Transform Yes! There is a transform called the Zak Transform , though it’s much less commonly mentioned than the Fourier or Z-transform. It comes up mostly in signal processing, time-frequency analysis, and communications . 1. Definition The Zak Transform of a signal x(t) (continuous-time) is defined as: $$ Z_x(t, f) = \sum_{n=-\infty}^{\infty} x(t - nT) e^{j 2 \pi n f T} $$ Where: x(t) is the input signal T is a fixed period t is a continuous variable within [0, T) f is the frequency variable It essentially maps a 1D signal into a 2D representation in time-frequency space . 2. Key Properties Periodicity: $ Z_x(t + T, f) = e^{j 2 \pi f T} Z_x(t, f) $ Time-Frequency Analysis: Converts time-domain signal into a representation showing both time and frequency structure . Often used in Gabor frames, multicarrier communications, and OFDM . Invertible: There’s an inverse Zak Transform to recover x(t) . 4. Discrete Zak ...