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LU Decomposition LU Decomposition is a method to find solutions of linear equations.   Using Gauss Elimination Method   Consider a matrix 𝐴. If all entries below the diagonal entries are zero, then the matrix is called “upper triangular.” If all entries above the diagonal entries are zero, then the matrix is called “lower triangular.”          And A = L*U L =   ;   U =    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 remaining components are zero. L matrix size will be ( m X m ). ...

How to Find the Fourier Transform of a Sine or Cosine Wave When we talk about the continuous-time Fourier transform (CTFT), it's a bit tricky to compute for pure sinusoids like sine and cosine because these functions extend infinitely in time and are not absolutely integrable. To make the Fourier transform well-defined, we truncate the sinusoid over a finite interval, say from \( -T/2 \) to \( T/2 \), by multiplying it with a rectangular window. Suppose you have a cosine signal: \[ x(t) = \cos(2\pi f t) \] Truncated over a finite interval using a rectangular window: \[ x(t) = \cos(2\pi f t) \cdot \text{rect}\left( \frac{t}{T} \right) \] where: \[ \text{rect}\left(\frac{t}{T}\right) = \begin{cases} 1 & \text{if } |t| \leq \frac{T}{2} \\ 0 & \text{otherwise} \end{cases} \] The rectangular function is 1 between \( -T/2 \) and \( T/2 \), and 0 elsewhere. Frequency-domain Result: Convolution By the Convol...

  MATLAB Script % The code is developed by SalimWireless.com clc; clear; close all; % INPUT TEXT input_text = 'www.salimwireless.com'; % STEP 1: Source Statistics — Get unique symbols and probabilities [unique_symbol, probability] = source_statistics(input_text); % STEP 2: Huffman Encoding — Generate code words for each symbol code_word = huffman_encoding(probability); % STEP 3: Encode Text to Bit Stream bit_stream = []; for i = 1:length(input_text) index = find(unique_symbol == input_text(i)); bit_stream = [bit_stream code_word{index}]; end bit_stream = bit_stream - '0'; % Convert to numeric % STEP 4: Decode Bit Stream to Text decoded_msg = huffman_decoding(unique_symbol, code_word, bit_stream); % OUTPUT disp('Original Message:'); disp(input_text); disp('Encoded Bit Stream:'); disp(num2str(bit_stream)); disp('Decoded Message:'); disp(decoded_msg); % STEP 5: Visualization figure; % Plot symbol probabilities subplot(2, 1, 1); ...

  Formula for Frequency Resolution (in general) The frequency resolution is the smallest frequency difference between two adjacent frequency points in your sampling range. It is determined by the total frequency range and the number of frequency samples  N . The formula for the frequency resolution (or step size)  Δf  is: Δf = (f max  - f min ) / (N - 1) Where: f min  is the minimum frequency in the range (in this case, -50 Hz). f max  is the maximum frequency in the range (in this case, 50 Hz). N  is the number of frequency points / frequency bins. Using the Given Values: From the function: f min  = -50 Hz f max  = 50 Hz N  = 1000 The frequency resolution is: Δf = (50 - (-50)) / (1000 - 1) = 100 / 999 ≈ 0.1001 Hz   Understanding Frequency Resolution in Signal Processing Alternative Formula Using Time Duration Another common way to define frequency resolution, especially in time-domain signal processing, is: Δf = 1 / T W...

  QPSK provides twice the data rate compared to BPSK. However, the bit error rate (BER) is approximately the same as BPSK at low SNR values when gray coding is used. On the other hand, QPSK exhibits similar spectral efficiency to 4-QAM and 16-QAM under low SNR conditions. In very noisy channels, QPSK can sometimes achieve better spectral efficiency than 4-QAM or 16-QAM. In practical wireless communication scenarios, QPSK is commonly used along with QAM techniques, especially where adaptive modulation is applied. Modulation Bits/Symbol Points in Constellation Usage Notes BPSK 1 2 Very robust, used in weak signals QPSK 2 4 Balanced speed & reliability 4-QAM 2 4 Equivalent to QPSK 16-QAM 4 16 Higher data rate, less robust in noise   QPSK vs BPSK and QAM: A Com...