Interactive DOA Simulator X-axis XY angle (deg): 45 XZ angle (deg): 30 Noise: 0.05 Y-axis XY angle (deg): 60 YZ angle (deg): 45 Noise: 0.05 Z-axis XZ angle (deg): 60 YZ angle (deg): 30 Noise: 0.05 Estimated DOA (deg): 0 Simulation Workflow and Mathematical Background This simulator demonstrates Direction of Arrival (DOA) estimation using three-axis sensor signals (X, Y, Z), Maximal Ratio Combining (MRC) , and the MUSIC algorithm . It allows interactive control of signal angles and noise for teaching purposes. 1. Signal Generation A pure sinewave signal of frequency f is projected onto three axes using user-defined angles in different planes: X-axis: θ XY , θ XZ Y-axis: θ XY , θ YZ Z-axis: θ XZ , θ YZ Mathematically, for each time sample t : x(t) = s(t) * cos(θ_xy_x) * cos(θ_xz_x) + n_x(t) y(t) = s(t) * sin(θ_xy_y) * cos(θ_yz_y) + n_y(t) z(t) = s(t) * sin(θ_xz_z) * sin(θ_yz_z) + n_z(t) w...

Direction of Arrival (DOA) Estimation Direction of Arrival (DOA) estimation is a fundamental problem in array signal processing. It involves determining the angle at which one or more incident signals arrive at a sensor array. This information is critical in applications such as radar, sonar, wireless communications, and acoustic source localization. 1. Signal Model Consider a uniform linear array (ULA) of \( M \) sensors receiving a narrowband signal \( s(t) \) from a direction \( \theta \). The received signal at the array can be modeled as: $$ \mathbf{x}(t) = \mathbf{a}(\theta) s(t) + \mathbf{n}(t) $$ where: \( \mathbf{x}(t) = [x_1(t), x_2(t), \dots, x_M(t)]^T \) is the received signal vector. \( \mathbf{a}(\theta) \) is the steering vector of the array, representing the relative phase shifts across sensors: $$ \mathbf{a}(\theta) = \begin{bmatrix} 1 \\ e^{-j 2 \pi \frac{d}{\lambda} \sin \theta} \\ e^{-j 2 \pi \frac{2d}{\lambda} \sin \th...

  MATLAB Code clc; clear; close all ; %% --- Numerical input arrays (example values) --- h_x = [1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1]; % x-axis channel h_y = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]; % y-axis channel h_z = [0.5, 0.6, 0.4, 0.7, 0.3, 0.8, 0.2, 0.9, 0.1, 1]; % z-axis channel r_x = [0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.1]; % received x r_y = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]; % received y r_z = [0.5, 0.4, 0.6, 0.3, 0.7, 0.2, 0.8, 0.1, 0.9, 0.1]; % received z % Ensure all arrays are the same length N = min([length(h_x), length(h_y), length(h_z), length(r_x), length(r_y), length(r_z)]); h_x = h_x(1:N); h_y = h_y(1:N); h_z = h_z(1:N); r_x = r_x(1:N); r_y = r_y(1:N); r_z = r_z(1:N); %% --- Maximal Ratio Combining (MRC) across 3 channels --- w_x = abs(h_x).^2 ./ (abs(h_x).^2 + abs(h_y).^2 + abs(h_z).^2); w_y = abs(h_y).^2 ./ (abs(h_x).^2 + abs(h_y).^2 + abs(h_z).^2); w_z = abs(h_z).^2 ./ (abs(h_x).^2 + abs(h_y).^2 + abs(h_z).^2); r...

  MATLAB Code M = 4; % QPSK numSymbols = 10000; numTrainingSymbols = 5000; chtaps = [1 0.5*exp(1i*pi/6) 0.1*exp(-1i*pi/8)]; % Generate QPSK symbols data = randi([0 M-1], numSymbols, 1); tx = pskmod(data, M, pi/4); % DFE setup eq = comm.DecisionFeedbackEqualizer; eq.ReferenceTap = 1; eq.NumForwardTaps = 5; eq.NumFeedbackTaps = 3; % SNR sweep snrVec = 0:5:25; % 0,5,10,...25 dB berBefore = zeros(size(snrVec)); berAfter = zeros(size(snrVec)); for k = 1:length(snrVec) % Apply channel and AWGN rx = awgn(filter(chtaps,1,tx), snrVec(k), 'measured' ); % BER before equalization rxDataNoEq = pskdemod(rx, M, pi/4); [~, berBefore(k)] = biterr(data, rxDataNoEq); % Equalize [y,~,~] = eq(rx, tx(1:numTrainingSymbols)); % Phase correction phaseOffset = angle(mean(conj(tx(1:numTrainingSymbols)) .* y(1:numTrainingSymbols))); yCorrected = y * exp(-1i*phaseOffset); % BER after equalization rxDataEq = pskdemod(yCorrected, M, pi/4); [...

  MATLAB Code %% BER vs SNR for SIMO Communication System % QPSK modulation % Single Sensor, Sensor Array, and Vector Sensor (x, y, z axes) % Maximum Ratio Combining (MRC) clc; clear; close all ; %% PARAMETERS SNRdB = 0:2:30; % SNR range Nbits = 1e6; % Large bit count for smooth BER curves M = 4; % QPSK modulation k = log2(M); Nr_cases = [1 3 4]; % Receiver configurations % 1 = Single Sensor % 3 = Vector Sensor (x,y,z axes) % 4 = Sensor Array + Vector Components BER_sim = zeros(length(Nr_cases), length(SNRdB)); %% GENERATE RANDOM BITS bits = randi([0 1], Nbits, 1); %% QPSK MODULATION symbols = pskmod(bits, M, pi/4, 'InputType' , 'bit' ); %% MAIN SIMULATION LOOP for r = 1:length(Nr_cases) Nr = Nr_cases(r); % Number of receiving elements for i = 1:length(SNRdB) snr = SNRdB(i); ...

Rayleigh and Weibull Distributions in Signal Processing Understanding statistical distributions in noise and signal analysis. Rayleigh Distribution from Vector Magnitudes Consider a two-dimensional vector: v = (x, y) If both components x and y follow Gaussian distributions, the magnitude of the vector becomes: |v| = √(x² + y²) The resulting distribution of magnitudes follows the Rayleigh distribution . Example in Signal Processing In signal processing, signals analyzed using the Fast Fourier Transform (FFT) often contain real and imaginary components that are Gaussian distributed. When the magnitude of the FFT output is calculated, the amplitude distribution typically follows a Rayleigh distribution. Relationship to the Weibull Distribution The Weibull distribution is closely related to the Rayleigh distribution and can be considered a more general form. It frequently appears in natural systems involving multiple scaling processes such as:...

Understanding Decibels and Hearing Loss How audiologists measure sound intensity and its impact on human hearing. How Sound is Measured Audiologists measure sound using two important parameters: frequency and decibels (dB) . Frequency determines the pitch of a sound. Decibels measure the intensity or loudness of a sound. The typical human hearing range extends from 0 dB to about 140 dB . Sounds at the lower end are extremely quiet, while sounds near the upper limit may cause pain or permanent hearing damage. Dangerous Noise Levels Prolonged exposure to sounds above 80 decibels can damage hearing. Everyday examples of loud sounds include: Ambulance sirens Fireworks Heavy machinery such as tractors Repeated exposure to these sounds may lead to noise-induced hearing loss . Levels of Hearing Loss Hearing Level Decibel Range Normal Hearing 0 – 25 dB Mild Hearing Loss 30 – 40 dB Moderate Hearing Loss 50 – 70 dB Individual...

Understanding Negative SNR and the Shannon Limit Understanding Negative SNR and the Shannon Limit An explanation of Signal-to-Noise Ratio (SNR), its behavior in decibels, and how Shannon's theorem defines the ultimate communication limit. Signal-to-Noise Ratio in Shannon’s Equation In Shannon's equation, the Signal-to-Noise Ratio (SNR) is defined as the signal power divided by the noise power: SNR = S / N Since both signal power and noise power are physical quantities, neither can be negative. Therefore, the SNR itself is always a positive number. However, engineers often express SNR in decibels: SNR(dB) When SNR = 1, the logarithmic value becomes: SNR(dB) = 0 When the noise power exceeds the signal power (SNR < 1), the decibel representation becomes negative. Behavior of Shannon's Capacity Equation Shannon’s channel capacity formula is: C = B log₂(1 + SNR) For SNR = 0: log₂(1 + SNR) = 0 When SNR becomes smaller (in...