OFDM FFT-Based Channel Estimation Simulator This simulator demonstrates FFT-based channel estimation in an OFDM system. The receiver estimates the channel using: H(k) = Y(k) / X(k) Subcarriers CP Length Channel Taps Noise Std Run Simulation Read More Understand the simulator workflow: How does it work?

FFT-Based Channel Estimation Using OFDM in MATLAB Modern wireless systems such as WiFi, LTE, and 5G rely heavily on Orthogonal Frequency Division Multiplexing (OFDM) . One major advantage of OFDM is that it simplifies channel equalization using the Fast Fourier Transform (FFT) . When a cyclic prefix (CP) is inserted before each OFDM symbol, the wireless channel behaves like a circular convolution . This allows the channel response to be estimated easily in the frequency domain. After applying FFT, the received signal can be written as: Y(k) = X(k)H(k) Therefore the channel response can be estimated using: H(k) = Y(k) / X(k) System Overview The following simulation demonstrates the basic OFDM transmission chain: Random data generation BPSK modulation OFDM modulation using IFFT Cyclic prefix insertion Multipath wireless channel FFT-based channel estimation Frequency-domain equalization MATLAB Simulation Code % --------------------------...

  MATLAB Code close all; clear; rng('shuffle'); clc; % Simulation parameters t = 32; % Number of Tx/Rx Antennas r = 32; % Number of Tx/Rx Antennas numRF = 6; % Number of RF Chains G = 64; % Grid Size L = 8; % Grid Size Ns = 6; % Sparsity level ITER = 100; % Number of iterations % Initializations H = zeros(r,t); % Channel matrix SNRdB = -5:1.5:15; % SNR in dB C_HYB = zeros(length(SNRdB), 1); % Capacity of Hybrid MIMO C_MIMO = zeros(length(SNRdB), 1); % Capacity of Conventional MIMO A_T = zeros(t, G); % Transmit response matrix A_R = zeros(r, G); % Receive response matrix % Generate transmit array response matrix for l = 1:G dirCos = 2/G*(l-1) - 1; % Direction cosine for each grid point for K = 1:t A_T(K, l) = 1/sqrt(t) * exp(-j*pi*(K-1)*dirCos); end end % Assume A_R = A_T for simplicity A_R = A_T; % Main simulation loop for iter1 = 1:ITER % ===== Channel Generation ===== A_T_genie = zeros(t, L); A_R_genie = zeros(r, L); % Generate AoD/AoA uniformly in grid AoDlist = ran...

  MATLAB Code clc; clear; close all; % Given measurement matrix Q (4x6) Q = [1 0 1 0 0 1; 0 1 1 1 0 0; 1 0 1 0 1 0; 0 1 0 1 1 1]; % Given measurement vector y (4x1) y = [0; 2; 3; 5]; % Sparsity level (number of nonzero entries to find) L = 2; % you can adjust this based on expected sparsity % --- OMP Algorithm --- r = y; % Initial residual support = []; % Index set x_hat = zeros(size(Q, 2), 1); % Sparse solution initialization for l = 1:L % Step 1: Find column most correlated with residual proj = abs(Q' * r); [~, idx] = max(proj); support = unique([support, idx]); % Step 2: Solve LS for chosen support Qs = Q(:, support); x_s = pinv(Qs) * y; % Step 3: Update residual r = y - Qs * x_s; % Optional stopping condition if norm(r) < 1e-6 break; end end % Step 4: Construct full sparse vector x_hat(support) = x_s; % Display results disp('Selected atom indices (support):'); disp(support); disp('Recovered sparse signal x_hat:'); disp(x_hat); disp('Res...

  MATLAB Code clc; close all; clear all; rng('shuffle'); % Simulation parameters t = 32;              % Number of Tx antennas r = 32;              % Number of Rx antennas numRF = 8;           % Number of RF Chains N_Beam = 24;         % Number of Pilot Symbols G = 32;              % Grid Size ITER = 10;           % Number of iterations L = 5;               % Sparsity level % Initializations omp_thrld = 1; kp = zeros(t * r, L); SNRdB = 10:10:50; mseOMP = zeros(length(SNRdB), 1); mseGenie = zeros(length(SNRdB), 1); % G-quantized Tx/Rx array response matrices A_T = zeros(t, G); A_R = zeros(r, G); % ------------------------------ % Generate Array Response Matrices % ------------------------------ for l = 1:G     dirCos = 2/G * (l - 1) - 1; ...

Orthogonal Matching Pursuit (OMP) in Compressive Sensing 1. Introduction Compressive Sensing (CS) aims to recover a sparse signal from a small number of measurements. In many systems, the measurement process can be expressed as: y = Q h b + n where: y : Measurement vector (observed data) Q : Known sensing matrix or dictionary h b : Sparse vector (unknown signal to estimate) n : Noise Orthogonal Matching Pursuit (OMP) is a greedy algorithm that reconstructs h b by iteratively selecting the columns of Q that best match the measurements y . 2. OMP Algorithm Overview The OMP process proceeds as follows: Initialization: Set residual r (0) = y and selected index set S = ∅ . Correlation: Compute correlations of all columns with the current residual: c i = q i T r (k) Select: Choose the column with the maximum absolute correlation: i(k+1) = argmax |c i | Update Support: Add the selected index to the set S . L...

This tutorial walks through the derivation of the Equivalent Sensing Matrix and how the mmWave MIMO channel estimation problem can be expressed as a compressed sensing model. A numerical example is included for clarity. The mmWave MIMO System Model In a hybrid beamforming setup, the received signal is: \[ \mathbf{y} = \sqrt{P}\, \mathbf{W}^H \mathbf{H} \mathbf{F} \mathbf{s} + \tilde{\mathbf{n}} \] Where: Symbol Meaning \( P \) Transmit power \( \mathbf{W} = \mathbf{W}_{RF}\mathbf{W}_{BB} \) Receiver hybrid combiner \( \mathbf{F} = \mathbf{F}_{RF}\mathbf{F}_{BB} \) Transmitter hybrid precoder \( \mathbf{s} \) Training / pilot vector \( \tilde{\mathbf{n}} \) Noise vector \( \mathbf{H} \) mmWave channel matrix Sparse Channel Representation The physical mmWave channel can be expressed using array response matrices: \[ \mathbf{H} = \mathbf{A}_R \mathbf{H}_b \mathbf{A}_T^* \] Where: \( \mathbf{A}_R \): receive array response matrix \( \math...

How Beamspace Fits into the Original MIMO Equation 1. Original MIMO Model In a standard MIMO system, the received signal at the antennas is given by: y = Hx + n where: y : Received signal vector (size N r ×1) H : Channel matrix (size N r ×N t ) x : Transmitted signal vector (size N t ×1) n : Noise vector This is the antenna-domain channel representation — it tells how each transmit antenna affects each receive antenna. 2. Why We Need Beamspace At mmWave frequencies, the signal propagation has only a few dominant paths (directions of arrival and departure). That means the channel is sparse in the angular domain , but not in antenna space. To expose that sparsity, we convert the channel into beamspace . 3. Beamspace Representation We express the physical channel as: H = A r H b A t H where: A t – Transmit array response (DFT matrix mapping antennas → transmit beams) A r – Receive array response (DFT matrix mapping antennas → re...

  MATLAB code for channel impulse response estimation using FFT-based channel estimation method % The code is developed by SalimWireless.com clc; clear all; close all; % Parameters N = 128; % Signal length epsilon = 1e-3; % Regularization term useWhiteNoise = true; % Set false to use delta function probe SNR_dB = 100; % Signal-to-Noise Ratio in dB % True channel (FIR with 3 taps) true_h = [0.8, 0.5+0.3j, 0.2, 0.1]; L = length(true_h); % Generate probe signal if useWhiteNoise x = randn(1, N); % White noise else x = zeros(1, N); % Delta function x(1) = 1; end % Pass through the channel (linear convolution) y_clean = conv(x, true_h, 'same'); % Add AWGN signal_power = mean(abs(y_clean).^2); noise_power = signal_power / (10^(SNR_dB/10)); noise = sqrt(noise_power/2) * (randn(1, N) + 1j * randn(1, N)); y_noisy = y_clean + noise; % FFTs Xf = fft(x); Yf = fft(y_noisy); % Estimate channel frequency response with regula...

Equalizers in Wireless Communication Typical wireless communication introduce multipath fading that leads to ISI. Estimating the channel is necessary to compensate for these effects. By sending a known prefix alongside the data, the channel response can be determined using Fourier-transform-based methods: H(f) = P r (f) / P t (f) h(t) = IFFT(H(f)) Here, P r (f) and P t (f) are the Fourier transforms of the received and transmitted prefixes. Although this approach is straightforward, it is sensitive to noise. Equalization and ISI Mitigation Pulses transmitted through underwater channels often get distorted, producing inter-symbol interference. Equalizers are used to counteract this effect. Adaptive equalizers, such as Recursive Least Squares (RLS) or Least Mean Squares (LMS) , adjust their parameters based on the channel's characteristics. Using TRM can simplify the channel by reducing eigenvalue spread, which improves equalizer convergence. Combining TR...

  Channel estimation using DFE is a similar process to a non-linear equalization process. In DFE (decision feed equalizer), equalization error bits/symbols between the feedforward tabs and feedback taps are calculated continuously. And equalizer's tap weights tap weights are updated correspondingly.  In plain language, the error between the received bits and known training bits is calculated, and tap weights are updated accordingly. The equalizer estimates the channel impulse response (CIR) .  Once we find the channel impulse response or channel information, we can easily retrieve the original message signal from the noisy data. In the communication process, the whole system is modeled as a linear time-invariant (LTI) system. And  y = h*x + n where, y = received signal            x = transmitted signal           n = additive white Gaussian noise [Read more about the Linear time-invariant (LTI) system and convolu...

Channel Estimation Techniques Channel Estimation is an auto‑regressive process that may be performed with a number of iterations. There are commonly three types of channel estimation approaches: Pilot estimation Blind estimation Semi‑blind estimation. For Channel Estimation, CIR [↗] is used. The amplitudes of the impulses decrease over time and are not correlated. For example: y(n) = h(n) * x(n) + w(n) where y(n) is the received signal, x(n) is the sent signal, and w(n) is the additive white Gaussian noise. At the next stage: h(n+1) = a * h(n) + w(n) The channel coefficient will be modified as stated above at the subsequent stage. The scaling factor “a” determines the impulse’s amplitude, whereas h(n+1) represents the channel coefficient at the following stage. Pilot Estimation Method To understand how a communication medium is currently behaving, a channel estimate is necessary...

5G Channel Estimation... For millimeter wave massive MIMO communication in 5G, we observe that the number of available multipath components (MPCs) is much smaller than the maximum possible connections between the transmitter (TX) and receiver (RX). Only a few MPCs reach the receiver with significant signal strength. For example, let the number of strong MPCs be $L$, while there are $N_t$ antennas at the transmitter and $N_r$ antennas at the receiver. From the channel matrix of the massive MIMO system, the total number of antenna-to-antenna connections is $N_r \times N_t$. In a beamspace representation, if we use a grid of $N_{tBeams}$ and $N_{rBeams}$, the total possible beam combinations is $N_{tBeams} \times N_{rBeams}$. The key condition for sparsity is: $L \ll (N_t \times N_r)$ If we look up the massive MIMO channel matrix , then, H= Because the number of strong MPCs ($L$) is much lower than the maximum connections, the beamspace c...