Fundamentals of Channel Estimation

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. In order to monitor a channel’s behavior in practical communication systems, channel estimation is performed at very short periodic intervals. For instance, a pilot bit is injected into an OFDM data frame at every ’n’‑th bit interval. The pilot signal is known to the receiver. The receiver analyzes the channel influence on the pilot signal and adjusts the channel estimate for the entire data packet accordingly.

This training sequence‑based strategy, also known as the pilot estimation method, is very well‑liked and also not overly complex. The frame contains pilot bit sequences sent over the channel and known to the receiver.

Blind Estimation Method

No training or pilot symbol is sent when using the blind estimation approach. This method makes use of some hidden mathematical operations or properties of the provided data.

MATLAB Code for OFDM Channel Estimation using LSE

% The code is developed by SalimWireless.com clc; clear; close all; %% ------------------- System Parameters ------------------- N_subcarriers = 256; % Total number of OFDM subcarriers pilot_spacing = 8; % Spacing between pilot subcarriers N_pilots = N_subcarriers / pilot_spacing; % Number of pilots N_data = N_subcarriers - N_pilots; % Number of data subcarriers guard_interval = N_subcarriers / 4; % Length of cyclic prefix mod_order = 2; % Modulation order (BPSK: 2 symbols) L_channel = 16; % Length of multipath channel pilot_power = 2; % Pilot power scaling factor n_iterations = 500; % Number of Monte Carlo runs SNR_dB_vector = 0:3:27; % SNR range (in dB) %% ------------------- Pilot & Subcarrier Indexing ------------------- pilot_indices = 1:pilot_spacing:N_subcarriers; % Positions of pilots data_indices = setdiff(1:N_subcarriers, pilot_indices); % Data positions %% ------------------- Preallocation ------------------- ber_results = zeros(length(SNR_dB_vector), n_iterations); % Store BER for each run %% ------------------- Main Loop over SNR ------------------- for snr_idx = 1:length(SNR_dB_vector) SNR_dB = SNR_dB_vector(snr_idx); for iter = 1:n_iterations %% ------------------- Channel Generation ------------------- power_profile = exp(-0.2*(0:L_channel-1)); % Exponentially decaying power h = sqrt(power_profile.') .* (randn(L_channel,1) + 1i*randn(L_channel,1)); h = h / norm(h); % Normalize channel impulse response %% ------------------- Transmitter ------------------- % Generate random data bits tx_bits = randi([0 mod_order-1], N_subcarriers, 1); % QAM Modulation tx_symbols = qammod(tx_bits, mod_order, 'UnitAveragePower', true); % Pilot insertion tx_symbols(pilot_indices) = sqrt(pilot_power) * tx_symbols(pilot_indices); % OFDM modulation (IFFT) tx_ofdm_time = ifft(tx_symbols, N_subcarriers); % Add cyclic prefix tx_with_cp = [tx_ofdm_time(end-guard_interval+1:end); tx_ofdm_time]; %% ------------------- Channel & Noise ------------------- % Pass through multipath channel rx_with_cp = filter(h, 1, tx_with_cp); % Compute noise power signal_power = mean(abs(rx_with_cp).^2); noise_power = signal_power / (10^(SNR_dB/10)); % Add AWGN noise noise = sqrt(noise_power/2) * (randn(size(rx_with_cp)) + 1i*randn(size(rx_with_cp))); rx_with_cp_noisy = rx_with_cp + noise; %% ------------------- Receiver ------------------- % Remove cyclic prefix rx_ofdm_time = rx_with_cp_noisy(guard_interval+1:end); % OFDM demodulation (FFT) rx_symbols = fft(rx_ofdm_time, N_subcarriers); % Extract received pilots rx_pilots = rx_symbols(pilot_indices); % Extract transmitted pilots tx_pilots = tx_symbols(pilot_indices); %% ------------------- Channel Estimation ------------------- % Least Squares Estimation at pilot locations H_pilot_est = rx_pilots ./ tx_pilots; % Smooth interpolation across all subcarriers (use 'spline' instead of 'linear') H_estimated = interp1(pilot_indices, H_pilot_est, 1:N_subcarriers, 'spline', 'extrap').'; % Optional: Apply simple moving average to smooth out noise smoothing_window = 3; % Must be odd number (e.g., 3, 5) H_estimated = movmean(H_estimated, smoothing_window); % Equalize received symbols rx_equalized = rx_symbols ./ H_estimated; %% ------------------- Demodulation ------------------- % Demodulate equalized data rx_bits = qamdemod(rx_equalized, mod_order, 'UnitAveragePower', true); %% ------------------- BER Calculation ------------------- % Only evaluate BER on data subcarriers (not pilots) [~, ber_results(snr_idx, iter)] = symerr(tx_bits(data_indices), rx_bits(data_indices)); end % End of iteration loop end % End of SNR loop %% ------------------- Plotting Results ------------------- figure; semilogy(SNR_dB_vector, mean(ber_results,2), 'b-o', 'LineWidth', 2); grid on; xlabel('SNR (dB)'); ylabel('Bit Error Rate (BER)'); title('OFDM Channel Estimation using LSE'); set(gca,'FontSize',12); axis([0 30 1e-4 1]); web('https://www.salimwireless.com/search?q=channel%20estimation', '-browser');

Output

Further Reading

1. Channel Impulse Response (CIR)
2. Beamforming & Channel Estimation
3. Decision Feedback Equalizer (DFE)
4. Orthogonal Frequency Division Multiplexing
5. OFDM Symbols and Subcarriers Explained