Fading : Slow & Fast and Large & Small Scale Fading

📘 Overview
📘 LARGE SCALE FADING
📘 SMALL SCALE FADING
📘 SLOW FADING
📘 FAST FADING
🧮 MATLAB Codes
📚 Further Reading

LARGE SCALE FADING

The term 'Large scale fading' is used to describe variations in received signal power over a long distance, usually just considering shadowing. Assume that a transmitter (say, a cell tower) and a receiver (say, your smartphone) are in constant communication. Take into account the fact that you are in a moving vehicle. An obstacle, such as a tall building, comes between your cell tower and your vehicle's line of sight (LOS) path. Then you'll notice a decline in the power of your received signal on the spectrogram. Large-scale fading is the term for this type of phenomenon.

SMALL SCALE FADING

Small scale fading is a term that describes rapid fluctuations in the received signal power on a small time scale. This includes multipath propagation effects as well as movement-induced Doppler frequency shifts. The statistics of small scale fading in industrial contexts can be described as Rician fading, and the Rician K-factor values for various factory conditions were estimated. We're talking abut the industrial environment and the Rician k-factor because there's a lot more signal reflection and refraction than in other contexts or environments.

Rayleigh fading is a perfect example of small-scale fading because it models rapid variations in signal amplitude due to multipath propagation without a dominant direct path. Rayleigh fading captures variations that occur over short times or distances, typically on the order of the wavelength of the signal. Here, The signal's amplitude varies randomly according to a Rayleigh distribution, with fluctuations that are typical of small-scale fading.

Slow fading

Because of the Doppler shift. When the signal bandwidth is much lesser than the Doppler spread. Slow fading occurs when the channel changes faster than the modulated symbol rate. What causes the channel to change? Doppler shift is responsible for that. Go through the formula of Doppler shift.

Fast fading

Because of Doppler shift, particularly when the Doppler spread is equal to or larger than the signal bandwidth. The additive or subtractive nature of waveforms with varying phases can also produce fast fading. In simpler words, fast fading occurs when the channel changes faster than the modulated symbol rate.

Effect of Muiti-path Fading on the Alamouti Scheme in 2x1 MIMO Communication

Effect of Minimal Fading on the Alamouti Scheme in 2x1 MIMO Communication with an Ideal Channel

Copy the MATLAB Code above from here

% Developed by SalimWireless.Com
clc;
clear;
close all;

% Parameters
numBits = 10000;          % Number of bits per trial
SNR_dB = 0:2:20;          % SNR range in dB
numSNR = length(SNR_dB); 
numTrials = 100;          % Number of Monte Carlo trials
ber = zeros(1, numSNR);   % BER initialization

% BPSK Modulation Function
bpskModulation = @(bits) 2*bits - 1;  % Maps 0 -> -1, 1 -> +1

% Simulation
for i = 1:numSNR
    totalErrors = 0;      % Initialize total errors for this SNR
    totalBits = 0;        % Initialize total bits for this SNR
    
    for trial = 1:numTrials
        % Generate random bitstream
        bits = randi([0 1], 1, numBits);
        symbols = bpskModulation(bits);  % BPSK modulation

% Alamouti Encoding
        s1 = symbols(1:2:end);  % Even symbols
        s2 = symbols(2:2:end);  % Odd symbols
        numSymbols = length(s1);

% Rayleigh fading coefficients
        h1 = (randn(1, numSymbols) + 1j*randn(1, numSymbols)) / sqrt(2);
        h2 = (randn(1, numSymbols) + 1j*randn(1, numSymbols)) / sqrt(2);

% Transmit signals through Rayleigh fading channel
        r1 = h1 .* s1 + h2 .* s2;                         % First time slot
        r2 = -conj(h1) .* s2 + conj(h2) .* s1;            % Second time slot

% Add AWGN noise
        noisePower = 10^(-SNR_dB(i)/10);                  % Noise power based on SNR
        noise1 = sqrt(noisePower/2) * (randn(1, numSymbols) + 1j*randn(1, numSymbols));
        noise2 = sqrt(noisePower/2) * (randn(1, numSymbols) + 1j*randn(1, numSymbols));
        r1 = r1 + noise1;
        r2 = r2 + noise2;

% Alamouti Decoding at Receiver
        s1_hat = conj(h1) .* r1 + h2 .* r2;               % Estimated s1
        s2_hat = conj(h2) .* r1 - h1 .* r2;               % Estimated s2

% Decision Making
        s1_dec = real(s1_hat) > 0;                        % Threshold at 0
        s2_dec = real(s2_hat) > 0;

% Reconstruct received bits
        bits_hat = zeros(1, numBits);
        bits_hat(1:2:end) = s1_dec;                       % Decoded even bits
        bits_hat(2:2:end) = s2_dec;                       % Decoded odd bits

% Calculate errors for this trial
        errors = sum(bits ~= bits_hat);
        totalErrors = totalErrors + errors;
        totalBits = totalBits + numBits;
    end
    
    % Average BER over trials
    ber(i) = totalErrors / totalBits;
end

% Plot BER vs. SNR
figure;
semilogy(SNR_dB, ber, 'o-', 'LineWidth', 2);
grid on;
title('BER vs. SNR for Alamouti Scheme with an Ideal Channel (Monte Carlo)');
xlabel('SNR (dB)');
xlim([0, 20]);
ylabel('Bit Error Rate (BER)');
legend('Alamouti Scheme');
web('https://www.salimwireless.com/search?q=salimwireless.com+fading', '-browser');

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