1. Introduction
Rayleigh fading coefficients and AWGN, or Additive White Gaussian Noise (AWGN) in Wireless Channels, are two distinct factors that affect a wireless communication channel. In mathematics, we can express it in that way.
Let's explore wireless communication under two common noise scenarios: AWGN (Additive White Gaussian Noise) and Rayleigh fading.
y = h*x + n ... (i)
Symbol '*' represents convolution.
The transmitted signal x is multiplied by the channel coefficient or channel impulse response (h) in the equation above, and the symbol "n" stands for the white Gaussian noise that is added to the signal through any type of channel (here, it is a wireless channel or wireless medium). Due to multi-paths the channel impulse response (h) changes. And multi-paths cause Rayleigh fading.
2. Additive White Gaussian Noise (AWGN)
The mathematical effect involves adding Gaussian-distributed noise to the modulated signal. The received signal y(t) is given by:
y(t) = x(t) + n(t)
Where:
- x(t) is the modulated signal.
- n(t) is the AWGN.
The effect of AWGN is to add random variations to the amplitude of the signal, which can lead to erroneous detection of the transmitted symbols. The SNR (signal-to-noise ratio) plays a crucial role in determining the quality of demodulation, with higher SNR values leading to better performance.
We measure SNR at the receiver side due to AWGN for a variety of reasons (Signal-to-Noise Ratio (SNR) Explained). For additional information about the Gaussian Noise and its PDF, click here to learn about Gaussian PDF in Wireless. Because the power spectrum density of this type of noise is frequency independent, the term "white Gaussian noise" has been used here.
3. Rayleigh Fading
Mathematically, Rayleigh fading can be represented as a complex Gaussian random variable with zero mean and a certain variance. The received signal y(t) in the presence of Rayleigh fading can be represented as:
y(t) = h * x(t) + n(t)
Where:
This symbol '*' represents convolution
- h is the complex fading coefficient, representing the channel gain and phase shift.
- x(t) is the modulated signal.
- n(t) is the noise.
The fading coefficient h introduces random amplitude and phase variations to the signal. Due to the randomness of h, the received signal's amplitude will experience fluctuations, impacting the detection of transmitted symbols. The actual fading distribution might vary depending on the specific channel characteristics.
We will now talk about Rayleigh fading. We'll start by talking about what fading actually is. Any sort of wireless communication uses many paths Line-of-Sight (LOS) vs Non-Line-of-Sight (NLOS) Propagation to carry the signal from the transmitter to the receiver. To learn more about multi-paths (MPCs) in wireless communication, click here [↗]. Due to various reflections or diffractions from building walls, vegetation, etc., as they pass through multi-paths, the resulting signal at the receiver may be additive or destructive. Diversity, which is achieved by multi-antenna transmission and reception, is the best method to deal with this scenario. The topic " Diversity" will be covered in a later article.
The Rayleigh fading coefficient, or h in equation (i) above, is a complex coefficient that depends on the signal's attenuation and delay spread.
The Rayleigh distribution describes how the amplitudes of channel coefficients vary over a range. If the amplitude of the channel coefficient, a = |h|, then the distribution of the channel coefficient,
fA(a) = 2a e-a^2, a>=0
On the other hand, the phases of the fading channel coefficient are distributed over the range of 0 degrees to 2Đ (or, 2*pi).
Simulator for the Effect of AWGN and Rayleigh Fading on a BPSK Signal
This simulation below represents a standard wireless communication system featuring 4 multipath components, each separated by 1 millisecond, and employing BPSK modulation at a data rate of 100 bps
MATLAB Code to demonstrate the effects of AWGN and Rayleigh fading on wireless communication channels
% The code is written by SalimWireless.Com
clc;
clear all;
close all;
% Parameters
SNR_dB = -10:2:20; % Range of SNR values to simulate
num_bits = 10^5; % Number of bits to transmit
% Generate random message
message = randi([0 1], 1, num_bits);
% BPSK modulation
modulated_signal = 2 * message - 1;
% Simulation
ber_awgn = zeros(size(SNR_dB));
ber_rayleigh = zeros(size(SNR_dB));
for i = 1:length(SNR_dB)
% AWGN channel simulation
received_signal_awgn = awgn(modulated_signal, SNR_dB(i), 'measured');
demodulated_signal_awgn = received_signal_awgn > 0;
ber_awgn(i) = sum(demodulated_signal_awgn ~= message) / num_bits;
% Rayleigh fading channel simulation
% Generate random fading coefficients
fade_coefficients = (randn(1, length(message)) + 1i * randn(1, length(message))) / sqrt(2);
% Apply fading coefficients
received_signal_rayleigh = fade_coefficients .* modulated_signal;
% AWGN noise is added after fading
received_signal_rayleigh = awgn(received_signal_rayleigh, SNR_dB(i), 'measured');
% Demodulation
demodulated_signal_rayleigh = real(received_signal_rayleigh) > 0;
ber_rayleigh(i) = sum(demodulated_signal_rayleigh ~= message) / num_bits;
end
% Plot results
figure;
semilogy(SNR_dB, ber_awgn, 'b-o', 'LineWidth', 2);
hold on;
semilogy(SNR_dB, ber_rayleigh, 'r-o', 'LineWidth', 2);
grid on;
xlabel('SNR (dB)');
ylabel('Bit Error Rate (BER)');
legend('AWGN Channel', 'Rayleigh Fading Channel');
title('Bit Error Rate Performance: AWGN vs Rayleigh Fading');
Output
Equalizer to reduce Rayleigh Fading or Multi-path Effects
(Get the MATLAB Code for the below)
MATLAB Code to overcome the effect of the Rayleigh Fading with Receiver Diversity Gain
% The code is written by SalimWireless.Com
clc; clear; close all;
N = 1e6; % Number of bits
data = randi([0,1], 1, N);
x = 2 * data - 1;
nRx_max = 20;
nRx = 1:nRx_max;
snr_dB = 1:0.5:10; % Adjusted SNR range for better resolution
ber_sim_EGC = zeros(length(snr_dB), nRx_max);
for j = 1:nRx_max
for k = 1:length(snr_dB)
h = randn(j, N) + 1i * randn(j, N);
x_kron = kron(ones(j, 1), x);
c_in = h .* x_kron;
y = awgn(c_in, snr_dB(k), 'measured');
y_rec = y .* exp(-1i * angle(h)); % removing the phase of the channel
y_rec = sum(y_rec, 1); % adding values from all the receive chains
% Demodulation and error counting
rxBits = real(y_rec) > 0; % BPSK demodulation
errors = sum(rxBits ~= data);
ber_sim_EGC(k, j) = errors / N;
end
end
% Plot BER vs SNR on logarithmic scale
figure;
semilogy(snr_dB, ber_sim_EGC);
xlabel('SNR (dB)');
ylabel('Bit Error Rate (BER)');
title('BER vs SNR for Equal Gain Combining');
legend('1 Rx', '2 Rx', '3 Rx', '4 Rx', '5 Rx', '6 Rx', '7 Rx', '8 Rx', '9 Rx', '10 Rx', '11 Rx', '12 Rx', '13 Rx', '14 Rx', '15 Rx', '16 Rx', '17 Rx', '18 Rx', '19 Rx', '20 Rx', 'Location', 'best');
grid on;
Output
Frequently Asked Questions
Q. Why does Rayleigh fading occur?
A. Due to multi-path
Q. Which kind of fading is Rayleigh fading, exactly?
A. Small-scale fading
Q. What other type of fading is there?
A. Large-scale fading
Q. When deep fade occurs?
You can notice a sudden drop in signal power while performing a signal analysis or spectrum analysis. If the signals that reach the receiver are fully destructive, as we have already discussed, this phenomenon is known as "deep fading." Such a condition may also arise as a result of signal shadowing, etc. [Read More about Fading: Slow & Fast Fading and Large & Small Scale Fading, etc.]