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Difference between AWGN and Rayleigh Fading


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. Mathematically, the received signal is represented as:

Rayleigh Fading due to multi-paths
Fig: Rayleigh Fading due to multi-paths

Let's explore wireless communication under two common noise scenarios: AWGN (Additive White Gaussian Noise) and Rayleigh fading.

y = h * x + n ... (i)

The symbol '*' represents convolution (in frequency-selective multipath channels) or multiplication (in frequency-flat fading channels).

The transmitted signal x is modified by the channel coefficient or channel impulse response (h), and the symbol "n" stands for the white Gaussian noise added to the signal. Due to multiple propagation paths (reflections, diffractions), the channel impulse response (h) changes, leading to the phenomenon of Rayleigh fading.

2. Additive White Gaussian Noise (AWGN)

The mathematical effect involves adding Gaussian-distributed noise to the modulated signal. In a channel where fading is negligible (h=1), the received signal y(t) is:

y(t) = x(t) + n(t)

Where:

  • x(t) is the modulated signal.
  • n(t) is the AWGN with zero mean and variance σ².

The effect of AWGN is to add random variations to the amplitude and phase of the signal, which can lead to erroneous detection. The SNR (signal-to-noise ratio) plays a crucial role in determining the quality of demodulation; in AWGN channels, the BER decreases exponentially as SNR increases.

We measure SNR at the receiver side to evaluate performance (Signal-to-Noise Ratio (SNR) Explained). Because the power spectral density (PSD) of this noise is frequency independent (flat), it is termed "white" Gaussian noise.

3. Rayleigh Fading

AWGN and Rayleigh Fading illustration

Mathematically, Rayleigh fading can be represented as a complex Gaussian random variable with zero mean. This occurs when there is no dominant Line-of-Sight (LOS) path. The received signal y(t) is:

y(t) = h * x(t) + n(t)

  • h is the complex fading coefficient, representing random attenuation and phase shift.
  • x(t) is the modulated signal.
  • n(t) is the noise.

Rayleigh fading causes random fluctuations in signal power (fades). Diversity, achieved through multiple antennas (MIMO), is the most effective method to mitigate these deep fades.

The Rayleigh distribution describes how the amplitudes a = |h| vary. For a normalized channel (average power = 1), the PDF is:

fA(a) = 2a e-a^2, a ≥ 0

The phases of the fading channel coefficient are uniformly distributed over the range 0 to 2Ï€.

Simulator for the Effect of AWGN and Rayleigh Fading on a BPSK Signal

Experiment for Students: Set the SNR value to 0 dB and the number of multipath components to 1. You will observe that the theoretical BER of the BPSK signal is approximately 0.078 at an SNR of 0 dB.

Access Full Simulation →

Wireless communication system simulation featuring multipath components and BPSK modulation.

Wireless communication system simulation

MATLAB Code: AWGN vs Rayleigh Fading 

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

% Parameters
SNR_dB = 0:2:20; 
num_bits = 10^6; 

% BPSK modulation
message = randi([0 1], 1, num_bits);
modulated_signal = 2 * message - 1;

ber_awgn = zeros(size(SNR_dB));
ber_rayleigh = zeros(size(SNR_dB));

for i = 1:length(SNR_dB)
    % 1. AWGN Channel
    rx_awgn = awgn(modulated_signal, SNR_dB(i), 'measured');
    ber_awgn(i) = sum((rx_awgn > 0) ~= message) / num_bits;
    
    % 2. Rayleigh Channel (Coherent Detection)
    h = (randn(1, num_bits) + 1i * randn(1, num_bits)) / sqrt(2);
    rx_rayleigh = h .* modulated_signal;
    rx_rayleigh = awgn(rx_rayleigh, SNR_dB(i), 'measured');
    
    % Coherent compensation (removing phase rotation)
    rx_comp = rx_rayleigh .* conj(h);
    ber_rayleigh(i) = sum((real(rx_comp) > 0) ~= message) / num_bits;
end

% Plotting
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('BER');
legend('AWGN', 'Rayleigh Fading');
title('BER Performance: AWGN vs Rayleigh Fading');

Output

Graph showing BER performance for AWGN vs Rayleigh Fading
Fig 1: Effects of AWGN and Rayleigh Fading in Wireless Communication

Equalizer to reduce Rayleigh Fading or Multi-path Effects

(Get the MATLAB Code for the below)

BER of Equalized BPSK Signal

Diversity Gain MATLAB Code 

% Written by SalimWireless.Com 
clc; clear; close all;
N = 1e5; data = randi([0,1], 1, N); x = 2 * data - 1;
nRx_max = 5; snr_dB = 0:2:15;                 
ber_sim = 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)) / sqrt(2);
        y = h .* repmat(x, j, 1);
        y = awgn(y, snr_dB(k), 'measured');
        
        % Equal Gain Combining (EGC)
        y_rec = sum(y .* exp(-1i * angle(h)), 1); 
        ber_sim(k, j) = sum((real(y_rec) > 0) ~= data) / N;
    end
end

figure; semilogy(snr_dB, ber_sim); grid on;
xlabel('SNR (dB)'); ylabel('BER'); title('BER with EGC Diversity');
legend('1 Rx', '2 Rx', '3 Rx', '4 Rx', '5 Rx');

Output

Graph showing BER vs SNR for Equal Gain Combining (EGC)
Fig 2: BER vs SNR for Equal Gain Combining (EGC)

Frequently Asked Questions

Q. Why does Rayleigh fading occur?

A. Due to multi-path propagation without a Line-of-Sight component.

Q. Which kind of fading is Rayleigh fading?

A. Small-scale fading.

Q. When does deep fade occur?

A deep fade occurs when multiple signals reach the receiver out-of-phase, interfering destructively and significantly dropping the signal power.

Further Reading 



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