## 1. Introduction

Rayleigh fading coefficients and AWGN, or additive white gaussian noise [↗], 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.Let's explore wireless communication under two common noise scenarios: AWGN (Additive White Gaussian Noise) and Rayleigh fading.

**y = hx + n ... (i)**

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)

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

**x(t)**is the modulated signal.

**n(t)**is the AWGN.

We measure SNR at the receiver side due to AWGN for a variety of reasons. For additional information about the Gaussian Noise and its PDF, click here. 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 (LOS or NLOS) [↗] 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.

**a = |h|**, then the distribution of the channel coefficient,

**f _{A}(a) = 2ae^{-a^2}_{,
a>=0}**

On the other hand, the phases of the fading channel coefficient are distributed over the range of **0 degrees to 2ÐŸ (or, 2*pi).**

** **

## MATLAB Code to demonstrate the effects of AWGN and Rayleigh fading on wireless communication channels

## Output

**Fig 1: Effects of AWGN and Rayleigh Fading in Wireless Communication**

## Equalizer to reduce Rayleigh Fading or Multi-path Effects

## MATLAB Code to overcome the effect of the Rayleigh Fading with Receiver Diversity Gain

## Output

## Q. Which kind of fading is Rayleigh fading, exactly?

## Q. What other type of fading is there?

## 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.]