Relationship between Gaussian and Rayleigh distributions

The Rayleigh distribution in classical fading models (like wireless communication) arises from modeling the real and imaginary parts of a complex baseband signal as independent, zero-mean Gaussian random variables — under specific assumptions.

1. Gaussian Distribution

The Gaussian distribution has a lot of applications in wireless communication. Since noise in wireless communication systems is unpredictable, we frequently assume that it has a Gaussian distribution. Any wireless communication diagram will show the addition of AWGN noise as the signal travels through the channel. Due to its independence from operating frequency, it is known as AWGN, or additive white Gaussian noise. To determine the noise in a signal, we compute noise power density, or noise power / Hz (here, bandwidth in Hz). It mostly serves to represent real-valued random variables whose distributions are unknown in the scientific and social sciences.

It has a bell shape. According to the theory of a Gaussian random variable, under certain circumstances, the average of numerous samples (observations) of a random variable with a finite mean and variance is itself a random variable, whose distribution tends to become more normal as the number of samples rises. [Read More] about Gaussian Random Variable and Its PDF (Probability Distribution Function)

2. Relationship between Gaussian & Rayleigh Distribution

To compute the distribution of two independent random variables, Rayleigh is essentially employed. Let me give you a typical wireless communication example. Multi-path is something we see in wireless communication. These multiple pathways are time-delayed variations of the identical signal that the receiver relayed. The distribution becomes Rayleigh when the receiver receives these signals with a different time delay. because the same signal's time-delayed received impulses are unrelated, independent by nature. Therefore, we see that the distribution of channel gains in wireless communication, especially for multi-antenna communication systems, is Rayleigh distributed. Keep in mind that the Rayleigh distribution is primarily Gaussian. Books typically describe channel noise as a Gaussian distribution with a zero mean and a specified standard deviation. The Rayleigh distribution typically represents the distribution of magnitudes of a two-dimensional vector whose components are independent and identically distributed Gaussian variables.

The mean of a Rayleigh distribution is not zero; it's actually related to a parameter σ (scale parameter), and it's equal to σ√(π/2). So, the mean of a Rayleigh distribution is finite and dependent on this parameter.

If you're implying that the mean changes from zero to a finite value due to the distribution involving at least two random variables, that's not entirely accurate. The mean of the Rayleigh distribution is not zero to begin with. It's a characteristic of the distribution itself, irrespective of the number of variables involved.

Fig 1: Effect of AWGN and Rayleigh Fading in Wireless Communication (MATLAB Code)

How to mitigate Rayleigh fading?

Mitigating Rayleigh fading in wireless communication involves various techniques designed to counter the rapid fluctuations in signal strength caused by multipath propagation. Some of the most common methods include: 1. Diversity Techniques (Antenna Diversity, Time Diversity, Frequency Diversity, and Space Diversity), 2. Equalization, 3. Channel Coding, etc.

Equalizer to reduce Rayleigh Fading (or Multi-path Effects)

Adaptive Equalization: Compensates for the effects of multipath fading by adjusting the signal at the receiver. Equalizers can dynamically change to combat time-varying channel conditions caused by Rayleigh fading. (Read more ...)

(Get MATLAB Code)

This is a simulation of a typical wireless communication system with 4 multipath components spaced 1 ms apart, using BPSK modulation at a bitrate of 100 bps

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