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## Summary

Probabilistic Channel Models are used to describe how a communications channel will likely behave given the impact of various impairments and distortions to the transmitted signal as it passes through its medium. Probabilistic or random models are used when inputs and impossible to precisely predict, so instead the model must consider the uncertainties and variations in a channel’s behavior such as with a probability distribution. By incorporating randomness or uncertainty they differ from deterministic models that have fixed inputs and outputs.

Elements of Probabilistic Channel Models Include:

**Channel Impairments**: These are factors that affect the quality of the transmitted signal. Common impairments include noise, interference, fading, attenuation, and other forms of distortion.

**Probability Distributions**: Instead of assuming a fixed or deterministic behavior for the channel, probabilistic models use probability distributions to describe the likelihood of different channel conditions. For example, the Gaussian or normal distribution is often used to model additive white Gaussian noise (AWGN) in wireless communication.

**Random Variables**: Parameters of the channel, such as signal-to-noise ratio (SNR) or fading coefficients, are treated as random variables with specific probability distributions.

**Statistical Properties**: Probabilistic channel models provide statistical properties of the channel, such as mean, variance, and higher-order moments. These properties offer insights into the expected behavior of the channel over time.

## Types of Channel Models

### Additive White Gaussian Noise (AWGN)

A Gaussian distribution or normal distribution assumes that thermal noise will be randomly distributed across the normal distribution, and each time instance will be independent from each other which simplifies the model without introducing correlations. The model is comparable to the randomness of television static noise.

This model is relatively straightforward and useful thanks to its simplicity but can potentially not account for real world complexities such as non-gaussian distributions, time-varying effects, multipath fading, and an assumption of infinite bandwidth.

### Rayleigh Fading

Rayleigh Fading models the effect of multipath propagation, or when the signal arrives at the receiver after following multiple paths. Depending on the path the distance traveled, and interference encountered will differ, resulting in a random amplitude and phase upon arrival.

A comparison is ripples produced by pebbles dropping into a pond, and how by the time they reach the shore their amplitudes and phases will vary due to interactions with other ripples and travel through the water.

### Rician Fading

Rican Fading extends the Rayleigh model with the incorporation of a dominant line-of-sight component in addition to scattered paths. This can be compared to a flashlight concentrating its light into a cone, but also spills light generally.

### Nakagami Fading

Nakagami fading is similar to the Rayleigh model but for the use of the variable m, used to characterize the severity of fading in a channel. A high value of m indicates less variation in amplitude and a more reliable channel. When m reaches its maximum of 1 the Nakagami model is essentially the same as a Rayleigh distribution. Lower m values correspond to more severe fading and more scattering. Higher values are generally used to represent urban areas and lower values rural settings.

### Gilbert-Elliot Channel (Binary Symmetric Channel)

This models communication channels with intermittent errors or periods of good and bad transmission. It measures the likelihood of transition from one state to the other and the error probability while in each state. It is most useful in systems where intermittent errors are a significant factor. This is an example of a Markov process.

## References

Dispersion of the Gilbert-Elliott channel | IEEE Conference Publication | IEEE Xplore