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Rayleigh multipath channel model

Posted By Krishna Sankar On July 14, 2008 @ 5:42 am In Channel | 196 Comments

The article gives a quick overview of a simple statistical multipath channel model called Rayleigh fading channel model.

## Multipath environment

In a multipath environment, it is reasonably intuitive to visualize that an impulse transmitted from transmitter will reach the receiver as a train of impulses.

Figure: Impulse response of a multipath channel

Let the transmit bandpass signal be,
$x(t) = \Re\left\{x_b(t)e^{j2\pi f_ct}\right\}$, where

$x_b(t)$ is the baseband signal,
$f_c$ is the carrier frequency and
$t$ is the time.

As shown above, the transmit signal reaches the receiver through multiple paths where the $n^{th}$ path has an attenuation $\alpha_n(t)$ and delay $\tau_n(t)$. The received signal is,

$r(t) = \sum_n\alpha_n(t)x$t-\tau_n(t)$$.

Plugging in the equation for transmit baseband signal from the above equation,

$r(t) = \Re\left\{\sum_n\alpha_n(t)x_b[t-\tau_n(t)]e^{j2\pi f_c[t-\tau_n(t)]}\right\}$.

The baseband equivalent of the received signal is,

$\begin{eqnarray}r_b(t)& = &\sum_n\alpha_n(t)e^{-j2\pi f_c\tau_n(t)}x_b[t-\tau_n(t)]\\& = &\sum_n\alpha_n(t)e^{-j\theta_n(t)}x_b[t-\tau_n(t)] \end{eqnarray}$,

where $\theta_n(t)=2\pi f_c \tau_n(t)$ is the phase of the $n^{th}$path.

The impulse response is,

$\begin{eqnarray}h_b(t)& = &\sum_n\alpha_n(t)e^{-j\theta_n(t)} \end{eqnarray}$.

The phase of each path can change by $2\pi$radian when the delay $\tau_n(t)$ changes by $\frac{1}{f_c}$. If $f_c$is large, relative small motions in the medium can cause change of $2\pi$radians. Since the distance between the devices are much larger than the wavelength of the carrier frequency, it is reasonable to assume that the phase is uniformly distributed between 0 and $2\pi$ radians and the phases of each path are independent (Sec 2.4.2 [WIRELESS-COMMUNICATION: TSE, VISWANATH] [1]).

When there are large number of paths, applying Central Limit Theorem [2], each path can be modelled as circularly symmetric complex Gaussian random variable with time as the variable. This model is called Rayleigh fading channel model.

A circularly symmetric complex Gaussian random variable is of the form,

$Z = X + jY$,

where real and imaginary parts are zero mean independent and identically distributed (iid) Gaussian random variables. For a circularly symmetric complex random variable $Z$,

$E[Z] = E[e^{j\theta}Z]=e^{j\theta}E[Z]$.

The statistics of a circularly symmetric complex Gaussian random variable is completely specified by the variance,

$\sigma^2=E[Z^2]$.

The magnitude $|Z|$ which has a probability density,

$p(z) = \frac{z}{\sigma^2}e^{\frac{-z^2}{2 \sigma^2}},\ \ \ z\ge 0$

is called a Rayleigh random variable.

This model, called Rayleigh fading channel model, is reasonable for an environment where there are large number of reflectors.

## Reference

[WIRELESS-COMMUNICATION: TSE, VISWANATH] [1]Fundamentals of Wireless Communication, David Tse, Pramod Viswanath [1]
Note:
In a future post, we will try and derive the probability density function of sum of squares of independent Gaussian random variables