Application of the Exponential integral in information theory

Question

the exponential integral is defined as follows: $$E_1\left(z\right)=\int _z^{\infty \:}\:\frac{e^{-t}}{t}dt$$ I found that this integrals has applications in information theory where it is used to calculate the Shannon capacity of a wireless communication system after fading (ie attenuation due to multipath-interference). This capacity is given by the following equation:$$C_{fading}=\frac{exp\left(\frac{1}{P}\right)}{\ln \:\left(2\right)}E_1\left(\frac{1}{P}\right)$$ where $P$ is upper bound of average transmitted power of the communication system. Also, this system is assumed to have additive Gaussian white noise only. The question is how is this expression derived? Also are there further sources to understand this expression better

Answer 1

This is the, so called, ergodic capacity for transmission over a Rayleigh fading channel with (perfect) channel state information at the receiver (CSIR).

With the transmitter sending a signal $x$, the receiver observes $$ y = h x + w, $$ where $h \in \mathbb{C}$ is the channel fading gain and $w \in \mathbb{C}$ is a sample of a white Gaussian noise, of zero mean and variance normalized to $1$.

Now, for a fixed and known value of $h$ at the receiver side, the channel capacity is $$ C(h)=\log_2(1 + |h|^2 P), $$ where $P \triangleq \mathbb{E}(|x|^2)$ is the (maximum) signal power. The, so called, ergodic capacity of a channel is the quantity $$ \label{1} C \triangleq \mathbb{E}(C(h)) $$ where the expectation is over the random variable $h$.

Now, under the standard Rayleigh fading model, $h$ has independent real and imaginary parts, identically distributed as normal of zero mean and variance $1/2$. This means that $|h|^2$ is distributed as an exponential random variable of mean $1$ (show it).

The result then follows by performing the integration.