Q$.4-$ a$-$ Obtain the expression for probability density function of the power in SIMO Rayleigh fading channel assuming independent and identically distributed channels from the transmit antenna to the $L$ receive antennas as

$f(x)=\frac{1}{(L-1)!}{x}^{L-1}{e}^{-x},x0$

Hint: Use the characteristic function approach.

b$)$ Obtain the cumulative distribution function of $x.$

c$)$ Show that the probability of outage for a target rate, $R,$ can be approximated for small $x$ as

$P_{\text{o}\text{u}\text{t}}(R)=\frac{(2^R-1{)}^L}{L!SN{R}^L}$

Hint: $,{\displaystyle \underset{l=1}{\overset{L}{}}}\frac{1}{(l-1)!}{x}^{l-1}$~~$(e^x-\frac{x^L}{L!})$