BPSK singlepath Rayleigh channel

[yongs90]

Homework Statement

The aim of this exercise is to derive and simulate the error rate performance
of binary PSK when the signal is transmitted over a single-path (or frequency
non-selective) slowly fading channel. The frequency-nonselective channel results
in multiplicative distortion of the transmitted signal s(t). In addition, the con-
dition that the channel fades slowly implies that the multiplicative process may
be regarded as a constant during at least one signaling interval. Therefore, the
received equivalent lowpass signal in one signaling interval is

r(t) = $\alpha$ * exp(j * $\phi$) * s(t) + z(t) 0 $\leq$ t $\leq$ ts

where z(t) represents the complex-valued white Gaussian noise process with
power spectral density $N_0$/2

Assume that the channel fading is sufficiently slow that the phase shift, $\phi$ can
be estimated from the received signal without error. Then, we can achieve ideal
coherent detection of the received signal. The received signal can be processed
by passing it through a matched filter. Show that for a fixed (time-invariant)
channel, i.e., for a fixed attenuation $\alpha$, the error rate of binary PSK as a function
of the received SNR, $\gamma$ is
$P_b$($\gamma$) = Q($\sqrt{}2\gamma$)
where $\gamma$ = $\alpha^2 E_b / N_0$ and $E_b$ is the energy of the transmitted signal per bit.

Homework Equations

The Attempt at a Solution

Can someone give me an idea on how to start..

 

[yongs90]

How do I go about it
i) Take the convolution of r(t) with the impulse response of a filter matched to s(t). Consider output signal and output noise separately.
ii) Remember that the modulation method is binary PSK, which means the phase shift is either 0 or pi. What is the possible output signal values?
iii) Before you compute error probability, you must obtain the pdf of the output noise.