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.