Suboptimum Receiver

A BPSK signal set is used to transmit equally likely signals over a channel which adds white Gaussian noise (autocorelation function $R_N(\tau) = \frac{N_0}{2} \delta(\tau)$ to the transmitted signal. Hence, the signal set consists of the signals

$$\begin{array} {cll} s_0(t) & = \sqrt{\frac{2E}{T}} \cos(2 \... ...}} \cos(2 \pi f_0 t) & \mbox{for $0 \leq t \leq T$}\end{array}$$

You may assume that f₀T is an integer.

1.

Sketch a block diagram of a receiver which minimizes the probability of error and uses a single (matched) filter followed by a sampler. Sketch the impulse response of the matched filter used in yor detector.

2.

Compute the probability of error your receiver achieves.

3.

To save some expenses, the following receiver is considered.

The switch in the receiver above is closed whenever $\cos(2 \pi f_0 t) \geq 0$ and is open when $\cos(2 \pi f_0 t) < 0$.
Find the distribution of the random variable R for both cases, s₀(t) was transmitted and s₁(t) was transmitted.

4.

Find the value of the threshold K which minimizes the probability of error.

5.

What is the minimum probability of error achievable with the receiver shown above?