A baseband digital communication system employs the signals shown in Figure P-10.15(a) for the transmission of two equiprobable messages. It is assumed that the communication problem studied here is a "one shot" communication problem, i.e., the messages are transmitted just once and no transmission takes place afterward. The channel has no attenuation and the noise is AWG with power spectral density .

1. Find an appropriate orthonormal basis for the representation of the signals.

2. In a block diagram, give the precise specifications of the optimal receiver using matched filters. Label the block diagram carefully.

3. Find the error probability of the optimal receiver.

4. Show that the optimal receiver can be implemented by using just one filter. [See the block diagram shown in Figure P-10.15(b).] What are the characteristics of the matched filter and the sampler and decision device?

5. Now assume the channel is not ideal, but has an impulse response of c(t) =  Using the same matched filter as the previous part, design an optimal receiver.

6. Assume that the channel impulse response is where a is a random variable uniformly distributed on [0, 1]. Using the same matched filter, design the optimal receiver.