Binary Receivers

The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel (autocorrelation function $R_N(\tau) = \frac{N_0}{2} \delta(\tau)$):

$$\begin{array} {cc} s_0(t) = \left\{ \begin{array} {cl} A ... ... t \leq T$}\\ 0 & \mbox{else.} \end{array} \right.\end{array}$$

1.

Draw and accurately label a block diagram of the receiver which minimizes the probability of error and utilizes a single matched filter followed by a sampler. Sketch the impulse response of the matched filter.

2.

Compute the minimum probability of error achievable with this signal set.

3.

For the remainder of the problem the following receiver is used:

Compute the impulse response of the RC-circuit at the front end of the receiver above, i.e., what signal $\tilde{R}_t$ would you observe if the input signal were $R_t = \delta(t)$?

4.

Assume now that the impulse response of the RC-circuit is

$$h(t) = \frac{1}{RC}\exp(-\frac{t}{RC}) \cdot u(t),$$

where u(t) denotes the unit step function. Find the distribution of the random variable R, i.e., the sample taken from $\tilde{R}_t$ at t=T, for both cases: s₀(t) was transmitted or s₁(t) was transmitted.

5.

Compute the probability of error for the receiver above and compare it to the probability of error for the optimum receiver.