Influence of Unknown Delays

The following binary signal set is used to transmit equally likely messages over an additive white Gaussian noise channel (spectral height $\frac{N_0}{2}$),

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

1.

Draw a block diagram of the optimum receiver which uses only a single correlator.

2.

Compute the probability of error for this receiver.

3.

Assume now that the receiver is not properly synchronized with the received signals. Specifically, the signals are subject to a an unknown delay $\tau$ during transmission. Hence, the received signals is $R_t=s_i(t-\tau)+N_t$, with i=0 or 1, but the receiver still correlates with s₀(t)-s₁(t). Find the probability of error as a function of the delay error $\tau$.

4.

For which value of of the delay $\tau$ does the probability of error become 50%?