A deterministic signal, sk, in discrete time, is given by We would like to determine whether this signal has been sent or not, when one receives the signal sk + Nk if the signal was sent and Nk if...

A deterministic signal, sk, in discrete time, is given by

We would like to determine whether this signal has been sent or not, when one receives the signal

s_k
+ N_k
if the signal was sent and N_k
if it was not. We assume that N_k
is standardized white Gaussian noise.

(a) We use a filter, defined by its impulse response:

and choose the decision time k = 3. If the output of the filter is larger than 1 at k = 3 we consider the signal to be present. What is the probability that we erroneously believe that the signal has been sent? (b) The filter used in (a) is not optimal. Design an optimal causal filter and specify the time of decision and the threshold level of the discriminator such that the probabilities of the two possible errors are equal. What is the probability of an error?