Consider the asynchronous communication system shown in Figure P16.15. The two receivers...

Consider the asynchronous communication system shown in Figure P16.15. The two receivers are not colocated, and the white noise processes n(1)(t) and n(2)(t) may be considered to be independent. The noise processes are identically distributed, with power spectral density σ2 and zero-mean. Since the receivers are not colocated, the relative delays between the users are not the same—denote the relative delay of user k at receiver i by τ(i )k . All other signal parameters coincide for the receivers, and the received signal at receiver i is

where sk has support on [0, T ]. You may assume that the receiver i has full knowledge of the waveforms, energies, and relative delays τ (i ) 1 and τ (i ) 2 . Although receiver i is eventually interested only in the data from transmitter i , note that there is a free communication link between the sampler of one receiver, and the postprocessing circuitry of the other.

Following each postprocessor, the decision is attained by threshold detection. In this problem, you will consider options for postprocessing and for the communication link in order to improve performance.

a. What is the bit error probability for users 1 and 2 of a receiver pair that does not utilize the communication link and does not perform postprocessing? Use the following

Notation:

b. Consider a postprocessor for receiver 1 that accepts y2(l −1) and y2(l) from the communication link and implements the following postprocessing on y1(l)

Determine an exact expression for the bit error rate for user 1.

c. Determine the asymptotic multiuser efficiency of the receiver proposed in (b), and compare with that in (a). Does this receiver always perform better than that proposed in (a)?