Multi-Path Channel and Tapped Delay-Line Model

A mobile communication system employs binary phase shift keying with rectangular pulses

$$u(t) = \left\{ \begin{array}{cl} 1 & \mbox{for $0\leq t \leq T$}\\ 0 & \mbox{else} \end{array}\right.$$

to transmit information at a rate of one bit per $T$ seconds over a channel with (time-invariant) baseband-equivalent impulse response

Furthermore, the channel adds white Gaussian noise $N_t$ of spectral height $\frac{N_0}{2}$. The receiver is designed under the assumption that $c(t)=\delta(t)$ and hence is simply a filter matched to the transmitted pulses $u(t)$. In summary, the communication system can be described by the following block diagram.

The objective of this problem is to determine the parameters of the equivalent tapped delay-line model between the input sequence $I_n$ and the output sequence $R_n$. A generic block diagram for a tapped delay-line model is given below.

1. Compute the convolution of the pulse shaping filter $u(t)$ with the channel impulse response $c(t)$.
2. How many memory elements $L$ are there in the tapped delay-line?
3. What are values for the coefficients in the tapped delay-line model?
4. Find the distribution (type, mean, variance, etc.) of the additive noise $N_n$ samples in the tapped delay line model.