Mobile Communication Channels

Assume the impulse response of a channel is given by

$$h(t) = \delta(t) + \alpha \cdot \delta(t-T_c).$$

The amplitude of the delayed component is constant.

1. Is the channel time-varying or time-invariant. Explain.
2. Find the response of the channel if the following signals are applied as the input to the channel
   1. $s(t)=1$ for all $t$,
   2. $s(t)=\sin(\frac{\pi t}{T_c})$ for all $t$,
   3. $s(t)=\sin(\frac{\pi t}{2 T_c})$ for all $t$.
3. Find a general expression for the response of the channel, when the input is $s(t)=\sin(2 \pi f t)$ for all $t$.
4. Plot the magnitude of the response as a function of the frequency $f$.
5. What conclusions you can draw from that plot?
6. Express the coherence-bandwidth of the channel in terms of the parameters.
7. The above channel is used to transmit digitally modulated data at a rate $\frac{1}{T_b}$. Data are transmitted in packets of $N$ symbols. For each of the following cases indicate if the channel is frequency selective or non-selective.
   1. $T_b \gg T_c$,
   2. $T_b \approx T_c$.
8. For each of the two cases, explain which provisions must be made to ensure reliable communication.