Time-Varying Channels

Assume the time-varying impulse response of a channel is given by

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

More precisely, $h(t,\tau)$ is the response of the channel to an impulse applied at time $\tau$. The time-varying amplitude of the delayed component is given by

$$\alpha(\tau) = A \cdot \cos(2 \pi f_d \tau).$$

1. Find the response of the channel if an impulse is applied at time
   1. $\tau =0$
   2. $\tau = \frac{1}{4f_d}$
   3. $\tau = \frac{1}{2f_d}$
2. Find the response of the channel to a constant signal, $s(t) =1$ for all $t$. I.e., compute
   
   $$s(t) \ast h(t,\tau) = \int s(\tau) h(t-\tau,\tau) d\tau.$$
   
   

3. Sketch the resulting signal.
4. A packet of $N$ bits is to be transmitted over the channel using BPSK modulation. The bit period is $T_b$ seconds. For each of the following cases explain qualitatively the influence of the channel on the transmitted signal:
   1. $T_b \gg T_c$ and $NT_b \ll \frac{1}{f_d}$
   2. $T_b \gg T_c$ and $NT_b \approx \frac{1}{f_d}$
   3. $T_b \approx T_c$ and $NT_b \ll \frac{1}{f_d}$
5. For each of the three cases, explain which provisions must be made in the receiver to ensure reliable communication.