Mobile Communication Channels

Assume the impulse response of a channel is given by

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

I.e., the response of the channel to a signal $s(t)$ is given by

$$s(t) \ast h(t,\tau) = \int s(\tau) h(t-\tau,\tau) d\tau.$$

The 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 the following impulses are applied as the input to the channel
   1. $\delta(t)$
   2. $\delta(t - \frac{1}{4f_d})$
   3. $\delta(t - \frac{1}{2f_d})$
   Is the channel time-varying or time-invariant.
2. Find the response $r(t)$ of the channel to a constant signal, $s(t) =1$ for all $t$. Then, sketch the resulting signal $r(t)$.
3. Compute the Fourier transform $R(f)$ of the signal $r(t)$ and explain what conclusions you can draw from $R(f)$.
4. Estimate the coherence-time and the coherence-bandwidth of the channel in terms of the parameters $T_c$ and $f_d$.
5. 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 and fast or slow fading.
   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 $NT_b \ll \frac{1}{f_d}$
6. For each of the three cases, explain which provisions must be made to ensure reliable communication.