Stochastic processes and boundary value problems

In communications systems, a communication channel may change with time. Coherence time is the time duration over which the channel impulse response is considered to be not varying. Such channel variation is much more significant in wireless communications systems, due to Doppler effects.

Simple model

In a simple model, a signal $x(t)$ transmitted at time $t_1$ will be received as

$y_{t_1}(t)=x(t-t_1)*h_{t_1}(t),$

where $h_{t_1}(t)$ is the channel impulse response (CIR) at time $t_1$. A signal transmitted at time $t_2$ will be received as

$y_{t_2}(t)=x(t-t_2)*h_{t_2}(t).$

Now, if $h_{t_1}(t)-h_{t_2}(t)$ is relatively small, the channel may be considered constant within the interval $t_1$ to $t_2$.

Coherence time ($T_c$) will therefore be given by

$T_c=t_2-t_1.$

Relation with Doppler frequency

Using Clarke's model, from the maximum Doppler frequency $f_d$ we can obtain 50% coherence time

$T_c=\sqrt{\frac{9}{16\pi f_d^2}}$

Usually, we use the following relation

$T_c=\sqrt{\frac{9}{16\pi }}\frac{1}{f_d}\simeq \frac{0.423}{f_d}$