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 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 Now, if 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

Relation with Doppler frequency

Coherence time T_c is the time-domain dual of Doppler spread and is used to characterize the time-varying nature of the frequency dispersiveness of the channel in the time domain. The Maximum Doppler spread and coherence time are inversely proportional to one another. That is, where (f_m) is the maximum Doppler spread or, maximum Doppler frequency or, maximum Doppler shift given by with f_c being the center frequency of the emitter. Coherence time is actually a statistical measure of the time duration over which the channel impulse response is essentially invariant, and quantifies the similarity of the channel response at different times. In other words, coherence time is the time duration over which two received signals have a strong potential for amplitude correlation. If the reciprocal bandwidth of the baseband signal is greater than the coherence time of the channel, then the channel will change during the transmission of the baseband message, thus causing distortion at the receiver. If the coherence time is defined as the time over which the time correlation function is above 0.5, then the coherence time is approximately, In practice, the first approximation of coherence time suggests a time duration during which a Rayleigh fading signal may fluctuate wildly, and the second approximation is often too restrictive. A popular rule of thumb for modern digital communications is to define the coherence time as the geometric mean of the two approximate values, also known as Clarke's model; from the maximum Doppler frequency f_m we can obtain 50% coherence time Usually, we use the following relation