Sometimes, we see some word that sound like simple, but hard to get clear understanding. To me, the word 'Coherence/Coherent' is one of those words. Without clear understanding of this words, I got confused with many other terms that came from this word, like 'Coherent time', 'Coherent Channel', 'Coherent frequency' etc.
What dose 'Coherent' mean ?
It mean 'Something stays same (does not change)'.
What does that 'Something' mean ?
It would mean different things depending on situation (context).
Based on this, you can derive the meaning of some of common terminology you would hear or see in various materials on communication theory.
Coherent time
It means a certain span of time during which 'something' stays same. As I said, the word 'something' would have different meaning depending on situation. For example, if you heard of this when you are talking about some traveling wave (propagating wave), the "something" may mean 'phase delay'. So in this case, the Coherent time would mean 'a certain time span in which the phase delay of traveling wave does not change. If you hear of this when you talking about fading stuff, 'coherent time' would mean 'a certain time span during which the fading property remain unchanged'.
The definition above says what coherent time is. It does not say how long it lasts, and that is the number an engineer actually needs. For a radio channel the answer comes from motion, because a channel changes when the transmitter, the receiver, or the things between them move.
Motion appears as a Doppler shift. A receiver moving at speed v relative to the transmitter sees the carrier shifted by fd = v / lambda, where lambda is the wavelength. Coherent time is roughly the inverse of that shift, so Tc is about 1 / fd. Faster motion means a larger shift and a shorter coherent time.
Two examples make the scale concrete. At a 3 GHz carrier the wavelength is 0.1 m. A walking user at 1 m/s produces a 10 Hz Doppler shift, so the coherent time is around 100 ms. The same user in a car at 30 m/s produces 300 Hz, and the coherent time falls to around 3 ms.
Treat those figures as orders of magnitude rather than exact values. Different textbooks define coherent time against different correlation thresholds, so the constant in front of 1 / fd varies from one source to the next. The relationship itself does not vary. Double the speed and the coherent time halves.
Coherent frequency
Same logic, It means a certain span of frequency during which 'something' stays same. As I said, the word 'something' would have different meaning depending on situation. For example, if you hear of this when you talking about fading stuff, 'coherent frequency' would mean 'a certain frequency span within which the fading property remain unchanged'.
The same question applies here, and the number comes from a different property of the channel. Coherent time follows from motion. Coherent frequency follows from the spread of path lengths instead.
A signal that reaches the receiver by several paths arrives several times over, because the longer paths take longer to travel. The gap between the earliest arrival and the latest one is the delay spread, usually quoted as an RMS value and written tau. Coherent frequency is roughly the inverse of it, so Bc is about 1 / tau.
Again two examples fix the scale. An indoor channel with an RMS delay spread near 50 ns gives a coherent frequency of roughly 20 MHz. An outdoor urban channel with 1 microsecond of delay spread gives roughly 1 MHz. A longer echo therefore narrows the band over which the channel stays flat.
One note on the name. Most textbooks call this quantity coherence bandwidth rather than coherent frequency, and a search on the second term returns far less than a search on the first. The quantity is the same either way.
Coherent channel, and why the two numbers matter
The introduction names coherent channel alongside the other two terms without defining it. It follows the same rule as the rest. A channel counts as coherent over a span of time and over a span of frequency, and those two spans are the numbers derived above.
The practical use is channel estimation. A receiver measures the channel from known pilot symbols, then applies that one measurement to the data around them. The measurement stays usable for about one coherent time and across about one coherent frequency, so the two numbers set how often pilots have to repeat.
That is why a fast moving user costs more overhead than a stationary one. The coherent time shrinks, the pilots have to repeat sooner, and the share of the transmission spent on pilots rather than on data rises. The same argument in the frequency direction sets how far apart pilot subcarriers can sit.
The two spans also decide whether a channel is called flat or frequency selective. A signal narrower than the coherent frequency fades as a whole, while one wider than it fades unevenly across its own band. The Channel Model page works through what that does to the received signal, and Channel Estimation covers the measurement itself.