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I come from Maths background, but recently I encountered problem where a channel is described as "flat fading channel". I would like to make some conclusions based on that fact. I suspect that for radio engineers this is self explanatory, but not for me. Do we prefer a channel to be flat fading vs selective fading? Can we even make the distinction here? If no what are the pros and cons of each?

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The names stem from the fact that communications engineers tend to think in the frequency domain.

A flat fading channel is one where the entire signal is scaled by the fading, i.e. the signal is multiplied by a gain $a(t)$ which (usually) varies more slowly than the bandwidth of the channel.

A selectively fading channel is one where some time-varying, frequency-selective process is at work. As an example of a baseband channel, this would be as if you're talking on the phone, and the frequencies from 1000Hz to 1200Hz were being randomly notched out and added back in.

Flat fading is easier to deal with, because the whole signal is unmolested other than being changed in amplitude.

Frequency selective fading is much harder to deal with (although using OFDM would make it easier), because the impulse response of the channel is constantly changing, so not only is the signal distorted, but the equalizer that you'd need to reassemble it is constantly changing.

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