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I have a two part related question. Channel gain is often being considered the side-information that the Rx and Tx usually know. Why is channel gain such an important side-information? And second, what exactly is channel gain?

I am reading many definitions of it, but I don't have a solid understanding. I've been told that it's a measurement of how well the power works given all the different impediments, such as hardware imperfections, noise, etc. This sounds a lot like SNR. Are they the same or is there a distinction between them?

Moreover, I would understand of the BER of the channel (w/o any error correction) is important side-information. Because that allows for correct arguments to be used during encoding and decoding.

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If you transmit signal $s(t)$ and receive $$r(t)=gs(t)+n(t),$$ where $n(t)$ is noise, then $g$ is the channel gain. Note that, when the signals are low-pass equivalents (that is, complex envelopes), then in general $g$ is complex.

The channel gain can also be written for a discrete channel. If you matched-filter $r(t)$ and sample the result at the symbol times, then you'll obtain a sequence of numbers $$r_i = g s_i +n_i,$$ where $s_i$ are the transmitted pulse amplitudes and again $g$ is the channel gain. Note that the channel gain has two main effects:

  1. It reduces the SNR, because when $|g|<1$, the received signal loses power compared to the noise.

  2. It changes the phase of the signal.

In other words, it scales and rotates the transmitted constellation. It is common to assume that the receiver knows $g$, because it can be learned by using a training sequence. If the receiver knows $g$, then it can compensate the constellation rotation. In the case of a wireless channel with diversity, the receiver can align and add coherently the different received signals (see maximal ratio combining).

It is less common for the transmitter to know the channel gain, but if it does, it can do beamforming and increase the SNR. Basically the idea is to transmit a signal that is "aligned" with the channel's phase $\angle g$.

You're correct that the BER (along with maybe other channel statistics) is useful side information. Wireless systems such as Wi-Fi include a control channel where the receiver can provide these statistics to the transmitter, so that it can select an appropriate modulation, code rate, and symbol rate.

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