What is "average recieved signal power over bandwidth"?
First to your question: The term "Average received signal power over bandwidth" is the power spectral density (PSD), usually in unit dBm/Hz or W/Hz. It cannot be used directly to calculate channel capacity, it's prime use is to calculate interference.

For discussion of channel capacity, it is important we are not dealing with "distortion" (which would be counteracted by pre-distortion and line coding), but with additional noise. There are different sources of noise with different spectra, but for discussion of channel coding it is sufficient to look at  additional white gaussian noise (AWGN), a memoryless stochastic process totally uncorrelated to your signal.

AWGN has in infinite bandwidth, but limited radiant intensity. This opens the door to reduction of total received noise power while keeping the full total received signal power, thus improving SNR.


Your signal will be received from a certain direction. By using directive antennae, you reduce the solid angle for noise while retaining all of the signal (antenna gain) 
Your signal will only occupy a limited bandwidth. By filtering, you reduce the total received noise power while retaining all of the received signal power. Noise spectral density over the "bandwidth of interest" will determine the noise power
Some processes/components in your receiver will add more noise than they amplify your signal. This is expressed as noise figure of this process/component.
Noise spectral density depends on temperature, cooling lowers the density and thus also the total power.
Some tricks allow you to reduce your signals occupied bandwidth while retaining its power, making bandwidth filtering even more efficient (spread-spectrum processing gain).


At each reference point in your receiver, you are able to divide total signal by noise power and tell a signal to noise ratio (SNR), typically in decibels. Important for channel capacity is the SNR before channel decoding. Now Shannon comes into play.

He states that information can be transferred as long as the SNR (in dB) is \$&gt;-\infty\$. A common misunderstanding is that SNR > 0 is required, but this is just the point where detection and demodulation of the signal become very easy. Detection is harder for SNR &lt; 0 but still any deviation from a pure stochastic process is detectable. The reasoning that information transfer with arbitrary low bit error rate is possible for every SNR is quite surprising and makes Shannons original article an interesting read.

The rate at which information can be transferred is however limited and Shannon is able to proove an upper bound for this rate. It depends on occupied bandwidth, which is related to maximum slew rate (think of Fourier decomposition) and minimum temporal spacing of symbols, and the SNR, requiring a minimum symbol distinctness and limiting bits per symbol.

The Shannon-Limit is a theoretical upper bound, channel codes with near maximum performance are still an active field of research. LPDC-Codes and Turbo-Codes are the best known solutions.