I came across to Shannon’s formula for calculating channel capacity and I don’t quite understand why we calculate average signal power… “over bandwidth”. Why over that "domain of interest"?

Here is my interpretation why but I’m not sure if I understood it right.

Imagine we are sending some signal f(t) from point A to point B. We take FT (Fourier transform) of signal f(t) at point A, and we get ourselves all the sinusoid components of that signal. For every component of that signal in frequency domain we have particular value of amplitude. Now, I’m not sure if this can be done but here is my interpretation of “power over bandwidth”: “Inside frequency domain” we calculate avg. power of signal in such fashion that we calculate power of every component of signal and by doing some manipulations with all those calculated “powers” (sum of them equals power of signal in time domain?) we can calculate average power of the signal in frequency domain.

So we have avg. power of “clear/undistorted signal” of value for example 40[W] “over bandwidth” that interest us (bandwidth of our signal).

Signal traveling from point A to receiver side B gets distorted. At point B we take FT of that distorted signal. Now, maybe we have some components of signal that are added within our signal, so all those components represent “noise signal”. We “separate” noise components and our signal components (in our head). Since signal is distorted maybe some of the amplitudes of signal’s components are now changed, so we again calculate (in a same fashion as one above) average power of that components and get… maybe some different average power which for example might be less than one at point A.

So, based on this interpretation do we by that mean average signal power “over bandwidth” (“over bandwidth of something that interests us, over our signal bandwidth to see if something has changed”)?

  • \$\begingroup\$ I suggest that you split the "wall of text" 2nd paragraph into a few paragraphs just by adding a few empty lines. That would make things so much more readable. \$\endgroup\$ Aug 22 '17 at 11:59
  • \$\begingroup\$ @Bimpelrekkie Hope it now looks better! \$\endgroup\$
    – Krushe
    Aug 22 '17 at 12:03
  • \$\begingroup\$ Well, if you didn't specify a BW then you'd need to take into account "everything" so also noise over an infinite BW. That makes things complicated so in order to "limit our view" we define a BW and look at the power in that BW. This does not so much concern itself with transmitting or distorting a signal as in your example, that's not needed. A certain (average) power in a certain BW is just a way of describing the power of a signal in a certain frequency band. This also helps describing the S/N ratio which needs to be good enough to extract the data. \$\endgroup\$ Aug 22 '17 at 12:03
  • \$\begingroup\$ Are you basically asking for an explanation of the shannon formula? \$\endgroup\$
    – Andy aka
    Aug 22 '17 at 12:06
  • \$\begingroup\$ @Bimpelrekkie Ok. So, if I understood you right that makes things much easier in a sense: if we define same BW at the receive end, then no components other than "signal" ones we don't need to get into account or should I say "interpret". We limit ourselves to the spectrum of a signal and see what happens with frequency components. Now, this might sound stupid but what is the point of "filters" at the receiver end if we can just manipulate signal in a such sense that we decrease/increase amplitudes of some components and get ourselves "orginal signal". PS: I'm newbie. \$\endgroup\$
    – Krushe
    Aug 22 '17 at 12:13

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 \$>-\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 < 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.

  • \$\begingroup\$ Nice! I have a couple of questions: part 1: „It cannot be used directly to calculate channel capacity, it's prime use is to calculate interference.“ how does it tells me if there was some interference? example: if I sent 40[W] signal and recieved 40[W] signal power but signal was distorted, how would it tell me if there was interference? \$\endgroup\$
    – Krushe
    Aug 24 '17 at 11:01
  • \$\begingroup\$ part 2: what does „signal is uncorellated with noise signal“ tells me about signal and that noise in some simpler terms? what is generalization of idea of „correlation“? what are the benefits if two signals are uncorellated if they somehow „meet“ with one another during transmission? \$\endgroup\$
    – Krushe
    Aug 24 '17 at 11:01
  • \$\begingroup\$ part 3: „AWGN has limited radiant intensity“. so if I „direct a beam of radiation“ in some „V shape“ away from antena (directional antena) then if I lower that „V angle“, AWGN's radiant intensity will be lower (they are directly proportional?). that will imply that Noise power will be lower and so S/N is bigger. so if we lower „V angle“ that means that „beam“ will be „less exposed to outside noises (AWGN)“ aka „AWGN has limited radiant intensity“ \$\endgroup\$
    – Krushe
    Aug 24 '17 at 11:01
  • \$\begingroup\$ part 4: „By filtering, you reduce the total received noise power while retaining all of the received signal power“ when you recieve signal with some frequency components then what are you trying to filter there in order to reduce total recieved noise power? imagine you've done Fourier deomposition of signal at recieve end. what are you trying to filter there? \$\endgroup\$
    – Krushe
    Aug 24 '17 at 11:01
  • \$\begingroup\$ @Krushe EMC regulation limits the allowed power spectral density (to put it short). This is because it is not know a priori which part of the spectrum causes the interference. "Density" and "total power" are mathematically related (total is the integral of density over bandwidth), but for calculation of SNR, you need total power. Note: Interference is the interaction of your mobile with your TV set, distortion is about the non-linearities between the TV station and your TV set and noise is what your TV set receives from cosmic background etc. Shannon capacity is about noise, not distortion. \$\endgroup\$
    – Andreas
    Aug 24 '17 at 12:37

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