There’s this nice calculation in a paper of Shannon’s on optimal codes for Gaussian channels which essentially provides a “back of the envelope” way to understand how noise that is correlated with the signal can affect the capacity. I used this as a geometric intuition in my information theory class this semester, but when I described it to other folks I know in the field, they said they hadn’t really thought of capacity in that way. Perhaps it’s all of the AVC-ing I did in grad school.
Suppose I want to communicate over an AWGN channel
where and satisfies a power constraint . The lazy calculation goes like this. For any particular message , the codeword is going to be i.i.d. , so with high probability it has length . The noise is independent and so with high probability, so is more-or-less orthogonal to with high probability and it has length with high probability. So we have the following right triangle:
Geometric picture of the AWGN channel
Looking at the figure, we can calculate using basic trigonometry:
which is the AWGN channel capacity.
We can do the same thing for rate-distortion (I learned this from Mukul Agarwal and Anant Sahai when they were working on their paper with Sanjoy Mitter). There we have Gaussian source with variance , distortion and a quantization vector . But now we have a different right triangle:
Geometry of the rate-distortion problem
Here the distortion is the “noise” but it’s dependent on the source . The “test channel” view says that the quantization is corrupted by independent (approximately orthogonal) noise to form the original source . Again, basic trigonometry shows us
Turning back to channel coding, what if we have some intermediate picture, where the noise slightly negatively correlated with the signal, so ? Then the cosine of the angle between and in the picture is and we have a general triangle like this:
Geometry for channels with correlated noise
Where we’ve calculated the length of using the law of cosines:
So now we just need to calculate again. The cosine is easy to find:
Then solving for the sine:
and applying our formula, for ,
If we plug in we get back the AWGN capacity and if we plug in we get the rate distortion function, but this formula gives the capacity for a range of correlated noise channels.
I like this geometric interpretation because it's easy to work with and I get a lot of intuition out of it, but your mileage may vary.