# Channel capacity with additive Gaussian noise with changing variance

As we all know the capacity for a AWGN channel with constant noise variance is just given by $$\frac{1}{2}\log(1+SNR)$$

What if the variance of the noise changes with the type (assuming the variance is known for any time index)? Consider the following discrete channel: $$y[n] = x[n] + w[n], \quad n=1,\dots,N$$ where $w[n]$ is the noise during the $n$-th symbol distributed as $\mathcal{N}(0,\sigma^2_n)$, with known $\{\sigma_n^2\}_{n=1}^N$.

• What is the capacity defined in such case?
• It should be related to some dynamic rate control, right?
• Could anyone give me some reference on related topic?
• I don't know the answer, but I suspect the capacity can be calculated with the same formula, with the smallest SNR in your sequence. – MBaz Mar 27 '15 at 20:56
• The standard capacity calculations are asymptotic results; we need to let $N$ increase without bound, which for your model means that we have to have complete knowledge of the future values of $\sigma^2_n$ for all integers $n$. – Dilip Sarwate Mar 28 '15 at 14:29
• @DilipSarwate Thanks for the reply. If we have the complete knowledge of the future values of $\sigma_n^2$ for all integers $n$, and the constant signal power $P$, how's the capacity defined? Or it's only meaningful for constant noise power? That's where I'm confusing. – Bo LI Mar 29 '15 at 5:24
• Search the literature for things like "arbitrarily varying channels" for the most general results. – Dilip Sarwate Mar 29 '15 at 13:29

I think you are looking for the information rate,

$$I(x_{1},x_{2},\cdots,x_{n}; y_{1}, y_{2}, \cdots, y_{n}) = \frac{1}{n} \sum_{i=1}^{n} I( x_{i};y_{i} )$$

when input achieves the capacity of the channel $x \sim N(0,P)$. Then,

$$C = \frac{1}{2n} \sum_{i=1}^{n} \log \left( 1 + \text{SNR}_{i} \right)$$

I think every time we use the channel we have a new random variable $y$ since we have a memoryless channel and $w$ changes. Then, we have a sequence of related random variables $\left\lbrace x_{i} \right\rbrace$ and $\left\lbrace y_{i} \right\rbrace$. With the information rate we compute the mutual information between this two sequences.