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Suppose I have a sequence of identical symbols carrying the same information at a sample rate equal to the symbol rate, why do we always talk about averaging them to average out noise, why not just adding them all together without averaging them? I suppose the improvement in SNR is going to be the same assuming that noise is uncorrelated from one sample or symbol to another, is this related to the limited dynamic range in DSP?

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Yes, they both give you the same SNR gain. I suppose there might be a dynamic range issue, as you say, if a LOT of samples are added, but I think the main reason that the literature speaks of "averaging" instead of "adding" is because for systems that allow a variable integration rate, we can be assured that other parts of the system work the same, like AGC.

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Adding $N$ independent random variables (finite mean $\mu$, finite variance $\sigma^2$ produces a result $\Sigma X$ with mean $N\mu$ and variance $N\sigma^2$. Averaging these same $N$ random variables produces a result $\bar{X}$ with mean $\ mu$ and variance $\frac{\sigma^2}{{N}}$. The average is thus a much more compact result -- very nearly equal to the mean $\mu$ when $N$ is large, whereas the sum will vary quite considerably about the large mean $N\mu$.

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