# Combining simultaneous measurements of the same signal for better SNR?

I have multiple data sets of the same signal measured using the same hardware and I want to know what sort of techniques I can use to improve the SNR.

I was thinking if I adjust and ensure they have the same phase, i could coherently add them, but I didn't really see an improvement in SNR. I'm guessing the reason for this is because the noise gets added together as well.

It makes me wonder two things:

• What are some valid ways to combine like signals, to improve SNR

• Is there any way to exploit the similarity in signals to reduce noise?

Thank you

• Coherent addition should reduce noise, provided the noise is iid across the datasets and is independent of the signal. eg. with additive Gaussian distributed noise, coherent addition of N observations should reduce noise variance by a factor of N. There are surely many ways to exploit similarity in signals, but you'll have to provide more details about your system and measurement model. – Atul Ingle Oct 5 '17 at 15:03
• I rather say what matters is noise model. – AlexTP Oct 5 '17 at 16:36

Coherent Gain happens when you have copies of the same signal but different noise. If the noise is independent in each copy, the gain should be $10log N$ dB where $N$ is the number of copies. For a well crafted measurement system, additive electronic noise should be independent. Things like ambient noise will have some correlation provided a reasonable distance between sensors and $5logN$ is a reasonable lower expected gain.

If you don’t see at least $5 log N$ dB, then there is probably something wrong. Coherent Gain works. If it didn’t, many here would not have jobs.

You might try cross correlating pairs of your copies as a diagnostic. Without knowing your data, I cant say with certainty that you should see a well defined peak but the absence of a well defined peak would be a cause for concern

Each of $C$ different collections, with signal mean $\mu$ and zero-mean iid noise of variance $\tau$ has SNR (in power) $\frac {\mu^2} {\tau}$

By linearity of expectation, the summation of these collections will have a signal level $C\mu$ and variance $C\tau$, with SNR $\frac {(C\mu)^2} {C\tau}$, yielding a power gain of $C$.

Converting power to dB, the gain of C similar signals is $10 log_{10}(C) \approx 3$ dB per doubling the number of collectors.

If your signals are not same-level with iid noise, then you might want to check out Maximal Ratio Combining, or Wiener Filters.