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If I have two SNR measurements (i.e SNR1 = 15, SNR2 = 20), how do I calculate a combined SNR?

In my example, would this be SNR_Total = 35?

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closed as unclear what you're asking by Marcus Müller, MBaz, Dilip Sarwate, Stanley Pawlukiewicz, lennon310 Jun 7 at 16:17

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    $\begingroup$ What do you mean by combining? What is the context? A diversity reception system? (SIMO) $\endgroup$ – Damien Jun 6 at 15:44
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    $\begingroup$ how do these measurements relate? Are they observing the same thing? $\endgroup$ – Marcus Müller Jun 6 at 16:11
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You can definitely not add the SNRs in logarithmic scale. This would be equivalent to multiplying the SNRs in linear scale and there is no good reason to do so.

As the others have pointed out, it depends a bit what you do. Let me use an example to explain it. Let's say we take two measurements of a scalar variable $x$ of interest, given by $y_1 = a_1 x + w_1$, $y_2 = a_2 x + w_2$, where $w_i$ are i.i.d. additive noise samples with zero mean and variance $P$ (more or less without loss of generality) and $a_i \in \mathbb{R}$ are gain factors to model different SNR settings. For instance, $a_1 = 10$ corresponds to an SNR of $a_1^2 / 1 = 100 = 20 {\rm dB}$ and $a_2 = 5.6$ corresponds to an SNR of about 15 dB. Before you can make any calculations, it is advisable to convert your SNRs to linear scale, i.e., 20 dB $\rightarrow$ 100, 15 dB $\rightarrow$ 31.6.

The question is, how do you combine the two measurements $y_1$ into one? There are several strategies:

  • Selection combining: Pick the one with the better SNR. Resulting SNR = max(SNR1,SNR2) = 100 = 20dB.
  • Equal gain combining: add the two quantities, i.e., $y = y_1 + y_2$. Resulting SNR = $(a_1+a_2)^2/(P+P) = \frac 12 (\sqrt{{\rm SNR1}}+\sqrt{{\rm SNR2}})^2 = 122 = 20.9$ dB.
  • Maximum ratio combining: $y = a_1 y_1 + a_2 y_2$. Resulting SNR = $(a_1^2+a_2^2)^2/(a_1^2 P + a_2^2 P) = a_1^2/P + a_2^2/P = {\rm SNR1} + {\rm SNR2} = 131.6 = 21.2$ dB.

So, if you do MRC, you can add the SNRs but only in linear scale, not in log scale. Also, MRC gives you provably the best SNR out of all linear combining strategies.

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As already discussed by @Florian and commentators, decibel level addition cannot be done without care. For pratical use, it is common to distinguish coherent from uncoherent sources. The following links provide computations, and graphical explanations, on how signals add:

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