Suppose I have y = conv (x,h) + n = s +n, where h is channel impulse response with length L and unnormalized. How do I define SNR? Is it SNR = power of x/power of n or SNR = power of s/power of n?
I was wondering this same question a few years ago. I came to a conclusion that the definition depends on your application. In my field of science which is room acoustics, the very first part of the impulse response, say 5 ms after the direct sound, is considered as signal. Then your SNR is "power of conv(x,h(0:5ms)) / (power of noise)". Thus, the rest of the impulse response, i.e., from 5 ms to infinity is considered as convolutive noise and is not presented in the traditional SNR figure. Instead the convolutive noise is presented by another figure called the Direct-to-reverberant ratio (DRR) and it is defined as "power of h(0:5ms) / power of h(5ms:infinity)". Sometimes, in room acoustics, also a parameter called the reverberation time is used as the indication of how much convolutive noise there is, as in
Champagne, B. Bedard, S. ; Stephenne, A "Performance of time-delay estimation in the presence of room reverberation" IEEE Transactions on Speech and Audio Processing, Volume:4 , Issue: 2 , Pages:148 - 152
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=486067
That is, the signal-to-noise ratio is defined by what is considered noise and what is considered signal in your application.
Normally, the SNR would be characterized in the same manner as the strength of the S and N signals (typically in some sort of dB). Obviously, if the values of S and N are not constant over a given time interval, neither is the SNR. Whether this has perceptual significance depends on how quickly things happen. For example, a very brief duration (e.g.,
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