# How exactly is analog Signal to Noise Ration (SNR) defined?

Say, my ADC reads noise signal only, not sure how to define a threshold. Let's say my noise is sitting at 0V (zero mean), oscillating between +5 mV to -5 mV, so Vnoise = 10 mVpp. I'm looking for 20 dB SNR with target present. Does it occur when ADC reads either 10^(20/20)(5 mV) = 50 mV or 10^(20/20)(-5 mV) = -50 mV OR when it's 10^(20/20)(10 mV) = 100 mV or 10^(20/20)(-10 mV) = -100 mV?

Now, a different scenario, what if my noise oscillates between -5 mV to +5 mV but there's a noise spike at +7 mV and -6 mV (just 2 values out of 10000 samples collected), do I change my threshold based on maximum noise level, not mean noise level?

• This answer may help you to understand how to quantify (define) SNR for an ADC signal: dsp.stackexchange.com/questions/40259/… Nov 19 '19 at 23:12
• What do you mean by "threshold"? Maybe a block diagram of your measurement and what you are referring to could help. Nov 20 '19 at 2:35
• Sorry if it wasn't clear, by "threshold" I meant noise and clutter with no target present. If I were to capture data with no target present for a long time, I want to set it it as my "base" clutter+noise and compare the target against it. So if my target is 20 dB above that "base" threshold, only then I count this as a target. Nov 20 '19 at 15:32

It is not clear from your question if you are referring to ADC measurements using volts (typical would be counts) or if you are referring to the ADC input. However your question on SNR can apply to either case. Your reference to a "threshold" is unclear.

The signal to noise ratio is typically referred to the signal power, which occupies a bandwidth, as a ratio to the noise power over that same bandwidth. Any other noise we have the opportunity to filter out in our subsequent processing.

Also, the noise power with reference to a voltage reading would proportional to the square of the rms voltage (1 sigma) not the peak-peak (which could be 6 sigma or more).

Further, this rms voltage would be representative of the (square root of the) power across the entire measurement bandwidth (in terms of an analog measurement), or the digital Nyquist bandwidth in terms of an ADC output (and again here I would use rms counts, where counts are increments of the lsb of the ADC).

So to compute SNR, if you are using rms quantities, it would be 20 Log the ratio of the rms signal to the rms noise in band. If you are using rms squared (power or variance) quantities, it would be 10 log the ratio of the variance of the signal to the variance of the noise is band (the portion of the noise that occupies the same bandwidth of the signal).

• Sorry if it wasn't clear, by "threshold" I meant noise and clutter with no target present. If I were to capture data with no target present for a long time, I want to set it it as my "base" clutter+noise and compare the noise+clutter+target against it. So if signal with target present is 20 dB above that "base" threshold, only then I count this as a target. Nov 20 '19 at 15:46
• I see sounds like a radar and your are doing the probability of false alarm versus probability of detection trade? Did I answer your SNR question? Nov 20 '19 at 20:23
• To answer your second question the detection above threshold would be based on a single sample occurrence I believe- Typically I’ve seen this as an “N-sigma” problem— you determine sigma with the long term average with no successful detections (throw away positive detections from polluting the noise average) and then set your threshold N times sigma above that based on the tolerable false alarm rate noting you are reducing the probability of detection as a trade (sigma is a magnitude quantity not power). The probability of false alarm is the tail PDF under N sigma. Nov 20 '19 at 20:28
• Yes, you answered it. I found the same answer in a radar textbook. Vrms ratio is the same as Vpeak ratio so I'll be using the latter because I need to count it an event when signal's peak over noise peak passes 20 dB, not when mean signal over mean noise is equal 20 dB. Nov 21 '19 at 21:14
• Super- you should then check off the answer so it in not in the unanswered question pile. Nov 21 '19 at 21:15