I'm facing a gap in my understanding of when a signal is a "voltage" signal and when it is a "power" signal that I've always managed to avoid resolving until now... First what I think I understand, then my question after the horizontal rule:
I understand SNR to be the ratio of average signal power to RMS noise power: $$ SNR=\frac{S_P}{N_P} $$
or in dB: $$ SNR_{dB}=10\cdot\log(S_P)-10\cdot\log(N_P) $$
When calculating SNR based on voltages, power is proportional to voltage squared ($P=V^2/R$) and the Rs drop out in the ratio so we get: $$ SNR=\frac{{S_V}^2/R}{{N_V}^2/R}=\left(\frac{{S_V}}{{N_V}}\right)^2 $$$$ SNR_{dB}=20\cdot\log(S_V)-20\cdot\log(N_V) $$
With the familiar 10log for power, 20log for voltage relation doing the squaring for voltage.
I have a radio signal that I receive at an antenna. For simplicity, I assume I have a perfect receiver and noiseless amplifier so that my received signal power is $P_{Rx}$ and noise is only Johnson thermal AWGN: $N_T=k_B\cdot T\cdot B$, where $k_B$ is Boltzman's constant, T is temperature and B is bandwidth-- none of which really figure in after this.
I believe my SNR at this point to be: $$ SNR_{RF}=\frac{P_{Rx}}{N_T} $$
Now I sample, and this is where I start to have questions.
Ignoring quantization noise and such, I believe the analog-digital conversion process results in voltage signals-- that is, the sample values are measurements of the voltage across a resistor ladder or some such meaning the SNR of the sampled signal follows the voltage law: $$ SNR_{ADC}=\left(\frac{S_{ADC}}{N_{ADC}}\right)^2 $$
Setting aside practical implementation losses, I believe $SNR_{ADC}=SNR_{RF}$.
Where I start to be less sure is when I start operating on the sampled signals. Let's say I multiply the sampled signal by a delayed version of itself. Ok, the noise terms get more complicated because I'm taking the product of two independent random variables with non-zero mean but, more fundamentally, is the result a "voltage" or a "power"? Is there a physical explanation that will help me understand this?
That is: in order to maintain consistency among my SNR estimates, is this a 10log or 20log calculation?
By a pure units analysis, I should have voltage-squared which implies power-- but these are still ADC levels. It would also seem odd to say that my signal is voltage-cubed if I multiply by two delayed copies...