# Difference in mathematical formula between signal to noise ratio (SNR) and signal to noise and distortion ratio (SINAD)

I am trying to get a grip on mathematical meaning of signal to noise ratio (SNR) and signal to noise and distortion ratio (SINAD). What I found out is that in terms of power:

$$SNR=\frac{P_{signal}}{P_{noise}}$$

and

$$SINAD=\frac{P_{signal}+P_{noise}+P_{distortion}}{P_{noise}+P_{distortion}}$$

This is fine, but I found an article here that suggest that in terms of decibels when dealing with amplitudes they are as follows:

$$SNR_{dB}=20\text{log}_{10}\bigg(\frac{S}{N}\bigg)$$

and

$$SINAD_{dB}=20\text{log}_{10}\bigg(\frac{S}{N+D}\bigg)$$

I can easily derive the SNR since power in terms of voltage and resistance is: $$P=\frac{V^2}{R}$$ So $$SNR_{dB}=10\text{log}_{10}\bigg(\frac{V^2_{signal}}{R}\frac{R}{V^2_{noise}}\bigg)$$

Hence:

$$SNR_{dB}=20\text{log}_{10}\bigg(\frac{V_{signal}}{V_{noise}}\bigg)$$

I can't however see how we can derive this for SINAD since we have the additional terms in the numerator. I have only just started to look at this so perhaps I am missing something crucial?

• Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Jun 20 at 13:40