While working on this problem, I started to have doubts that my initial definition of

$$SINAD = 10 \log_{10} \left( \frac{p_f} {\sum_i{(p_i)} - p_0 - p_f} \right)dB$$

is correct. In this equation, $p_x$ is the power the FFT bin at frequency $x$, $p_f$ is the power of the frequency bin containing the signal frequency $f$ and $p_0$ is the DC component. The sum over $i$ accumulates all frequency components, before removing the DC component $p_0$ and the the signal frequency $p_f$.

More specifically, I am unsure about the $\sum_i(p_i)$ part, which I interpreted from the Wikipedia description

The ratio of (a) the power of original modulating audio signal, i.e., from a modulated radio frequency carrier to (b) the residual audio power, i.e., noise-plus-distortion powers remaining after the original modulating audio signal is removed. With this definition, it is possible to have a SINAD level less than one.

Comparing with the equation, the "original modulating audio signal" is at frequency $f$, which is accounted for in the $p_f$ term from the FFT. The $p_0$ term I got from the following paper, which says to remove the DC component:

Signal-to-Noise-and Distortion (SINAD, or S/(N + D) is the ratio of the rms signal amplitude to the mean value of the root-sum-square (rss) of all other spectral components, including harmonics, but excluding dc

Looking at these definitions, I can think of another possible definitions of SINAD, namely

$$SINAD = 10 \log_{10} \left( \frac{p_f} { \sqrt{ \sum_i{(p_i^2)} } - p_0 - p_f} \right)dB$$

which uses the RSS (root-sum-square) of the noise and distortion bins of the FFT result. But then, what exactly is meant by "mean value" in that document?

• Could you elaborate more on how you've come up with the sum over $i$ and what $p_f$ and $p_0$ are? – Phonon Feb 15 '13 at 18:31
• @Phonon, I tried to clarify. Let me know if it needs some more. Thanks. – FriendFX Feb 17 '13 at 23:05
• @FriendFX I have been struggling with the same questions and uncertainties about SINAD, one thing that I identified is the second SINAD definition is equivalent if your FFT output is a Voltage spectrum, not the power spectrum. – Mark Omo Feb 10 at 18:27

If SINAD can be determined from only expected value and variance then it is possible to determine how SINAD transforms. Variance $\sigma^2$ is preserved whereas expected value $\mu$ grows as $\sqrt{N}$ where N is the sampling set size. Noise + distorsion is assumed to have variance $\sigma^2$.

Thus the SINAD value would then be determined to become $${\mathrm {SINAD}}={\frac {P_{{\mathrm {signal}}}+P_{{\mathrm {noise}}}+P_{{\mathrm {distortion}}}}{P_{{\mathrm {noise}}}+P_{{\mathrm {distortion}}}}} = {\frac {N |\mu|^2+\sigma^2}{\sigma^2}} = N \cdot SNR + 1$$

I can explain this in more detail if needed.

• Interesting approach. Could you add how this relates to the bins of the FFT calculated from a real-world A/D conversion of a sine signal? – FriendFX Feb 17 '15 at 23:41

what exactly is meant by "mean value" in that document?

In the time domain, SINAD is calculated as a ratio of the RMS value of the signal to the RMS value of the noise + distortion, so I believe the mean value in the context of the AD document refers to the mean in the RMS measurement. Doing the calculation in the frequency domain conceals the mean operation because the magnitude of the DFT coefficients are already conditioned to be proportional to the time domain RMS value. RMS values are summed as squares and then the square root is taken of the result to obtain a composite RMS value. The RSS achieves the necessary arithmetic operation.

• Could you post the correct equation for calculating SINAD from the FFT result (and the conditions like power/amplitude spectrum etc.)? The main reason for my question was that I couldn't find such an equation anywhere, only textual descriptions which I found rather difficult and error-prone to interpret. If I were to interpret your current description, my second equation seems to be the one to use. – FriendFX Feb 19 '13 at 3:11
• Go to this link: fhnw.ch/technik/ime/publikationen Download the paper ""How to use the FFT for signal and noise simulations and measurements". I'll try to follow up as soon as I have some free time. – user2718 Feb 20 '13 at 23:50

Look in your ADC datasheet, most of the time they provide a formula and even explain how to calulate it.

Mine says:

SINAD is the ratio of the power of the fundamental (PS) to the power of all the other spectral components including noise (PN) and distortion (PD), but excluding dc.

hence the formula is :

$$10 \log_{10} \left( \frac{P_S} {P_N + P_D} \right)$$

• I changed the formula to use the LaTeX display for clarity, hope I translated it correctly. Could you elaborate on how to calculate PS, PN and PD from the bins which are the result of the FFT calculation? – FriendFX Apr 30 '15 at 0:40
• Thanks for editing my answer. Well, in fact if your bins are correctly mapped to the components of the power spectrum, you can do it as follows: Ps: get the power of the signal at the fundamental frequency ( I believe you should know what your fundamental is) it should be easy. PN: PS + P(harmonics) - DC As for the PD i'm not really sure. – Kevin.hammet Apr 30 '15 at 9:05
• I think I already tried to explain this in more detail in my question (e.g. see that my question has your formula with PS, PN and PD substituted). What I really need is some equations which put what you just said (and which I have read multiple times in different variants across datasheets, Wikipedia, papers, etc.) into a mathematical form which can be applied to any FFT of an A/D converted sine wave. Maybe what is missing is the "if your bins are correctly mapped to the components of the power spectrum", but I don't know how to make sure it is correct. – FriendFX Apr 30 '15 at 23:48
• Okay it's like this: the first FFT bin coreesponds to DC at 0 Hz, the following bin is 1*Fs/Nfft, the thrid is 2*Fs/Nfft and so on... Where Fs is your sampling frequency and Nfft is the number of FFT points. – Kevin.hammet May 4 '15 at 6:21
• Okay, so which one of the equations in my question is correct? Or is it an entirely different one? As a side note, I already know how to find the bins of specific frequencies (or ranges thereof), so the main part of my question is about the correct summation of those bins in the context of the SINAD calculation. – FriendFX May 4 '15 at 10:58