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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:

In the "Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR" document, it says

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?

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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

2 Answers 2

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.

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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

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.

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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 at 23:41

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