FFT and decibel scales

Question

If I take audio data on an iPhone (i.e. real) data, perform an FFT and then take the magnitudes (Re^2 + Im^2).

These vary from >0 to some large numbers, so I do 10log(n) to get it in dB.

This gives me outputs that are negative (for the inputs that were < 1) to positive.

But the examples I've seen for this (and also drawing the spectrum in Sonic Visualiser) always have positive spectrums when measured in dB.

So what have I missed?!

On a wider note, as I understand it decibels are a ratio, so in this context when turning the FFT magnitudes into dB, what are they a ratio to?

Answer 1

The simple answer is that, for the most part, you can add an arbitrary number to the dB value to make the values all positive, or all negative, or whatever you prefer. With an uncalibrated microphone, like on the iPhone, this is all that makes sense anyway, since all you know are relative values.

For a more advanced technical approach, using a calibrated microphone, you could reference everything using dB (SPL), as a reasonable standard, but this is a hassle, and not meaningful in your use case anyway.

Rationale:
The main reason that shifting by an arbitrary amount is that the log doesn't report the units of measurement. For example, even if you know the input amplitude is 0.1 Pascal, it's completely valid to say this is 100 milliPascal, where you'd be taking the log of 100 rather than 0.1 (so the log values are either 2 or -1). Both are completely valid and the choice is entirely arbitrary. When comparing to a standard reference, as in dB SPL, note that it's done as a ratio, log(P/P_ref), removing the impact of changing the units.