Perfect symmetry

When working on the last series of posts, I stumbled on a signal that caused an FFT analysis to look a little strange to me. This post is about that strangeness.

If I make a sine wave that sits perfectly on an FFT bin, and I do an FFT of it, the noise floor that I see is the result a lack of precision of the calculations that were used to make the signal. An example of this is shown in Figure 1.

Figure 1. Two plots of the same analysis. The top plot has a logarithmic frequency scale and a range of 20 Hz to 20 kHz. The bottom plot is on a linear frequency scale, and has a range of 0 Hz to 2 kHz.

As can be seen there, the noise floor in the fit is typically at least 300 dB down from the signal level. This means that if the signal has a peak amplitude of 1, then each bin in my FFT has a peak amplitude of less than 0.000 000 000 000 001, which is very, very, low.

If I dither and quantise the sine wave with, say, a 24-bit LPCM precision, then the result would be different, as shown in Figure 2.

Figure 2. Two plots of the same analysis. The top plot has a logarithmic frequency scale and a range of 20 Hz to 20 kHz. The bottom plot is on a linear frequency scale, and has a range of 0 Hz to 2 kHz.

Now the noise floor seen in the FFT analysis is the noise that is intentionally generated as dither to randomise the quantisation error when converting to LPCM.

However, what happens if the signal is quantised but not dithered? Then the result looks like Figure 3.

Figure 3.

This is interesting because, starting the first bin, every second bin has nothing in it, so on a decibel scale, the value is -∞ dB. Why does this happen?

The short answer is symmetry. By quantising the sine wave, I made it perfectly symmetrical.

This removes the DC content, since the positive-going portion of the waveform is identical to the negative-going portion. Therefore there’s nothing at 0 Hz (which is DC) or any of its “harmonics” (at least in the world of FFT bins…).

There is content of some kind in the other bins because our sine wave is not perfectly sinusoidal. All those steps that I put in it are an error that generates information at frequency centres other than the sine tone’s itself.

So, if you do an FFT on a sinusoidal signal and you see a result where half of the bins have nothing in them, one possible reason is that you’re dealing with a perfectly symmetrical signal.