Why are my frequency bins oscillating?

Question

I am working on a personal project that maps bass notes to colors in real-time. However, I'm encountering some issues with oscillations in my frequency bins.

I visualized my frequency bins to illustrate the issue:

• 30 Hz tone
• 45 Hz tone
• 75 Hz tone

Here is an outline of my signal processing chain:

1. Capture 2205 samples from my input device at a sample rate of 44.1 kHz (allowing me to identify frequencies down to 20 Hz)
2. Apply the Hann function to the samples
3. Calculate frequency bins from 20 Hz to 150 Hz in 1 Hert increments using the Goertzel algorithm

It seems like the oscillations occur at all frequencies, but become more noticeable the lower I go. I assume this is because given some period, the lower the frequency the fewer the cycles captured in that period. I had expected that to result in more leakage, but not necessarily oscillating values. Increasing the sampling period helps, but necessarily increases perceived latency (because I'm using samples further back in time).

Does my assumption about the sampling period seem correct? Are there ways to mitigate this oscillation without increasing my sampling period significantly?

30 Hz tone graphed from real data:

Answer 1

This is indeed spectral leakage and specifically the dependency of the leakage on phase. Here is an example of 30 Hz sine with with a random phase. These are 50 frames with a randomized phase.

The "flickering" is easy to see and it's also easy to see why it happens. At 2205 sample the window will include 1.5 periods are 30 Hz. For a phase of $\varphi = 0$ the extra half period will be entirely positive so you will see a very significant DC component. For a phase of $\varphi = -\pi/2$ the extra half period will be symmetric, so the DC component will be 0.

The easy fix is to simply make your frames larger. This way the "extra" fractional periods become a much smaller part of the overall signal and you get better frequency resolution as well since the main lobe of the hanning window becomes much narrower.

Below is the same example but with 4 times the frame size

Answer 2

The idea that you're looking for here is spectral leakage. As alluded to in the comments, the DFT bins will undergo different amounts of spectral leakage depending on the initial phase of the tone going into the DFT.

The DFT is defined for integer multiples of the fundamental frequency, which is $\frac{f_{s}}{N}$. If your tone frequency is not an integer multiple of the fundamental frequency, you will end up with spectral leakage, which will spread out your spectrum. There's not really a good way to predict how much spectral leakage a given input to the DFT will suffer, but if your tone is not an integer multiple of the fundamental frequency, you will suffer spectral leakage as there will be a discontinuity in the assumed periodicity of the DFT.

Here's a Matlab script to demo this:

fs = 44.1e3;
N = 2205;
f1 = 30;
f2 = 40;
t = (0:N-1)/fs;
c1 = cos(2pif1t);
c2 = cos(2pif1t + pi/2);
c3 = cos(2pif2t);
c4 = cos(2pif2t + pi/2);
w = ((0:N-1)-N/2)*fs/N;
win = hanning(2205).';
figure;
plot(w,fftshift(abs(fft(c1.*win))));
title('30 Hz Aperiodic No Phase Offset')
xlabel('Frequency (Hz)')
ylabel('FFT Magnitude')
axis([-3000 3000 -100 600])
figure;
plot(w,fftshift(abs(fft(c2.*win))));
title('30 Hz Aperiodic Phase Offset')
xlabel('Frequency (Hz)')
ylabel('FFT Magnitude')
axis([-3000 3000 -100 600])
figure;
plot(w,fftshift(abs(fft(c3.*win))));
title('40 Hz Periodic No Phase Offset')
xlabel('Frequency (Hz)')
ylabel('FFT Magnitude')
axis([-3000 3000 -100 600])
figure;
plot(w,fftshift(abs(fft(c4.*win))));
title('40 Hz Periodic Phase Offset')
xlabel('Frequency (Hz)')
ylabel('FFT Magnitude')
axis([-3000 3000 -100 600])