FFT in Maxima

Maxima has a basic fast Fourier transform package called fft. Here's are really basic introduction to using it on music data. We define a basic sine wave form with frequency 440 Hz and amplitude 40 as our input and then run fft on it. The interpretation of the results as frequency depends on coordinating together the sample rate, the total number of samples (N), and the sample index (i).

wave(amp, freq, t) := amp*sin(2*float(%pi)*freq*t); sampleRate: 44100.0$ timeStep: 1/sampleRate$ N: 2^14; inputFFT: makelist(wave(40, 440, t*timeStep), t, 1, N)$ transformed: fft(inputFFT)$ wxplot2d([discrete, makelist(i*sampleRate/N, i, 1, 200), firstn(map(lambda([u], abs(u)), transformed), 200)]);

Fig 1. Looks like about 440 Hz

The thing that most threw me for a loop in trying to make sense of the various sources I read was that the x axis needs to be inferred in order to be comprehensible. I wasn't really clear on what that x-axis was. If you dig into the details of the output numbers, you will find the 164th value, has the maximum value. This implies 164 * 44100 / 2^14 = 441 Hz. So, a bit of error has crept into our interpretation of the result because of the discrete nature of the intermediate result. What this means in theory is that there is a theoretical peak that "should" exist somewhat before the 164th index, but because we aren't working with continuous data, we can't see that peak (at the ~163.47th index 😜--or maybe we should be considering it at the 164.47 "index" and subtract 1, interpreting it as zero based, since the peak is probably between the 164th and 165th index).

If we change the function to have two wave forms at different frequencies (340 Hz and 440 Hz), we get both of these in the result:

Fig. 2. wave(20, 340, t*timeStep) + wave(40, 440, t*timeStep)

Next step, in search of the true peak.