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pythonfftdftspectrum

Maximum value of a direct fourier transform

I have a list of samples of a wave with all values between -1 and +1. Those values have been read from a music file. I will now,

  1. apply the direct fourier transform, (scipy.fftpack.rfft)
  2. normalize the values by dividing them by the square root of the number of samples,
  3. calculate the power for each item in the list. (sqrt(real^2 + imag^2))

What are the maximum values I can expect to be in this list after all of this? I would have expected the maximum power to be 1, as the maximum amplitude in the input data is also 1. However, this is only the case for a simple sine wave. As soon as I start doing this with real music, I get higher values.

How would I "normalize" the power to get values between 0and 1? Is it even possible to find out the maximum value? If not, what is the best practice to scale the results?

Solution

  • If you look at a DC signal 1 1 1 1, its DFT is 4 0 0 0.

    To normalize that back into [0,1], you need to divide by 4, i.e. the number of points.