# Understanding why spectral flatness cannot be computed

I have a real signal recorded 512Hz for 1s. After resampling the signal at 256Hz, I would like to compute its spectral flatness. To do so, I start by taking its discrete Fourier transform with fft = scipy.fft.rfft(x) and then I compute the power spectrum with pw = np.abs(fft). When I compute the geometric mean of pw an error is raised because that the last coefficient of fft is equal to 0. I am trying to understand why.

This last coefficient returned by fftpack.rfft is defined as: \begin{align*} \mathrm{Re}\big( y(n/2)\big) & = {} \mathrm{Re}\Big( \sum_{k=0}^{n-1} x_k e^{-jk\pi}\Big) \\[2mm] & = x_0 - x_1 + x_2 - x_3 + \ldots \end{align*} where $(x_k)_{0 \leq k \leq n-1}$ is my resampled signal. A quick check that this quantity is close to 0 : np.sum([x[k]*((-1.) ** (k % 2)) for k in range(x.size)]) gives me 5.68e-14, which is not 0 but quite close.

What does that say about my signal ? Usually the spectral flatness is defined without assuming that some coefficients in pw could be zero. So, am I doing something wrong (or not doing something I should do) ?

Edit : is there a convention which consists in saying that the flatness of the signal is 0 whenever a single "coefficient" returned by rfft is zero ?

• Wikipedia: "Note that a single (or more) empty bin yields a flatness of 0, so this measure is most useful when bins are generally not empty." You can just drop the last coefficient, though, which is Nyquist frequency. – endolith Sep 26 '17 at 13:31