I feel like I am having a brainfart over here and can't seem to remember what's going on with STFT outputs.

Consider these two lines of code from the Python library Librosa:

# Window the time series.
y_frames = util.frame(y, frame_length=n_fft, hop_length=hop_length)

# Pre-allocate the STFT matrix
stft_matrix = np.empty((int(1 + n_fft // 2), y_frames.shape[1]),

In the first line, we have a function that creates a matrix with a window length of n_fft (2048).

Then in the next line, we pre-allocate our STFT, but our window length is now 1025 instead of 1024 as dictated by the 1+n_fft // 2? Where does this extra frequency bin come from? Why is not just 1024?


1 Answer 1


The continuous Fourier transform possesses symmetries when computed on real signals (Hermitian symmetry). The discrete version, an FFT (of even length) possesses a slighty twisted symmetry.

The DC coefficient ($F(0)$) is real, as well as the Nyquist one ($F(N/2)$). In between, you get $\frac{2048-2}{2}=1023$ "complex" coefficients, "duplicated" in positive and negative frequencies.

So for real signal, each STFT frame can be represented by $1023+2$ frequency bins, the remaining 1023 being recovered by Hermitian symmetry.

You can get complementary information at waybackmachine version of Pure Real Sequences broken FFT of :

As a result, we can see that if N is even, both $F(0)$ and $F(N/2)$ must be real. Given these two values and the complex values $F(1)...F(N/2-1)$, (I.E. N numbers in total) the sequence is completely characterized.

  • $\begingroup$ Hello, the link is broken, anybody has complementary information for this? Thanks $\endgroup$
    – Tom
    Jan 7, 2023 at 18:12
  • 1
    $\begingroup$ You can get a look using Waybackmachine. web.archive.org/web/20180202152348/http://… I'll try to find another link $\endgroup$ Jan 8, 2023 at 8:55
  • 1
    $\begingroup$ Thanks, I did not know that trick. $\endgroup$
    – Tom
    Jan 12, 2023 at 13:17

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