STFT Window Size is 2048, Why is the output 1025?

Question

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]),
                       dtype=dtype,
                       order='F')

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?

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 FFT of Pure Real Sequences:

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 characterised.