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I also see that for my data (audio data, real valued), np.fft.fft returns a 2 dimensional array of shape (number_of_frames, fft_length) containing complex numbers.

For np.fft.rfft returns a 2 dimensional array of shape (number_of_frames, ((fft_length/2) + 1)) containing complex numbers. I am led to believe that this only contains nonredundant FFT bins .

Can someone explain in more depth the difference between the commands and why the shape of the returned array is different. Thank you.

When the DFT is computed for purely real input, the output is Hermitian-symmetric, i.e. the negative frequency terms are just the complex conjugates of the corresponding positive-frequency terms, and the negative-frequency terms are therefore redundant. This function does not compute the negative frequency terms, and the length of the transformed axis of the output is therefore n//2 + 1.

As a consequence, the algorithm is optimized and rfft is twice as fast. Furthermore, the spectrum is easier to plot :

In [124]: s=abs(sin(arange(0,2**13,3)))
In [125]: sp=rfft(s)
In [126]: plot(abs(sp))
                I like the plot addition to the answer and so upvoted you. If we were to plot the complex conjugates, would that just be symmetrical over the amplitude axis?
– MichaelAndroidNewbie
                Sep 18, 2018 at 14:19
                since  the frequency are proportional to  -n,-n+1,.....-1,0,1,... n-1,n. the +1 is for the central term which is is own symetric.
– B. M.
                Feb 26, 2021 at 7:53
FFT output
 [ 4.        +0.j         -2.11803399-1.53884177j  0.11803399+0.36327126j
  0.11803399-0.36327126j -2.11803399+1.53884177j]
RFFT output
 [ 4.        +0.j         -2.11803399-1.53884177j  0.11803399+0.36327126j]
  
Notice how the final element of the fft output is the complex
  conjugate of the second element, for real input. For rfft, this
  symmetry is exploited to compute only the non-negative frequency
  terms.
        
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