Bug summary

If my understanding is correct:

• NFFT ofmlab._spectral_helper corresponds tonperseg of spectral funtions ofscipy.signal.
  It's the actual length of signal (segment) for DFT.
• pad_to ofmlab._spectral_helper corresponds tonfft of spectral funtions ofscipy.signal.
  It's the length of DFT, i.e. parametern offft funtion.
  

  matplotlib/lib/matplotlib/mlab.py

  Line 385 in3418bad

  result=np.fft.fft(result,n=pad_to,axis=0)[:numFreqs, :]

But the implementation of Matplotlib is really confusion:

• If length of signal is less than NFFT, signal will pad to length of NFFT.
  

  matplotlib/lib/matplotlib/mlab.py

  Lines 342 to 351 in3418bad

  	# zero pad x and y up to NFFT if they are shorter than NFFT
  	iflen(x)<NFFT:
  	n=len(x)
  	x=np.resize(x,NFFT)
  	x[n:]=0
  	ifnotsame_dataandlen(y)<NFFT:
  	n=len(y)
  	y=np.resize(y,NFFT)
  	y[n:]=0

  And the window is determined by NFFT instead of actual length of signal. This would cause wrong window correction.
  

  matplotlib/lib/matplotlib/mlab.py

  Lines 376 to 380 in3418bad

  	ifnotnp.iterable(window):
  	window=window(np.ones(NFFT,x.dtype))
  	iflen(window)!=NFFT:
  	raiseValueError(
  	"The window length must match the data's first dimension")

• The Nyquist frequency isNFFT/2 instead ofpad_to/2.
  

  matplotlib/lib/matplotlib/mlab.py

  Lines 411 to 416 in3418bad

  	# if we have a even number of frequencies, don't scale NFFT/2
  	ifnotNFFT%2:
  	slc=slice(1,-1,None)
  	# if we have an odd number, just don't scale DC
  	else:
  	slc=slice(1,None,None)

Code for reproduction

importnumpyasnpfrommatplotlibimportmlabfs=1000t=np.arange(0,1,1/fs)rng=np.random.default_rng(42)s=np.sin(2*np.pi*100*t)+0.5*rng.normal(size=t.shape)nfft=2048nsignal=len(s)#1000psd,freqs=mlab.psd(s,NFFT=nfft,Fs=fs,window=mlab.window_none)mag=np.abs(np.fft.rfft(s,n=nfft))psd1= (mag)**2/nsignal/fsifnotnfft%2:psd1[1:-1]*=2else:psd1[1:]*=2relative_error=np.abs(2* (psd-psd1)/(psd+psd1) )print(relative_error.max())

Actual outcome

0.6876640419947717

Expected outcome

0