Prefactors computing PSD of a signal with numpy.fft VS. scipy.signal.welch

Question

The power spectral density St of a signal u may be computed as the product of the FFT of the signal, u_fft with its complex conjugate u_fft_c. In Python, this would be written as:

import numpy as np

u = # Some numpy array containing signal
u_fft = np.fft.rfft(u-np.nanmean(u))

St = np.multiply(u_fft, np.conj(u_fft))

However, the FFT definition in Numpy requires the multiplication of the result with a factor of 1/N, where N=u.size in order to have an energetically consistent transformation between u and its FFT. This leads to the corrected definition of the PSD using numpy's fft:

St = np.multiply(u_fft, np.conj(u_fft))
St = np.divide(St, u.size)

On the other hand, Scipy's function signal.welch computes the PSD directly from input u:

from spicy.signal import welch

freqs_st, St_welch = welch(u-np.nanmean(u), 
     return_onesided=True, nperseg=seg_size, axis=0)

The resulting PSD, St_welch, is obtained by performing several FFTs in segments of the array u with size seg_size. Thus, my question is:

Should St_welch be multiplied by a factor of 1/seg_size to give an energetically consistent PSD? Should it be multiplied by 1/N? Should it not be multiplied at all?

PD: Comparison by performing both operations on a signal is not straightforward, since the Welch method also introduces smoothing of the signal and changes the display in the frequency domain.

Information on the necessity of the prefactor when using numpy.fft :

Journal article on the matter

Answer 1

The definition of the paramater scale of scipy.signal.welch suggests that the appropriate scaling is performed by the function:

scaling : { ‘density’, ‘spectrum’ }, optional Selects between computing the power spectral density (‘density’) where Pxx has units of V^2/Hz and computing the power spectrum (‘spectrum’) where Pxx has units of V^2, if x is measured in V and fs is measured in Hz. Defaults to ‘density’

The correct sampling frequency is to be provided as argumentfs to retreive the correct frequencies and an accurate power spectral density. To recover a power spectrum similar to that computed using np.multiply(u_fft, np.conj(u_fft)), the length of the fft frame and the applied window must respectively be provided as the length of the frame and boxcar (equivalent to no window at all). The fact that scipy.signal.welch applies the correct scaling can be checked by testing a sine wave:

import numpy as np
import scipy.signal

import matplotlib.pyplot as plt


def abs2(x):
    return x.real**2 + x.imag**2

if __name__ == '__main__':
    framelength=1.0
    N=1000
    x=np.linspace(0,framelength,N,endpoint=False)
    y=np.sin(44*2*np.pi*x)
    #y=y-np.mean(y)
    ffty=np.fft.fft(y)
    #power spectrum, after real2complex transfrom (factor )
    scale=2.0/(len(y)*len(y))
    power=scale*abs2(ffty)
    freq=np.fft.fftfreq(len(y) , framelength/len(y) )

    # power spectrum, via scipy welch. 'boxcar' means no window, nperseg=len(y) so that fft computed on the whole signal.
    freq2,power2=scipy.signal.welch(y, fs=len(y)/framelength,window='boxcar',nperseg=len(y),scaling='spectrum', axis=-1, average='mean')
    
    for i in range(len(freq2)):
        print i, freq2[i], power2[i], freq[i], power[i]
    print np.sum(power2)
    
    
    plt.figure()
    plt.plot(freq[0:len(y)/2+1],power[0:len(y)/2+1],label='np.fft.fft()')
    plt.plot(freq2,power2,label='scipy.signal.welch()')
    plt.legend()
    plt.xlim(0,np.max(freq[0:len(y)/2+1]))


    plt.show()

For a real to complex transform, the correct scaling of np.multiply(u_fft, np.conj(u_fft)) is 2./(u.size*u.size). Indeed, the scaling of u_fft is 1./u.size. Furthermore, real to complex transforms only report half of the frequencies, because the magnitude of the bin N-k would be the complex conjugate of that of the bin k. The energy of that bin is therefore equal to that of bin k and it is to be summed to that of bin k. Hence the factor 2. For the tested sine wave signal of amplitude 1, the energy is reported as 0.5: it is indeed, the average of a squared sine wave of amplitude 1.

Windowing is useful if the length of the frame is not a multiple of the period of the signal or if the signal is not periodic. Using smaller fft frames is useful if the signal is made of damped waves: the signal could be considered as periodic on a characteristic time: choosing a fft frame smaller than that characteristic time but larger than the period of the waves seems judicious.