Amplitude and phase spectrum. Shifting the phase leaving amplitude untouched

Question

I have the data that has equivalent intervals and corresponding measurements at relevant points. As an example, here is the excerpt of the data I have:

y =[2.118, 2.1289, 2.1374, 2.1458, 2.1542, 2.1615, 2.1627, 2.165 2.1687...]

interval between the points is 0.1

So, what I need to get out of the data is the Amplitude spectrum (amplitude vs frequency) and also phase spectrum (phase angle vs frequency). In addition I should shift the phase of the data by negative 90 degrees (-pi/2).

Upon shifting the phase and leaving the amplitude untouched, I need to do the inverse fft and get the new signal. I want to do this in Python.

Could you please give me an example of performing this.

The code that I have used, was taken from another SO question, but I have done some modifications

## Perform FFT WITH SCIPY
signalFFT = np.fft.fft(y)

## Get Power Spectral Density
signalPSD = np.abs(signalFFT) ** 2
signalPhase = np.angle(signalFFT)

## Shift the phase py +90 degrees
new_signalPhase =(180/np.pi)*np.angle(signalFFT)+90 


## Get frequencies corresponding to signal 
fftFreq = np.fft.fftfreq(len(signalPSD), 0.1)

## Get positive half of frequencies
i = fftFreq>0

##
plt.figurefigsize=(8,4)
#plt.plot(fftFreq[i], 10*np.log10(signalPSD[i]));

plt.plot(fftFreq[i], new_signalPhase[i]);
plt.ylim(-200, 200);
plt.xlabel('Frequency Hz');
plt.ylabel('Phase Angle')
plt.grid()
plt.show()

The problem is that I want to regenerate the signal, with the same amplitudes but shifted phase. I know that the answer is smth related to ifft, but how should I prepare data for it? Could you please advice me the further steps.

output

Hello Kamran, and welcome to SO! When asking a question here, it is important to show us what you have tried to fix your problems already, and to be very specific about what it is that you are stuck with. Read how do I ask a good question?. This will help us help you. Right now, it is hard for us to tell if you do not understand how the FFT works, or if you simply do not know how to code it. — Floris
Kamran, please see my answer and code below. The phase shifted signal is produced in three lines of code, Fourier transform, multiply by exp( i phase), and inverse transform. — DrM
@DrM great thank you very much. Correct me if I am wrong, multiplication by the exp(i phase) is simply addition in the power of exp(i w t), I mean in the DFT expression. Am I getting it right? — akkab
exp( i w t ) x exp( i phase ) is equal to exp[ i (w t + phi ) ], i.e., it shifts the phase. In other words, after multiplying the Fourier transformed signal by exp( i phase ), what you have, is the FT of a signal shifted by that same phase. This works for any input signal. — DrM

DrM DrM · Accepted Answer · 2018-09-12T13:53:11

Here is your code with some modifications. We apply a Fourier transform, phase shift the transformed signal, and then perform the inverse Fourier transform to produce the phase shifted time domain signal.

Notice that the transforms are done with rfft() and irfft(), and that the phase shift is done by simply multiplying the transformed data by cmath.rect(1.,phase). The phase shift is equivalent to multiplying the complex transformed signal by exp( i * phase ).

In the graphics, in the left panel, we show the original and shifted signals. The new signal is advanced by 90 degrees. In the right panel we show the power spectrum on the left axis. In this example we have power at a single frequency. Phase is graphed on the right axis. But again, since we have power at only one frequency, the phase spectrum shows noise everywhere else.

#!/usr/bin/python

import matplotlib.pyplot as plt
import numpy as np
import cmath

# Generate a model signal
t0 = 1250.0
dt = 0.152
freq = (1./dt)/128

t = np.linspace( t0, t0+1024*dt, 1024, endpoint=False )
signal = np.sin( t*(2*np.pi)*freq )

## Fourier transform of real valued signal
signalFFT = np.fft.rfft(signal)

## Get Power Spectral Density
signalPSD = np.abs(signalFFT) ** 2
signalPSD /= len(signalFFT)**2

## Get Phase
signalPhase = np.angle(signalFFT)

## Phase Shift the signal +90 degrees
newSignalFFT = signalFFT * cmath.rect( 1., np.pi/2 )

## Reverse Fourier transform
newSignal = np.fft.irfft(newSignalFFT)

## Uncomment this line to restore the original baseline
# newSignal += signalFFT[0].real/len(signal)


# And now, the graphics -------------------

## Get frequencies corresponding to signal 
fftFreq = np.fft.rfftfreq(len(signal), dt)

plt.figure( figsize=(10, 4) )

ax1 = plt.subplot( 1, 2, 1 )
ax1.plot( t, signal, label='signal')
ax1.plot( t, newSignal, label='new signal')
ax1.set_ylabel( 'Signal' )
ax1.set_xlabel( 'time' )
ax1.legend()

ax2 = plt.subplot( 1, 2, 2 )
ax2.plot( fftFreq, signalPSD )
ax2.set_ylabel( 'Power' )
ax2.set_xlabel( 'frequency' )

ax2b = ax2.twinx()
ax2b.plot( fftFreq, signalPhase, alpha=0.25, color='r' )
ax2b.set_ylabel( 'Phase', color='r' )


plt.tight_layout()

plt.show()

And here is the graphical output.

Amplitude and phase spectrum. Shifting the phase leaving amplitude untouched

1 Answers