How to get phase DC offset and amplitude of sine wave in Python

Question

I have a sine wave of the known frequency with some noise with uniform samples near Nyquist frequency. I want to get approximate values of amplitude, phase, and DC offset. I searched for an answer and found a couple of answers close to what I needed, but still was unable to write a proper code that achieves what I need. When I run the code below, I get the wrong phase and amplitude. Would be happy to get some help.

import sys
import numpy
import pylab as plt

def cosfunc(time, amplitude, omega, phase, offset):
    ''' Function to create sine wave. Phase in radians '''
    return amplitude * numpy.cos(omega*time + phase) + offset

def get_cosine_approx(timeline,sine_data):
    points_num=len(timeline)
    
    fft_freq = numpy.fft.fftfreq(points_num-1, timeline[1]-timeline[0])   # assume uniform spacing
    fft_result=numpy.fft.fft(sine_data)
    
    #Remove negative frequencies
    for i in range(len(fft_freq)):
        if fft_freq[i]<0:
            fft_result[i]=0

    ampl=numpy.abs(fft_result)/points_num*2
    max_index=numpy.argmax(ampl)

    guess_amplitude=ampl[max_index]
    phase_unwrapped=numpy.unwrap(numpy.angle(fft_result))
    
    guess_phase=phase_unwrapped[max_index]
    guess_phase_dig=guess_phase*180./numpy.pi

    print("freq",fft_freq[max_index])
    print("amplitude",guess_amplitude)
    print("phase",guess_phase_dig)

    plt.plot(timeline, sine_data, "ok", label="sine")
    new_timeline=numpy.linspace(timeline[0], timeline[-1], len(timeline)*1000)
    plt.plot(new_timeline, cosfunc(new_timeline,guess_amplitude,2.*numpy.pi*56e9,guess_phase,0), "r-", label="fit")
    plt.legend(loc="best")
    plt.show()

    return {"amp":guess_amplitude, "ph":guess_phase,"ph_dig":guess_phase_dig}

N = 256  # Sample points
f=56e9 #56GHz
t = numpy.linspace(0.0, 100./f, N) # Time
omega = 2.*numpy.pi*f
offset=0
phase=0
A=1.

cos=cosfunc(t,A,omega,phase,offset)
result=get_cosine_approx(t,cos)

Answer 1

You are catching the phase at an inflection point, where the phase is suddenly transitioning from +pi/2 to -pi/2, and the bin you are looking at is just partway through the downhill slide. This is just because the FFT results are not continuous. A single bin spans a range of frequencies.

Notice when we plot the phase and the amplitude:

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

def cosfunc(time, amplitude, omega, phase, offset):
    ''' Function to create sine wave. Phase in radians '''
    return amplitude * np.cos(omega*time + phase) + offset

def get_cosine_approx(timeline,sine_data):
    points_num=len(timeline)
    
    fft_freq = np.fft.fftfreq(points_num, timeline[1]-timeline[0])
    fft_result=np.fft.fft(sine_data)
    fft_freq = np.fft.fftshift(fft_freq)
    fft_result = np.fft.fftshift(fft_result)
    
    ampl = np.abs(fft_result) * 2 / points_num
    phase = np.angle(fft_result)

    plt.plot(fft_freq, ampl, label='ampl' )
    plt.plot(fft_freq, phase, label='phase' )
    plt.legend(loc="best")
    plt.show()

    return 0

N = 256  # Sample points
f=56e9 #56GHz
t = np.linspace(0.0, 100./f, N) # Time
omega = 2.*np.pi*f
offset=0
phase=0
A=1.

cos=cosfunc(t,A,omega,phase,offset)
result=get_cosine_approx(t,cos)

The plot shows the inflection point right at the peak frequency bin.