moble
commented
3 years ago Converting to the frequency domain is very tricky. You have to take a few more steps:
- Ignore junk radiation (data before t=0)
- Interpolate to uniform timesteps
- Evaluate in a particular direction
- Take the real part (like a single detector would measure)
- Multiply by a window function to reduce Gibbs effects
Putting these together, I would do something more like this:
import numpy as np
from scipy import signal
import sxs
import matplotlib.pyplot as plt
waveform = sxs.load('SXS:BBH:0123/Lev/rhOverM', extrapolation_order=2)
# Ignore junk radiation (data before t=0)
t_start = 0.0
# Interpolate to uniform time steps. Here, I'm just choosing the smallest
# timestep in the input data. If you want more physical results, you'll
# need to scale to the appropriate total mass, and a corresponding physical
# timestep.
t_end = waveform.t[-1]
dt = np.min(np.diff(waveform.t))
t_uniform = np.arange(t_start, t_end, dt)
h = waveform.interpolate(t_uniform)
# Evaluate in a particular direction. I'm arbitrarily choosing a direction
# (and implicitly a polarization angle).
theta, phi = 0.0, 0.0
h = h.evaluate(theta, phi)
# Take the real part (like a single detector would measure)
h = h.real
# Multiply by a window function. I like Tukey windows because, if you make
# them narrow enough, they leave the inspiral and merger data unchanged.
# Adjust the 0.12 below as needed for a particular waveform.
window = signal.windows.tukey(h.shape[0], alpha=0.12, sym=True)
h = h * window
# Now we can convert to the frequency domain
h_tilde = np.fft.fft(h)
freq = np.fft.fftfreq(h.size, dt)
# And plot it
plt.loglog(freq, np.abs(h_tilde))
plt.xlabel('Frequency (units of $1/M$)')
plt.ylabel(r'$|\tilde{h}|$');The result looks like this:
The actual data starts around a frequency of 6e-3. A physical binary system would extend to earlier times and frequencies, but our numerical waveforms are finite, and that happens to be where this system starts. At smaller frequencies, you basically have nonsense that you should ignore (or you could try harder to window the data to make it smaller). Then, the around 6e-3, we start to see frequencies that are in the actual data. At first, you see some oscillations due to remaining Gibbs effects, but then it becomes smoother and more physically realistic. (Note that lower frequencies have larger amplitude in the frequency domain, even though they have smaller amplitude in the time domain, because the signal spends longer at lower frequencies, so more power gets built up there.) You get a Fourier transform that slopes downward to around 1e-1, where it sort of flattens out a little, and then drops steeply. That little flat part around 1e-1 is the merger, and the drop is because of ringdown — the physical system just never really oscillates at any higher frequencies. Above about 2e-1, you just have numerical noise that you could maybe reduce with better windowing.
Sunnnsh
I think I'm doing bad with interpolate the waveform, and I also read your code about interpolate, spherical and so on, these codes are really hard for me to understand with my python level.
[A user asked this question through a separate channel recently. It'll be easier to answer it here.]
I need to do the fft to transform time domain waveform to the frequency domain. But I tried many times and I can't get the correct waveform of frequency domain. For the python code to do fft, I used:
For the results of doing fft of (2, 2) mode, it have much higher oscillation in the high frequency domain with the loglog plot of absolute value of h(f), and I think the trend of waveform with increasing of frequency is also incorrect.