<p>It turns out there are many ways to do this. A, conceptually, straight forward way is to differentiate the phase, <span class="math-container">$\Phi_{lm}$</span>, of the gravitational wave. Expanding the strain of the gravitational wave with spherical harmonics, each mode is complex-valued, we have</p>
<p><span class="math-container">$$ h_{lm}(t) = h_+(t) -ih_x(t) = A_{lm}(t) e^{i\Phi_{lm}(t)}$$</span></p>
<p>where <span class="math-container">$A_{lm}$</span> is the amplitude of the wave.</p>
<p>The frequency is then,</p>
<p><span class="math-container">$$ f_{lm}(t) = \frac{1}{2\pi}\frac{d}{dt}\Phi_{lm}(t)$$</span>.</p>

CC BY-SA 4.0
2021-12-23T13:10:01.673Z
2021-12-23T13:10:01.673Z
Daddy Kropotkin
<p>Suppose I have a time-series of the gravitational-wave strain amplitude as a (discrete, i.e., an array of numbers) function of time. The figure below is just illustrative. I am not using measured LIGO/Virgo data, as that would require windowing the data and many other steps; rather, I'm getting my strain vs time from a waveform model such as surrogate models.</p>
<p><strong>Question</strong>:
how does one derive the gravitational-wave frequency as a function of time?</p>
<p>I think that this involves the discrete Fourier transform, for example with python using the fast Fourier transform provided by the scipy library's fft function, but I'm unsure how to arrive at the frequency as a function of time in theory. Any help is greatly appreciated, and sources are very much welcome.</p>
<p>As shown in the figure below, in essence I'm wondering how to start from the above time series and arrive at the bottom time series.</p>
<p>Thanks!</p>
<p><a href="https://i.sstatic.net/VJNDt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/VJNDt.png" alt="enter image description here" /></a>
<a href="https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves" rel="nofollow noreferrer">source</a></p>

CC BY-SA 4.0
2023-06-09T21:18:21.963Z
for programming , suggest that look at these videos (look at python section for gravitational-waves) channel link:
https://youtube.com/@gwopendataworkshop384
Hamidreza Abdollahi
<modeling><time-domain-astronomy><gravitational-waves>
How to derive gravitational-wave frequency vs time from strain vs time
Daddy Kropotkin
# How to derive gravitational-wave frequency vs time from strain vs time

(Asked by: username_0 on 2021-12-19)

Suppose I have a time-series of the gravitational-wave strain amplitude as a (discrete, i.e., an array of numbers) function of time. The figure below is just illustrative. I am not using measured LIGO/Virgo data, as that would require windowing the data and many other steps; rather, I'm getting my strain vs time from a waveform model such as surrogate models.

**Question**:
how does one derive the gravitational-wave frequency as a function of time?

I think that this involves the discrete Fourier transform, for example with python using the fast Fourier transform provided by the scipy library's fft function, but I'm unsure how to arrive at the frequency as a function of time in theory. Any help is greatly appreciated, and sources are very much welcome.

As shown in the figure below, in essence I'm wondering how to start from the above time series and arrive at the bottom time series.

Thanks!

[![enter image description here](https://i.sstatic.net/VJNDt.png)](https://i.sstatic.net/VJNDt.png)
[source](https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves)

**Tags:** modeling, time-domain-astronomy, gravitational-waves
> **Comments:**
> **username_1**: for programming , suggest that look at these videos (look at python section for gravitational-waves) channel link: https://youtube.com/@gwopendataworkshop384  

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## 1 answers

### Answer by username_0 on 2021-12-23. Score: 1 | Accepted answer

It turns out there are many ways to do this. A, conceptually, straight forward way is to differentiate the phase, $\Phi\_{lm}$, of the gravitational wave. Expanding the strain of the gravitational wave with spherical harmonics, each mode is complex-valued, we have

$$ h\_{lm}(t) = h\_+(t) -ih\_x(t) = A\_{lm}(t) e^{i\Phi\_{lm}(t)}$$

where $A\_{lm}$ is the amplitude of the wave.

The frequency is then,

$$ f\_{lm}(t) = \frac{1}{2\pi}\frac{d}{dt}\Phi\_{lm}(t)$$.


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