Numerical integration along constant and collapsing spectrum

[nkinar]

Hello---

I am reading a paper which describes a somewhat-unfamiliar mathematical procedure. The paper asks for a 2D spectrum $$U(t, \omega)$$ of a signal $$s(t)$$ which is calculated using the short-time Fourier transform (Gabor transform). This is reasonably straight-forward.

However, the paper then asks for the 2D spectrum $$U(t, \omega)$$ to be collapsed into a 1D spectrum $$U(t * \omega) = U(\chi)$$, where $$\chi = t * \omega$$.

In the above, $$t$$ is the time and $$\omega = 2 \pi f$$ is the angular frequency.

Apparently the 2D spectrum needs to be converted into a 1D spectrum by integration along constant $$\chi$$.

To me, this appears to be a form of line integral, evaluated numerically.

I can't find a good reference on how to do this, and I don't really know what is meant by integration along constant $$\chi$$. Perhaps this means "averaging along constant chi"?

I've tried to simply multiply each element in the rows of $$U(t, \omega)$$ with each element in the columns of $$U(t, \omega)$$ using two nested for loops, but a plot of the resulting 1D spectrum "jumps around."

The paper says that the 1D spectrum will decrease monotonically along the $$\chi$$ axis, and the figures in the paper clearly show this monotonic decrease.

Has anyone seen anything similar, or would someone be able to point me in the right direction?

 

[jvicens]

Hello,

Thank you for sharing your question with the forum. It sounds like you are working with a complex mathematical procedure and are having trouble understanding how to convert the 2D spectrum into a 1D spectrum. I understand your confusion and would be happy to offer some guidance.

First, let's clarify what is meant by "integration along constant \chi." In this context, integration means finding the area under a curve. So, when the paper mentions integrating along constant \chi, it is referring to finding the area under the curve at different values of \chi.

To convert the 2D spectrum into a 1D spectrum, you will need to integrate along the \chi axis. This means taking the values at each point along the \chi axis and finding the area under the curve. This process can also be thought of as averaging along constant \chi, as you suggested.

In terms of implementation, there are a few different methods you could use to integrate along the \chi axis. One approach would be to use the trapezoidal rule, where you divide the \chi axis into small sections and approximate the area under the curve in each section. Another option would be to use a numerical integration method, such as Simpson's rule or Gaussian quadrature.

It's possible that the nested for loop approach you tried previously was not accurate enough, resulting in the "jumpy" 1D spectrum. I would recommend trying one of the numerical integration methods mentioned above to see if that produces a smoother and more accurate 1D spectrum.

I hope this helps to clarify the process of converting the 2D spectrum into a 1D spectrum. If you have any further questions or need more guidance, please don't hesitate to ask. Good luck with your research!