- #1

arcTomato

- 105

- 27

I am thinking about the power spectrum when observing X-rays.

We are trying to obtain the power spectrum by applying a window function ##w(t)## to a light curve ##a(t)## and then Fourier transforming it.

I have seen the following definition of power spectrum ##P(\omega)##. Suppose the Fourier transformation of light curve is ##A## and the Fourier transformation window function is ##W##,

$$P(\omega)= |A(\omega)|^{2} *|W(\omega)|^{2}$$

In other words, you are squaring the absolute value of each and then doing convolution integration.

But my actual calculations are as follows.

The Fourier transformation of light curve and window function is

$$

\int_{-\infty}^{\infty} a(t)w(t) e^{i \omega t} d t \\ = A(\omega) * W(\omega) .

$$

so Power spectrum is

$$

P(\omega)= | A(\omega) * W(\omega) | ^2.

$$

In my own calculations, I convolution-integrate each and then square the absolute value. (The order of the calculations is different.)

I have two questions.

First, I think these are very different, but am I doing my calculations wrong?

Second, under certain conditions, can these be approximated?

Thank you.