So I found the following discussion about finding the pdf of a square of a Rician RV somewhere on the internet, which involves using the dirac delta function in an integration. I am paraphrasing, but it goes as follows:
Let $x \sim Rice(v, \alpha)$. Then the pdf is given by:
$$ f_{x}(x) = \frac{x}{\alpha^{2}} e^{\frac{-(x^{2} + v^{2})}{2 \alpha^{2}}} I_{0} \big( \frac{x v}{\alpha^{2}} \big)$$
Where $I_{0} (\cdot)$ is the modified Bessel function of the first kind of order 0. To find the pdf of the square, the following was done.
First, we let $y = x^{2}$. Then we use the 'dirac delta method' to find the resulting pdf. So we have:
$$ f_{y}(y) = \displaystyle\int\limits_{0}^{\infty} \frac{x}{\alpha^{2}} e^{\frac{-(x^{2} + v^{2})}{2 \alpha^{2}}} I_{0} \big( \frac{x v}{\alpha^{2}} \big) \delta(y - x^{2}) dx$$
$$ f_{y}(y) = \frac{1}{\alpha^{2}} \displaystyle\int\limits_{0}^{\infty} x e^{\frac{-(x^{2} + v^{2})}{2 \alpha^{2}}} I_{0} \big( \frac{x v}{\alpha^{2}} \big) \frac{\delta(x - \sqrt{y})}{2 \sqrt{y}} dx$$
$$ f_{y}(y) = \frac{1}{\alpha^{2}} \displaystyle\int\limits_{0}^{\infty} \frac{x}{2 \sqrt{y}} e^{\frac{-(x^{2} + v^{2})}{2 \alpha^{2}}} I_{0} \big( \frac{x v}{\alpha^{2}} \big) \delta(x - \sqrt{y}) dx$$
$$ f_{y}(y) = \frac{1}{2\alpha^{2}} e^{\frac{-(y + v^{2})}{2\alpha^{2}}} I_{0}\big( \frac{ v \sqrt{y}}{\alpha^{2}} \big)$$
I've tried looking on the web for explanations of this method, however I can't seem to find any.
My questions:
- Can someone give me some references about this method, and/or explain why it works?
- Where does the $\frac{\delta(x - \sqrt{y})}{2 \sqrt{y}}$ comes from?
Thanks!