# Hilbert transform of Sinc

Tags:
1. May 3, 2016

### roam

1. The problem statement, all variables and given/known data

Show that the Hilbert transform of $\frac{\sin(at)}{at}$ is given by

$$\frac{\sin^2(at/2)}{at/2}.$$

2. Relevant equations

The analytic signal of a function is given by $f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,$ where $F(\nu)$ is the Fourier transform of the function. We have $f_a (t) = f(t) +j\hat{f}(t),$ where $\hat{f}(t)$ is the Hilbert transform.

$FT(sinc(at)) = \frac{1}{a} \Pi (\frac{\nu}{a})$, where $\Pi$ denotes the rectangular function.

3. The attempt at a solution

This is the analytic signal whose imaginary part would be the Hilbert transform:

$$f_a(t) = 2 \int^\infty_0 FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu$$

I tried to rewrite this as:

$$f_a(t) = 2 \int^\infty_{-\infty} u(t) \ FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu$$

So, it looks like an inverse Fourier transform, so this becomes $\left( \frac{1}{j2 \pi \nu} + \frac{\delta(\nu)}{2} \right) \frac{\sin(at)}{at}.$ So this is not the correct solution. So here is another approach:

$$f_a(t) = 2 \int^\infty_0 FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu = 2 \int^\infty_0 \frac{1}{|a|} \Pi \left( \frac{\nu}{a} \right) \exp(j2 \pi \nu t) \ d\nu$$

So, how do I continue from here? What method should I use?

Last edited: May 3, 2016
2. May 8, 2016