# Hilbert transform of Sinc

## Homework Statement

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

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

## Homework 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.

## 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?

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## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?