# Fourier Transform of Hann Window

• Apr 12th 2006, 12:48 PM
malibupete
Fourier Transform of Hann Window
Hi all,

I'm trying to work out the Fourier transform for a Hann window , defined by f(x)=0.5(1-cos((2pi/T)x)), and really having no luck. Ultimately I want to find the transform of cos((2pi/T)x))f(x)... Any help would be very much appreciated- Thanks!
• Apr 13th 2006, 01:04 AM
CaptainBlack
Quote:

Originally Posted by malibupete
Hi all,

I'm trying to work out the Fourier transform for a Hann window , defined by f(x)=0.5(1-cos((2pi/T)x)), and really having no luck. Ultimately I want to find the transform of cos((2pi/T)x))f(x)... Any help would be very much appreciated- Thanks!

First thing to sort out is the definition of the Hann window:

$
f(x)=\left\{ \begin{array}{cc} 0.5(1-\cos(2\pi x /T)), & x \in [0,T]\\0, & \mbox{otherwise} \end{array} \right
$

This is the product of a rectangular window and a cosine function (plus a
constant), and its FT is the convolution of the FT's of the rectangular
window and the function (which is the sum of two delta functionals and so
trivial to evaluate).

RonL
• Apr 13th 2006, 02:46 AM
malibupete
Ok, transforming the first part of the Hann window, I've got

F(k)=0.5 (delta(k) + 0.5 (delta (k-2pi/T) + delta (k+2pi/T).

and for the rectangular window I have

W(k)= (sin(0.5kT))\k

Am I right in thinking that the convolution of the two is

S(k)= 0.5 ((sin(0.5kT))\k + 0.5 ((sin (0.5 (k-2pi/T))) + (sin (0.5 (k+2pi/T)))))?

Would this then make the transform of the window multiplied by cos (2pi/T)x

0.5(S(k-2pi/T) + (S(k+2pi/T))?

Thanks ever so much for your help, and sorry about the mess- I've tried to make things as clear as possible, but I don't know how to input mathematical symbols on here...

Thanks :)
• Apr 13th 2006, 04:12 AM
CaptainBlack
Quote:

Originally Posted by malibupete
Ok, transforming the first part of the Hann window, I've got

F(k)=0.5 (delta(k) + 0.5 (delta (k-2pi/T) + delta (k+2pi/T).

and for the rectangular window I have

W(k)= (sin(0.5kT))\k

Am I right in thinking that the convolution of the two is

S(k)= 0.5 ((sin(0.5kT))\k + 0.5 ((sin (0.5 (k-2pi/T))) + (sin (0.5 (k+2pi/T)))))?

Would this then make the transform of the window multiplied by cos (2pi/T)x

0.5(S(k-2pi/T) + (S(k+2pi/T))?

Thanks ever so much for your help, and sorry about the mess- I've tried to make things as clear as possible, but I don't know how to input mathematical symbols on here...

Thanks :)

It looks OK on a quick scan, I will check it in detail when I have a bit more
time.

RonL
• Apr 13th 2006, 04:20 AM
malibupete
Thanks :)

Though I'm not sure it can be quite right- I should get two peaks and small values/0 everywhere else.. I can't see it cancelling down at the moment, but I'm probably missing something obvious. Either way, thanks for taking the time and effort to have a look.
• Apr 13th 2006, 06:27 AM
CaptainBlack
Quote:

Originally Posted by malibupete
Thanks :)

Though I'm not sure it can be quite right- I should get two peaks and small values/0 everywhere else.. I can't see it cancelling down at the moment, but I'm probably missing something obvious. Either way, thanks for taking the time and effort to have a look.

No the peaks actualy overlap in such a way that with the aid of nearby
zeros they merge into a single peak (I had a peek :D )

RonL
• Apr 13th 2006, 06:35 AM
malibupete
So am I right, then? :)
• Apr 13th 2006, 07:31 AM
malibupete
No it doesn't mean I'm right, as I got several things wrong in my initial typing...

W(k)=sin(0.5kT)/0.5k (Meaning that S(k) is different, as I forgot to put in the last two denominators, and obviously everything is 2x bigger...)