# Thread: Fourier Transform of Hann Window

1. ## 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!

2. 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

3. 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

4. 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

5. 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.

6. 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 )

RonL

7. So am I right, then?

8. 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...)