how to derive IFFT of boxcar filter

Started by February 22, 2008
```Hi,

We can very well represent brickwall filter in FFT domain.

Also we know that IFFT of brickwall will look like
sin(x)/x function extending towards +/- infinity.

How to derive this using IFFT equation??

Regards
Bharat Pathak

Arithos Designs
www.Arithos.com
```
```"bharat pathak" <bharat@arithos.com> wrote in message
news:kNidndDZBvps-CPanZ2dnUVZ_hisnZ2d@giganews.com...
> Hi,
>
>  We can very well represent brickwall filter in FFT domain.
>
>  Also we know that IFFT of brickwall will look like
>  sin(x)/x function extending towards +/- infinity.

Bharat,

The IFFT of a brickwall will look like a Dirichlet - periodic.

The Inverse Fourier Transform of a perfect lowpass will look like sinx/x
with infinite extent.

Fred

```
```>Bharat,
>
>The IFFT of a brickwall will look like a Dirichlet - periodic.
>
>The Inverse Fourier Transform of a perfect lowpass will look like sinx/x

>with infinite extent.
>
>Fred

Fred,

In that case, why don't we make the periodic - dirichlet as our
start point for window based FIR designs??

Instead of using sinx/x as our start point.

Will using periodic dirichlet, be better?

Also in this case will the Gibbs oscillations completely disappear,
thus not needing any windowing approach.

I am just thinking loud.......or am I just moving in circles??

Regards
Bharat
```
```On Feb 21, 10:04 pm, "bharat pathak" <bha...@arithos.com> wrote:
> Hi,
>
>   We can very well represent brickwall filter in FFT domain.
>
>   Also we know that IFFT of brickwall will look like
>   sin(x)/x function extending towards +/- infinity.

The FT of a rectangle is of sin(x)/x shape.  But this
isn't precisely true for a finite length IFFT of sampled
data containing a rectangle.  What you have instead is
the sum of two Dirichlet (periodic Sinc) functions.

>   How to derive this using IFFT equation??

...thus you can't.  What you can do is to take the definite
integral of complex exponentials of unit magnitude, which
constitute the basis functions of the FT.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
```
```On Feb 21, 10:38 pm, "bharat pathak" <bha...@arithos.com> wrote:
> >Bharat,
>
> >The IFFT of a brickwall will look like a Dirichlet - periodic.
>
> >The Inverse Fourier Transform of a perfect lowpass will look like sinx/x
> >with infinite extent.
>
> >Fred
>
> Fred,
>
>     In that case, why don't we make the periodic - dirichlet as our
>     start point for window based FIR designs??
>
>     Instead of using sinx/x as our start point.
>
>     Will using periodic dirichlet, be better?

The Sinc and the Dirichlet are usually identical to around
3 or 4 decimal places for the first several lobes of typical
width filters, the Sinc being the already very slightly
windowed version of the Dirichlet for the first half of the
transform.  Thus the difference is negligible compared to
the effects of any window applied.

>     Also in this case will the Gibbs oscillations completely disappear,
>     thus not needing any windowing approach.

Also, the un-windowed Dirichlet has more high frequency
content than the Sinc, so it might be even more likely to
contribute to sharp transients in the other domain.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
```
```On 22 Feb, 07:04, "bharat pathak" <bha...@arithos.com> wrote:
> Hi,
>
> &#4294967295; We can very well represent brickwall filter in FFT domain.
>
> &#4294967295; Also we know that IFFT of brickwall will look like
> &#4294967295; sin(x)/x function extending towards +/- infinity.
>
> &#4294967295; How to derive this using IFFT equation??

If you want to do that formally, you need to dig rather
deep into the theory of complex analytic functions. The
only place I have attempted to do that stuff is in a
non-compulsary (for EE students) class on complex maths.

If you have the 4th edition of the Proakis & Manolakis book,
check out ch. 3.1.2, particuarly eq.3.1.13 to see what you
are up against.

Rune
```
```On Fri, 22 Feb 2008 00:04:33 -0600, "bharat pathak"
<bharat@arithos.com> wrote:

>Hi,
>
>  We can very well represent brickwall filter in FFT domain.
>
>  Also we know that IFFT of brickwall will look like
>  sin(x)/x function extending towards +/- infinity.
>
>  How to derive this using IFFT equation??
>
>Regards
>Bharat Pathak
>
>Arithos Designs

Hi Bharat,
If I remember correctly, you have copy of
the 2nd edition of my DSP book.  Perhaps the
discussion on page 107 will be of some help to you.

Regards,
[-Rick-]
```
```On Feb 22, 1:04&#4294967295;am, "bharat pathak" <bha...@arithos.com> wrote:
> Hi,
>
> &#4294967295; We can very well represent brickwall filter in FFT domain.
>
> &#4294967295; Also we know that IFFT of brickwall will look like
> &#4294967295; sin(x)/x function extending towards +/- infinity.
>
> &#4294967295; How to derive this using IFFT equation??
>
> Regards
> Bharat Pathak
>
> Arithos Designswww.Arithos.com

Hello Bharat,

Seems you have use for easy prove equation (can prove by induction if
maths geek):

sum (for i=0..N-1) of (ALPHA^i) = (1-ALPHA^N)/(1-ALPHA), good for
APLHA != 0 if N finite.

Regards,

Kamar Ruptan
DSP Guru
```
```On Feb 22, 9:53&#4294967295;am, DSPGURU <krup...@gmail.com> wrote:
> On Feb 22, 1:04&#4294967295;am, "bharat pathak" <bha...@arithos.com> wrote:
>
> > Hi,
>
> > &#4294967295; We can very well represent brickwall filter in FFT domain.
>
> > &#4294967295; Also we know that IFFT of brickwall will look like
> > &#4294967295; sin(x)/x function extending towards +/- infinity.
>
> > &#4294967295; How to derive this using IFFT equation??
>
> > Regards
> > Bharat Pathak
>
> > Arithos Designswww.Arithos.com
>
> Hello Bharat,
>
> Seems you have use for easy prove equation (can prove by induction if
> maths geek):
>
> sum (for i=0..N-1) of (ALPHA^i) = (1-ALPHA^N)/(1-ALPHA), good for
> APLHA != 0 if N finite.
>
> Regards,
>
> Kamar Ruptan
> DSP Guru

Also, don't use if ALPHA=1. Could use L'Hospitals rule or just use
obvious result of N for ALPHA=1. Rest of ALPHA (not 0, not 1) is OK if
N finite.

Kamar Ruptan
DSP Guru
```
```>
>Hi Bharat,
>  If I remember correctly, you have copy of
>the 2nd edition of my DSP book.  Perhaps the
>discussion on page 107 will be of some help to you.
>
>Regards,
>[-Rick-]

Hi Rick,

On page 107 i have 3.12.1 Processing Gain of single DFT

Can u tell me what section num and heading r u referring to?

Thanks and Regards
Bharat
```