Started by July 1, 2008
```Hi All,

I read in this book DSP with Computer Applications that the rectangular
window (used for windowing )  gives the best approximation to the desired
frequency response in a least square sense. That is, it gives the least
squared, integrated, error between the Hd(W) and Ha(W) for a given number
of impulse response terms.
where Hd(W) is the ideal response of the filter.
Ha(W) is the response of the filter after the window is used.

```
```On Jul 1, 10:15 am, "vasindagi" <vish...@gmail.com> wrote:
> Hi All,
>
> I read in this book DSP with Computer Applications that the rectangular
> window (used for windowing )  gives the best approximation to the desired
> frequency response in a least square sense. That is, it gives the least
> squared, integrated, error between the Hd(W) and Ha(W) for a given number
> of impulse response terms.
> where Hd(W) is the ideal response of the filter.
>       Ha(W) is the response of the filter after the window is used.
>
>

The rectangular window gives the minimum integrated squared error from
the ideal response. (The ideal response has constant gain in a
passband and zero response elsewhere.) This integral applies the same
importance to error, regardless of where it occurs. Whether this is
'best' will vary with one's application. Flattop windows (flattop in
the frequency domain sense) are used to minimize error in the passband
at the cost of greater error in the nearby rejection band. Minimum
sidelobe and Maximum sidelobe rolloff windows produce less error in
the passband than the rectangular but more in the close rejection band
and much less error in the distant rejection band. The prolate
spheroidal windows are the solution to an optimization problem that
minimizes mainlobe width against total energy outside the mainlobe.

There are also other error criteria than squared error such as
absolute value of error. The Dolph-Chebychev window minimizes mainlobe
width for a given maximum stopband absolute error (no summation
involved).

Many windows are designed as easier to calculate approximations to
more nearly ideal windows.

Dale B. Dalrymple
```