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Rectangular Window Advantage

Started by vasindagi 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.

Can anyone explain more about this. What this actually means and why so ?

Thanks in advance.
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. > > Can anyone explain more about this. What this actually means and why so ? > > Thanks in advance.
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