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From: Fred Marshall on 21 Nov 2009 14:41
Richard Owlett wrote:
> In an *UNRELATED* post Chris Bore said:
> "
> The way I sometimes think of it, you choose a window function
> whose kernel best matches the shape that either you think the
> signal's spectrum has, or that you want the signal's spectrum
> to have.
> "
>
> My question --- Is there any ESSENTIAL difference between a "window" and
> a "filter"?
>
> I suspect that the answer is "No."
>
> Explanation:
> If in "time domain" one refers to a "window".
> If in "frequency domain" one refers to a "filter".
>
> Am I close?
>

Is there any essential difference between a window and a filter? Yes.
Unless you want to call them both "functions" and leave it at that.

A window function refers first to a time-limited or bandwidth-limited or
space-limited situation. An antenna or a lens is a pretty good example
that's easy to visualize and leads one immediately to the notion of
"window". Think of a window in a building... same thing. The latter are
all spatially limited in extent.

So, the simplest window is one that doesn't weigh what comes through it.
Thus the gate function, uniform window that we use all the time.

Antennas again are a good example of more complicated window functions
where the aperture (the physical extent of the window) is weighted in
order to reduce sidelobes.
In signal processing window functions are used for the same thing - to
reduce spectral sidelobes *AS APPLIED TO OR RELATIVE TO SPECTRAL LINES*
(because of the convolution in frequency).
So, just about all window functions are nonzero except maybe at the
edges and perhaps with a rather longish taper at the edges and extend
for the entire limited extent.

A filter refers first to what one wants to do with a function such as
spectral content which may be of infinite extent (but not need be).
Except for equalizers, most filters have stop bands and lowpass filters
attempt to be as close to zero as possible above a certain frequency.
In this sense they are very different from windows. In fact, maybe
you'd call them "black out screens" if taken in the same context.

You can take any function and multiply with it in one domain and see the
result as a convolution in the Fourier Transformed domain and vice
versa. So, this mathematical assertion has nothing to do with the type
of functions being used. In general though:
- we design a window function according to its convolution kernel and
most often implement a window function by multiplying in the opposite
domain. The objective is reduction of sidelobes or spectral leakage.
- we design a filter function according to its multiplicative effect and
very often implement a filter by convolving in the opposite domain and
sometimes by multiplying in the design domain. The objective is band
pass and band stop.
And, we use the two terms because they have specific meaning as above.

If you look at the windowing method of filter design then it should be
clear how the two fit together in distinct ways.
An idealized filter function is first defined - likely with sharp /
brick wall band transsitions
Then a window function is used to conceptually convolve the ideal filter
function to achieve a real filter function with realizable band
transitions and with acceptable band edge trillies and often to
time-limit the filter unit sample response as the original filter spec
likely doesn't represent a FIR filter.

I hope this helps.

Fred


From: brent on 22 Nov 2009 07:42
On Nov 20, 1:50 pm, Richard Owlett <rowl...(a)pcnetinc.com> wrote:
> In an *UNRELATED* post Chris Bore said:
> "
> The way I sometimes think of it, you choose  a window function
> whose kernel best matches the shape that either you think the
> signal's spectrum has, or that you want the signal's spectrum
> to have.
> "
>
> My question --- Is there any ESSENTIAL difference between a
> "window" and a "filter"?
>
> I suspect that the answer is "No."
>
> Explanation:
> If in "time domain"      one refers to a "window".
> If in "frequency domain" one refers to a "filter".
>
> Am I close?

There are four flash tutorials here on the DFT. The last one goes
over why windowing is used:

http://www.fourier-series.com/fourierseries2/DFT_tutorial.html

There is a flash tutorial on digital convolution here:

http://www.fourier-series.com/fourierseries2/convolution.html

This gives background to how an FIR filter coefficients are calculated
to get the proper answer at each point in time

This page points to two flash tutorials on how to generate low pass
FIR filters and FIR BP filters. It allows you to either apply
windowing or not applt windowing.


http://www.fourier-series.com/fourierseries2/FIR-filter.html
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