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Next: Triangular Input Balanced Realizations Up: Maximally Flat Impulse Response Previous: Properties of the Maximally

Comparison With a Rectangular Window

The rectangular window of length T has impulse response tex2html_wrap_inline1192, that is
equation341
The transfer function of is
eqnarray347
We would like to approximate a rectangular window with a maximally flat window which means choosing tex2html_wrap_inline1150 and d appropriately. The complication is that the values of
equation357
is independent of T, but our derivation of the maximally flat impulse response only uses information local to t=0, so it cannot determine T from this information.

One approach is to match the tex2html_wrap_inline1204 moments of the functions tex2html_wrap_inline1206 and tex2html_wrap_inline1186,
eqnarray366
where we impose
equation376
Observe that since
equation380
we must have
equation392
Appealing to a rather technical theory, we find that this means that
equation399
at all points of continuity of tex2html_wrap_inline1206, (i.e. everywhere except for at t=T).

This approach successfully determines T from tex2html_wrap_inline1150 and d, but there is still one free parameter if only T is given. When we compare higher moments of the functions tex2html_wrap_inline1206 and tex2html_wrap_inline1186 we find
eqnarray407
We can evaluate the term
equation425
by means of the Stirling numbers of the first kind, giving
equation436
so the relative error of the tex2html_wrap_inline1226 moment is tex2html_wrap_inline1228, and in particular the first moment has relative error tex2html_wrap_inline1230. Note that when k>0, the tex2html_wrap_inline1226 moment of tex2html_wrap_inline1186 is greater than that of tex2html_wrap_inline1206, and for k=1 this means that the maximally flat impulse response uses older data than a corresponding rectangular window. It also shows that no choice of d will match higher moments than another choice.

We can also use this comparison of moments to determine how large d needs to be for tex2html_wrap_inline1186 to approximate tex2html_wrap_inline1206 to a specified degree.




Fri Jun 27 03:10:38 EDT 1997

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