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The rectangular window of length T has impulse response , that is
The transfer function of is
We would like to approximate a rectangular window with a maximally flat
window which means choosing and d appropriately. The
complication is that the values of
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 moments of the functions and ,
where we impose
Observe that since
we must have
Appealing to a rather technical theory, we find that this means that
at all points of continuity of , (i.e. everywhere except for at t=T).
This approach successfully determines T from and d, but there
is still one free parameter if only T is given. When we compare higher
moments of the functions and we find
We can evaluate the term
by means of the Stirling numbers of the first kind, giving
so the relative error of the moment is , and in particular the first moment has
relative error . Note that when k>0, the moment of is greater than that of , 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 to approximate to a specified degree.
Fri Jun 27 03:10:38 EDT 1997