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Comparison With a Rectangular Window

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