Recursive Windowing of Time Series

Windowing a time series means computing another time series by the formula

where the numbers define the window. In the case where the input series is a unit impulse

the output time series is called the impulse response of the windowing filter. Alternatively, we can obtain the series by passing through any filter with the same impulse response. Different filters with the same impulse response are called different realizations of each other. Although all realizations of the a filter have the same signal characteristics, not all realizations of the same filter have the same properties when approximated by finite precision (fixed or floating point) arithmetic, and not all realizations require the same amount of computational cycles. This note explains how to obtain particularly advantageous realizations.

One practical difficulty in using the representation (1) is that it requires storage of all the values of for which , and when used sequentially, in storage of all the values of for which and . In particular, it is not practical to realize an IIR (infinite impulse response) filter in this way.

The most common IIR impulse response used for windowing is the case

which admits the recursive realization

It is frequently useful to refer to the DC gain of the filter, which is the sum of the impulse response. In this case, the DC gain is

Another filter which is used for windowing is the `modified Barnwell' window suggested by Strobach. This second order filter generalizes (4) is given by
eqnarray55
The impulse response of this window is

It is a little easier to see this if we recast the filter in the state space form

so that the impulse response is
eqnarray87
Note that we are numbering the indices of the vector starting at 0, hence the first standard basis vector is , etc.

The DC gain is
equation108

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Fri Jun 27 03:10:38 EDT 1997