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Recursive Windowing of Time Series

Windowing a time series tex2html_wrap_inline1092 means computing another time series tex2html_wrap_inline1094 by the formula
 equation12
where the numbers tex2html_wrap_inline1096 define the window. In the case where the input series tex2html_wrap_inline1092 is a unit impulse
equation22
the output time series tex2html_wrap_inline1094 is called the impulse response of the windowing filter. Alternatively, we can obtain the series tex2html_wrap_inline1094 by passing tex2html_wrap_inline1092 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 tex2html_wrap_inline1106 for which tex2html_wrap_inline1108, and when used sequentially, in storage of all the values of tex2html_wrap_inline1110 for which tex2html_wrap_inline1112 and tex2html_wrap_inline1108. 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
equation38
which admits the recursive realization
 equation44
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
equation51

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
equation63
It is a little easier to see this if we recast the filter in the state space form
equation68
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 tex2html_wrap_inline1118, etc.

The DC gain is
equation108


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Next: Recursive Implementation of General Up: Recursive Windowing of Time Previous: Recursive Windowing of Time


Fri Jun 27 03:10:38 EDT 1997

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