Further, let hj denote the impulse response of the system function of the j '-stage up to and including the last stage. The functional representation of hj is derived and discussed in detail in the appendix. The error introduced at the jth-stage propagates to the output shaped by the impulse response, hj. In particular, the mean of the error at the output of the SCIC filter due to truncation at the j '-stage is
is the "mean error gain" for the jth error source due to truncation.
Similarly, the variance at the output of the SCIC due to truncation at the j '-stage is
where the sub-T stands for "total" and
is the 'variance error gain' corresponding to the jth error source.
Combining the results above, the total mean and variance of the error at the output are given by
Based on (23) and (24), the design criterion for the register length of each of the 6N filter stages is:
i) Make the variance due to truncation of the first 6N error sources at least as small as the error introduced by truncating the output.
ii) Distribute the accumulating truncation error equally among the first 6N sources.
Use of these criteria implies,
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