On the Interaction of Roundoff Noise and Dynamic Range in Digital Filters*

Abstract

The interaction between the roundoff-noise output jrom a digital filter and the associated dynamic-range limitations is investigated for the case of uncorrelaled rounding errors from sample to sample and from one error source to another. The required dynamic-range constraints are derived in terms of L, norms of the input-signal spectrum and the transfer responses to selected nodes within the filter. The concept of "transpose configurations" is introduced ond is found to be quite useful in digitd-fiUer synthesis; for although such configurations have identical transfer functions, their round off-noise outputs and dynamic-range limitations can be quite different, in general. Two transpose configurations for the direct form of a digital filler are used to illustrate these results.

Delta operator realizations of direct-form IIR filters

“…The key simplification is to model the quantization error statistically as an additive noise sequence [24], [25]. The error sequence is modeled as a white noise process uncorrelated with the input signal It is also assumed that roundoff noise introduced at one node is uncorrelated with that introduced at any other node in the system.…”

“…The validity of this stochastic noise model is discussed, e.g., in [26]. If quantization is done by rounding, the process has a zero mean and the variance is [24] (6) where is the total word length, including the sign bit. It was mentioned earlier that in delta realizations using fixedpoint arithmetic, enhanced precision inverse delta operations are required.…”

“…The optimum value of the parameter can be found using the procedure presented in the Appendix. In the examples we have looked at, the optimum turned out to be (24) where . Thus, we conjecture that this will be optimal for the narrow-band low-pass filters using scaling.…”

Wordlength optimization for linear digital signal processing

“…p i ; n j 0 p j ) + p o . We may express this as (25) and (26), since prequantization wordlengths only appear with negative coefficient in the error terms, so the error constraints can be relied upon to reduce n q o to achieve equality n q o n i 0 p i + p o (25) n q o n j 0 p j + p o : (26) …”

“…It should be noted that even for the uniform wordlength structures, Synoptix has been used to automatically insert power-of-two scaling [26], which is good practice in DSP design. Also, note that for both uniform and multiple wordlength structures, these circuits represent a completely unpipelined implementation of the specification, in order to aid direct comparison of maximum clock rate f clk reported.…”