Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision

Abstract

Abstract-The worst-case peak gain (WCPG) of an LTI filter is an important measure for the implementation of signal processing algorithms. It is used in the error propagation analysis for filters, thus a reliable evaluation with controlled precision is required. The WCPG is computed as an infinite sum and has matrix powers in each summand. We propose a direct formula for the lower bound on truncation order of the infinite sum in dependency of desired truncation error. Several multiprecision methods for complex matrix operations are developed and their error analysis performed. We present a multiprecision complex matrix inversion algorithm using Newton-type iteration, along with its error analysis and proof of convergence. A multiprecision matrix powering method is presented. All methods yield a rigorous solution with an absolute error bounded by an a priori given value. The results are illustrated with numerical examples.

Towards Hardware IIR Filters Computing Just Right: Direct Form I Case Study

“…where sign(x) returns ±1 or 0 depending on the sign of x. This work computes the WCPGs with arbitrary precision using the reliable algorithm presented in [2] and its fast but rigorous implementation 3 . It also builds worst-case signals implementing 7to test the resulting architectures.…”

“…In other words, the internal format adds 1 + log 2 H ε LSB guard bits to the output format. The implementation of this error analysis actually uses a guaranteed overestimation of H ε [2]. This ensures that rounding errors in the the computation of H ε itself do not jeopardize the accuracy.…”

“…Equation (1) or (2), along with a definition of each coefficient a i and b i , constitute the mathematical specification of the filter. This article deals with the implementation of such a specification as fixed-point hardware operating on low-to moderate-precision data (typically 8 to 24 bits).…”

“…For this, the precision of the output must also specify its accuracy. In this work, the following specification is used: An implemented filter must behave as if each result was computed with infinite accuracy with respect to (2), then converted only once to the low-precision output format, with an error smaller than the weight of its least significant bit (LSB).…”

“…The main contribution of this article is to show that for such last-bit accurate implementations, near-optimal values of architectural parameters such as coefficient quantization and datapath bit-widths can be automated. This is achieved thanks to a complete and rigorous analysis of the rounding errors and their amplification through the feedback loop, building upon recent work on reliably bounding error propagation through filters [2], [3].…”