Floating-Point-Roundoff Accumulation in Digital-Filter Realizations
Irwin W. Sandberg · Bell System Technical Journal · 1967
In this paper, several results are presented concerning the effects of roundoff in the floating-point realization of a general discrete filter governed ideally by a stable difference equation of the form$w_{n}\ =\ \sum_{k=0}^{M}\ b_{k}_{n-k}\ -\ \sum_{k=1}^{N}\ a_{k}w_{n-k}, \quad n \geqq N \qquad \eqno{\hbox{(1)}}$in which {wn} and {xn} are output and input sequences, respectively. In particular, for a large class of filters it is proved that there is a function f(K) with f(K) → 0 as K → ∞ and a constant c, both dependent on the bk, the ak, the order in which the products on the right side of (1) are summed in the machine, and t, the number of bits allotted to the mantissa, such that$\langle e \rangle _{k} \leqq c \langle y \rangle_{k}+f(K)$for all K ≧ N, in which, with {yn} the computed output sequence of the realized filter,$\langle y \rangle_{k}=\left({1 \over k+1} \sum_{n=0}^{k}\vert y_{n}\vert^{2}\right)^{{1 \over 2}}$and$\langle e \rangle_{k}=\left({1 \over k+1} \sum_{n=0}^{k}\vert w_{n}- y_{n}\vert^{2}\right)^{{1 \over 2}}$Bounds on f(K) and c are given that are not difficult to evaluate, and which, in many realistic cases, are informative. For example, for the second-order bandpass filter:$w_{n}=x_{n}a_{1}w_{n-1}\ - \ a_{2}w_{n-2}, \quad n \geqq 2 \qquad \eqno{\hbox{(2)}}$with a1and a2chosen so that its poles are at approximately ± 45° and at distance approximately (but not less than) 0.001 from the unit circle, we find that c, an upper bound on the “asymptotic output error-to-signal ratio”, is not greater than 0.58 × 10−4assuming that t = 27, that the terms on the right side of (2) are summed in the machine in the order indicated from right to left), and that the x_{n} in (2) are machine numbers. I f the x_{n} are not machine numbers, and hence must be quantized before processing, then c ≦ 0.76 × 10−4. In addition to error bounds, an inequality is derived which, if satisfied, rules out certain types of generally undesimble behavior such as self-sustained output limit cycles due to roundoff effects. This inequality is satisfied for the example described above.