Realization of Digital Filters Using Block- Floating-point Arithmetic

Recently, statistical models for the effects of roundoff noise in fixedpoint and floating-point realizations of digital filters have been proposed and verified, and a comparison between these realizations presented. In this paper a structure for implementing digital filters using block-floating-point arithmetic is proposed and a statistical analysis of the effects of roundoff noise is carried out. On the basis of this analysis, block-floating-point is compared to fixed-point and floatingpoint arithmetic with regard to roundoff noise effects. A Structure for Block-Floating-point Realization In block-floating-point arithmetic the input and filter states (Le., the outputs of the delay registers) are jointly normalized before the multiplications and adds are performed using fixed-point arithmetic. The scale factor obtained during the normalization is then applied to the final output to produce a fixed-point result. To illustrate, consider a first-order filter described by the difference equation Yn = 5, + a1yn-1. (1) For convenience we will treat all numbers as fixed-point fractions. To perform the computation in a block-floatingpoint manner, we define 1 A -~ ~. n ( 2 ) ~ ~ [ m a x { I X, 1 I yn-11 1 I where IP[M] is used to denote the integer power of two such that M<IP(M)_<2M, i.e., with M written as M=2m .P with P between + and 1, IP(M) = 2m. For M a fraction, 2’, is less than or equal to unity so that A, is greater than or equal to unity. Thus A , represents the power-of-two scaling which will jointly normalize x, and yn-l. Thus with block-floating-point we can compute y , as Manuscript received August 8, 1969. This work was sponsored by the Department of the Air Force. out in a fixed-point manner. 130 KEEE TRANSACTIONS OX AUDIO AND EL,ECTROACOCSTICS VOL. A W I 8, NO. 2 JUNE 1970 where the multiplications and addition in (3) are carried Because of the recursive nature of the computation for a digital filter, it is advantageous to modify (3) as 9, = Anz, + alAnWln (4)