Finite word effects in pipelined recursive filters

High sample rate recursive filtering can be achieved by transforming the original filters to higher-order filters using the scattered look-ahead computation technique (which relies upon pole-zero cancellation). Finite word-length implementation of these filters leads to inexact pole-zero cancellation. This necessitates a thorough study of finite word effects in these filters. Theoretical results on roundoff and coefficient quantization errors in these filters are presented. It is shown that to maintain the same error at the filter output, the word length needs to be at most increased by log₂ log₂ 2M bit for a scattered look-ahead decomposed filter (where as M is the level of loop pipelining). This worst case corresponds to the case when all poles are close to zero. For M between two and eight, the word length needs to be increased only by 1 or 2 bit. Contrary to common beliefs, it is concluded that pole-zero canceling scattered look-ahead pipelined recursive filters have good finite word error properties