Digital Filter Structures and Their Implementation

Abstract

In this chapter we discuss some important issues regarding the design, selection of filter structure, and implementation of FIR and IIR digital filters. First we discuss some basic properties of digital filters and our design target. Next, we discuss the design of FIR and IIR filters as well as common FIR structures. Low-sensitive IIR filter structures are based on their analog counterparts. Hence, we first discuss the corresponding analog approximation problem. We comparison of the standard approximations for lowpass filters and we find that Chebyshev II and Cauer approximations are the most favorable for realization using doubly resistively terminated lossless networks. We explain why these structures may have optimal sensitivity in the passband using the concept of maximal power transfer design. In addition, we discuss the errors in the elements in doubly terminated filters structures. The two most common structures are ladder and lattice structures. However, these lumped element structure cannot directly be converted to digital filter structures. We need first to map the lumped structure to a corresponding structure with only commensurate-length transmission lines. The corresponding digital filter structures are referred to as wave digital filters since voltages and currents in the analog filter are represented by incident and reflected waves in the digital filter. The wave digital filter inherits the sensitivity properties of its analog counterpart and it can therefore be implemented with simple coefficient values. We discuss in detail the basic concepts and the design procedure for design of wave digital filters. We also compare the sensitivities of ladder and lattice structures. Very high orders are required for FIR filters if the transition band is very small. An efficient approach, i.e., frequency response masking (FRM), to alleviate this problem is to use cascaded two FIR filters. The basic idea is to first design a low-order FIR filter with a transition band that is M times larger than the desired bandwidth. Next, each delay element is replaced with M delay elements. The filter has high order, but only every Mth coefficient is nonzero. The resulting frequency response will have M copies of the original frequency response. The undesired copies are the removed by the second filter. We discuss the design of both single and multistage FRM structures as well as special cases like half-band filters and Hilbert transformers. In addition, we discuss FRM structures for multirate filters, filter banks, and two-dimensional FRM structures. Next we discuss some computational properties of filter algorithms, e.g., latency, throughput, and the maximal sample rate bound. We show how this bound can be achieved using cyclic scheduling over several sample periods. We also discuss the realization of arithmetic operations, e.g., sum-of-products (SOP), multiple-constant multiplication (MCM), and distributed arithmetic. Finally, we discuss some power reduction techniques that are available for the filter designer. Many of the design examples are supplemented with MATLAB programs and references to additional available toolboxes are also provided.