Applied Digital Signal Processing Theory and Practice

As we discussed in Chapter 2, any LTI can be implemented using three basic computational elements: adders, multipliers, and unit delays. For LTI systems with a rational system function, the relation between the input and output sequences satisfies a linear constant-coefficient difference equation. Such systems are practically realizable because they require a finite number of computational elements. In this chapter, we show that there is a large collection of difference equations corresponding to the same system function. Each set of equations describes the same input-output relation and provides an algorithm or structure for the implementation of the system. Alternative structures for the same system differ in computational complexity, memory, and behavior when we use finite precision arithmetic. In this chapter, we discuss the most widely used discrete-time structures and their implementation using Matlab. These include direct-form, transposed-form, cascade, parallel, frequency sampling, and lattice structures.

Study objectives

After studying this chapter you should be able to:

• Develop and analyze practically useful structures for both FIR and IIR systems.

• Understand the advantages and disadvantages of different filter structures and convert from one structure to another.

• Implement a filter using a particular structure and understand how to simulate and verify the correct operation of that structure in Matlab.

Block diagrams and signal flow graphs

Every practically realizable LTI system can be described by a set of difference equations, which constitute a computational algorithm for its implementation.