This chapter introduces and reviews the important properties of the z-transform and shows how this transform is used to solve difference equations. From this development we obtain the system function and the discrete-time convolution, both useful for design, analysis, and synthesis. Three design methods are discussed - two for recursive filters and one for nonrecursive filters. This chapter then examines sources of error in filter responses, introduces the fast Fourier transform, and discusses its application to digital filtering.

Chapter Contents:

• 9.1 The Uniform Sampling Theorem
• 9.1.1 Theorem and Proof
• 9.1.2 Reconstruction of Time Function
• 9.1.3 Physical Interpretation of the Sampling Theorem
• 9.1.4 Interpolation Functions
• 9.2 Definition of a Digital Filter
• 9.3 The Difference Equation
• 9.3.1 Example of a Difference Equation Computation
• 9.3.2 Digital Filter Simulation of a First-Order Difference Equation
• 9.4 The z-Transform
• 9.4.1 Definitions and Properties
• 9.4.2 Mapping the s-Plane into the z-Plane
• 9.4.3 Frequency-Domain Characteristics
• 9.4.4 The Inverse z-Transform
• 9.5 Application of z-Transforms to Difference Equations
• 9.5.1 Classical Solution of Difference Equations
• 9.5.2 Solution by z-Transforms
• 9.5.3 System Function and Unit-Sample Response
• 9.5.4 Example of System Response Calculation
• 9.5.5 Frequency-Domain Functions
• 9.6 Introduction to Design Techniques
• 9.6.1 The Impulse-Invariant Method
• 9.6.2 Example Illustrating Impulse-Invariant Method
• 9.6.3 The Bilinear z-Transform
• 9.6.4 Example of Bilinear z-Transform Design
• 9.6.5 Nonrecursive Filters
• 9.6.6 Example of Nonrecursive Filter Computation
• 9.6.7 Fourier Series Approach to Nonrecursive Filtering
• 9.6.8 Example of the Fourier Series Design Technique
• 9.6.9 Example of Nonrecursive Filter Realizations
• 9.6.10 Window (Weighting) Functions
• 9.6.11 Example of MTI Filter Design
• 9.7 Digital Networks, Error Sources, and the FFT
• 9.7.1 Digital Networks
• 9.7.2 Example of Filter Realization in Parallel and Cascade Forms
• 9.7.3 Errors Due to Practical Hardware
• 9.7.4 The Fast Fourier Transform
• 9.7.5 Summary
• References
• Problems

Inspec keywords: Z transforms; recursive filters; difference equations; convolution; digital filters; fast Fourier transforms

Other keywords: nonrecursive filters; recursive filters; FFT; filter responses; system function; fast Fourier transform; discrete-time convolution; difference equations; digital filtering; z-transform

Subjects: Integral transforms; Signal processing theory; Integral transforms; Filtering methods in signal processing