Although the differential equation is a basic system description, obtaining this equation can be tedious and time-consuming. Consequently for a time-invariant system this approach is avoided in practice, except in special cases. The Fourier and Laplace transforms offer an alternative approach for characterizing and analyzing these systems. Insight into system behavior is often obtained by the transform method. These transforms change a function of one variable into a function of another variable, and, when applied to problems in the physical sciences, the transform pairs and variables may correspond to physical quantities. We assign time and frequency as the transform variables because these are the variables associated with the filtering devices considered in this text.

Chapter Contents:

• 2.1 The Fourier Transform
• 2.1.1 Real Time Functions
• 2.1.2 Causal Time Functions
• 2.1.3 Symmetric Time Functions
• 2.1.4 Parseval's Theorem
• 2.2 The Laplace Transform
• 2.3 The Inverse Transform
• 2.4 Solution of Differential Equations by Laplace Transforms
• 2.5 The Transfer Function
• 2.5.1 Derivation
• 2.5.2 Poles and Zeros
• 2.5.3 Steady-State Responses
• 2.5.4 s-Plane Geometry
• 2.6 Group Delay and Phase Delay
• 2.6.1 Definitions
• 2.6.2 System Delay and Signal Distortion
• 2.6.3 Phase-Intercept Distortion
• 2.6.4 Modulated-Signal Delay
• 2.7 The Hilbert Transform
• References
• Problems

Inspec keywords: differential equations; Fourier transforms; filtering theory; time-frequency analysis; Laplace transforms

Other keywords: Fourier transform; time invariant system; Laplace transform; time-domain analysis; frequency-domain analysis; differential equation; filtering device

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