Contents 3 THE Z-TRANSFORM AND ITS APPLICATION TO THE ANALYSIS OF LTI SYSTEMS 151 3.1 The :-Transform 151 3.1.1 The Direet :-Transform.152 3.1.2 The inverse :-Transform,160 3.2 Properties of the z-Transform 161 3.3 Rational <-Transforms 172 3.3.1 Poles and Zeros,172 3.3.2 Pole Location and Time-Domain Behavior for Causal Signals.178 3.3.3 The System Function of a Linear Time-Invariant System.181 3.4 Inversion of the =-Transform 184 3.4.1 The Inverse :-Transform by Contour Integration.184 3.4.2 The Inverse :-Transform by Power Senes Expansion.186 3.4.3 The Inverse :-Transform by Partial-Fraction Expansion.188 3.44 Decomposition of Rational =-Transforms.195 3.5 The One-sided :-Transform 197 3.5.1 Definition and Properties.197 3.5.2 Solution of Difference Equations.201 3.6 Analysis of Linear Time-Invariant Systems in the :-Domain 203 3.6.I Response of Systems with Rational System Functions.203 3.6.2 Response of Pole-Zero Systems with Nonzero Initial Conditions.204 3.6.3 Transient and Steady-State Responses.206 3.6.4 Causality and Stabilty.208 3.6.5 Pole-Zero Cancellations.210 3.6.6 Multiple-Order Poles and Stabihty.211 3.6.7 The Schur-Cohn Stability Test.213 3.68 Stability of Second-Order Systems.215 3.7 Summary and References 219 Problems 220 4 FREQUENCY ANALYSIS OF SIGNALS AND SYSTEMS 230 4.1 Frequency Analysis of Continuous-Time Signals 230 4.1.1 The Fourier Series for Continuous-Time Periodic Signals.232 4.12 Power Density Spectrum of Periodic Signals.235 4.1.3 The Fourier Transform for Continuous-Time Aperiodic Signals,240 4.1.4 Energy Density Spectrum of Aperiodic Signals.243 4.2 Frequency Analysis of Discrete-Time Signals 247 4.2.1 The Fourier Series for Discrete-Time Periodic Signals.247