Fundamentals of Signals, Systems and Filtering

Transcript

1 Fundamentals of Signals, Systems and Filtering Brett Ninness c , Brett Ninness, School of Electrical Engineering and Computer Science The University of Newcastle, Australia.

2 2 c Brett Ninness

3 Contents 1 Introduction 7 2 Signal Types and Operations Types of Signals Constant Signal Unit Step Signal Ramp Signal Pulse Signal (Dirac) Delta Function Periodic Signal Even and Odd Signals Exponential Signals Exponential Signals: Complex Representations Operations on Signals Magnitude Scaling Time Shifting (Translation) Time Reversal (Flipping) Time Scaling Concluding Summary Exercises A Some Mathematical Background A.1 The limit of a function A.2 Continuous Functions Modelling, Differential Equations and System Properties Electrical Circuit Modelling Examples Mechanical System Modelling Examples General Differential Equation Models Effect of constants Solution of First Order Differential Equations State Space Descriptions A 2nd Order Case The General Case State Transformations and Canonical Forms The presence of a Feed-through term Direct State-Space Modelling

4 4 c Brett Ninness 3.6 Differential Equation Solution via State-Space Formulation Initial Conditions Observability Controllability Forced and Natural Response Impulse Response Invariance of Impulse Response Convolution Properties of Convolution Signal Convolution Examples System Properties Linearity/Non-linearity Causality Memory Time Invariance System Dimension Stability Concluding Summary Exercises A Some Mathematical Background A.1 Fundamental Results of Calculus A.2 Eigen-analysis and Matrix Decomposition Laplace Transform Analysis Laplace Transform Definition and Interpretation Geometric Interpretation Computation of Laplace Transforms Dirac Delta δ(t) Step 1(t) Ramp r(t) Exponential e αt Polynomial times Exponential t n e αt Cosine cos ωt Sinusoid sin ωt Exponentially Damped Cosine e αt cos ωt and Sine e αt sin ωt Properties of Laplace Transforms Linearity Time Shifting Time Scaling Convolution Differentiation Integration Transform of a Product Multiplication by e αt Multiplication by t Initial Value Theorem Final Value Theorem

5 DRAFT & INCOMPLETE Version Summary of Properties Inversion of Laplace Transforms Contour Integral Formulation Evaluation by Residue Calculation The Case of Isolated Poles The Case of Repeated Poles Dealing with the Bi-proper Proper Case Causality of the Inverse Transform Dealing with Time Translations Inversion by Partial Fraction Expansion Differential Equation Solution by Laplace Transform Methods More on Initial Conditions The System Transfer Function Stability and the Transfer Function Transfer Function and Impulse Response Relationship Direct Transform Modelling of Electrical Circuits Kirchoff s Laws Parallel and Series Combinations Voltage Division Examples of Direct Laplace Transform Modelling Concluding Summary Exercises Frequency Response, Fourier Analysis and Filter Design System Frequency Response and System Transfer function The Fourier Transform Inverse Fourier Transform Properties of the Fourier Transform Fourier Series Relationship between Fourier Series and the Fourier Transform Properties of Fourier Series Co-efficients Bode Frequency Response Diagrams Straight line approximation of Bode magnitude Straight Line Approximation of Bode Phase Filter Design Constraints on Achievable Performance Filter Types and Brick Wall Specification Butterworth Low-Pass Approximation Chebychev Low-pass Approximation High-pass Filter Approximation Band-Pass and Band-Stop filter design Concluding Summary Exercises A Mathematical Background A.1 Dirac Delta Interpretation of Integrated Exponential A.2 Spectral Factorisation for Chebychev Filter A.3 The Law of Sines

6 6 c Brett Ninness 6 Sampling, Reconstruction and the Discrete Time Fourier Transform Sampling Spectral relationship between a signal and its sampled version The Nyquist Sampling Criterion Reconstruction Aliasing The Discrete Time Fourier Transform The Inverse Discrete Time Fourier Transform Normalised Frequency Properties of the Discrete Time Fourier Transform Properties inherited from the continous time Fourier Transform Further Properties The Discrete Fourier Transform (DFT) The Inverse Discrete Fourier Transform (IDFT) Relationship between DFT and DTFT Properties of the DFT Properties equivalent to those of DTFT Relationship of DFT to Fourier Transform - Data Windowing The Fast Fourier Transform (FFT) Exercises A Mathematical Background A.1 Summation of a Geometric Series A.2 Inverse of Ω N Z Transforms Introduction Relationship between Z Transforms and Laplace Transforms Calculation of Z-Transforms Properties of ZTransforms Inverting ZTransforms Using Contour Integration