Course Description

The objectives of this course are to cover the modelling and analysis of continuous and discrete signals using linear techniques. Topics covered include: a review of Laplace transforms; methods for the basic modelling of physical systems; discrete and continuous convolution; impulse and step response; transfer functions and filtering; the continuous Fourier transform and its relationship to the Laplace transform; frequency response and Bode plots; sampling; the Z-transform.

Topics

This skeleton outline is the core that is expected to be covered by the instructor in the actual course offering. Note that the order and emphasis may differ (within reasonable limits) from one offering to another.

• Basic System Properties: causality, memory, time invariance, discrete time, continuous time, linearity, concept of state. Test inputs. Some comments on modeling.
• Time-domain Analysis of Systems (Discrete time and Continuous time): Superposition, unit step response, unit pulse response, response to arbitrary inputs. Convolution. Steady state and transient response. Discrete time approximation of continuous time systems by impulse response invariance, digital filters.
• Finite Order Systems in Time Domain: differential and difference equation representation of systems -- mechanical, thermal, electric, financial. Review of differential and difference equation solution (homogeneous and particular solution). Zero input and zero state responses, impulse (pulse) response a sum of complex exponentials.
• Frequency Domain (Fourier) Analysis of Discrete and Continuous Systems: Review of the complex exponential and rectangular and polar representations of sinusoids, phasors, negative frequency. Basis sets and linearity, the complex exponential as eigenfunction of convolution. The continuous-time and discrete-time Fourier Series -- Complex Fourier series and Fourier pair, frequency makeup of periodic signals, synthesis of periodic signals, existence and Gibbs phenomenon, response of linear systems to periodic inputs. Multiplication/convolution, symmetries, representation of impulse train in time, time/frequency inverse scaling, resolution/bandwidth.
• The Continuous-Time and Discrete-Time Fourier Transform --- Relation to Fourier Series. Fourier transform pairs, frequency content of signals, frequency response linked to impulse response, multiplication/convolution, duality, rectangular pulses, sinc functions, linear phase shifts (delays, offset tones), time/frequency scaling, windowing, resolution/extent, symmetries, transform of periodic signals, Parseval, impulse train, introduction to spectral analysis using the DFT.
• Sampled Signals and Fourier Representations -- Sampling by multiplication with impulse train, periodic spectra, sampling theorem, aliasing, antialiasing and reconstruction filters, interpretation through duality. Relationships between continuous and discrete frequencies in sampled systems.
• Laplace Transform Analysis: Motivation as variant of Fourier transform. Region of Convergence. Forced convergence of Fourier transform, generalization to s-plane, region of convergence for exponentials, sinusoids. Inversion of Laplace Transforms. Initial and final value theorems. Convolution/Transfer Functions. Laplace transform of differential equations, rational polynomial transfer functions. The effects of feedback on transfer functions.
• Z-Transform Analysis: Definition as a generalization of the DTFT. Pole locations and the region of convergence. Basic Properties of Z-Transform. Frequency response from z-transform; normalized frequency. Initial and final value theorems. Inversion of Z-Transforms -- power series method, long division method, residue method, partial fractions. Solving Difference Equations by Z-Transform. Zero state and zero input responses. Links Between Laplace Transform s and Z-Transform z. Mapping between planes. The effects of feedback in the discrete context.

Software Tools

• MatLab is used heavily in this course.

Labs

The lab component is intended to reinforce the analytical topics and give you direct experience with concepts such as linearity, impluse response, spectral analysis and filtering. There are two labs in this course, one on continuous time signals/systems and another on discrete-time signals/systems.

Prerequisites

Successful completion of MATH 310 is required for students wishing to take this course. Students with credit for ENSC 281 or 382 cannot take this course for further credit. This course must be taken in conjunction with ENSC 320-3.

Additional Information For ENSC 380-3