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The Mathematics of Signal Processing - an Innovative

Approach

Peter Driessen

Faculty of Engineering

University of Victoria

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Outline

Introduction Traditional course curriculum Context and motivation New course curriculum Software Project Conclusions

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Introduction

complex variables and z transforms may seem irrelevant to students

Context and motivation are needed Thus a new approach: teach CV/ZT in

context of digital filter design

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Outline

Introduction Traditional course curriculum Context and motivation New course curriculum Software Project Summary

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Traditional course curriculum - signals and systems (discrete-time)

Z-transform definition and properties Methods of taking inverse z-transforms

– Long division– Partial fractions and tables

Solution of difference equations using z-transforms

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Traditional course curriculum - complex variables

Properties of functions of complex variable Complex line and contour integrals Convergence of sequences and series Power series expansions Residue theory

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Recall: complex inversion integral

Inverse z-transform using inversion integral h[k]= int H(z)z^{k-1} dz Different integral for each k

This is the connection between z transforms and complex variable theory

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Complex variable methods for taking inverse z-transforms

Inversion integral– Line integral along path– Residue theory

Series expansions– Laurent series in negative powers of z– Defined radius of convergence

» Find using ratio test or root test used to test the convergence of series

These methods incorporate most of the traditional complex variables course material

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Outline

Introduction Traditional course curriculum Context and motivation New course curriculum Software Project Summary

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Complex variables and digital filters

Digital filter design – Select poles and zeros for desired transfer

function H(z)– Take inverse z-transform to obtain impulse

response h[k] Complex variable theory is applied to

taking inverse z-transforms and thus is motivated in context of digital filter design

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Context and motivation for complex variable theory

Design digital filter Find impulse response using

– Complex line integral– Residue theory– Laurent series expansion

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Context and motivation 2

Obtain numerical results for different values of k for each of these 3 methods

Thus complex variable theory is used to obtain a useful and practical result: the impulse response of a digital filter

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Outline

Introduction Traditional course curriculum Context and motivation New course curriculum Software Project Summary

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New course curriculum Intro to applications of DSP Discrete time systems

– Linearity, time-invariance, difference equations, FIR/IIR, convolution

Z-transform– transfer function, solution of difference equations

inverse z-transforms– Complex variable methods: inversion integral, power

series– Other methods: partial fractions, tables

Software project– Application to digital filter design

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Intro to applications of DSP

Digital audio and video– CD, DVD, MP3, MP4

Digital control systems Digital processing of images Audio and video special effects

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Inverse z-transforms

Via definition: inversion integral» motivates complex contour integrals, integration

along a path

Practical methods to simplify calculation– Residue theory– Power series expansion

» Motivates sequences, series, convergence properties

– Partial fractions, tables, long division

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Outline

Introduction Traditional course curriculum Context and motivation New course curriculum Software Project Summary

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Software project

Everything about a 2-pole 2-zero digital filter– Design: choose pole-zero locations– Analyze: find impulse response– Implement in software– Test and compare results with analysis

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Digital filter design software

Implemented by 4th year project students

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Project task list 1

Design filter: bandpass 2-pole 2-zero Choose pole-zero locations for desired

response and find H(z) Plot frequency response (amplitude&phase) Find difference equations from H(z) Find impulse response by computer

– IDFT of sampled frequency response– Iteration of difference equations

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Project task list 2

Find impulse response by analysis– Inversion integral, integration along path– Inversion integral, residue theory– Laurent series expansion

» Find ROC using ratio and root test

– Long division– Partial fractions

» First order factors, quadratic factors

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Project task list 3 Prepare table with 9 columns for k and 8 methods of

finding h[k]» Observe that the algebraic formulas for h[k] may be different for each

method, but the numbers h[k] are the same

Test bandpass filter: – sinusoidal input

» Observe amplitude and phase shift

– Multiple sine waves» Observe only one sine wave output

– Sine wave above Nyquist rate» Observe aliasing

– Audio input: voice, music» Observe qualitative change in sound

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Project task list 4

Take DFT of impulse response to get frequency response– Choose DFT size to get desired freq resolution

Find filter output with given initial conditions and given input– Z-transform analysis and computer simulation

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Project task list 5

Adaptive filter for which the center frequency changes linearly in response to a control signal input– Application: audio special effects

Tests understanding of the relationship between– the filter coefficients a1,a2,b0,b1,b2 in the difference

equation and

– the pole-zero locations p1,p2,z1,z2 in the transfer funcction

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Outline

Introduction Traditional course curriculum Context and motivation New course curriculum Software Project Summary

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Summary

Innovative approach to teaching complex variable theory:

Motivate the theory by digital filter design, and use the theory to analyze a digital filter

Project unifies all theory of the entire course in a single context

Students love the project