The Mathematics of Signal Processing - an Innovative Approach

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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
• hk int H(z)zk-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 hk
• 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 (amplitudephase)
• 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 hk
• Observe that the algebraic formulas for hk may
  be different for each method, but the numbers
  hk 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