# The Mathematics of Signal Processing - an Innovative Approach

Document Sample

The Mathematics of Signal
Processing - an Innovative
Approach
Peter Driessen
Faculty of Engineering
University of Victoria

1
Outline
 Introduction
curriculum
 Context and motivation
 New course curriculum
 Software Project
 Conclusions

2
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

3
Outline
 Introduction
curriculum
 Context and motivation
 New course curriculum
 Software Project
 Summary

4
signals and systems (discrete-time)
 Z-transform definitionand properties
 Methods of taking inverse z-transforms
– Long division
– Partial fractions and tables
 Solution of difference equations using z-
transforms

5
complex variables
 Properties offunctions of complex variable
 Complex line and contour integrals
 Convergence of sequences and series
 Power series expansions
 Residue theory

6
Recall: complex inversion integral
 Inverse z-transform using inversion integral
 h[k]= int H(z)z^{k-1} dz
 Different integral for each k

 Thisis the connection between z transforms
and complex variable theory

7
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
» 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

8
Outline
 Introduction
curriculum
 Context and motivation
 New course curriculum
 Software Project
 Summary

9
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

10
Context and motivation for
complex variable theory
 Design digital filter
 Find impulse response using
– Complex line integral
– Residue theory
– Laurent series expansion

11
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

12
Outline
 Introduction
curriculum
 Context and motivation
 New course curriculum
 Software Project
 Summary

13
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                  14
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

15
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

16
Outline
 Introduction
curriculum
 Context and motivation
 New course curriculum
 Software Project
 Summary

17
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

18
Digital filter design software
 Implemented by   4th year project students

19
 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

20
 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

21
   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                         22
 TakeDFT of impulse response to get
frequency response
– Choose DFT size to get desired freq resolution
filter output with given initial
 Find
conditions and given input
– Z-transform analysis and computer simulation

23
   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

24
Outline
 Introduction
curriculum
 Context and motivation
 New course curriculum
 Software Project
 Summary

25
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

26

DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 6 posted: 9/8/2010 language: English pages: 26