1. 1 The Mathematics of Signal Processing - an Innovative Approach Peter Driessen Faculty of Engineering University of Victoria

2. 2 Outline u Introduction u Traditional course curriculum u Context and motivation u New course curriculum u Software Project u Conclusions

3. 3 Introduction u complex variables and z transforms may seem irrelevant to students u Context and motivation are needed u Thus a new approach: teach CV/ZT in context of digital filter design

4. 4 Outline u Introduction u Traditional course curriculum u Context and motivation u New course curriculum u Software Project u Summary

5. 5 Traditional course curriculum - signals and systems (discrete-time) u Z-transform definition and properties u Methods of taking inverse z-transforms –Long division –Partial fractions and tables u Solution of difference equations using z- transforms

6. 6 Traditional course curriculum - complex variables u Properties of functions of complex variable u Complex line and contour integrals u Convergence of sequences and series u Power series expansions u Residue theory

7. 7 Recall: complex inversion integral u Inverse z-transform using inversion integral u h[k]= int H(z)z^{k-1} dz u Different integral for each k u This is the connection between z transforms and complex variable theory

8. 8 Complex variable methods for taking inverse z-transforms u Inversion integral –Line integral along path –Residue theory u 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 u These methods incorporate most of the traditional complex variables course material

9. 9 Outline u Introduction u Traditional course curriculum u Context and motivation u New course curriculum u Software Project u Summary

10. 10 Complex variables and digital filters u Digital filter design –Select poles and zeros for desired transfer function H(z) –Take inverse z-transform to obtain impulse response h[k] u Complex variable theory is applied to taking inverse z-transforms and thus is motivated in context of digital filter design

11. 11 Context and motivation for complex variable theory u Design digital filter u Find impulse response using –Complex line integral –Residue theory –Laurent series expansion

12. 12 Context and motivation 2 u Obtain numerical results for different values of k for each of these 3 methods u Thus complex variable theory is used to obtain a useful and practical result: the impulse response of a digital filter

13. 13 Outline u Introduction u Traditional course curriculum u Context and motivation u New course curriculum u Software Project u Summary

14. 14 New course curriculum u Intro to applications of DSP u Discrete time systems –Linearity, time-invariance, difference equations, FIR/IIR, convolution u Z-transform – transfer function, solution of difference equations u inverse z-transforms –Complex variable methods: inversion integral, power series – Other methods: partial fractions, tables u Software project –Application to digital filter design

15. 15 Intro to applications of DSP u Digital audio and video –CD, DVD, MP3, MP4 u Digital control systems u Digital processing of images u Audio and video special effects

16. 16 Inverse z-transforms u Via definition: inversion integral »motivates complex contour integrals, integration along a path u Practical methods to simplify calculation –Residue theory –Power series expansion »Motivates sequences, series, convergence properties –Partial fractions, tables, long division

17. 17 Outline u Introduction u Traditional course curriculum u Context and motivation u New course curriculum u Software Project u Summary

18. 18 Software project u 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

19. 19 Digital filter design software u Implemented by 4th year project students

20. 20 Project task list 1 u Design filter: bandpass 2-pole 2-zero u Choose pole-zero locations for desired response and find H(z) u Plot frequency response (amplitude&phase) u Find difference equations from H(z) u Find impulse response by computer –IDFT of sampled frequency response –Iteration of difference equations

21. 21 Project task list 2 u 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

22. 22 Project task list 3 u 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 u 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

23. 23 Project task list 4 u Take DFT of impulse response to get frequency response –Choose DFT size to get desired freq resolution u Find filter output with given initial conditions and given input –Z-transform analysis and computer simulation

24. 24 Project task list 5 u Adaptive filter for which the center frequency changes linearly in response to a control signal input –Application: audio special effects u 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

25. 25 Outline u Introduction u Traditional course curriculum u Context and motivation u New course curriculum u Software Project u Summary

26. 26 Summary u Innovative approach to teaching complex variable theory: u Motivate the theory by digital filter design, and use the theory to analyze a digital filter u Project unifies all theory of the entire course in a single context u Students love the project