Digital Signal Processing

1. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory Digital Signal Processing Prof. George Papadourakis, Ph.D.

2. Z TRANSFORM AND DFTZ-Transform • Fourier Transformof a discrete time signal: Given a sequencex(n),its z transform is defined: Where zis a complex variable z=ejω • Thez transform does not converge for all sequences or for all values of z • The set of values ofz for which thez transform converges is calledregion of convergence • The properties of the sequencex(n)determines theregion of convergence ofX(z)

3. Z TRANSFORM AND DFTZ-Transform Finite-Length Sequences : FIR filters Convergencerequires : zmay take all values except : Region of convergence : Compute X(z) : z = 1 (ω=0) z = j (ω = π/2) z = -1 (ω = π) Unit circle inside Region of convergence

4. Introduction to Neural Networks Z TRANSFORM AND DFTZ-Transform In many casesX(z)is arational function: Ratio of polynomials Values ofzfor whichX(z)=0 Zeros of X(z) Values ofzfor whichX(z)=infinity Poles of X(z) • No poles of X(z)can occur within theregion of convergence (is bounded by poles) • Graphically display z transform by pole-zero plot Example: Compute theztransform of the sequencex(n)=anu(n)

5. Z TRANSFORM AND DFTZ-Transform • If |a|<1the unitcircleis included in the regionofconvergence, X(z)converges • ForcausalsystemsX(z)convergeseverywhere outside a circle passing through the polefarthestfrom the origin of the z plane.

6. Z TRANSFORM AND DFTZ-Transform

7. Z TRANSFORM AND DFTProperties of the Z-Transform Linearity: Shifting: Time scaling by a Complex Exponential Sequence : Convolution: Differentiation:

8. Z TRANSFORM AND DFTRelationship between Z-Transform and Laplace • If z=esT, s=d+jω • z=e(d+jω)T= edT ejωT • Then, • Stability: Polesshould be inside the unit circle • Stabilitycriterion: Finding the poles of the system • FIR digital filtersalways stable: Poles in origin

9. Z TRANSFORM AND DFTGeometric Evaluation of Fourier Transform • X(z)has M zeros at z=z1,z2,…,zM • X(z) has N poles at z=p1,p2,…,pN • We can writeX(z) in factored form: • Multiplying factorsX(z) can be written as a rational fraction: • This form is often used forgeneral filter design

10. Z TRANSFORM AND DFTGeometric Evaluation of Fourier Transform TheFourier transform or system function : • EvaluatingX(z) on the unit circle, z=ejω

11. Z TRANSFORM AND DFTGeometric Evaluation of Fourier Transform • From the pointz= ejωdrawvectorsto zerosandpoles • Magnitudesof vectors determinemagnitudeatω • Anglesdetermine phase Example :

12. Z TRANSFORM AND DFTInverse Z-Transform From the inverse z transform we get x(n) • Power series (long division) • Partial fraction expansion • Residue Theorem • Power series (long division) • X(z) can be writtenas rational fraction: • It can be extended into an infinite series in z-1by long division :

13. Z TRANSFORM AND DFTInverse Z-Transform Example: Find thefirst 4values of the sequence f(k) f(k)={0,2,6,14….} • The long division approach can be reformulated so x(n)can be obtained recursively:

14. Z TRANSFORM AND DFTInverse Z-Transform Partial fraction expansion : • If poles of X(z) first order (distinct) and N=M, • p(k): distinct poles, Ck partial fraction coef. • B0=a0/b0 • If N<M then B0= 0 • If N>M then by long division make N<=M

15. Z TRANSFORM AND DFTInverse Z-Transform • The coefficient Ck can be derived as: • If X(z) contains multiple poles extra terms are required - X(z) contains mth-order poles:

16. Z TRANSFORM AND DFTInverse Z-Transform Example: Find theinversez-transform :

17. Z TRANSFORM AND DFTInverse Z-Transform Residue Theorem IZT obtained by evaluating the contour integral: • Where Cis the path of integration enclosing all the poles of X(z). Cauchy’s residue theorem: • Sum of the residues of z n-1X(z) at all the poles inside C • Every residueCk, is associated with a pole atpk • m is the order of the pole at z=pk • For a first-orderpole:

18. Z TRANSFORM AND DFTInverse Z-Transform Example: Find theinverseztransform : Singlepole @ z=0.5, second-order pole @ z=1

19. Z TRANSFORM AND DFTInverse Z-Transform Combining the results we have: x(n)=2[(n-1)+(0.5)n] No need to use inverse tables!!! Comparison of the inverse z-transform • Power series: • Does not lead to a closed form solution, it is simple, easy computer implementation • Partial fraction, residue: • Closed form solution, • Need to factorize polynomial (find poles of X(z)) • May involve high order differentiation (multiple poles) • Partial fraction : Useful in generating the coefficients of parallel structures for digital filters. • Residuemethod : widely used in the analysis of quantization errors in discrete-time systems.

20. Z TRANSFORM AND DFTSolving Difference Equations Using Z-Transform The difference equation of interest (IIRfilters) is: The z-transform is: Transfer function is: If coefficients ai=0 (FIRfilter):

21. Z TRANSFORM AND DFTSolving Difference Equations Using Z-Transform Example: Find the output of the following filter: y(n) = x(n) + a y(n-1) Initial condition: y(-1) = 0 Input: x(n) = ejωn u(n) Using z transform: Y(z) = X(z) + a z-1 Y(z) Using partial fraction expansion:

22. Z TRANSFORM AND DFTDiscrete Fourier Transform (DFT) • Techniques for representing sequences: • Fourier Transform • Z-transform • Convolution summation • Threegood reasons to study DFT • It can be efficiently computed • Large number of applications • Filter design • Fast convolution for FIR filtering • Approximation of other transforms • Can be finitely parametrized • When a sequence is periodic orof finite duration, the sequence can be represented in a discrete-Fourier series • Periodic sequencex(n),periodN,

23. Z TRANSFORM AND DFTDiscrete Fourier Transform (DFT) • Remember: ejωperiodic with frequency2π • 2π kn / N = 2π n  k = N • N distinct exponentials • ,1/N just a scale factor • The DFTis defined as: • DFTcoefficientscorrespond to Nsamples of X(z) :

24. Z TRANSFORM AND DFTProperties of the DFT Linearity Ifx(n)andy(n)are sequences(N samples) then: a x(n) + b y(n)  a X(k) + b Y(k) Remember: x(n) andy(n) must beN samples, otherwise zerofill Symmetry Ifx(n)is a real sequence of N samplesthen: Re[X(k)] = Re [X(N-k)] Im[X(k)] = -Im[X(N-k)] |X(K)| = |X(N-k)| Phase X(k) = - Phase X(N-k) Ifx(n) is real and symmetric x(n) = x(N-n) then: X(K) ispurely real

25. Z TRANSFORM AND DFTProperties of the DFT Shifting Property Ifx(n) isperiodic then x(n)  X(k), x(n-n0)  X(k) e-j(2π/N) n0k Ifx(n)isnotperiodicthen time-shiftis created byrotatingx(n) circularlybyn0samples.

26. Z TRANSFORM AND DFTConvolution of Sequences Ifx(n), h(n)areperiodicsequencesperiod N, DFTs: y(n): circular convolution of x(n), h(n) Y(k) N-point DFT ofy(n) Linear convolutionhasinfinitesum.

27. Z TRANSFORM AND DFTConvolution of Sequences • Imagine onesequence around a circle Npoints. • Secondsequence around a circle Npoints but timedreversed Convolution: multiply values of2circles,shift multiply, shift, …… N times Example:

28. Z TRANSFORM AND DFTConvolution of Sequences

29. Z TRANSFORM AND DFTSectioned Convolution • Fast Convolution: Using DFT for 2 finite sequences • Evaluated Rapidly, efficientlywith FFT • N1+N2 > 30FastConvolutionmoreefficient • Direct Convolution: direct evaluation • L > N1+N2Addzeros to achieveL power of 2 • Sectioned Convolution • N1 >> N2,what to do? • L > N1+N2 ,inefficient andimpractical. Why? • Long sequencemust be availablebeforeconvolution • Practicalwaveforms : Speech,Radar - not available • No processing before entire sequence - Long delays • Solution: Sectioned Convolution • Overlap – Add • Overlap – Save

30. Z TRANSFORM AND DFTSectioned Convolution Overlap – Add • Long sequencex(n) infinite duration • Short sequenceh(n) N2duration • x(n) is sectionedN3 or L orM • Duration of each convolutionN3 + N2 – 1 (overlap)

31. Z TRANSFORM AND DFTSectioned Convolution

32. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory