Digital Signal Processing Applications( 310253)

1. Digital Signal ProcessingApplications(310253) UNIT-III Z-Transform Prof. Vina M. Lomte RMDSSOE,Warje 1/4/2020

2. 310253 Digital Signal Processing Applications • Teaching Scheme: Examination Scheme: • Theory: 3 Hrs/Week • In Semester Assessment: 30 Marks • End Semester Assessment: 70 Marks 1/4/2020

3. Course Objectives: · Study and understanding of representation of signals and systems. · To learn and understand different Transforms for Digital Signal Processing · Design and analysis of Discrete Time signals and systems · To Generate foundation for understanding of DSP and its applications like audio, Image, telecommunication and real world 1/4/2020

4. Syllabus • Definition of Z-Transform, ZT and FT, ROC, ZT properties, pole-zero plot, • Inverse Z-Transform, Methods, System function H(Z), Analysis of DT LTI • systems in Z-domain: DT system representation in time and Z domain. • Relationship of FT and ZT 1/4/2020

5. Teaching Plan 1/4/2020

6. Session 1 • Introduction • Why z-Transform? • Definition of Z-Transform, • Relationship ZT and FT 1/4/2020

7. What is ZT Real What is Z ? It is Z= x + iy Real

8. Whyz-Transform? • It is very simple method for analyzing system(by ROC properties) ex. LTI system • A generalization of Fourier transform • Why generalize it? • FT does not converge on all sequence Notation good for analysis • Bring the power of complex variable theory deal with the discrete-time signals and systems • The z-transform is a very important tool in describing and analyzing digital systems. •  It offers the techniques for digital filter design and frequency analysis of digital signals.

9. Definition • The z-transform of sequence x(n) is defined by Time Domain Frequency Domain Convert Fourier Transform • Let z = ej.

10. Relationship Between FT and ZT • The following Eq.(1) and (2) are FT and ZT, respectively. • Replacing Z with , ZT will become FT 1/4/2020

11. Session 2 • ROC • ZT properties, • pole-zero plot 1/4/2020

12. Definition of ROC • The region in which Z is valid • Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<, is called the region of convergence. ROC is centered on origin and consists of a set of rings.

13. Im r Re Example: Region of Convergence ROC is annual ring centered an on the origin.

14. Im 1 Re Stable Systems • A stable system requires that its Fourier transform is uniformly convergent. • Fact: Fourier transform is to evaluate z-transform on a unit circle. • A stable system requires the ROC of z-transform to include the unit circle.

15. x(n) . . . n -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 Example: A right sided Sequence A right hand sequence x(n) is one for which x(n)>=0 for all n<no where no is +ve or –ve but finite . If n0>=0 the resulting sequence is causal sequence . For such type of sequence ROC is entire z-plane except at z=0 All positive values

16. Example: A right sided Sequence For convergence of X(z), we require that

17. Im Im 1 1 a a a a Re Re Example: A right sided Sequence ROC for x(n)=anu(n) Which one is stable? ROC includes unit circle

18. -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 n . . . x(n) Example: A left sided Sequence A left hand sequence x(n) is one for which x(n)>=0 for all n<no where no is +ve or –ve but finite . If n0<=0 the resulting sequence is anticausal sequence . For such type of sequence ROC is entire z-plane except at z=∞ Negative Values

19. Example: A left sided Sequence For convergence of X(z), we require that

20. Im Im 1 1 a a a a Re Re Example: A left sided Sequence ROC for x(n)=anu(n1) Which one is stable?

21. Properties of ROC • A ring or disk in the z-plane centered at the origin. • The Fourier Transform of x(n) is converge absolutely iff the ROC includes the unit circle. • The ROC cannot include any poles • Finite Duration Sequences: The ROC is the entire z-plane except possibly z=0 or z=. • Right sided sequences: The ROC extends outward from the outermost finite pole in X(z) to z=. • Left sided sequences: The ROC extends inward from the innermost nonzero pole in X(z) to z=0.

22. 1/4/2020

23. if you need stability then the ROC must contain the unit circle. • If you need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. • If you need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. • If you need both, stability and causality, all the poles of the system function must be inside the unit circle. 1/4/2020

24. Pole and Zeros Represent z-transform as a Rational Function where P(z) and Q(z) are polynomials in z. Zeros: The values of z’s such that X(z) = 0 Poles: The values of z’s such that X(z) = 

25. Im a Re Example: A right sided Sequence ROC is bounded by the pole and is the exterior of a circle.

26. Im a Re Example: A left sided Sequence ROC is bounded by the pole and is the interior of a circle.

27. Im 1/12 Re 1/3 1/2 Example: Sum of Two Right Sided Sequences ROC is bounded by poles and is the exterior of a circle. ROC does not include any pole.

28. Im 1/12 Re 1/3 1/2 Example: A Two Sided Sequence ROC is bounded by poles and is a ring. ROC does not include any pole.

29. Im Re Example: A Finite Sequence N-1 zeros ROC: 0 < z <  ROC does not include any pole. N-1 poles Always Stable

30. BIBO Stability Bounded Input Bounded Output Stability If the input is bounded, we want the output is bounded too For limited input sequence its output should respectively limited

31. Sequence z-Transform ROC All z All z except 0 (if m>0) or  (if m<0) Z-Transform Pairs

32. Sequence z-Transform ROC Z-Transform Pairs

33. Signal Type ROC Finite-Duration Signals Causal Entire z-plane Except z = 0 Anticausal Entire z-plane Except z = infinity Two-sided Entire z-plane Except z = 0 And z = infinity Causal Infinite-Duration Signals |z| > r2 Anticausal |z| < r1 Two-sided r2 < |z| < r1

34. Some Common z-Transform Pairs Sequence Transform ROC 1. [n] 1 all z 2. u[n] z/(z-1) |z|>1 3. -u[-n-1] z/(z-1) |z|<1 4. [n-m] z-m all z except 0 if m>0 or ฅif m<0 5. anu[n] z/(z-a) |z|>|a| 6. -anu[-n-1] z/(z-a) |z|<|a| 7. nanu[n] az/(z-a)2 |z|>|a| 8. -nanu[-n-1] az/(z-a)2 |z|<|a| 9. [cos0n]u[n] (z2-[cos0]z)/(z2-[2cos0]z+1) |z|>1 10. [sin0n]u[n] [sin0]z)/(z2-[2cos0]z+1) |z|>1 11. [rncos0n]u[n] (z2-[rcos0]z)/(z2-[2rcos0]z+r2) |z|>r 12. [rnsin0n]u[n] [rsin0]z)/(z2-[2rcos0]z+r2) |z|>r 13. anu[n] - anu[n-N] (zN-aN)/zN-1(z-a) |z|>0

35. Z-Transform Properties: 1.Linearity • Notation • Linearity • Note that the ROC of combined sequence may be larger than either ROC • This would happen if some pole/zero cancellation occurs • Example:

36. Proof: According to defination of ZT Here x(n)=a1x1(n) + a2x2(n) Writing two terms separately we get, Here a1 & a2 are constants se we can take it outside the summation sign By comparing eqn 1 & 3 we get X(z) =a1X1(z)+a2X2(z) Hence proved ROC : the combined ROC is overlapped or intersection of the individual ROCs of X1(z) & X2(z)

37. 2. Time Shifting • Here no is an integer • If positive the sequence is shifted right • If negative the sequence is shifted left • The ROC can change the new term may • Add or remove poles at z=0 or z= • Example Here x(n-no) indicates that the sequence is shifted in the time domain by (-no) samples corresponds to multiplication by in the frequency domain

38. Proof Statement : if X(n) z Z(z) Then x(n-k) ) z Z-k X(z) ----- 1 Then Z{x(n-k)} = -----2 Now Z-n can be written as Z-(n-k) Z(x(n-k) = Since the limits of summation are in terms of n we can take Z-k outside of the summation Z(x(n-k) = --------3 Now put n-k=m on RHS the limit will change as follows At n=-∞ , -∞-k = m m=-∞ At n= +∞, ∞-k=m , m= ∞ Z{x(n-k)} = -----------4

39. Compare eqn 1 & 4 Z{x(n-k) = Z-k X(z) hence X(n) z Z(z) Similarly it can be shown that x(n-k) z Z-k X(z) = x(n-k) z z +k X(z) Here x(n-k) indicates that the sequence is shifted in time domain by (-k) samples corresponding to multiplication by z-k in the frequency domain ROC of z-k is same as that X(Z) except z=0 if k>0 and z=∞ if k>0

40. Example Find ZT of x( n) = (n-k) That means (n) Z 1 x(n-k) ) Z Z-1 X(z) hence (n-k) Z Z-k . 1

41. 3. Scaling in Z-domain (Multiplication by Exponential) • ROC is scaled by |zo| • All pole/zero locations are scaled • If zo is a positive real number: z-plane shrinks or expands • If zo is a complex number with unit magnitude it rotates • Example: We know the z-transform pair • Let’s find the z-transform of

42. 4. Differentiation • Example: We want the inverse z-transform of • Let’s differentiate to obtain rational expression • Making use of z-transform properties and ROC Multiplying the sequence in time domain by n is equivalent to multiplying the sequence the derivation of its ZT by –Z in the Z-domain

43. 5. Conjugation • Example

44. 6. Time Reversal • ROC is inverted • Example: • Time reversed version of Here x(-n) is the folded version of x(n) so,x(-n) is the time reverse signal thus the folding of signal in time domain is equivalent to replacing z by z-1 in the z-domain Replacing z by z-1 in the z-domain is called as inversion hence folding in the time domain is equivalent to the inversion in z-domain

45. 7. Convolution • Convolution in time domain is multiplication in z-domain • Example:Let’s calculate the convolution of • Multiplications of z-transforms is • ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a| • Partial fractional expansion of Y(z)

46. Linearity Overlay of the above two ROC’s

47. Shift

48. Multiplication by an Exponential Sequence

49. Differentiation of X(z)

50. Conjugation