6.003: Signals and Systems

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1 6.003: Signals and Systems Z Transform September 22,

2 2 Concept Map: Discrete-Time Systems Multiple representations of DT systems. Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

3 3 Concept Map: Discrete-Time Systems Relations among representations. Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

4 4 Concept Map: Discrete-Time Systems Two interpretations of Delay. Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

5 5 Concept Map: Discrete-Time Systems Relation between System Functional and System Function. Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... R 1 z Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

6 Check Yourself What is relation of System Functional to Unit-Sample Response Block Diagram Delay R System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Difference Equation y[n] = x[n] + y[n 1] + y[n 2] 6 System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

7 7 Check Yourself Expand functional in a series: Y 1 X = H(R) = 1 R R 2 H(R) = 1 +R +2R 2 +3R 3 +5R 4 +8R R R R R 2 R +R 2 R R 2 R 3 2R 2 +R 3 2R 2 2R 3 2R 4 3R 3 +2R 4 3R 3 3R 4 3R R R 2 = 1 + R + 2R2 + 3R 3 + 5R 4 + 8R R 6 +

8 8 Check Yourself Coefficients of series representation of H(R) 1 H(R) = 1 R R 2 = 1 + R + 2R2 + 3R 3 + 5R 4 + 8R R 6 + are the successive samples in the unit-sample response! h[n] : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... If a system is composed of (only) adders, delays, and gains, then H(R) = h[0] + h[1]r + h[2]r 2 + h[3]r 3 + h[4]r 4 + = h[n]r n n We can write the system function in terms of unit-sample response!

9 9 Check Yourself What is relation of System Functional to Unit-Sample Response? Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 H(R) = h[n]r n index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

10 Check Yourself What is relation of System Function to Unit-Sample Response? Block Diagram Delay R System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 H(R) = h[n]r n index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Difference Equation y[n] = x[n] + y[n 1] + y[n 2] 10 System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

11 11 Check Yourself Start with the series expansion of system functional: H(R) = h[n]r n n 1 Substitute R : z n H(z) = h[n]z n

12 12 Check Yourself What is relation of System Function to Unit-Sample Response? Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 H(R) = h[n]r n index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... H(z) = h[n]z n Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

13 13 Check Yourself Start with the series expansion of system functional: H(R) = h[n]r n n 1 Substitute R : z n H(z) = h[n]z n Today: thinking about a system as a mathematical function H(z) rather than as an operator.

14 14 Z Transform We call the relation between H(z) and h[n] the Z transform. n H(z) = h[n]z n Z transform maps a function of discrete time n to a function of z. Although motivated by system functions, we can define a Z transform for any signal. n X(z) = x[n]z n= Notice that we include n < 0 as well as n > 0 bilateral Z transform (there is also a unilateral Z transform with similar but not identical properties).

15 15 Simple Z transforms Find the Z transform of the unit-sample signal. δ[n] n x[n] = δ[n] n X(z) = x[n]z = x[0]z 0 = 1 n=

16 16 Simple Z transforms Find the Z transform of a delayed unit-sample signal. x[n] n x[n] = δ[n 1] n 1 1 X(z) = x[n]z = x[1]z = z n=

17 17 Check Yourself What is the Z transform of the following signal. x[n] = ( 7 8 ) n u[n] n z z 1 3. z z 4. z z 1 5. none

18 18 Check Yourself What is the Z transform of the following signal. x[n] = ( 7 8 ) n u[n] n n n 7 n 7 n 1 X(z) = z u[n] = z = n= n=0 8 z

19 19 Check Yourself What is the Z transform of the following signal. 2 x[n] = ( 7 8 ) n u[n] n z z 1 3. z z 4. z z 1 5. none

20 20 Z Transform Pairs The signal x[n], which is a function of time n, maps to a Z transform X(z), which is a function of z. ( ) n 7 1 x[n] = u[n] X(z) = z For what values of z does X(z) make sense? The Z transform is only defined for values of z for which the defining sum converges. ( n n 7 ) ( n 7 ) n 1 X(z) = z u[n] = z = 8 8 n= 1 7 n=0 8 z 1 7 Therefore 8 z 1 < 1, i.e., z > 7 8.

21 21 Regions of Convergence The Z transform X(z) is a function of z defined for all z inside a Region of Convergence (ROC). ( ) n x[n] = u[n] X(z) = z > ; 8 8 z 1 7 ROC: z > 8

22 22 Z Transform Mathematics Based on properties of the Z transform. Linearity: if x 1 [n] X 1 (z) for z in ROC 1 and x 2 [n] X 2 (z) for z in ROC 2 then x 1 [n] + x 2 [n] X 1 (z) + X 2 (z) for z in (ROC 1 ROC 2 ). Let y[n] = x 1 [n] + x 2 [n] then n Y (z) = y[n]z n= n = (x 1 [n] + x 2 [n])z n= n n = x 1 [n]z + x 2 [n]z n= n= = X 1 (z) + X 2 (z)

23 23 Delay Property If x[n] X(z) for z in ROC then x[n 1] z 1 X(z) for z in ROC. We have already seen an example of this property. δ[n] 1 δ[n 1] z 1 More generally, n X(z) = x[n]z n= Let y[n] = x[n 1] then n n Y (z) = y[n]z = x[n 1]z n= n= Substitute m = n 1 Y (z) = x[m]z m 1 = z 1 X(z) m=

24 24 Rational Polynomials A system that can be described by a linear difference equation with constant coefficients can also be described by a Z transform that is a ratio of polynomials in z. b 0 y[n] + b 1 y[n 1] + b 2 y[n 2] + = a 0 x[n] + a 1 x[n 1] + a 2 x[n 2] + Taking the Z transform of both sides, and applying the delay property b 0 Y (z)+b 1 z 1 Y (z)+b 2 z 2 Y (z)+ = a 0 X(z)+a 1 z 1 X(z)+a 2 z 2 X(z)+ a 0 + a 1 z 1 + a 2 z 2 + H(z) = Y (z) = X(z) b 0 + b 1 z 1 + b 2 z 2 + a 0 z k + a 1 z k 1 + a 2 z k 2 + = b0 z k + b 1 z k 1 + b 2 z k 2 +

25 Rational Polynomials Applying the fundamental theorem of algebra and the factor theorem, we can express the polynomials as a product of factors. H(z) = a 0z k + a 1 z k 1 + a 2 z k 2 + b 0 z k + b 1 z k 1 + b 2 z k 2 + (z z 0 ) (z z 1 ) (z z k ) = (z p0 ) (z p 1 ) (z p k ) where the roots are called poles and zeros. 25

26 26 Rational Polynomials Regions of convergence for Z transform are delimited by circles in the Z-plane. The edges of the circles are at the poles. Example: x[n] = α n u[n] n n X(z) = α n u[n]z = α n z n= n=0 1 = αz 1 < 1 1 αz 1 ; z = z α ; z > α x[n] = α n u[n] n z z α ROC z-plane α

27 27 Check Yourself What DT signal has the following Z transform? z-plane z z 7 8 ; z < 7 8 ROC 7 8

28 28 Check Yourself Recall that we already know a function whose Z transform is the outer region. x[n] = ( 7 8 ) n u[n] n z z 7 8 ROC z-plane 7 8 What changes if the region changes? The original sum ( n 7 ) n X(z) = z 8 n=0 7 does not converge if z < 8.

29 29 Check Yourself The functional form is still the same, z H(z) = Y (z) = X(z) z 7 8. Therefore, the difference equation for this system is the same, y[n + 1] 8 7 y[n] = x[n + 1]. Convergence inside z = 7 8 corresponds to a left-sided (non-causal) response. Solve by iterating backwards in time: 8 y[n] = (y[n + 1] x[n + 1]) 7

30 Check Yourself Solve by iterating backwards in time: 8 y[n] = (y[n + 1] x[n + 1]) 7 Start at rest : n x[n] y[n] > n 8 n 7 ( ) n ( ) n 8 7 y[n] = ; n < 0 = u[ 1 n]

31 31 Check Yourself Plot y[n] = ( 7 8 ) n u[ 1 n] n z z 7 8 ROC z-plane 7 8

32 32 Check Yourself What DT signal has the following Z transform? z-plane z z 7 8 ; z < 7 8 ROC 7 8 y[n] = ( 7 8 ) n u[ 1 n] n

33 33 Check Yourself Two signals and two regions of convergence. ROC z-plane x[n] = ( 7 8 ) n u[n] n z z y[n] = ( 7 8 ) n u[ 1 n] n z z 7 8 ROC z-plane 7 8

34 34 Check Yourself Find the inverse transform of X(z) = 3z 2z 2 5z + 2 given that the ROC includes the unit circle.

35 35 Check Yourself Find the inverse transform of 3z X(z) = 2z 2 5z + 2 given that the ROC includes the unit circle. Expand with partial fractions: 3z 1 2 X(z) = = 2z 2 5z + 2 2z 1 z 2 Not a standard form!

36 36 Check Yourself Standard forms: ROC z-plane x[n] = ( 7 8 ) n u[n] n z z y[n] = ( 7 8 ) n u[ 1 n] n z z 7 8 ROC z-plane 7 8

37 37 Check Yourself Find the inverse transform of 3z X(z) = 2z 2 5z + 2 given that the ROC includes the unit circle. Expand with partial fractions: 3z 1 2 X(z) = = 2z 2 5z + 2 2z 1 z 2 Not a standard form! Expand it differently: as a standard form: 3z 2z z z z X(z) = = = 2z 2 5z + 2 2z 1 z 2 z 1 z 2 2 Standard form: a pole at 1 2 and a pole at 2.

38 Check Yourself Ratio of polynomials in z: 3z z z X(z) = = 2z2 5z + 2 z 1 2 z 2 a pole at 1 2 and a pole at 2. z-plane ROC Region of convergence is outside pole at 1 ( ) 2 n 1 x[n] = u[n] + 2 n u[ 1 n] 2 38 but inside pole at 2.

39 39 Check Yourself Plot. ( ) n 1 x[n] = u[n] + 2 n u[ 1 n] 2 x[n] n

40 40 Check Yourself Alternatively, stick with non-standard form: 3z 1 2 X(z) = = 2z 2 5z + 2 2z 1 z 2 Make it look more standard: 1 1 z 1 z X(z) = 2z 2 z z 1 z 2 2 Now ) n 1 1 x[n] = R u[n] + 2R {+2 n u[ 1 n]} 2 2 ( ) n = u[n 1] n 1 u[ n] 2 2 = ) n 1 u[n 1] 2 + {+2 n u[ n]} x[n] n

41 41 Check Yourself Alternative 3: expand as polynomials in z 1 : 3z 3z 1 X(z) = = 2z 2 5z z 1 + 2z = = 2 z 1 1 2z z 1 1 2z 1 Now ( ) n 1 x[n] = u[n] + 2 n u[ 1 n] 2 x[n] n

42 42 Check Yourself Find the inverse transform of X(z) = 3z 2z 2 5z + 2 given that the ROC includes the unit circle. x[n] n

43 43 Solving Difference Equations with Z Transforms Start with difference equation: y[n] 2 1 y[n 1] = δ[n] Take the Z transform of this equation: Y (z) 1 2 z 1 Y (z) = 1 Solve for Y (z): 1 Y (z) = z 1 Take the inverse Z transform (by recognizing the form of the transform): ( ) n 1 y[n] = u[n] 2

44 44 Inverse Z transform The inverse Z transform is defined by an integral that is not particularly easy to solve. Formally, 1 x[n] = X(z)z n 1 dz 2πj C where C represents a closed contour that circles the origin by running in a counterclockwise direction through the region of convergence. This integral is not generally easy to compute. This equation can be useful to prove theorems. There are better ways (e.g., partial fractions) to compute inverse transforms for the kinds of systems that we frequently encounter.

45 45 Properties of Z Transforms The use of Z Transforms to solve differential equations depends on several important properties. Property x[n] X(z) ROC Linearity ax 1 [n] + bx 2 [n] ax 1 (z) + bx 2 (z) (R 1 R 2 ) Delay x[n 1] z 1 X(z) R Multiply by n nx[n] dx(z) z dz R Convolve in n x 1 [m]x 2 [n m] X 1 (z)x 2 (z) (R 1 R 2 ) m=

46 46 Check Yourself ( ) 2 z Find the inverse transform of Y (z) = z 1 ; z > 1.

47 Check Yourself ( ) 2 z Find the inverse transform of Y (z) = ; z > 1. z 1 y[n] corresponds to unit-sample response of the right-sided system ( ) 2 ( ) 2 ( ) 2 Y z 1 1 = = 1 = X z 1 1 z 1 R = 1 + R + R 2 + R R + R 2 + R R R 2 R 3 1 R R 2 R 3 1 R R 2 R 3 R R 2 R 3 R 4 R 2 R 3 R 4 R 5 R 3 R 4 R 5 R 6 Y = 1 + 2R + 3R 2 + 4R 3 + = (n + 1)R n X n=0 y[n] = h[n] = (n + 1)u[n] 47

48 48 Check Yourself Table lookup method. ( ) 2 z Y (z) = y[n] =? z 1 z u[n] z 1

49 49 Properties of Z Transforms The use of Z Transforms to solve differential equations depends on several important properties. Property x[n] X(z) ROC Linearity ax 1 [n] + bx 2 [n] ax 1 (z) + bx 2 (z) (R 1 R 2 ) Delay x[n 1] z 1 X(z) R Multiply by n nx[n] dx(z) z dz R Convolve in n x 1 [m]x 2 [n m] X 1 (z)x 2 (z) (R 1 R 2 ) m=

50 50 Check Yourself Table lookup method. ( ) 2 z Y (z) = z 1 z z 1 z d ( ) ( ) 2 z 1 = z dz z 1 z 1 ( z z d ( )) ( ) 2 z z = dz z 1 z 1 y[n] =? u[n] nu[n] (n + 1)u[n + 1] = (n + 1)u[n]

51 51 Concept Map: Discrete-Time Systems Relations among representations. Delay R Block Diagram System Functional X + + Y Y Delay Delay X = H(R) = 1 1 R R 2 index shift Unit-Sample Response h[n]: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... Z transform Difference Equation y[n] = x[n] + y[n 1] + y[n 2] System Function H(z) = Y (z) X(z) = z 2 z 2 z 1

52 MIT OpenCourseWare Signals and Systems Fall 2011 For information about citing these materials or our Terms of Use, visit:

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing Lecture 4: Inverse z Transforms & z Domain Analysis Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner