Chapter 7: The z-transform

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Archibald Silas Manning
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1 Chapter 7: The -Transform

2 ECE352 1 The -Transform - definition Continuous-time systems: e st H(s) y(t) = e st H(s) e st is an eigenfunction of the LTI system h(t), and H(s) is the corresponding eigenvalue. Discrete-time systems: x[n] = n h[n] y[n] y[n] = x[n] h[n] = = k= k= h[k] n k = n h[k]x[n k] k= h[k] k = n H() n is an eigenfunction of the LTI system h[n], and H() is the corresponding eigenvalue.

3 ECE352 2 The -Transform - definition (cont.) The transfer function: H() = Generally, let = re jω. Then, k= h[k] k. H(re jω ) = n= ( h[n] n ) e jωn. Thus, H() is the DTFT of h[n]r n. The inverse DTFT of H(re jω ) must be h[n]r n.

4 ECE352 3 The -Transform - definition (cont.) So we may write h[n]r n = 1 2π π π H(re jω )e jωn dω. = re jω d = jre jω dω. dω = 1 j 1 d. As Ω goes from π to π, traverses a circle of radius r in a counterclockwise direction. Thus, we may write h[n] = 1 2πj H() n 1 d

5 ECE352 4 The -Transform - definition (cont.) For an arbitrary signal x[n], the -transform and inverse -transform are expressed as X() = n= x[n] = 1 2πj x[n] n X() n 1 d We express this relationship between x[n] and X() as x[n] X()

6 ECE352 5 The -Transform - convergence A necessary condition for convergence: (absolute summability) n= x[n]r n < The range of r for which this condition is satisfied is termed the region of convergence (ROC). Complex number is represented as a location in a complex plane, termed the -plane. If x[n] is absolutely summable, then the DTFT of x[n] is obtained as X(e jω ) = X() =e jω The contour = e jω is termed the unit circle.

7 ECE352 6 The -Transform - poles and eros The most commonly encountered form of the -transform is a ratio of two polynomials in 1, as shown by the rational function X() = b 0 + b b M M a 0 + a a N N = b M k=1 (1 c k 1 ) N k=1 (1 d k 1 ) b = b 0 /a 0. c k : eros of X(). Denoted with the symbol in the plane. d k : poles of X(). Denoted with the symbol in the plane.

8 ECE352 7 The -Transform - Review of commonly used series Geometric series: Let s n = a + ar + ar ar n, then Proof: s n = a(1 rn+1 ) 1 r lim s n = n a, if r < 1 1 r s n = a + ar + ar ar n rs n = ar + ar ar n + ar n+1 s n rs n = a(1 r n+1 ) s n = a(1 rn+1 ) 1 r

9 ECE352 8 The -Transform - Review of commonly used series (cont.) Arithmetic progression: Let s n = a + (a + r) + (a + 2r) + + (a + nr), then Proof: s n = (n + 1)(a + (a + nr)) 2 lim n n =, if r > 0 lim n n =, if r < 0 s n = a + (a + r) + (a + 2r) + + (a + nr) s n = (a + nr) + (a + (n 1)r) + + (a + r) + a 2s n = (n + 1)(a + (a + nr)) s n = (n + 1)(a + (a + nr))/2

10 ECE352 9 The -Transform - Review of commonly used series (cont.) n=1 1 2 n = = π = n= = π n 2 = π2 6

11 ECE The -Transform - Convergence of commonly used series n=1 1 n p for p > 0: Convergent, if p > 1 Divergent, if p 1. Example: n + : divergent n 2 + : convergent

12 ECE The -Transform - Convergence of commonly used series (cont.) Ratio test: Suppose lim n a n+1 a n = r. r > 1: divergent r < 1: convergent r = 1: test gives no information Comparison test: Assume 0 a n b n, n. If b n is convergent a n is convergent (For convenience, we use b n to represent an infinite series in the notes) Example: Let a n = 2n 3n 3 1, b n = 1 n 2.

13 ECE bn is convergent. Thus, a n is convergent because n 1 n 3 1 3n 3 1 2n 3 a n b n.

14 ECE The -Transform - Convergence of commonly used series (cont.) Corollary of comparison test (limiting form): Suppose that a n a n > 0, b n > 0 and then lim = k > 0. n b n an convergent iff b n convergent Example: Let a n = a n n n 2 + 1, b n = 1 n. n 2 n = 1. lim = lim n b n n Because b n is divergent a n is divergent.

15 ECE The -Transform - Convergence of commonly used series (cont.) Necessary condition for convergence of a n : lim = a n = 0 n It is not a sufficient condition. For example, a n = 1 n, a n is divergent.

16 ECE The -Transform - Examples Determine the -transform of the following signals and depict the ROC and the locations of the poles and eros of X() in the -plane: x[n] = α n u[n] (causal signal) x[n] = α n u[ n 1] (anticausal signal) For signal x[n] = α n u[n]: X() = = α n u[n] n n= n=0 ( α ) n.

17 ECE The -Transform - Examples (cont.) This infinite series converges to X() = 1 1 α 1 =, for > α. α For signal x[n] = α n u[ n 1]: X() = = = n= = 1 1 n= k= 1 ( α n u[ n 1] n) ( α ) n ( α ) k ( α ) k = 1 k=0 1 1 α 1 = α, for < α.

18 ECE The -Transform - Examples (cont.) Observations: As bilateral Laplace transform, the relationship between x[n] and X() is not unique. The ROC differentiates the two transforms. We must know the ROC to determine the correct inverse -transform.

19 ECE The -Transform - Examples (cont.) Read Example 7.4, p 560. Problem 7.1(c), p 561 : Determine the -transform, the ROC, and the locations of poles and eros of X() for the following signal x[n] = ( ) n ( 3 u[ n 1] + 1 n u[n] 4 3) Using the results given in the previous two slides: ( ) n 3 u[ n 1] 4 ( 1 n u[n] 3) 3/4 + 1/3. Thus, X() = 3/ /3 = (2 5/12) ( 3/4)( + 1/3)

20 ECE Properties of the ROC As the Laplace transform, the ROC cannot contain any poles. ROC for a finite-duration signal includes the entire -plane, except possibly = 0 or = or both. Left-sided sequence: x[n] = 0 for n 0 (notice the difference between the left-sided signal for Laplace transform). Right-sided sequence: x[n] = 0 for n < 0 Two-sided sequence: a signal that has infinite duration in both the positive and negative directions.

21 ECE Properties of the ROC RSS: ROC is of the form > r + LSS: ROC is of the form < r TSS: ROC if of the form r + < < r where the boundaries r + and r are determined by the pole locations. See the figure next page.

22 ECE Properties of the ROC (cont.)

23 ECE Properties of the ROC - Examples Example 7.5, identify the ROC associated with the -transform for each of the following signals. x[n] = ( 1/2) n u[ n] + 2(1/4) n u[n] y[n] = ( 1/2) n u[n] + 2(1/4) n u[n] w[n] = ( 1/2) n u[ n] + 2(1/4) n u[ n]

24 ECE Properties of the ROC - Examples (cont.) For x[n], the -transform is written as X() = = 0 n= ( ) n ( 2) k + 2 k=0 n=0 n=0 ( 1 4 ( 1 4 ) n ) n The first sum converges for < 1 2. The second sum converges for > 1 4. Thus, the ROC is 1 4 < < 1 2. Summing the two geometric series: X() = /4.

25 ECE Observations: Properties of the ROC - Examples (cont.) The first term on the right side of [n] is a left-sided sequence. Its ROC is < r, where r is determined by its pole location. The second term on the right side of [n] is a right-sided sequence. Its ROC is > r +, where r + is determined by its pole location.

26 ECE Properties of the ROC - Examples (cont.) For y[n], both terms are right-sided sequences. Thus, the ROC is > r +, where r + is determined by the pole locations. Y () = n=0 ( ) n n=0 ( ) n 1 4 The first series converges for > 1/2 and the second series converges for > 1/4. Thus, the ROC is > 1/2, and we write Y () as Y () = + 1/ /4.

27 ECE Properties of the ROC - Examples (cont.) For w[n], both terms are left-sided sequences. Thus, the ROC is < r, where r is determined by the pole locations. W () = = 0 n= ( ) n ( 2) k + 2 k=0 0 n= (4) k k=0 ( ) n 1 4

28 ECE Properties of the ROC - Examples (cont.) The first series converges for < 1/2. The second series converges for < 1/4. Thus, the ROC is < 1/4, and we write W () as W () = The pole locations of sequences [n], y[n], w[n] are shown in the figure next slide.

29 ECE Properties of the ROC (cont.)

30 ECE Properties of the -transform Linearity Let x[n] X() (ROC R x ) and y[n] Y (). ax[n] + by[n] ax() + by (), with ROC at least R x Ry The ROC can be larger than the intersection if one or more terms in x[n] or y[n] cancel each other in the sum. In the -plane, this corresponds to a ero canceling a pole that defines one of the ROC boundaries.

31 ECE Properties of the -transform (cont.) Example: Example 7.5, p 567. Suppose x[n] = ( ) n ( ) n 1 3 u[n] u[ n 1] 2 2 X() = ( 1/2)( 3/2) with ROC 1/2 < < 3/2, and y[n] = ( ) n 1 u[n] 4 ( ) n 1 u[n] 2 Y () = 1 4 ( 1/4)( 1/2) with ROC > 1/2. Evaluate the -transform of ax[n] + by[n], where a and b are constants.

32 ECE Properties of the -transform (cont.) Using the linearity property, we have ax[n] + by[n] a ( 1/2)( 3/2) + b 1 4 ( 1/4)( 1/2) Must be careful in determining ROC. In general, the ROC is the intersection of individual ROCs. For some special cases, however, the ROC could be larger. For instance, let a = b = 1.

33 ECE Properties of the -transform (cont.) Then, ax() + by () = = = ( 1/2)( 3/2) ( 1/4)( 1/2) 5 4( 1/2) ( 1/4)( 1/2)( 3/2) 5 4 ( 1/4)( 3/2) The ROC can be verified to be 1/4 < < 3/2 because the pole-ero cancellation ( = 1/2), and the (1/2) n u[n] no longer presents.

34 ECE Properties of the -transform (cont.) Time reversal x[ n] X ( ) 1 with ROC 1 R x. If R x is of the form a < < b, the ROC of the reflected signal is 1/b < < 1/a. Time shift x[n n 0 ] n 0X() with ROC R x, except possibly = 0 and =. If n 0 > 0, n 0 introduces a pole = 0. If n 0 < 0, n 0 introduces a pole = ±.

35 ECE Properties of the -transform (cont.) Multiplication by an exponential sequence α n x[n] X ( α) with ROC α Rx. α R x implies that the ROC boundaries are multiplied by α. If α = 1, then the ROC is unchanged. Convolution x[n] y[n] X()Y () with ROC at least R x Ry. The ROC may be larger than the intersection of R x and R y if a pole-ero cancellation occurs in the product of X()Y ().

36 ECE Properties of the -transform (cont.) Differentiation in the -domain nx[n] d d X(), with ROC R x. Read Example 7.8, p 570. Example: Example 7.7, p 570. Find the -transform of x[n] = ( n ( ) n 1 u[n]) 2 ( ) n 1 u[ n]. 4

37 ECE Properties of the -transform: example Basic signal of first term: ( ) 1 n 2 u[n] > 1/2. +1/2, with ROC Applying the -domain differentiation property: n ( ) n 1 u[n] 2 d d + 1/2 = 1 2 ( + 1/2) 2, with ROC > 1/2

38 ECE Properties of the -transform: example (cont.) Applying time reversal property: ( 1 > 1/4. Thus, ( 1 4 < 4 4 ) n u[n] ) n u[ n] 1/ 1/ 1/ 1/4 = 4 4 1/4, with ROC with ROC Applying convolution property: x[n] ROC 1 2 < < 4. 2 ( 4)(+1/2) 2, with

Chapter 13 Z Transform

Chapter 13 Z Transform 1. -transform 2. Inverse -transform 3. Properties of -transform 4. Solution to Difference Equation 5. Calculating output using -transform 6. DTFT and -transform 7. Stability Analysis