UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).

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1 Page no: 1 UNIT-II Z-TRANSFORM The Z-Transform The direct -transform, properties of the -transform, rational -transforms, inversion of the transform, analysis of linear time-invariant systems in the - domain, block diagrams and flow graph representation of digital network, matrix representation. 2.1 DEFINITION OF Z TRANSFORMS: The -transform of a discrete is defined as the power series, [x n ] = X = x n n n= Where is a complex variable. This is generally referred to as two sided -transform. If is a causal sequence, = 0, for n < 0, then its -transform is X = x n n n= This expression is called a one sided -transform. This causal sequence produces negative powers of in X (). Generally we assume that is a causal sequence, unless it is otherwise stated. If is non- ausal se ue e, x = fo, the its -transform is X = x n n n= This expression is also called a one sided -transform. This non causal sequence produces positive powers of in X (). 2.2 REGION OF CONVERGENCE: The alues of fo hi h X is fi ite a d lie ithi the egio alled as egio of o e ge e ROC. Therefore, the condition for X () to be finite is > 1. In other words, the ROC for X () is the area outside the unit circle in the plane. The ROC of a rational transform is bounded by the location of its pole. For Example, the transform of the unit step response u(n) is X = extending all the way to, as shown in figure 1 hi h has a e o at = a d a pole at = a d the ROC is > a d Im Unit circle Z plane Re ROC Pole at =1 Zero at = 0 Figure 1: Pole Zero plot and ROC of Unit Step Response

2 Page no: 2 PROPERTIES OF ROC OF Z-TRANSFORMS: X () converges uniformly if and only if the ROC of the transform X () of the sequence includes the unit circle. The ROC of X () consists of a ring in the plane centered about the origin. That is the ROC of transform of has values of for which x(n) r -n is absolutely summable. X = x n r n n= ROC does not contain any poles. If is a finite duration causal sequence or right sided sequence, then the ROC is entire -plane except at = 0. If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire -plane except at =. If is an infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. > a. If is an infinite duration anti-causal sequence, ROC is interior of the circle with radius a. i.e. < a. If is a finite duration two sided sequence, then the ROC is entire -plane except at = 0 & =. EXAMPLE (A): Determine the transform of the following finite duration s. (a) = { 1, 2, 5, 4, 0, 1} (b) x = δ (c) x = δ -k) (d) Determine the transform of a n, 0 x(n) = 0, n<0 SOLUTION: (a) = { 1, 2, 5, 4, 0, 1} Taking transform, we get X() = (b) x = δ X() = 1 (c) x = δ -k) x = δ -k), k > 0 hence X() = k d) ROC: Entire plane except = 0 x(n) = a n, 0 0, n<0 The transform of the given x(n) is X = [ n ] = n n = n n n= n=

3 Page no: 3 We know that, n = Hence, a for a < 1 X = 1 1 This o e ges he a -1 < o > a. Values fo fo hi h X () = 0 are called of X (), and values of for which X () tends to infinity are called poles of X (). 2.3 PROPERTIES OF Z TRANSFORM: 1) LINEARITY The linearity property states that if x1(n) X1() and x2(n) X2() a1 x1(n) + a2 x2(n) a1 X1() + a2 X2() Where a1 and a2 are arbitrary constants. It implies that the - transform of a linear combination of s is the same linear combination of transforms. EXAMPLE (B): Determine the transform of the x = δ + + δ + 6 δ -3) - δ -4) SOLUTION: From the linearity property, we have X = Z [δ + ] + Z [δ ] + 6 Z [δ -3)] Z [δ -4)] Using transform pair, we obtain = { 1, 3, 6, 0, 0, -1} 2) TIME REVERSAL The Time reversal property states that if x(n) X() : ROC r1 < < r2 x (-n) X ( -1 ) : ROC 1/ r1 < < 1/ r2 EXAMPLE (C): Find the transform of the = u (-n) SOLUTION:

4 We know that [u n ] = By using time reversal property, we obtain [u n ] = 1 1 ROC: > ROC: < 3) TIME SHIFTING The Time shifting property states that if X () x(n-k) X () k The ROC of k X () is the same as that of X() except for = 0 if k > 0 and = if k < 0. EXAMPLE (D): Find the transform of the X()= -1 /( ) SOLUTION: X()= -1 /( ) = -1 x1() Where X1()= 1 /( ) Here, from the time shifting property we have k=1 and x(n) = (3) n u(n) Hence x(n) (3) n-1 u(n-1) 4) SCALING x(n) X () : ROC r1 < < r2 : ROC a r1 < < a r2 a n x(n) X (a -1 ) where a is arbitrary constant, which can be real or complex. EXAMPLE (E): Find the -transform of x(n)=2 n u(n-2) SOLUTION: x(n)=2 n u(n-2) Z[u(n)] = 1/(1- -1 ) Z[u(n-2)] = -2 /(1- -1 ) Z[2 n u(n-2)] = -2 /(1- -1 ) limit -1 -->2-1 Z[2 n u(n-2)] = (2-1 ) 2 /( ) = (4-2 )/( ) 5) DIFFERENTIATION x1(n X() n x(n) Page no: 4 - dx ()/d

5 Page no: 5 EXAMPLE (F): Find the -transform of =n 2 u(n) SOLUTION: X () = Z [n 2 u (n)] = Z [n (n u (n)] d X () = -1 [ Z (nu(n))] d X () = -1 d d X () = -1 [ [ ] X () = (1- -1 ) ( ) / (1- -1 ) 4 X () = + 6) CONVOLUTION If x1(n) X1() and x2(n) X2() x1(n) * x2(n) X1(). X2() EXAMPLE (G): Compute the convolution of the s x 1 (n) = { 4, -2, 1 } x 2 (n) = a n, 0, n<0 SOLUTION: The transform of the given s are written as X1() = X2() = Therefore X () = X1(). X2() = Taking inverse transform, we obtain = { 4, 2, 3, 3, 3, 3, -1, 1} 7) CORRELATION PROPERTY The Correlation of two sequences states that if x1(n) X1() and x2(n) X2()

6 Page no: 6 x1 (l) x2(-l) X1() X2( -1 ) n=- 8) INITIAL VALUE THEOREM If x(n) is a causal sequence with transform X(), the initial value can be determined by using the expression, X() x (0) lim x(n) n 0 lim X() 9) FINAL VALUE THEOREM If X () = Z [] and the poles of X () are all inside the unit circle, then the final value of the sequence, x, can be determined by using the expression, x = lim x(n) = lim (-1) X() n 1 EXAMPLE (H): If X() = , find the initial value and final value of the corresponding sequence, x(n). SOLUTION: x(0) lim [ ] = 2 x = lim [ ] = = EVALUATION OF INVERSE Z TRANSFORM The three basic methods of performing the inverse transform, vi. 1. Partial fraction expansion Method 2. Power series expansion Method 1. PARTIAL FRACTION EXPANSION METHOD In this method X () is first expanded into sum of simple partial fraction. X () = a0 m + a1 m am b0 n + b1 n n n fo First find the roots of the denominator polynomial X () = a0 m + a1 m am

7 Page no: 7 (- p1 ) (- p2 - pn ) 2. POWER-SERIES EXPANSION METHOD The transform of a discrete time is given as X() = x n n=- Expanding the above terms we have x() =.. + x(-2) Z 2 + x(-1) Z + x(0) + x(1) Z -1 + x(2) Z This is the expansion of transform in power series form. Thus sequence x(n) is given as x(n) = {.., x(-2), x(-1), x(0), x(1), x,..}. Power series can be obtained directly or by long division method. 2.5 ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS IN THE Z- DOMAIN The transform, plays an important role in the analysis and representation of discrete time LTI systems. The convolution property of transform states that, y(n) = h(n) x(n) y (n) = h(n) x(n) Y() = H().X() Where Y (), H () and X () are the transforms of system input, output and impulse response respectively. H () is known as the transfer function of discrete time LTI systems. Sometimes, it is also called system function. Important properties of the system: 1) Causality of discrete time LTI system: The condition for causality of discrete time LTI system is that the impulse response of a causal discrete time LTI system is given as h (n) = 0 for n < 0 This means that h (n) is right sided. Also transfer function H () is the transform of h (n). The ROC of H () of a causal discrete time system will be unity i.e., H () = 1 A discrete time LTI system is causal if and only if the ROC of its transfer function is the exterior of a circle, including infinity. A discrete time LTI system which has a rational transfer function H () will be causal if and only if, i) The ROC is exterior of a circle outside the outermost pole, and ii) With H () expressed as a ratio of polynomials in, the order of the numerator should be smaller than order of denominator. 3. Stability Criteria for a Discrete time LTI systems: The stability of a discrete time LTI system is equivalent to its impulse response h (n) being absolutely summable, i.e.,

8 Page no: 8 h k < A discrete time LTI system is stable if and only if the ROC of its transfer function H () includes the unit circle = BLOCK DIAGRAM REPRESENTATION FOR DISCRETE TIME LTI SYSTEM To analye discrete time block diagrams such as series or cascade, parallel and feedback interconnections, the transfer function algebra if exactly the same as that for corresponding continuous time LTI system. 1. Series or Cascade inter connection: The series connection of two discrete time LTI system is shown below: Input h 1(n) H 1() h 2(n) H 2() Output y (n) Input The overall impulse response of series connection of two discrete time systems can be determined by taking convolution sum between h1(n) and h2(n).here, h1(n) and h2(n) are the impulse responses of two discrete time LTI systems. 2. Parallel interconnection: h (n) = h 1(n) h 2(n) H () = H 1() H 2() The parallel connection of two discrete time LTI system is shown below: h 1(n) Output y (n) Input H 1() h 2(n) H 2() Output y (n) The overall impulse response of parallel connection of two discrete time systems can be determined by addition of individual impulse responses of two discrete time LTI systems.

The Z transform (2) 1

The Z transform (2) 1 Today Properties of the region of convergence (3.2) Read examples 3.7, 3.8 Announcements: ELEC 310 FINAL EXAM: April 14 2010, 14:00 pm ECS 123 Assignment 2 due tomorrow by 4:00 pm