Chapter 13 Z Transform

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1 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 Sections: ,

2 Z-transform Analysis Equation: Synthesis Equation: x k k ( ) x[. 1 2πj k 1 [] k ( ) d C Activity 1: Calculate the -transform for the following DT sequences: 1. x[ α k u[. 2. x[ α k u[ k 1]. 3. x [ kα k u[. 4. Time-limited Sequence: 1, k 0,1 x[ 2, k 2,5 0, otherwise.

3 Z-tranform and DTFT k ( Ω) x[] k ( ) jω. k e 1. DTFT is a special case of the -transform with e jω. 2. Since the equality e jω corresponds to the circle of unit radius ( 1) in the complex -plane, the above equation implies that the DTFT is obtained by computing the -transform along the unit circle in the complex -plane. 3. For the above expression to be valid, the ROC in the - plane must contain the unit circle.

4 Properties of ROC 1. The ROC consists of a 2D plane of concentric circles of the form > 0, or < The ROC does not include any poles of the - transform. 3. The ROC of a finite length sequence contains the entire -plane except possible The ROC of a right-hand sided sequence (x[ 0, for k < k 0 ) is defined by the region outside of a circle, i.e., > The ROC of a left-hand sided sequence (x[ 0, for k > k 0 ) is defined by the inside region of a circle, i.e., < The ROC of a double sided (or bilateral) sequence, which extends to infinite values of k in both directions, is confined to a ring with a finite area and has the form ( 1 < < 2 ).

5 Z-transform Table (1)

6 Z-transform (2)

7 Inverse Z-transform Partial Fraction Method: Activity 2: Calculate the inverse -transform for the following rational function.. i) 1( ) ii) iii) 2 ( ) ( ) 1 ( 0.1)( 0.5)( + 0.2) 2(3 + 17) ( 1)( 6+ 25) 3 2

8 Inverse Z-transform (1) Partial Fraction Method: Activity 2: Calculate the inverse -transform for the following rational Function for partial fraction method.. i) 1( ) ii) iii) 2 ( ) ( ) 1 ( 0.1)( 0.5)( + 0.2) 2(3 + 17) ( 1)( 6+ 25) 3 2

9 Inverse Z-transform (2) Power Series Method: Activity 3: Calculate the inverse -transform for the following rational Function using the power series method.. i) 1( ) ii) iii) 2 ( ) ( ) 1 ( 0.1)( 0.5)( + 0.2) 2(3 + 17) ( 1)( 6+ 25) 3 2

10 Z-transform Properties (1) Linearity: If x 1 [ and x 2 [ are DT sequences with - transforms x then the linearity property states that a ( ) : R1 and x2[ 2( ) : [ R ( ) + a2 2( ) R1 2 DTFT 1 x1 + a2x2[ a 11 : [ R for constants a 1 and a 2, which may be complex valued. Time Scaling: If x[ ( ) with ROC R x, then the -transform (m) () of its interpolated version x (m) [ is given by x [ ( ) Z ( m) Z ( m) m ( ) with ROC (R x ) 1/m for (2 m < ).

11 Z-transform Properties (2) Time Shifting: If Z ( ) Rx x [ :, then Z m ( x) Rx x [ k k ] : 0 Time Differencing: If Z ( ) Rx x [ :, then [ ] 1 ( ) Rx Z 1 x [ x[ k 1] : Z-domain Differentiation: If Z ( ) Rx x [ :, then Time Accumulation: If k n 0 x[ n] Z d kx [ : d DTFT 1 Z ( 1 1 ) R x ( ) Rx x [ : Initial and Final Value Theorems: x[ ], then ( ) : R ( > 1) x[ 0] lim ( ) provided x[ 0 for k < 0. lim x[ k lim( 1) ( ) 1 x provided x[ ] exists.

12 DTFT Properties (4) Time Convolution: If x 1 [ and x 2 [ are DT sequences with -transforms then x x ( ) : R1 and x2[ 2( ) : [ R ( Ω) 2( Ω) R1 I 2 Z 1 x2[ 1 : [ R Activity 4: The exponential decaying sequence x[ a k u[, (0 a 1) is applied at the input of a LTID system with the impulse response h[ b k u[, (0 b 1). Using the DTFT approach, calculate the output of the system. Solution: y[ k ( k + 1) a u[ a b [ ] 1 k k a b u[ a b, a b

13 Z-transform Properties Activity 5: Determine the -transform of the following sequences using the specified property. i) x[ (5/6) k u[k 6] based on the -transform pair 4 in Table 13.1 and the time-shifting property. ii) x[ k(2/9) k u[ based on -transform pair 4 in Table 13.1 and the -domain differentiation property. iii) x[ ramp(k) based on the -transform pair 3 in Table 13.1 and the accumulation property. iv) x[ e k sin(k)u[ based on -transform pairs 4 and 9 in Table 13.1 and the linearity property.

14 Difference Equations Activity 6: A causal system is represented by the difference equation. y [ k + 2] 5y[ k + 1] + 6y[ 3x[ k + 1] + 5x[ Calculate y[ for input x[ 2 k u[ and initial conditions y[ 1] 11/6, y[ 2] 37/36. Activity 7: A causal system is represented by the difference equation. [ 1 y k + 2] 3 y[ k + 1] + y[ 2x[ k + 2] 4 8 Determine the transfer function and the impulse response of the system. Activity 8: Determine the poles and eros of the following LTID system. Determine if a causal and stable implementation is possible. H1( ) H 2 ( ) ( 0.1)( 0.5)( + 0.2)

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