Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part48

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part48

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 37 Copyright © Orchard Publications The Inverse Z Transform Next, we use the MATLAB collect function to expand the denominator to a polynomial. syms z; den=collect((z 0.25)*(z 0.5)*(z 0.75)) den = z^3-3/2*z^2+11/16*z-3/32 Thus, (9.101) Now, we perform long division as shown in Figure 9.10. Figure 9.10. Long division for the polynomials of Example 9.9 We find that the quotient is (9.102) By definition of the Z transform, (9.103) Equating like terms in (9.102) and (9.103), we obtain (9.104) F z ( ) z 3 z 2 2z 3 + + + z 0.25 ( ) z 0.5 ( ) z 0.75 ( ) --------------------------------------------------------------------- = F z ( ) z 3 z 2 2z 3 + + + z 3 3 2 ( ) z 2 11 16 ( ) z + 3 32 -------------------------------------------------------------------------------- = z 3 3 2 --z 2 11 16 -----z + 3 32 ----- z 3 z 2 2z 3 + + + 1 5 2 -- z 1 81 16 ----- z 2 + + + Divisor Dividend Quotient z 3 3 2 --z 2 11 16 -----z + 3 32 ----- 5 2 --z 2 21 16 ----- z 35 32 ----- + + 1st Remainder 5 2 --z 2 15 4 ----- z 55 32 ----- 15 64 -----z 1 + 81 16 ----- z + 2nd Remainder Q z ( ) Q z ( ) 1 5 2 -- z 1 81 16 ----- z 2 + + + = F z ( ) f n [ ] z n n 0 = f 0 [ ] f 1 [ ] z 1 f 2 [ ] z 2 + + + = = f 0 [ ] 1 f 1 [ ] , 5 2 and f 2 [ ] 81 16 = = =
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Chapter 9 Discrete Time Systems and the Z Transform 9 38 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications We will use the MATLAB dimpulse function to verify the answers, and to obtain the sequence of the first 15 values of . num=[1 1 2 3]; den=[1 3/2 11/16 3/32]; fn=dimpulse(num,den,15),... dimpulse(num,den,16) fn = 1.0000 2.5000 5.0625 8.9688 10.2070 9.6191 8.2522 6.7220 5.3115 4.1195 3.1577 2.4024 1.8189 1.3727 1.0338 Figure 9.11. Impulse response for Example 9.9 Table 9.4 lists the advantages and disadvantages of the three methods of evaluating the Inverse Z transform. 9.7 The Transfer Function of Discrete Time Systems The discrete time system of Figure 9.12, can be described by the linear difference equation f n [ ]
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