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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 5 Copyright © Orchard Publications Properties and Theorems of the Z Transform and this is the same as (9.11). For , (9.11) reduces to (9.12) and for , reduces to (9.13) 9.2.4 Left Shift in the Discrete Time Domain (9.14) that is, if is a discrete time signal, and is a positive integer, the left shift of is . Proof: We let ; then, , and when , . Then, When , , and when , . Then, by substitution into the last sum- mation term of the above expression, we obtain and this is the same as (9.14). For , the above expression reduces to (9.15) m1 = fn 1 [] z 1 Fz () f1 + m2 = fn 2 z 2 f2 z 1 ++ fn m + z m + z n nm = 1 + fn mm t h f n + Z + {} + z n n0 = = + k = nk m = = km = Z + fk z = z k z m = == z m z k k0 = z k = z m z k = = = = = 1 = n1 = Z + z m + z + = 1 + z m + z n = 1 + = Z + zF z f0 z =

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Chapter 9 Discrete Time Systems and the Z Transform 9 6 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications and for , reduces to (9.16) 9.2.5 Multiplication by in the Discrete Time Domain (9.17) Proof: 9.2.6 Multiplication by in the Discrete Time Domain (9.18) Proof: 9.2.7 Multiplication by and in the Discrete Time Domain (9.19) Proof: By definition, and taking the first derivative of both sides with respect to , we obtain m2 = Z fn 2 + [] {} z 2 Fz () f0 z 2 f1 z = a n a n fn F z a --   Z a n a n z n k0 = 1 a n ------fn z n = z a n = F z a == = = e naT e Fe aT z Z e e z n = e z n = z === nn 2 nf n z d dz -----Fz n 2 z d z 2 d 2 2 -------Fz + z n n0 = = z d n z n 1 = z 1 nf n z n =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 7 Copyright © Orchard Publications Properties and Theorems of the Z Transform Multiplication of both sides by yields Differentiating one more time, we obtain the second pair in (9.19). 9.2.8 Summation in the Discrete Time Domain (9.20) that is, the Z transform of the sum of the values of a signal, is equal to times the Z transform of the signal. This property is equivalent to time integration in the continuous time domain since integration in the discrete time domain is summation. We will see on the next sec- tion that the term is the Z

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