EE3TP4_33_DT_Z_Transform_v4_LECTURE 34

# EE3TP4_33_DT_Z_Transform_v4_LECTURE 34 - The Z Transform Z...

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Z Transform does for DT systems what the Laplace Transform does for CT systems Z Transform is used to Solve difference equations with initial conditions Solve zero-state systems using the transfer function We will: - Define the ZT - See its properties - Use the ZT and its properties to analyze D-T systems Where does the Z Transform come from? The Z Transform These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.

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X = −∞ x t e j t dt Fourier Transform X s = 0 x t e st dt Laplace Transform
X = n =−∞ x [ n ] e j n Discrete-Time Fourier Transform X z = ? Z Transform

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DTFT: X = n =−∞ x [ n ] e j n Periodic in Ω with period 2π Section 7.1 Z-Transform definitions Given a D-T signal x [ n ] - < n < we’ve already seen how to use the DTFT: Recall : For C-T case, the FT doesn’t converge for some signals … the LT mitigates this problem by including decay in the transform e jωt vs . e − σ t e st Controls decay of integrand So, for D-T signals we include decay into the transform; but in a slightly different way: e j n vs . α n e j n ≡ αe j n z n Controls decay of summand
So for the Laplace transform we looked at: s = σ + jω which is in rect. form But, for Z-transform we use: z = α e which is in polar form Q: Why the change? A: Suffice to say… it has to do with the periodic nature of the DTFT. Remember that the DTFT is a periodic function of Ω … and by using z = α e we stick Ω in as an angle which forces the periodic dependence on Ω. Two sided Z-transform X 2 z = n =−∞ x [ n ] z n z is complex-valued Just like for Laplace … there are two forms of the Z-Transform: One sided Z-transform X 1 z = n = 0 x [ n ] z n z is complex-valued If x [ n ] is a causal signal: X 1 ( z ) = X 2 ( z ) Our Focus is Here

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## This note was uploaded on 02/01/2011 for the course ECE 3TP4 taught by Professor Drterrencetodd during the Fall '10 term at McMaster University.

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EE3TP4_33_DT_Z_Transform_v4_LECTURE 34 - The Z Transform Z...

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