Bioen_303_2_10_2010_ZTransforms

Bioen_303_2_10_2010_ZTransforms - 2/10/2010 The ZTransform...

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2/10/2010 1 The Z–Transform (ZT) • The objectives of this session are: – To develop the z–transform – To investigate its properties To apply the z transform to IIR filter 1 – To apply the z–transform to IIR filters • Like the Discrete Time Fourier Transform (DTFT) is the discrete version of the continuous Fourier transform, the Z–Transform is the discrete version of the continuous Laplace transform; it allows the Laplace transform to be implemented on digital systems. Z–Transform Development • Recall the impulse train sampling process, where we can express the sampled sequence as: +∞ 2 • The one–sided (unilateral) Laplace transform of the function is: −∞ = δ = n s ) nT t ( ) nT ( f ) t ( f snT 0 n sam e ) nT ( f ) s ( F = = Z–Transform Development • Let’s define a new variable: interval. time sampling the is T where , e z sT = 3 • Then we have: = = 0 n n 1 z ) nT ( f ) z ( F Z–Transform Development • This is the z–transform corresponding to the one– sided Laplace transform. We can define a more general z–transform that spans all discrete values of n (from – to + ) and it corresponds to the 4 of n (from to + ) and it corresponds to the bilateral Laplace transform, in which case its form will be: • This is the Z–transform we will be dealing with. −∞ = = n n z ) n ( h ) z ( H Z–Transform Development • If we wish to consider the frequency response of a discrete system, e.g., an IIR filter, we can relate the variable z to the 5 complex frequency used in the Laplace transform. Recall the complex frequency is: s= σ ’+j ω ’, then . T ' angle the and , e r magnitude the where , re e e e z T ' j ' jT T ' sT ω = ω = = = = σ ω ω σ Z–Transform Development . T ' angle the and , e r magnitude the where , re e e e z T ' j ' jT T ' sT ω = ω = = = = σ ω ω σ 6 •No t e : ω ’ is the real frequency and it is related to the Z–transform angle via ω ’= ω /T= f s ω . If we use normalized frequency by assuming the sampling frequency f s =1 (then T=1), then r=e σ , and ω = ω ’.
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2/10/2010 2 Z–Transform Development • Using this representation of z, the z– transform equation takes the form: ω j r H H 7 −∞ = ω −∞ = ω = = = = n n j n n n j ) e ( r ) n ( h ) re )( n ( h ) re ( ) z ( Z–Transform Development −∞ = ω = n n j n ) e ( r ) n ( h ) z ( H 8 • An interpretation of the function above is that it is the Discrete Time Fourier Transform of the product of h(n) and r –n .
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Bioen_303_2_10_2010_ZTransforms - 2/10/2010 The ZTransform...

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