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Bioen_303_2_10_2010_ZTransforms

# Bioen_303_2_10_2010_ZTransforms - The ZTransform(ZT The...

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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 filters 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 • Note: 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 ) re ( H ) ( H 7 −∞ = ω −∞ = ω = = = = n n j n n n j ) e ( r ) n ( h ) re )( n ( h 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 - The ZTransform(ZT The...

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