lecture24 - – Make impulse response of a discrete-time...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Z and Laplace Transforms
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18 - 2 Z and Laplace Transforms Transform difference/differential equations into algebraic equations that are easier to solve Are complex-valued functions of a complex frequency variable Laplace: s = σ + j 2 π f Z : z = r e j ϖ Transform kernels are complex exponentials Laplace: e s t = e σ t + j 2 π f t = e σ t e j 2 π f t Z : z k = ( r e j ϖ ) k = r k e j ϖ k dampening factor oscillation term
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18 - 3 Z and Laplace Transforms No unique mapping from Z to Laplace domain or from Laplace to Z domain Mapping one complex domain to another is not unique One possible mapping is impulse invariance
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Unformatted text preview: – Make impulse response of a discrete-time linear time-invariant (LTI) system be a sampled version of the continuous-time LTI system. H ( z ) y [ k ] f [ k ] Z H ( s ) ( 29 t f ~ ( 29 t y ~ Laplace T s e z z H s H | ) ( ) ( = = 18 - 4 Impulse Invariance Mapping • Impulse invariance mapping is z = e s T Laplace Domain Z Domain Left-hand plane Inside unit circle Imaginary axis Unit circle Right-hand plane Outside unit circle 1 Im{ z } Re{ z } s = -1 ± j ⇒ z = 0.198 ± j 0.31 ( T = 1 s) s = 1 ± j ⇒ z = 1.469 ± j 2.287 ( T = 1 s) π 1 2 1 1 max max ⇒ = ⇒ = s f f ω 1 1-1-1 Im{ s } Re{ s } f j s 2 = lowpass, highpass bandpass, bandstop allpass, or notch filter?...
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