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Unformatted text preview: Lesson [07] Inverse zTransform Challenge 06 Lesson 07: Inverse ztransform Challenge 07 Whats it all about What is an inverse ztransform? How can it be computed? What type of MATLAB support is available? Practice Exam to be online by 12:00. Consists of 5 question (actual Exam will have 2 questions) Lesson [07] News Audio Signal Processing Lesson [07] Challenge 06 The discretetime (a.k.a., digital control) system shown below is to be studied and interpreted as a discretetime transfer function G(z)=Y(z)/X(z). Q: What is G (z)? and for the overachiever, what is G(z)? x(t) T=1 Zeroorder hold G (s)=(1esT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Challenge 06 x(t) T=1 Zeroorder hold G (s)=(1esT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler In the discretetime domain, the ZOH becomes a Kronecker impulse [k]. 0 T s In the continuoustime domain the impulse captures the signal and holds its value for a sample period (i.e., ZOH) Lesson [07] Challenge 06 x(t) T=1 Zeroorder hold G (s)=(1esT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler A Kronecker impulse What is G (z)? Since T s =1, G (z)=Z(G (s))=Z(( 1esT ) 1/s ) =( 1z1 ) ( z/(z1 )) =(1z1 ) (1/(1z1 ) = 1 Lesson [07] Challenge 06 What is G(z)? G(s) = G (s) G p (s) = ((1esT )/s) (1/(s(s+1)))= (1esT ) 1/(s 2 (s+1)) What now? compute Z(1/(s 2 (s+1))) x(t) T=1 Zeroorder hold G (s)=(1esT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Challenge 06 G(z) = Z [(1esT ) 1/(s 2 (s+1)) ] Trick: Perform a partial fraction expansion (assumed legacy knowledge) = (1z1 ) Z ( 1/s 21/s +1/(s+1) ) x(t) T=1 Zeroorder hold G (s)=(1esT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Challenge 06 Manipulating G(z) = Z [(1esT )/(s 2 (s+1))] = (1z1 ) Z ( 1/s 21/s +1/(s+1) ) = (z1)/z ( z/(z1) 2 z/(z1) + z/(ze1 )) (Simplify as needed) x(t) T=1 Zeroorder hold G (s)=(1esT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Inverse z Transform Assume that the z transform of elementary signals are known and cataloged. What is common to their numerators? Table 1: Primitive Signals and their zTransform Timedomain ztransform [ k ] 1 [ k m ] z m u [ k ] z /( z 1) ku [ k ] z /( z 1) 2 k 2 u [ k ] z ( z +1)/( z 1) 3 a k u [ k ] z /( z a ) ka k u [ k ] az /( z a ) 2 k 2 a k u [ k ] az ( z + a )/( z a ) 3 sin[ bk ] u [ k ] cos[ bk ] u [ k ] exp[ akT s ]sin[ bkT s ] u [ kT s ] exp[ akT s ]cos[ bkT s ] u [ kT s ] a k sin( bkT s ) u [ kT s ] a k cos( bkT s ) u [ kT s ] 1 ) cos( 2 ) sin( 2 + b z z b z 1 ) cos( 2 )) cos( ( 2 + b z z b z z S S S aT S aT S aT e bT ze z bT ze 2 2 ) cos( 2 ) sin( + S S S aT S aT S aT e bT ze z bT e z z 2 2 ) cos( 2 )) cos( ( + 2 2 ) cos( 2 ) sin(...
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 Spring '09
 HORTON

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