Lesson 07-Inverse z_Transform-1

Lesson 07-Inverse z_Transform-1 - Lesson[07 Inverse...

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Unformatted text preview: Lesson [07] Inverse z-Transform Challenge 06 Lesson 07: Inverse z-transform Challenge 07 What’s it all about • What is an inverse z-transform? • How can it be computed? • What type of MATLAB support is available? Practice Exam to be on-line by 12:00. Consists of 5 question (actual Exam will have 2 questions) Lesson [07] News – Audio Signal Processing Lesson [07] Challenge 06 The discrete-time (a.k.a., digital control) system shown below is to be studied and interpreted as a discrete-time 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 Zero-order hold G (s)=(1-e-sT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Challenge 06 x(t) T=1 Zero-order hold G (s)=(1-e-sT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler In the discrete-time domain, the ZOH becomes a Kronecker impulse δ [k]. 0 T s In the continuous-time 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 Zero-order hold G (s)=(1-e-sT )/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(( 1-e-sT ) × 1/s ) =( 1-z-1 ) × ( z/(z-1 )) =(1-z-1 ) × (1/(1-z-1 ) = 1 Lesson [07] Challenge 06 What is G(z)? G(s) = G (s) G p (s) = ((1-e-sT )/s) (1/(s(s+1)))= (1-e-sT ) × 1/(s 2 (s+1)) What now? – compute Z(1/(s 2 (s+1))) x(t) T=1 Zero-order hold G (s)=(1-e-sT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Challenge 06 G(z) = Z [(1-e-sT ) × 1/(s 2 (s+1)) ] Trick: Perform a partial fraction expansion (assumed legacy knowledge) = (1-z-1 ) × Z ( 1/s 2-1/s +1/(s+1) ) x(t) T=1 Zero-order hold G (s)=(1-e-sT )/s Plant G p (s) G p (s)=1/(s(s+1)) y(t) Ideal sampler Lesson [07] Challenge 06 Manipulating G(z) = Z [(1-e-sT )/(s 2 (s+1))] = (1-z-1 ) × Z ( 1/s 2-1/s +1/(s+1) ) = (z-1)/z × ( z/(z-1) 2- z/(z-1) + z/(z-e-1 )) (Simplify as needed) x(t) T=1 Zero-order hold G (s)=(1-e-sT )/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 z-Transform Time-domain z-transform δ [ k ] 1 δ [ 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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Lesson 07-Inverse z_Transform-1 - Lesson[07 Inverse...

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