(i) Find the inverse z-transform of the system H(z) = for the following regions of convergence. Write in the final answer for each case in the allocated rectangular box below (a) |z|<1 (b) |z|> 3 (c) 1
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- Q1) Find the inverse z-transform f(n) for the following using long division method. 3z2(2z – 4) X(Z) : %3D (z – 2)(z – 0.7)(z – 1)need help with this hw problem..thanks !1. If a system function has zeros at the origin of the z-plane then the system function is a properrational function. Why? 2. Can you obtain the z-transform of u[n] at z = 0? If you can, what is its value? If you cannot,why not? 3. What is the preferred method for obtaining the inverse z-transform of rational functions?Describe this method. 4. Explain why a one-sided z-transform is able to determine response to a LCCDE with initialconditions while the two-sided z-transform cannot. 5. Can any system with a rational system function be decomposed into a product of an allpasssystem and a minimum-phase system? If yes, then explain how?
- H.W. Prove the z-transforms for common sequences summarized in Table 1 except the last sequence (15). Table 1 Table of z-Transform Pairs Region of Convergence Line No. x(n), n20 z-Transform X(z) x(n) 2 8(n) 1 kl>0 az au(n) kl>1 Z-1 nu(n) kl>1 (z – 1)? z(z+1) nu(n) kl>1 (z – 1)3 a u(n) Iz|> la| Z-a e-na u(n) kl>ea (z -ea) 8. az na" u(n) kl> la| (2- a) z sin(a) z2 – 2z cos(a) +1 sin(an)u(n) kl>1 z[z – cos(a)] z2 – 2z cos(a) +1 10 cos(an)u(n) kl>1 [a sin(b)]z z2 - [2a cos(b)]z +a? zz - a cos(b)] z2 - [2a cos(b)]z +a 11 a" sin(bn)u(n) z| > la| 12 a" cos(bn)u(n) kl> la| (e sin(b)]z 2a z? - [2e "cos(b)z +e 13 e an sin(bn)u(n) kl>ea zz -e cos(b)] z2 - [2e-a cos(b)]z +e-2a 14 an cos(bn)u(n) kl>e • e Az A*Z 2|A||P|"cos(n0 + o)u(n) where P and A are complex constants defined by P = |P|Z0, A = |AZ 15 Z-P Z- P* 6,Q3) Find the inverse Z transform for X(z): X(z)= = z²+0.5 Z-0.32 (1-0.5 Z-¹)(Z -0.5)²If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then what is the z-transform of correlation between the two signals?a) X1(z).X2(z-1)b) X1(z).X2(z-1)c) X1(z).X2(z)d) X1(z).X2(-z)
- 2. Given 1 – 3z-2 Y(z) : (1– 3z-1)(1+z-1) (i) Expand Y (z) into partial fractions. (ii) Determine the sequences y, [n], y2[n], and y3[n] whose z-transform is given by the above expression, together with the region of convergence (in each case).Given the Z-transform 5z-4 X(z) = z2+z-2 What are the 3 possible regions of convergence for this X(z)? a) Using contour integration in the region of convergence that has a shape |z|> R (R depends on X(2)), invert X(z) to obtain the time domain signal x(n) b) Using contour integration in the region of convergence that has a shape of a disk, invert X(z) to obtain the time domain signal x(n) c) Using contour integration in the ROC that has a shape |z|3.3. Determine the z-transform of each of the following sequences. Include with your answer the ROC in the z-plane and a sketch of the pole-zero plot. Express all sums in closed form; a can be complex. (a) xa[n] = a»l. (b) xb[n] = 0< la] < 1. [ 1, 0sn< N 1, 0, otherwise. 0snsN- 1, (c) xe[n] = { 2N – 1 - n, NSEE MORE QUESTIONSRecommended textbooks for youIntroductory Circuit Analysis (13th Edition)Electrical EngineeringISBN:9780133923605Author:Robert L. BoylestadPublisher:PEARSON
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