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H.W. Prove the z-transforms for common sequences summarized in Table except the last sequence (15). Table 1 Table of z-Transform Pairs Region of Convergence Line No. x(n), n20 z-Transform X(z) 1 x(n) 2 ô(n) 1 kl>0 az au(n) kl>1 2-1 nu(n) kl>1 (z – 1)? 5 z(z+1) nu(n) kl>1 (z – 1)8 a u(n) kl> la| Z-a 7 e n u(n) (z-ea) z|>ea 8. az na" u(n) kl> la| (z - a)? z sin(a) z2 – 2z cos(a) +1 9 sin(an)u(n) kl>1 zz – cos(a)] z2 – 2z cos(a) +1 10 cos(an)u(n) z|>1 [a sin(b)]z z2 - [2a cos(b)]z +a² zz - a cos(b)] z2 - [2a cos(b)|z +a 11 a" sin(bn)u(n) zl> la| 12 a" cos(bn)u(n) kl> la| (e" sin(b)]z z? - |2e a cos(b)]z +e-2a zlz - eacos(b)] z2 - [2e" cos(b)]z+e-2a 13 e an sin(bn)u(n) kl>ea 14 e an cos(bn)u(n) kl>e a Az A'Z 2|A||P|"cos(n0 +9)u(n) where P and A are complex constants defined by P = |P|L0, A = JA20 15 Z-P Z- P* 3.

H.W. Prove the z-transforms for common sequences summarized in Table except the last sequence (15). Table 1 Table of z-Transform Pairs Region of Convergence Line No. x(n), n20 z-Transform X(z) 1 x(n) 2 ô(n) 1 kl>0 az au(n) kl>1 2-1 nu(n) kl>1 (z – 1)? 5 z(z+1) nu(n) kl>1 (z – 1)8 a u(n) kl> la| Z-a 7 e n u(n) (z-ea) z|>ea 8. az na" u(n) kl> la| (z - a)? z sin(a) z2 – 2z cos(a) +1 9 sin(an)u(n) kl>1 zz – cos(a)] z2 – 2z cos(a) +1 10 cos(an)u(n) z|>1 [a sin(b)]z z2 - [2a cos(b)]z +a² zz - a cos(b)] z2 - [2a cos(b)|z +a 11 a" sin(bn)u(n) zl> la| 12 a" cos(bn)u(n) kl> la| (e" sin(b)]z z? - |2e a cos(b)]z +e-2a zlz - eacos(b)] z2 - [2e" cos(b)]z+e-2a 13 e an sin(bn)u(n) kl>ea 14 e an cos(bn)u(n) kl>e a Az A'Z 2|A||P|"cos(n0 +9)u(n) where P and A are complex constants defined by P = |P|L0, A = JA20 15 Z-P Z- P* 3.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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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,
Transcribed Image Text: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,
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