[amp_mcq option1=”Is $$\\left| z \\right| > {1 \\over 9}$$” option2=”Is $$\\left| z \\right| < {1 \\over 3}$$\" option3=\"Is $${1 \\over 3} > \\left| z \\right| > {1 \\over 9}$$” option4=”Does not exist” correct=”option4″]<\/p>\n
The correct answer is $\\boxed{{1 \\over 3} > \\left| z \\right| > {1 \\over 9}}$.<\/p>\n
The region of convergence (ROC) of the z-transform of a sequence $x[n]$ is the set of values of $z$ for which the z-transform converges. The z-transform of $x[n]$ is defined as<\/p>\n
$$X(z) = \\sum_{n=-\\infty}^{\\infty} x[n] z^{-n}$$<\/p>\n
The ROC of $X(z)$ is the set of values of $z$ for which the infinite series in the definition of $X(z)$ converges.<\/p>\n
In this case, $x[n] = {\\left( { – {1 \\over 9}} \\right)^n}u\\left( n \\right) – {\\left( { – {1 \\over 3}} \\right)^n}u\\left( { – n – 1} \\right)$.<\/p>\n
The z-transform of $x[n]$ is<\/p>\n
$$X(z) = {1 \\over 1 – {1 \\over 9} z^{-1}} – {1 \\over 1 + {1 \\over 3} z^{-1}}$$<\/p>\n
The poles of $X(z)$ are at $z = 9$ and $z = -3$.<\/p>\n
The ROC of $X(z)$ is the set of values of $z$ for which the distance between $z$ and any pole is greater than 1.<\/p>\n
In this case, the distance between $z$ and $z = 9$ is $\\sqrt{100} = 10$. The distance between $z$ and $z = -3$ is $\\sqrt{10} = 3.16$.<\/p>\n
Therefore, the ROC of $X(z)$ is the set of values of $z$ such that $10 > \\left| z \\right| > 3.16$, or $${1 \\over 3} > \\left| z \\right| > {1 \\over 9}$$<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”Is $$\\left| z \\right| > {1 \\over 9}$$” option2=”Is $$\\left| z \\right| < {1 \\over 3}$$\" option3=\"Is $${1 \\over 3} > \\left| z \\right| > {1 \\over 9}$$” option4=”Does not exist” correct=”option4″]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[959],"tags":[],"class_list":["post-59831","post","type-post","status-publish","format-standard","hentry","category-signal-processing","no-featured-image-padding"],"yoast_head":"\n