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Problem 1 (Problem 7.2-4 from the textbook) 7.2-4. Consider a sampled-data system with 7 = 0,5 s and the characteristic equation given by (z – 0.9)(z – 0.8)(z – 1.9z + 1.0) = 0 (a) Find the terms in the system natural response. (b) A discrete LTI system is stable, unstable, or marginally stable. Identify the type of stability for this system. (c) The natural response of this system contains an undamped sinusoidal response term of the form . Find the frequency w of this term. A cos(wkT + 0) Solution: T = 0.5, char. eq.: (z–0.9)(z–0.8)(=² –1.9z +1) = 0 Complex roots: = = 0.95± j0.3122 =1Z+18.19° = 1Z±0.3175 rad (a) k,(0.9)*, k,(0.8)*, A cos (0.3175k +0) (b) marginally stable 0.3175 (c) root = 1Z± mT, .. mT = 0.3175 = m= 0.5 = 0.635 rad/s %3D

Problem 1 (Problem 7.2-4 from the textbook) 7.2-4. Consider a sampled-data system with 7 = 0,5 s and the characteristic equation given by (z – 0.9)(z – 0.8)(z – 1.9z + 1.0) = 0 (a) Find the terms in the system natural response. (b) A discrete LTI system is stable, unstable, or marginally stable. Identify the type of stability for this system. (c) The natural response of this system contains an undamped sinusoidal response term of the form . Find the frequency w of this term. A cos(wkT + 0) Solution: T = 0.5, char. eq.: (z–0.9)(z–0.8)(=² –1.9z +1) = 0 Complex roots: = = 0.95± j0.3122 =1Z+18.19° = 1Z±0.3175 rad (a) k,(0.9)*, k,(0.8)*, A cos (0.3175k +0) (b) marginally stable 0.3175 (c) root = 1Z± mT, .. mT = 0.3175 = m= 0.5 = 0.635 rad/s %3D

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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Problem 1 (Problem 7.2-4 from the textbook) 7.2-4. Consider a sampled-data system with 7 = 05 s and the characteristic equation given by (z – 0.9)(z – 0.8)(z – 1.9z + 1.0) = 0 (a) Find the terms in the system natural response. (b) A discrete LTI system is stable, unstable, or marginally stable. Identify the type of stability for this system. (c) The natural response of this system contains an undamped sinusoidal response term of the form A cos(wkT + 0) - Find the frequency w of this term. Solution: T =0.5, char. eq.: (z–0.9)(z–0.8)(z² -1.9z+1) = 0 Complex roots: z = 0.95+ j0.3122 = 1Z+18.19° =1Z±0.3175 rad (a) k(0.9)*, k¿(0.8)*, A cos (0.3175k +0) (b) marginally stable 0.3175 (c) root = 1Z±mT, ..oT = 0.3175 = 0= 0.5 0.635 rad/s
Transcribed Image Text:Problem 1 (Problem 7.2-4 from the textbook) 7.2-4. Consider a sampled-data system with 7 = 05 s and the characteristic equation given by (z – 0.9)(z – 0.8)(z – 1.9z + 1.0) = 0 (a) Find the terms in the system natural response. (b) A discrete LTI system is stable, unstable, or marginally stable. Identify the type of stability for this system. (c) The natural response of this system contains an undamped sinusoidal response term of the form A cos(wkT + 0) - Find the frequency w of this term. Solution: T =0.5, char. eq.: (z–0.9)(z–0.8)(z² -1.9z+1) = 0 Complex roots: z = 0.95+ j0.3122 = 1Z+18.19° =1Z±0.3175 rad (a) k(0.9)*, k¿(0.8)*, A cos (0.3175k +0) (b) marginally stable 0.3175 (c) root = 1Z±mT, ..oT = 0.3175 = 0= 0.5 0.635 rad/s
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