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Complex exponentials obey the expected rules of algebra when doing integrals and derivatives. Consider the complex signal z(t) = Zejлt/2 where Z = e¯jπ/³¸ (a) Show that the first derivative of z(t) with respect to time can be represented as a new com- plex exponential Qejлt/², i.e., ¼z(t) = Qеjлt/2. Determine the value for the complex amplitude Q. How much greater (or smaller) is the angle of Q than the angle of Z. (b) Evaluate the definite integral of z(t) over the range 0 < t < 1: 1 z(t)dt = ? Note that integrating a complex quantity follows the expected rules of algebra: you could integrate the real and imaginary parts separately, but you can also use the integration formula for an exponential directly on z(t). (c) Evaluate the integral of the magnitude squared |z(t)|² over the range -1 ≤ t ≤ 1: 1 [ \z (1)³ dt = ? -1

Complex exponentials obey the expected rules of algebra when doing integrals and derivatives. Consider the complex signal z(t) = Zejлt/2 where Z = e¯jπ/³¸ (a) Show that the first derivative of z(t) with respect to time can be represented as a new com- plex exponential Qejлt/², i.e., ¼z(t) = Qеjлt/2. Determine the value for the complex amplitude Q. How much greater (or smaller) is the angle of Q than the angle of Z. (b) Evaluate the definite integral of z(t) over the range 0 < t < 1: 1 z(t)dt = ? Note that integrating a complex quantity follows the expected rules of algebra: you could integrate the real and imaginary parts separately, but you can also use the integration formula for an exponential directly on z(t). (c) Evaluate the integral of the magnitude squared |z(t)|² over the range -1 ≤ t ≤ 1: 1 [ \z (1)³ dt = ? -1

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Complex exponentials obey the expected rules of algebra when doing integrals and derivatives. Consider the complex signal z(t) = Zejлt/2 where Z = e¯jπ/³¸ (a) Show that the first derivative of z(t) with respect to time can be represented as a new com- plex exponential Qejлt/², i.e., ¼z(t) = Qеjлt/2. Determine the value for the complex amplitude Q. How much greater (or smaller) is the angle of Q than the angle of Z. (b) Evaluate the definite integral of z(t) over the range 0 < t < 1: 1 z(t)dt = ? Note that integrating a complex quantity follows the expected rules of algebra: you could integrate the real and imaginary parts separately, but you can also use the integration formula for an exponential directly on z(t). (c) Evaluate the integral of the magnitude squared |z(t)|² over the range -1 ≤ t ≤ 1: 1 [ \z (1)³ dt = ? -1
Transcribed Image Text:Complex exponentials obey the expected rules of algebra when doing integrals and derivatives. Consider the complex signal z(t) = Zejлt/2 where Z = e¯jπ/³¸ (a) Show that the first derivative of z(t) with respect to time can be represented as a new com- plex exponential Qejлt/², i.e., ¼z(t) = Qеjлt/2. Determine the value for the complex amplitude Q. How much greater (or smaller) is the angle of Q than the angle of Z. (b) Evaluate the definite integral of z(t) over the range 0 < t < 1: 1 z(t)dt = ? Note that integrating a complex quantity follows the expected rules of algebra: you could integrate the real and imaginary parts separately, but you can also use the integration formula for an exponential directly on z(t). (c) Evaluate the integral of the magnitude squared |z(t)|² over the range -1 ≤ t ≤ 1: 1 [ \z (1)³ dt = ? -1
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