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Problem 2 (Time-invariance). Determine whether the following systems are time-invariant or time-varying. Note the first two are CT and the second two are DT. (a) y(t) = sin(x(t)) (b) y(t) = t sin(x(t)) %3D (c) y[k] = 3(x[k] – ¤[k – 2) %3D (d) y[k] = kx[k]

Problem 2 (Time-invariance). Determine whether the following systems are time-invariant or time-varying. Note the first two are CT and the second two are DT. (a) y(t) = sin(x(t)) (b) y(t) = t sin(x(t)) %3D (c) y[k] = 3(x[k] – ¤[k – 2) %3D (d) y[k] = kx[k]

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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**Problem 2 (Time-invariance).** Determine whether the following systems are time-invariant or time-varying. Note the first two are CT and the second two are DT. (a) \( y(t) = \sin(x(t)) \) (b) \( y(t) = t \sin(x(t)) \) (c) \( y[k] = 3(x[k] - x[k-2]) \) (d) \( y[k] = kx[k] \) --- **Explanation:** - **Continuous-Time (CT) Systems:** Refers to systems with continuous input values over time. - (a) \( y(t) = \sin(x(t)) \) evaluates the sine of the continuous input \( x(t) \). - (b) \( y(t) = t \sin(x(t)) \) multiplies time \( t \) with \( \sin(x(t)) \), suggesting a dependency on time. - **Discrete-Time (DT) Systems:** Refers to systems with discrete input values at specific time intervals. - (c) \( y[k] = 3(x[k] - x[k-2]) \) processes two previous discrete-time inputs, weighted by a factor of 3. - (d) \( y[k] = kx[k] \) involves a multiplicative factor of the current index \( k \) with \( x[k] \). This problem explores the concepts of time invariance, identifying changes in output with shifts in input over time.
Transcribed Image Text:**Problem 2 (Time-invariance).** Determine whether the following systems are time-invariant or time-varying. Note the first two are CT and the second two are DT. (a) \( y(t) = \sin(x(t)) \) (b) \( y(t) = t \sin(x(t)) \) (c) \( y[k] = 3(x[k] - x[k-2]) \) (d) \( y[k] = kx[k] \) --- **Explanation:** - **Continuous-Time (CT) Systems:** Refers to systems with continuous input values over time. - (a) \( y(t) = \sin(x(t)) \) evaluates the sine of the continuous input \( x(t) \). - (b) \( y(t) = t \sin(x(t)) \) multiplies time \( t \) with \( \sin(x(t)) \), suggesting a dependency on time. - **Discrete-Time (DT) Systems:** Refers to systems with discrete input values at specific time intervals. - (c) \( y[k] = 3(x[k] - x[k-2]) \) processes two previous discrete-time inputs, weighted by a factor of 3. - (d) \( y[k] = kx[k] \) involves a multiplicative factor of the current index \( k \) with \( x[k] \). This problem explores the concepts of time invariance, identifying changes in output with shifts in input over time.
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