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Find the Continuous Time Fourier Transform of the following signal

Find the Continuous Time Fourier Transform of the following signal

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Find the Continuous Time Fourier Transform of the following signal

### Transcription for Educational Website: **Equation:** \[ c) \, x(t) = 10^3 e^{-10^3 t} u(t) \] **Explanation:** This equation represents a time-domain signal \( x(t) \) where: - \( 10^3 \) is a constant multiplication factor. - \( e^{-10^3 t} \) is an exponential decay function, where the rate of decay is determined by the exponent \(-10^3 t\). This implies a rapid decrease in the signal over time. - \( u(t) \) is the unit step function, which is 0 for \( t < 0 \) and 1 for \( t \geq 0 \). This function is used to ensure that the signal is defined only for non-negative values of time, effectively "turning on" the signal at \( t = 0 \). This kind of signal is commonly used in systems and control theory to model processes that initiate at a certain point in time and decay exponentially thereafter.
Transcribed Image Text:### Transcription for Educational Website: **Equation:** \[ c) \, x(t) = 10^3 e^{-10^3 t} u(t) \] **Explanation:** This equation represents a time-domain signal \( x(t) \) where: - \( 10^3 \) is a constant multiplication factor. - \( e^{-10^3 t} \) is an exponential decay function, where the rate of decay is determined by the exponent \(-10^3 t\). This implies a rapid decrease in the signal over time. - \( u(t) \) is the unit step function, which is 0 for \( t < 0 \) and 1 for \( t \geq 0 \). This function is used to ensure that the signal is defined only for non-negative values of time, effectively "turning on" the signal at \( t = 0 \). This kind of signal is commonly used in systems and control theory to model processes that initiate at a certain point in time and decay exponentially thereafter.
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