Determine the Fourier transform for the signal x(t) given in Figure Q1(a). An arbitrary system has impulse response given h(t) = e¯²tu(t). If the signal x(t) is passed through the system, determine the output signal y(t) using Fourier analysis.
Determine the Fourier transform for the signal x(t) given in Figure Q1(a). An arbitrary system has impulse response given h(t) = e¯²tu(t). If the signal x(t) is passed through the system, determine the output signal y(t) using Fourier analysis.
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![**Fourier Transform and Signal Processing**
### Problem Statement
Determine the Fourier transform for the signal \( x(t) \) given in **Figure Q1(a)**.
An arbitrary system has an impulse response given by \( h(t) = e^{-2t}u(t) \). If the signal \( x(t) \) is passed through the system, determine the output signal \( y(t) \) using Fourier analysis.
**Hint:**
\[ \text{sinc}\left(\frac{\omega}{2}\right) = \frac{e^{ \frac{j\omega}{2}} - e^{ -\frac{j\omega}{2}}}{j \omega} \]
### Figure Explanation
**Figure Q1(a)** presents a graphical representation of the signal \( x(t) \) as a function of time \( t \).
- The signal \( x(t) \) is defined as follows:
- For \( t < 0 \), \( x(t) = 0 \).
- At \( t = 0 \), \( x(t) = 1 \).
- For \( 0 < t \leq 1 \), \( x(t) \) maintains a constant value of 1.
- For \( 1 < t \leq 2 \), \( x(t) \) linearly decreases from 1 to 0.
- For \( t > 2 \), \( x(t) = 0 \).
The \( x \)-axis represents time ( \( t \) ), and the \( y \)-axis represents the signal amplitude ( \( x(t) \) ).
**Figure Q1(a):**
Graph of the signal \( x(t) \)
This problem requires finding the Fourier transform of \( x(t) \), determining how an arbitrary system with impulse response \( h(t) = e^{-2t}u(t) \) affects the signal, and then computing the output signal \( y(t) \) via Fourier analysis. The hint provides a useful mathematical identity involving the sinc function to facilitate the Fourier transform calculation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07af7f94-fcc4-4c2d-8626-5a5eb17664d6%2F276b9e56-57a7-4d68-b536-084b8df5017a%2Ff2llg53_processed.png&w=3840&q=75)
Transcribed Image Text:**Fourier Transform and Signal Processing**
### Problem Statement
Determine the Fourier transform for the signal \( x(t) \) given in **Figure Q1(a)**.
An arbitrary system has an impulse response given by \( h(t) = e^{-2t}u(t) \). If the signal \( x(t) \) is passed through the system, determine the output signal \( y(t) \) using Fourier analysis.
**Hint:**
\[ \text{sinc}\left(\frac{\omega}{2}\right) = \frac{e^{ \frac{j\omega}{2}} - e^{ -\frac{j\omega}{2}}}{j \omega} \]
### Figure Explanation
**Figure Q1(a)** presents a graphical representation of the signal \( x(t) \) as a function of time \( t \).
- The signal \( x(t) \) is defined as follows:
- For \( t < 0 \), \( x(t) = 0 \).
- At \( t = 0 \), \( x(t) = 1 \).
- For \( 0 < t \leq 1 \), \( x(t) \) maintains a constant value of 1.
- For \( 1 < t \leq 2 \), \( x(t) \) linearly decreases from 1 to 0.
- For \( t > 2 \), \( x(t) = 0 \).
The \( x \)-axis represents time ( \( t \) ), and the \( y \)-axis represents the signal amplitude ( \( x(t) \) ).
**Figure Q1(a):**
Graph of the signal \( x(t) \)
This problem requires finding the Fourier transform of \( x(t) \), determining how an arbitrary system with impulse response \( h(t) = e^{-2t}u(t) \) affects the signal, and then computing the output signal \( y(t) \) via Fourier analysis. The hint provides a useful mathematical identity involving the sinc function to facilitate the Fourier transform calculation.
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