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3. Consider the continuous-time function $x_1(t) = 2 \left[ u(t) - u(t - 6) \right]$ (Note that here, $u(t)$ represents the unit step function). a) Find the Fourier transform of $x_1(t)$ using the Fourier integral. $X_1(\omega) = 12e^{-j3\omega} \text{sinc} (3\omega)$ (unnormalized)* $X_1(\omega) = 12e^{-j3\omega} \text{sinc} \left( \frac{3\omega}{\pi} \right)$ (normalized)* b) Produce a MATLAB plot of the magnitude response of $X_1(\omega)$ over the linear frequency range $\omega = 0$ to 5 rad/sec.

          3. Consider the continuous-time function $x_1(t) = 2 \left[ u(t) - u(t - 6) \right]$ (Note that here, 
$u(t)$ represents the unit step function).
a) Find the Fourier transform of $x_1(t)$ using the Fourier integral.
$X_1(\omega) = 12e^{-j3\omega} \text{sinc} (3\omega)$ (unnormalized)*
$X_1(\omega) = 12e^{-j3\omega} \text{sinc} \left( \frac{3\omega}{\pi} \right)$ (normalized)*
b) Produce a MATLAB plot of the magnitude response of $X_1(\omega)$ over the linear
frequency range $\omega = 0$ to 5 rad/sec.

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3. Consider the continuous-time function x1(t) = 2 [ u(t) - u(t - 6) ] (Note that here,
u(t) represents the unit step function).
a) Find the Fourier transform of x1(t) using the Fourier integral.
X1(ω) = 12e^-j3ωsinc (3ω) (unnormalized)*
X1(ω) = 12e^-j3ωsinc( (3ω)/(π)) (normalized)*
b) Produce a MATLAB plot of the magnitude response of X1(ω) over the linear
frequency range ω = 0 to 5 rad/sec.

Added by Melissa A.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Ameer Said

10/21/2021

Step 1

This represents a rectangular pulse of amplitude 2 and duration 6. The Fourier Transform is given by: $X_1(\omega) = \int_{-\infty}^{\infty} x_1(t)e^{-j\omega t} dt = \int_{0}^{6} 2e^{-j\omega t} dt$ $X_1(\omega) = 2 \left[ \frac{e^{-j\omega t}}{-j\omega} Show more…

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3. Consider the continuous-time function $x_1(t) = 2[u(t) - u(t - 6)]$ (Note that here, $u(t)$ represents the unit step function). a) Find the Fourier transform of $x_1(t)$ using the Fourier integral. $X_1(\omega) = 12e^{-j3\omega}\text{sinc}(3\omega)$ (unnormalized)* $X_1(\omega) = 12e^{-j3\omega}\text{sinc}(\frac{3\omega}{\pi})$ (normalized)* b) Produce a MATLAB plot of the magnitude response of $X_1(\omega)$ over the linear frequency range $\omega = 0$ to 5 rad/sec.

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