Question

The Fourier transform of the continuous time signal $x(t)$ as shown in the figure is

• A. $e^{-j2\pi f\tau} A \tau \text{ sinc}(f\tau)$
• B. $e^{-j\pi f\tau} A \tau \text{ sinc}(f\tau)$
• C. $e^{-j\pi f\tau} 2A \tau \text{ sinc}(f\tau)$
• D. $e^{j\pi f\tau} A \tau \text{ sinc}(f\tau)$

Hint:

Alternatively, use the time-shifting property.
A symmetric pulse centered at \( t = 0 \) of width \( \tau \) and height \( A \) has the transform \( A \tau \text{sinc}(f \tau) \).
Shifting this pulse to the right by \( \frac{\tau}{2} \) introduces a phase shift of \( e^{-j 2 \pi f (\tau/2)} = e^{-j \pi f \tau} \).
This avoids performing the full integration.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question requires computing the Continuous-Time Fourier Transform (CTFT) of a rectangular pulse of amplitude \(A\) defined over the time interval \([0,\tau]\). 

Step 2: Key Formula or Approach:
The Continuous-Time Fourier Transform (CTFT) of a signal \(x(t)\) is given by \[ X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}\,dt. \] 
Step 3: Detailed Explanation: 

• The given rectangular pulse can be written as \[ x(t)= \begin{cases} A, & 0\le t\le\tau,\\ 0, & \text{otherwise}. \end{cases} \] 
• Substitute \(x(t)\) into the CTFT formula: \[ X(f)=\int_{0}^{\tau}Ae^{-j2\pi ft}\,dt. \] 
• Evaluate the integral: \[ X(f) = A\left[\frac{e^{-j2\pi ft}}{-j2\pi f}\right]_{0}^{\tau} = \frac{A}{-j2\pi f} \left(e^{-j2\pi f\tau}-1\right). \] Hence, \[ X(f) = \frac{A}{j2\pi f} \left(1-e^{-j2\pi f\tau}\right). \] 
• Factor out \(e^{-j\pi f\tau}\): \[ X(f) = \frac{A}{j2\pi f} e^{-j\pi f\tau} \left(e^{j\pi f\tau}-e^{-j\pi f\tau}\right). \] 
• Using Euler's identity, \[ e^{j\theta}-e^{-j\theta}=2j\sin\theta, \] we get \[ X(f) = e^{-j\pi f\tau} \frac{A}{\pi f} \sin(\pi f\tau). \] 
• Multiply and divide the expression by \(\tau\): \[ X(f) = e^{-j\pi f\tau} A\tau \frac{\sin(\pi f\tau)}{\pi f\tau}. \] 
• Using the normalized sinc function, \[ \operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}, \] the Fourier Transform becomes \[ \boxed{ X(f)=A\tau\,\operatorname{sinc}(f\tau)\,e^{-j\pi f\tau}. } \]