Question

Signals and their Fourier Transforms are given in the table below. Match LIST-I with LIST-II and choose the correct answer.

LIST-I	LIST-II
A. \( e^{-at}u(t), a>0 \)	I. \( \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] \)
B. \( \cos \omega_0 t \)	II. \( \frac{1}{j\omega + a} \)
C. \( \sin \omega_0 t \)	III. \( \frac{1}{(j\omega + a)^2} \)
D. \( te^{-at}u(t), a>0 \)	IV. \( -j\pi[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] \)

• A. A-I, B-II, C-III, D-IV
• B. A-II, B-I, C-IV, D-III
• C. A-I, B-II, C-IV, D-III
• D. A-III, B-IV, C-I, D-II

Hint:

Memorizing the basic Fourier Transform pairs (for exponential, sine, cosine, rectangular pulse, and impulse functions) and properties (linearity, time shift, frequency differentiation) is essential for solving these types of problems quickly.

Answer 1

The Correct Option is B

Step 1: Determine the Fourier Transform for each signal.

• A. \(e^{-at}u(t), a>0\): This represents a decaying exponential function defined for positive time. Its Fourier Transform is: \[ \mathcal{F}\{e^{-at}u(t)\} = \frac{1}{a+j\omega} \] This corresponds to II.
• B. \(\cos \omega_0 t\): Utilizing Euler's formula, we can express cosine as: \(\cos \omega_0 t = \frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2}\). The Fourier Transform is: \[ \mathcal{F}\{\cos \omega_0 t\} = \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] \] This corresponds to I.
• C. \(\sin \omega_0 t\): Similarly, using Euler's formula, sine is expressed as: \(\sin \omega_0 t = \frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j}\). The Fourier Transform is: \[ \mathcal{F}\{\sin \omega_0 t\} = \frac{\pi}{j}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] = -j\pi[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] \] This corresponds to IV.
• D. \(te^{-at}u(t), a>0\): Applying the frequency differentiation property, which states \(\mathcal{F}\{t \cdot x(t)\} = j \frac{d}{d\omega}X(\omega)\), where \(x(t) = e^{-at}u(t)\), we get: \[ \mathcal{F}\{te^{-at}u(t)\} = j \frac{d}{d\omega}\left(\frac{1}{a+j\omega}\right) = \frac{1}{(a+j\omega)^2} \] This corresponds to III.

Step 2: Summarize the signal-transform pairings.

The correct matches are:

A-II, B-I, C-IV, D-III

Therefore, the answer is option (B).