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: Analyze each signal and find its Fourier Transform pair.

• A. \(e^{-at}u(t), a>0\): This is a standard one-sided decaying exponential. \[ \mathcal{F}\{e^{-at}u(t)\} = \frac{1}{a+j\omega} \] Matches II.
• B. \(\cos \omega_0 t\): Using Euler's identity: \(\cos \omega_0 t = \frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2}\). Fourier Transform: \[ \mathcal{F}\{\cos \omega_0 t\} = \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] \] Matches I.
• C. \(\sin \omega_0 t\): Using Euler's identity: \(\sin \omega_0 t = \frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j}\). Fourier Transform: \[ \mathcal{F}\{\sin \omega_0 t\} = \frac{\pi}{j}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] = -j\pi[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] \] Matches IV.
• D. \(te^{-at}u(t), a>0\): Using the frequency differentiation property: \(\mathcal{F}\{t \cdot x(t)\} = j \frac{d}{d\omega}X(\omega)\), where \(x(t) = e^{-at}u(t)\). \[ \mathcal{F}\{te^{-at}u(t)\} = j \frac{d}{d\omega}\left(\frac{1}{a+j\omega}\right) = \frac{1}{(a+j\omega)^2} \] Matches III.

Step 2: Consolidate the matches.

The correct pairings are:

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

This corresponds to option (B).