Question

Fourier transform of \( a_1 f_1(t) + a_2 f_2(t) \) is ________.

• A. \( a_1 F_1(\omega) + a_2 F_2(\omega) \)
• B. \( a_1 F_1(\omega) - a_2 F_2(\omega) \)
• C. \( a_1 F_1^2(\omega) + a_2 F_2^2(\omega) \)
• D. \( a_1 F_1(\omega) * a_2 F_2(\omega) \)

Hint:

Fourier transform is linear: addition and scalar multiplication in time domain maps directly to frequency domain.

Answer 1

The Correct Option is A

Step 1: Linearity Property of Fourier Transform
One of the most important properties of the Fourier Transform is linearity. It states that: \[ \mathcal{F}\{a_1 f_1(t) + a_2 f_2(t)\} = a_1 \mathcal{F}\{f_1(t)\} + a_2 \mathcal{F}\{f_2(t)\} \] This means that if you scale and add functions in the time domain, the same scaling and addition appear in the frequency domain.
Step 2: Apply this to our given expression
Let: \[ \mathcal{F}\{f_1(t)\} = F_1(\omega), \mathcal{F}\{f_2(t)\} = F_2(\omega) \] Then: \[ \mathcal{F}\{a_1 f_1(t) + a_2 f_2(t)\} = a_1 F_1(\omega) + a_2 F_2(\omega) \] Step 3: Eliminate incorrect options
- Option (2): subtracts instead of adding. Violates linearity.
- Option (3): squares the transforms. Not linear.
- Option (4): uses convolution. Convolution occurs for multiplication in time domain, not addition.
Therefore, Option (1) is correct.