Question

Fourier transform of \( f(t) = 1 \) is ________.

• A. \( 2\pi\delta(\omega) \)
• B. \( \pi\delta(\omega) \)
• C. \( 3\pi\delta(\omega) \)
• D. \( n\pi\delta(\omega) \)

Hint:

A constant function in time domain always maps to a delta function in frequency: \( f(t) = 1 \Rightarrow F(\omega) = 2\pi\delta(\omega) \)

Answer 1

The Correct Option is A

Step 1: Understand the function
The function given is \( f(t) = 1 \), a constant function defined for all time \( t \in (-\infty, \infty) \). This is also known as a unit constant signal.
Step 2: Apply the Fourier transform definition
The Fourier transform of a function \( f(t) \) is: \[ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt \] Substitute \( f(t) = 1 \): \[ F(\omega) = \int_{-\infty}^{\infty} 1 \cdot e^{-j\omega t} dt = \int_{-\infty}^{\infty} e^{-j\omega t} dt \] Step 3: Result of this integral
This integral does not converge in the usual sense, but in the theory of distributions (generalized functions), it equals: \[ \int_{-\infty}^{\infty} e^{-j\omega t} dt = 2\pi\delta(\omega) \] Step 4: Interpretation
This means a constant function in time corresponds to a delta function in frequency. All the energy is concentrated at \( \omega = 0 \).
Therefore, the correct answer is Option (1).