Question

The Fourier transform of a signum function is

• A. \(j\pi f\)
• B. \(\dfrac{1}{j\pi f}\)
• C. \(j\pi f+a\)
• D. \(\dfrac{1}{j\pi f+a}\)

Hint:

Remember the standard pair: \[ \operatorname{sgn}(t) \leftrightarrow \frac{1}{j\pi f}. \] It is one of the most frequently used Fourier Transform results.

Answer 1

The Correct Option is B

Concept: The signum function is defined as \[ \operatorname{sgn}(t)= \begin{cases} 1, & t>0 \\ 0, & t=0 \\ -1, & t<0 \end{cases} \] A standard Fourier Transform pair is \[ \operatorname{sgn}(t) \;\xleftrightarrow{\mathcal F}\; \frac{1}{j\pi f}. \]

Step 1: Recall the relation between signum and unit-step functions.
\[ \operatorname{sgn}(t) = 2u(t)-1. \] This relation is commonly used in transform analysis.

Step 2: Use the standard transform result.
The known Fourier Transform is \[ \boxed{ \operatorname{sgn}(t) \;\xleftrightarrow{\mathcal F}\; \frac{1}{j\pi f} } \] Therefore the required answer is \[ \boxed{\frac{1}{j\pi f}}. \]