Question

The Fourier transform of the signal u (t) is ____________

(a) π δ(ω)

(b) $\frac{1}{jω}$

(c) π δ(ω) + $\frac{1}{jω}$

(d) –$\frac{1}{jω}$

This question was addressed to me in my homework.

The question is from Common Fourier Transforms in portion Fourier Transform of Signals and Systems

Answer 1

The correct choice is (c) π δ(ω) + $\frac{1}{jω}$

To elaborate: Fourier transform of 1 = 2π δ(ω)

And Fourier transform of $\frac{1}{2}$ = π δ(ω)

Hence, Fourier transform of sgn (t) = $\frac{2}{jω}$

Or, $\frac{1}{2}$ sgn (t) = $\frac{1}{jω}$

∴ $\frac{1}{2} + \frac{1}{2}$ sgn (t) = w (t)

Fourier transform of w (t) = π δ(ω) + $\frac{1}{jω}$.