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) + \(\frac{1}{jω}\).