Question

The function which has its Fourier transform, Laplace transform, and Z transform unity is __________

(a) Gaussian

(b) Impulse

(c) Sinc

(d) Pulse

This question was posed to me during an online interview.

Question is from Fourier Series Analysis using Circuits in section Fourier Analysis on Complex Spaces of Signals and Systems

Answer 1

Right option is (b) Impulse

For explanation I would say: The definition of Fourier transform is \(\int_{-∞}^∞ f(x) e^{2πjxf} \,dx\)

The definition of Laplace transform is \(\int_0^∞ f(t) e^{-st} \,dt\)

The definition of Z-transform is \(∑_{n=-∞}^∞ x[n]z^{-n}\)

Substituting f (t) = δ (t) in the definitions of Fourier Laplace and Z-transform, we get the transforms in each case as 1. Hence an impulse function has unity Fourier Laplace and Z-transform.