Question

Find the Fourier transform of  f(t)=te^-at u(t).

(a) \(\frac{1}{(a-jω)^2} \)

(b) \(\frac{1}{(a+jω)^2} \)

(c) \(\frac{a}{(a-jω)^2} \)

(d) \(\frac{ω}{(a-jω)^2} \)

I had been asked this question during an interview.

The above asked question is from Properties of Fourier Transforms in division Fourier Transform of Signals and Systems

Answer 1

The correct option is (b) \(\frac{1}{(a+jω)^2} \)

To explain I would say: Using frequency differentiation property, \(tx(t) \leftrightarrow j \frac{d}{dω} \,X(ω)\)

\(F[te^{-at} u(t)] = j \frac{d}{dω} F[te^{-at} \,u(t)] = j \frac{d}{dω} \frac{1}{a+jω} = j \frac{-1(j)}{(a+jω)^2} = \frac{1}{(a+jω)^2} \)

\(te^{-at} \,u(t) \leftrightarrow \frac{1}{(a+jω)^2} \).