Question

If \(Fc\{e^{-ax}\} =  \frac{p}{a^2+p^2}\), find the \(Fs\{-a \, e^{-ax}\}.\)

(a) \(4 \frac{p}{a^2+p^2} \)

(b) \(\frac{-p^2}{a^2+p^2} \)

(c) \(4 \frac{p^2}{a^2+p^2} \)

(d) \(\frac{p}{a^2+p^2} \)

The question was posed to me in a job interview.

My question is based upon Fourier Transform and Convolution in chapter Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

The correct answer is (b) \(\frac{-p^2}{a^2+p^2} \)

Easy explanation: \(-a \, e^{-ax} = \frac{d}{dx}(e^{-ax}) = F’(x) \)

\(Fs\{F’(x)\} = -pfc(p) \)

\( = \frac{-p^2}{a^2+p^2} \).