GATE 2022 PH – Question 26

Correct answer: (A) $g(-k)=g^*(k)$ implies f real; (B) $g(-k)=-g^*(k)$ implies f purely imaginary

Explanation

The Fourier transform pair is $g(k)=\int f(x)e^{-ikx}dx$.

**If $f(x)$ is real:** $g^*(k)=\int f(x)e^{+ikx}dx=g(-k)$. So $g(-k)=g^*(k)$ (Hermitian symmetry) characterises a **real** $f$. ✓ (A)

**If $f(x)$ is purely imaginary,** $f=i\,h$ with $h$ real. Let $\hat h$ be the transform of $h$, so $g=i\hat h$. Since $h$ is real, $\hat h(-k)=\hat h^*(k)$, hence $g(-k)=i\,\hat h(-k)=i\,\hat h^*(k)=-\left[i\,\hat h(k)\right]^*=-g^*(k)$. So $g(-k)=-g^*(k)$ characterises a **purely imaginary** $f$. ✓ (B)

The statements C and D swap the two cases, so they are false. Answer **A and B**.