Question

A continuous real-valued signal $x(t)$ has finite positive energy and $x(t)=0,\ \forall\,t<0$. From the list below, select ALL signals whose continuous-time Fourier transform is purely imaginary.

• A. $x(t)+x(-t)$
• B. $x(t)-x(-t)$
• C. $j\,(x(t)+x(-t))$
• D. $j\,(x(t)-x(-t))$

Hint:

Real even $\rightarrow$ real spectrum; real odd $\rightarrow$ purely imaginary spectrum. Multiplying by $j$ swaps real and imaginary parts.

Answer 1

The Correct Option is B, C

To identify which signals have a purely imaginary continuous-time Fourier transform, we first analyze the Fourier transform properties of the given functions:

• $x(t)+x(-t)$: This is an even function, since $x(t)=x(-t)$ for even functions. The Fourier transform of an even function is real.
• $x(t)-x(-t)$: This is an odd function, since $x(t)=-x(-t)$ for odd functions. The Fourier transform of an odd function is purely imaginary.
• $j\,(x(t)+x(-t))$: The multiplication by $j$ (imaginary unit) will turn the real Fourier transform of the even function $x(t)+x(-t)$ into an imaginary one.
• $j\,(x(t)-x(-t))$: As $x(t)-x(-t)$ already has a purely imaginary Fourier transform, multiplying it by $j$ results in a real Fourier transform, since $j^2 = -1$.

Therefore, the expressions for which the continuous-time Fourier transform is purely imaginary are:

• $x(t)-x(-t)$
• $j\,(x(t)+x(-t))$