How would I find the function given the magnitude plot and the phase response?

Question

I'm wondering how I'd find the Fourier Transform X(jw) given the following information:

My understanding is that the expression for the continuous time fourier transform (CTFT) is magnitude(CTFT)exp(jw*phase), but I'm unsure how to use this expression here, since the phase is different for different values of w. I tried to do something piecewise:

from (-pi to 0): X(jw)=rect(w/(pi))*exp(pi/2*j) = j*rect(w/pi)

from (0 to pi): X(jw)=rect(w/(pi))*exp(-pi/2*j) = -j*rect(w/pi)

This doesn't agree with the answer key: X(jw)=jrect((w+pi/2)/pi) - jrect((w-pi/2)/pi).

Would appreciate any thoughts on how to better approach this problem, since I'm very confused. Thanks!

***please note, here, j is used instead of i, the imaginary number (sqrt(-1))

Answer 1

There is more than one way to write the same thing. You can use $rect()$ functions or define it piece-wise, but in the end it is the same thing, since the $rect()$ is just a shorthand for a piece-wise definition.

If you do it in sections, you don't need to use $rect()$ function at all, you could simply write is as

$$X(j \omega) = \begin{cases} & j \text{ if } -\pi < \omega < 0\\ & -j \text{ if } 0 < \omega < \pi \\ & 0 \text{ otherwise} \end{cases} $$

The answer just puts it together as two rectangle functions: one for negative frequencies and one for the positive ones.

Either way would be correct and I don't think one is "better" than the other unless there are some clear criteria what "better" means.