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

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))

• I don't think you need to specify that "j" is the imaginary number. People on this forum will use i or j interchangeably. – Ben Mar 18 at 1:51

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