Inverse Fourier Transform From Plots (2019 edition) Hello I borrowed the title for another post. 

I cannot figure out how to find the inverse fourier transform from this spectrum.


I know what the transform is 

I'm sorry for the plot being hard to read, it is how we were given it and I understand the frustration. The magnitude plot goes from -w0 to w0 with a amplitude 1. The phase plot has the same interval but alternates from pi/2 to -pi/2. Apologies for formatting I am new and on mobile. 
Can someone help me I am totally lost.
That's actually a fairly a tricky one. Two ways you can go about it

Method 1


Magnitude is that of a ideal lowpass filter. Inverse of that is sinc function
Phase is that of an hilbert transformer (times -1). Impulse response is (roughly) $1/\pi t$
convolve the two


Method 2


Write out the equation for inverse Fourier transform
Pop in values from the graph and simplify a bit
You get either sum or integral for unity amplitude sine waves up to $\omega_{0}$


Neither one is pretty.