Need help on signals and systems, Fourier representation of a signal

Question

I've got a homework question for my signals and systems class.

In the question, we're given the graph of the magnitude and the phase of a continuous signal. Here are the graphs:

I have to find the real continuous function x(t) based on the magnitude and phase given.

I found this formula on the internet:

I figure I just need to find the function for the magnitude and phase then I insert it into the formula to figure out the real continuous signal.

My problem is that I'm confused as to how to find the function for the magnitude and phase.

I figured out how to find the function for the magnitude, but I still don't know if it's correct or not:

As you can see, I already wrote the function for |X(jw)|, but I haven't figured out the function for the phase. For the function |X(jw)|, I figured it out from this formula:

To sum up:

1. Am I already on the correct path to find the real continuous signal x(t)?
2. If I'm already on the correct path, how do I find the function for the phase?
3. Is the function for the magnitude already correct?

Answer 1

The expression you wrote for the magnitude is indeed correct.

In order to find the function of the the phase, assuming its slope is linear of course, use the start and end points: $$(-2,\pi/2), (2, -\pi/2)$$

To compute the slope of the function. Then, it should be trivial to derive the function (This Wikipedia article should help) and solve the bounded integral.