inverse fourier transform of magnitude and phase

Question

I stuck this question. Frequency response is written as magnitude and phase and I don't find inverse fourier given signal which as magnitude and phase.How can I solve it ? Can you explain the solution way in order? magnitude of signal is : y=−ω while −1≤ω<0, y=ω while 0≤ω<1 phase of signal is: y=−3ω

What is the Fourier transform? I mean I have phase and magnitude information of a signal that has been transformed from fourier. I want to find inverse fourier transform. And phase and magnitude information are above

Answer 1

You can express a complex signal $y$ as $|y|.e^{i \phi_y}$ where $|y|$ is the magnitude and $\phi_y$ is the phase. You can use Inverse the Fourier transform formula: $$\displaystyle y(t) = \frac{1}{2\pi}\int_{-\infty}^\infty Y(\omega) e^{j\omega t} d\omega$$ You can express $Y(\omega)$ in $|y|.e^{i \phi_y}$ form and then perform the IFFT. But I am not sure if a minus comes in magnitude unless its in dB.

Answer 2

What you have as the magnitude response is $|Y(e^{j\omega})| = \left| \omega \right|$, for $\omega \in [-1,1]$ and phase response is $\angle{Y(e^{j\omega})} = -3\omega$. So, $Y(e^{j\omega}) = |Y(e^{j\omega})|. e^{j\angle{Y(e^{j\omega})} }$.