What are quadrature and in-phase component of a signal?

Question

I read the Wikipedia page but couldn't take much out of it. How do you calculate the in-phase and quadrature components of a modulated signal? From what I got quadrature is the imaginary part of the Fourier transformed of the signal after it has been shifted to the baseband and the in-phase is the real part. Is it right?

Answer 1

• It often happens that you have a signal $s(t) = \sum_n c_n \cos(\omega_n t + \phi_n)$ and you want instead the so-called analytic signal $s_a(t) = \sum_n c_n e^{i\omega_n t + \phi_n}$.

  Then $s(t) = Re(s_a(t))$ is the in-phase component, and $Im(s_a(t)) = \sum_n c_n \sin(\omega_n t + \phi_n)$ is the quadrature phase component, which is a completely different signal in time domain, but exactly the same up to a phase shift in the Fourier domain.

• A real-life application : you have a FM modulated signal $s(t) = \cos(\phi(t))$ and you want instead $s_a(t) = e^{i\phi(t)}$ such that $$-is_a'(t)\overline{s_a(t)} = \phi'(t)$$ which is the signal you want to isolate (for example your favorite FM radio, or the wifi data, or some bunch of data in a narrow frequency band in optic fibers).

All those require approximations of ideal unstable filters (Hilbert transform) and lead to complicated algorithms.