Let $$g(t) = g_r(t) + j g_i(t)$$ be a complex signal with $g_r(t)$ and $g_i(t)$ real signals.

We can write the Fourier transform $$G(f) = G_r(f) + jG_i(f),$$ where $G_r(f)$, $G_i(f)$ and $G(f)$ are Fourier transforms of $g_r(t)$, $g_i(t)$ and $g(t)$, respectively and are complex functions in general.

How can I write $G_r(f)$ and $G_i(f)$ in terms of $G(f)$? Any proof along with the answer would be appreciated.

  • $\begingroup$ Are you aware of what is called analytic signal ? $\endgroup$ – Jean Marie Mar 6 '16 at 19:05
  • $\begingroup$ @JeanMarie Yes I am. However, I guess I just found the answer. We can write $g_r(t) = (g(t) + g^*(t))/2$, meaning $G_r(f) = (G(f)+G^*(-f))/2$. $\endgroup$ – ubaabd Mar 6 '16 at 19:10

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