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if you have a signal say x(t) in continuous time and you transform it using the Fourier transform for continuous time you get X(w) which is the frequency domain representation of this signal x(t). From this point, you can get magnitude and phase and you now have a greater representation of the system outside of just the time domain.

My question is, what is X(w) just by itself? I understand that from X(w) you can get magnitude and phase and these are important in system analysis, but what is the significance of the X(w) term?

Can you plot X(w)? If you plot X(w), the x axis is frequency but what does the y axis denote? Not having a firm understanding of this is really fogging my intuition behind transforms.

Thank you all!

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    $\begingroup$ $x$ is recovered from its Fourier transform via $x(t) = \int X(\omega) e^{2\pi i\omega t} \, d \omega$. This formula expresses $x$ as a linear combination of "waves" $e^{2 \pi i \omega t}$, and $X(\omega)$ tells you how much of the wave with frequency $\omega$ we need. $\endgroup$
    – littleO
    Apr 21, 2014 at 6:13

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The way the fourier transform is usually defined, $X(\omega)$ is a complex number, which contains information about the magnitude and phase of each frequency component. More specifically, $$ X(\omega) = M_\omega e^{i\varphi_\omega} = M_\omega\left(\cos \varphi_\omega + i\sin\varphi_\omega\right) $$ where $M_\omega$ is the magnitude and $\varphi_\omega$ the phase of the frequency $\omega$.

If you want to plot $X(\omega)$, you have two choices

  1. You can plot separately plot $M_\omega$ over $\omega$, i.e the magnitude over the frequency, and $\varphi_\omega$ over $\omega$, i.e. the phase over the frequency. This is called a bode plot.

  2. You can treat $\omega$ as a *parameter, and plot $X(\omega)$ in the two-dimensional plane as $t$ varies. This is called a nyquist plot.

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  • $\begingroup$ ok thats helpful thank you. Another question, if I plot for magnitude or phase, what does the y axis mean? $\endgroup$
    – Billy Thompson
    Apr 21, 2014 at 9:48
  • $\begingroup$ @BillyThompson You can plot frequency (x-axis) vs. magnitude (y-axis) or frequency (x-axis) vs. phase (y-axis). Two more options would be frequency vs. real or imaginary as well. $\endgroup$
    – AnonSubmitter85
    Apr 23, 2014 at 2:51

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