Editing Complex number (section)

====Signal analysis====
Complex numbers are used in [[signal analysis]] and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a [[sine wave]] of a given [[frequency]], the absolute value {{math|{{!}}''z''{{!}}}} of the corresponding {{mvar|z}} is the [[amplitude]] and the [[Argument (complex analysis)|argument]] {{math|arg ''z''}} is the [[phase (waves)|phase]].

If [[Fourier analysis]] is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

<math display=block>x(t) = \operatorname{Re} \{X( t ) \} </math>

and

<math display=block>X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } </math>

where ω represents the [[angular frequency]] and the complex number ''A'' encodes the phase and amplitude as explained above.

This use is also extended into [[digital signal processing]] and [[digital image processing]], which use digital versions of Fourier analysis (and [[wavelet]] analysis) to transmit, [[Data compression|compress]], restore, and otherwise process [[Digital data|digital]] [[Sound|audio]] signals, still images, and [[video]] signals.

Another example, relevant to the two side bands of [[amplitude modulation]] of AM radio, is:

<math display=block>\begin{align}
  \cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right)
    & = \operatorname{Re}\left(e^{i(\omega + \alpha)t} + e^{i(\omega - \alpha)t}\right) \\
    & = \operatorname{Re}\left(\left(e^{i\alpha t} + e^{-i\alpha t}\right) \cdot e^{i\omega t}\right) \\
    & = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \cos\left(\omega t\right).
\end{align}</math>