section_path: Applications/In applied mathematics/Signal analysis
section_name: Signal analysis
paragraph_text: 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 | z | of the corresponding z is the amplitude and the argument arg z is the 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
  
  x (t) = Re { X (t) } {\displaystyle x(t)=\operatorname {Re} \{X(t)\}}
  
  and
  
  X (t) = A e i ω t = a e i ϕ e i ω t = a e i (ω t + ϕ) {\displaystyle X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi)}}
  
  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, compress, restore, and otherwise process digital audio signals, still images, and video signals.
  
  Another example, relevant to the two side bands of amplitude modulation of AM radio, is:
  
  cos ((ω + α) t) + cos ((ω − α) t) = Re (e i (ω + α) t + e i (ω − α) t) = Re ((e i α t + e − i α t) ⋅ e i ω t) = Re (2 cos (α t) ⋅ e i ω t) = 2 cos (α t) ⋅ Re (e i ω t) = 2 cos (α t) ⋅ cos (ω t). {\displaystyle {\begin{aligned}\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{aligned}}}