Editing Quadrature amplitude modulation (section)

== Fourier analysis ==

Applying [[Euler's formula]] to the sinusoids in {{EquationNote|Eq.1}}, the positive-frequency portion of {{math|''s''{{sub|c}}}} (or [[analytic representation]]) is:

:<math>
  s_c(t)_+ = \tfrac{1}{2} e^{i2\pi f_c t}[I(t) + i Q(t)]
  \quad\stackrel{\mathcal{F}}{\Longrightarrow}\quad
  \tfrac{1}{2}\left[\widehat{I\ }(f - f_c) + e^{i\pi/2} \widehat Q(f - f_c)\right],
</math>

where <math>\mathcal{F}</math> denotes the Fourier transform, and {{math|{{overset|︿|I}}}} and {{math|{{overset|︿|Q}}}} are the transforms of {{math|''I''(''t'')}} and {{math|''Q''(''t'').}} This result represents the sum of two DSB-SC signals with the same center frequency. The factor of {{math|1='''i''' (= ''e''{{sup|''iπ''/2}})}} represents the 90° phase shift that enables their individual demodulations.