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Recall Euler's Identity,
Multiplying this equation by
and setting
, where
is time in seconds,
is radian frequency, and
is a phase offset, we obtain what we call the complex sinusoid:
Thus, a complex sinusoid consists of an ``in-phase'' component for its
real part, and a ``phase-quadrature'' component for its imaginary
part. Since
, we have
That is, the complex sinusoid has a constant modulus (i.e.,
a constant complex magnitude). (The symbol
``
'' means ``identically equal to,'' i.e., for all
.) The
instantaneous phase of the complex sinusoid is
The derivative of the instantaneous phase of the complex sinusoid
gives its instantaneous frequency
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