Complex Sinusoids

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Complex Sinusoids

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Complex Sinusoids

Recall Euler’s Identity,

\

Multiplying this equation by $A \ and setting $\, we obtain the definition of 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 is constant modulus. (The symbol “$\” means “identically equal to,” i.e., for all $t$.) The phase of the complex sinusoid is

\

The derivative of the phase of the complex sinusoid gives itsfrequency

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