Generalized Complex Sinusoids

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

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Generalized Complex Sinusoids

We have defined sinusoids and extended the definition to include complex sinusoids. We now extend one more step by allowing for exponentialamplitude envelopes:

\

where ${\ and $s$ are complex, and further defined as
\


When $\, we obtain

\

which is the complex sinusoid at amplitude $A$, radian frequency $\, and phase $\.

More generally, we have

\


Defining $\, we see that the generalized complex sinusoid is just the complex sinusoid we had before with an exponential envelope:

\


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