NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page  TOC  INDEX  Next page >>

Complex Sinusoids

Recall Euler's Identity,

Multiplying this equation by 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 .) The phase of the complex sinusoid is

The derivative of the phase of the complex sinusoid gives itsfrequency

---

Subsections

• Circular Motion
• Projection of Circular Motion
• Positive and Negative Frequencies
• The Analytic Signal and Hilbert Transform Filters
• Generalized Complex Sinusoids
• Sampled Sinusoids
• Powers of
• Why (Generalized) Complex Sinusoids are Important
• Comparing Analog and Digital Complex Planes

<< Previous page  TOC  INDEX  Next page >>