Mathematics of the Discrete Fourier Transform (DFT)

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The Inner Product
The inner product (or ``dot product'') is an operation on two vectors which produces a scalar. Adding an inner product to a Banach space produces a Hilbert space (or ``inner product space''). There are many examples of Hilbert spaces, but we will only need $$\$ for this course (complex length vectors with complex scalars).
The inner product between two (complex) -vectors and is defined by

The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:

As a result, the inner product is conjugate symmetric:

Note that the inner product takes $${\$ to $${\$ . That is, two length complex vectors are mapped to a complex scalar.
Example:For we have, in general,

Let

Then

Subsections
Linearity of the Inner Product
Norm Induced by the Inner Product
Cauchy-Schwarz Inequality
Triangle Inequality
Triangle Difference Inequality
Vector Cosine
Orthogonality
The Pythagorean Theorem in N-Space
Projection

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"Music 320 Background Reader" by Julius O. Smith III, (Course Background Reader, Music 320). Copyright © 2001-01-02 by Julius O. Smith III. - Center for Computer Research in Music and Acoustics (CCRMA), Department of Electrical Engineering, Stanford University. This is a modified HTML version reproduced by permission.

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