The Inner Product

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. The Inner Product

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

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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 $N$ vectors with complex scalars).

The inner product between two (complex) $N$-vectors $x$ and $y$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 $N$ complex vectors are mapped to a complex scalar.

Example:For $N=3$ we have, in general,

\

Let

\



Then

\


Subsections

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