Signals as Vectors

For the DFT, all signals and spectra are length
. A length
sequence
can be denoted by
,
, where
may be
real (
) or complex (
). We now wish to regard
as a
*vector*^{5.1}
in an
dimensional *vector space*. That is,
each sample
is regarded as a *coordinate* in that space.
A *vector*
is mathematically a single *point* in
-space represented by a list of coordinates
called an *
-tuple*. (The
notation
means the same thing as
.) It can be interpreted
geometrically as an arrow in
-space from the origin
to the point
.

We define the following as equivalent:

where is the th sample of the signal (vector) . From now on, unless specifically mentioned otherwise,

The reader comfortable with vectors, vector addition, and vector subtraction may skip to §5.6.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University