## 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:

*all signals are length*.

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

### An Example Vector View:

Consider the example two-sample signal graphed in Fig.5.1.

Under the geometric interpretation of a length signal, each sample is a
*coordinate* in the dimensional space. Signals which are only two
samples long are not terribly interesting to hear,^{5.2} but they are easy to
plot geometrically.

**Next Section:**

Vector Addition

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The DFT