## 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

**Previous Section:**

The DFT