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 vector5.1 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:
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.