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 .
<11124>> Another notation commonly used for vectors is matrix notation which is covered in any course on linear algebra . A point in -space is normally expressed as a column vector
as opposed to a row vector . However, when working with Matlab, using row vectors by default saves screen space when typing them out. For that reason, we will adopt the row-vector convention. In state space analysis of dynamic systems, the column-vector convention is always used.
We define the following as equivalent:
where is the th sample of the signal (vector) . From now on, unless specifically mentioned otherwise, all signals are length .
The reader comfortable with vectors, vector addition, and vector subtraction may skip to §5.6.