Signals as Vectors

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Signals as Vectors

For the DFT, all signals and spectra are length . A length sequence can be denoted by , $n=0,1,2,\ldots N-1$ , where may be real ( $x\in{\bf R}^N$ ) or complex ( $x\in{\bf C}^N$ ). We now wish to regard as a vector^5.1 $\underline{x}$ in an dimensional vector space. That is, each sample is regarded as a coordinate in that space. A vector $\underline{x}$ is mathematically a single point in -space represented by a list of coordinates $(x_0,x_1,x_2,\ldots,x_{N-1})$ called an -tuple. (The notation means the same thing as .) It can be interpreted geometrically as an arrow in -space from the origin $\underline{0} \isdef (0,0,\ldots,0)$ to the point $\underline{x}\isdef (x_0,x_1,x_2,\ldots,x_{N-1})$ .

We define the following as equivalent:

$\displaystyle x \isdef \underline{x}\isdef x(\cdot) \isdef (x_0,x_1,\ldots,x_{N-1}) \isdef [x_0,x_1,\ldots,x_{N-1}] \isdef [x_0\; x_1\; \cdots\; x_{N-1}]$

where $x_n \isdef x(n)$ 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.

Subsections

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
Geometric Signal Theory
Signals as Vectors