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norm

$\displaystyle \left\Vert\,x\,\right\Vert _1 \mathrel{\stackrel{\Delta}{=}}\sum_{i=0}^{N-1} \vert x_i\vert$
- Sum of the absolute values of the elements
- ``City block'' distance
norm

$\displaystyle \left\Vert\,x\,\right\Vert _2 \mathrel{\stackrel{\Delta}{=}}\sqrt{ \sum_{i=0}^{N-1} \vert x_i\vert^2 }$
- ``Euclidean'' distance
- Minimized by ``Least Squares'' techniques
-norm

$\displaystyle \left\Vert\,x\,\right\Vert _\infty \mathrel{\stackrel{\Delta}{=}} \lim_{p\to\infty} \left( \sum_{i=0}^{N-1} \vert x_i\vert^p \right)^\frac{1}{p}$

In the limit as $p \rightarrow \infty$ , the norm of is dominated by the maximum element of .

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About the Author: 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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