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Mathematics of the DFT
    Geometric Signal Theory
       Signal Metrics
          Other Lp Norms

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Other Lp Norms

Since our main norm is the square root of a sum of squares,

$\displaystyle \Vert x\Vert \isdef \sqrt{{\cal E}_x} = \sqrt{\sum_{n=0}^{N-1}\left\vert x_n\right\vert^2}$   $\displaystyle \mbox{(norm of $x$)}$$\displaystyle , $

we are using what is called an $ L2$ norm and we may write $ \Vert x\Vert _2$ to emphasize this fact.

We could equally well have chosen a normalized $ L2$ norm:

$\displaystyle \Vert x\Vert _{\tilde{2}} \isdef \sqrt{{\cal P}_x} = \sqrt{\frac{... ...N-1} \left\vert x_n\right\vert^2} \qquad \mbox{(normalized $L2$\ norm of $x$)} $

which is simply the ``RMS level'' of $ x$ (``Root Mean Square'').

More generally, the (unnormalized) $ Lp$ norm of $ x\in{\bf C}^N$ is defined as

$\displaystyle \Vert x\Vert _p \isdef \left(\sum_{n=0}^{N-1}\left\vert x_n\right\vert^p\right)^{1/p}. $

(The normalized case would include $ 1/N$ in front of the summation.) The most interesting $ Lp$ norms are
  • $ p=1$: The $ L1$, ``absolute value,'' or ``city block'' norm.
  • $ p=2$: The $ L2$, ``Euclidean,'' ``root energy,'' or ``least squares'' norm.
  • $ p=\infty$: The $ L-infinity$, ``Chebyshev,'' ``supremum,'' ``minimax,'' or ``uniform'' norm.
Note that the case $ p=\infty$ is a limiting case which becomes

$\displaystyle \Vert x\Vert _\infty = \max_{0\leq n < N} \left\vert x_n\right\vert. $

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Next: Norm Properties
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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