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{...
\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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