Other Lp Norms

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

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

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:

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

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

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

(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

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