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 $\ensuremath{L_2}$ norm and we may write $\Vert x\Vert _2$ to emphasize this fact.

We could equally well have chosen a normalized $\ensuremath{L_2}$ norm:

$$\Vert x\Vert _{\tilde{2}} \isdef \sqrt{{\cal P}_x} = \sqrt{\frac{1}{N}\sum_{n=0}^{N-1} \left\vert x_n\right\vert^2} \qquad \mbox{(normalized $\ensuremath{L_2}$\ norm of $x$)}$$

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

More generally, the (unnormalized) $\ensuremath{L_p}$ norm of $x\in\mathbb{C}^N$ 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 $\ensuremath{L_p}$ norms are

• $p=1$ : The $\ensuremath{L_1}$ , ``absolute value,'' or ``city block'' norm.
• $p=2$ : The $\ensuremath{L_2}$ , ``Euclidean,'' ``root energy,'' or ``least squares'' norm.
• $p=\infty$ : The $\ensuremath{L_\infty}$ , ``Chebyshev,'' ``supremum,'' ``minimax,'' or ``uniform'' norm.

Note that the case $p=\infty$ is a limiting case which becomes

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