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In signal analysis, students have no qualms about associating the $L_2$ norm of a square integrable function $f(t)$ as the energy associated with that signal.

A good understanding of whether a function $f(t)$ is a square integrable function is to picture whether it has finite amount of energy. I find this to be an extremely useful knowledge.

However, when I plot the $L_1$ norm of a given function, all I'm getting is the total positive area lying under the curve. Which leads to two important questions:

  1. How did $L_2$ norm get associated with being the energy of a signal? I.e. what is the physics involved?

  2. What is the physical interpretation of the $L_1$ norm? I find "area" not very convincing not only given the enormous physical intuition behind the $L_2$ norm but an area is still more of a construction from a mathematical perspective.

Rodrigo de Azevedo
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Fraïssé
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2 Answers2

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While I can't satisfyingly answer your question about the $L_1$ norm, the $L_2$ norm has an easy answer.

Remember that a signal is generally, physically speaking, some sort of electronic oscillation in voltage (or current, if you prefer to think of it that way). It is well known that instantaneous power of an electronic signal is given by $$ \frac{dE}{dt} = R[I(t)]^2 = \frac{1}{R} [V(t)]^2 $$ Where $I$ refers to the current, $V$ refers to voltage, and the resistance $R$ is some time-independent coefficient. It follows then that the total energy expended, assuming a signal of finite energy is transmitted between times $\pm \infty$, is given by $$ E_{total} = \frac 1R \int_{-\infty}^\infty |V(t)|^2\,dt $$ The utility of the $L_2$ norm as being representative of "energy" does extend beyond this particular context. However, at the very least, it does make physical sense here.

Overall, the $L_2$ norm will always have a bit more mathematical utility than the $L_1$ norm since the $2$-norm is derived from an inner product, whereas the $L_1$ norm is not.

A useful property of the $L_1$ norm in the context of signal analysis is that an LTI-system is BIBO-stable iff its transfer function has a finite $L_1$ norm.

Ben Grossmann
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The explanation of Omnomnomnom of the $L_2$ norm is very good alreay.

One possible interpretation of the $L_1$ norm, or rather the ration between $L_1$ and the $L_2$ norm is that it can be used as a measure of sparsity or compressability of a signal. Therefore it has often been used as a regularization term in optimization problems. Today this is a research field on its own called compressed sensing. You might want to have a look at the linked Wiki page for reading up on the used techniques and the intuition behind the usage of the $L_1$ norm.

Chris
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