L2justitication

This page needs to be discussed and rewritten
Many signals we deal with in this course are energy-limited signals which belong to $\mathcal{L}_2$. However, $\mathcal{L}_2$ is not an inner product space and, hence, it becomes tricky to talk about inner-products. This can be easily seen by the fact it is possible for non-zero signals $s(t)$ to have $<s(t),s(t)>=0$. An example of such a signal is any signal $s(t)$ which is non-zero only over a set of measure zero.

Instead, it would be useful to make sure the signals we are interested in lie in an inner product space. One way to construct such a space is to consider equivalence classes of signals that all differ from each other over a set of measure zero. If we take space of such signals, then such a space is an inner product space called $L_2$. For this space, inner products are properly defined, only the zero-signal has zero energy and Fourier transforms can be defined if Dirichlet conditions are satisfied, or for periodic signals, Fourier transforms can be defined allowing delta-functions.

Lapidoth insists that it is better to stick to $\mathcal{L}_2$ inspite of the fact that it is not an inner product space because if $s(t)$ only represents an equivalence class, then the value taken by $s(t)$ at any $t_0$ is not well-defined! My engineering answer to this is that the value taken by $s(t)$ at any $t$ is not a practically relevant quantity. It is not possible to measure this using any physically realizable circuit. We can only measure the integral of such as signal over a set of who measure is not zero. So, we can simply think of $s(t)$ as a density function - This needs to be discussed.

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