signal basis functions

Any real energy signal space can be represented by a set of orthonormal basis functions \(\phi_0(t) \dots \phi_N(t)\).

Clearly if the signal are linearly independent, then \(N=M\) where \(M\) is the number of signals in the set of signals we are trying to represent.

The decomposition onto the basis functions is then

\begin{equation} s_i(t) = \sum_{j = 1}^{N} s_{ij}\phi_j(t) \end{equation}

where each \(s_{ij}\) is the projection of \(s_i(t)\) onto basis function \(\phi_j(t)\). The whole signal is a linear combination of projections and basis functions, a result from the previous idea. Each coefficient of the signal then makes up the signal constellation.

The whole idea is to take an infinite function space of signals and turn them into a space of vectors with axes given by the basis functions since we can easily compute things like distance with them.

The minimum number of basis function to represent a signal is rougly \(2BT\) where \(B\) is the bandwidth and \(T\) is duration. If so, then we can say the signal's signal space is dimension \(2BT\).

All basis functions should be orthonormal:

\begin{equation} \int_0^T \phi^2(t) = 1 \end{equation}

An example of basis functions are the sine and cosine functions for any linearly modulated signal. Each basis function may be also multiplied by a pulse shaping filter, as long as the pulse shaper \(g(t)\) maintains the orthonormality of the function.