Theory#
Introduction#
Basics#
Block pulse functions#
Define an \(m\)-set of block pulse functions as
\[\begin{split}\phi_i(t) = \begin{cases} 1 & , \ (i-1)h \leq t < ih \\ 0 &
\text{otherwise} \end{cases} \qquad (i=1,\ldots,m)\end{split}\]
for \(t \in [0,T)\) and an interval width of \(h=\frac{T}{m}\). The BPFs are disjoint, i.e.
\[\phi_i(t)\phi_j(t) = \delta_{ij} \phi_i(t)\]
for \(i,j = 1, \ldots, m\) and \(\delta_{ij}\) the Kronecker delta and orthogonal, i.e.
\[\int\limits_0^T \phi_i(t) \phi_j(t) dt = h \delta_{ij}\]
for \(i,j=1,\ldots,m\).