Recovery of one-dimensional input signals in discrete time-varying linear and bilinear systems by the least-squares method (Q1913844)

The authors consider one dimensional linear input-output discrete time-varying systems on the observation interval \(T = \{0,1, \dots, N - 1\}\) described by \[ B (E,T) y_T = A(E,T) u_T \tag{1} \] where \(u_T = [u_0, u_1, \dots, u_{N - 1}]^T\), \(y_T = [y_0, y_1, \dots, y_{N - 1}]^T\), and \[ \begin{aligned} A(E,T) & = \sum^m_{i = 0} A_{m - i} E^i,\;A_{m - i} = \text{diag} \{a_{(m - i) l};\;l = 0,\;1, \dots, N - 1\},\\ B(E,T) & = \sum^m_{j = 0} B_{n - j} E^j,\;B_{n - j} = \text{diag} \{b_{(n - j) l};\;l = 0,\;1, \dots, N - 1\}.\end{aligned} \] \(E\) is the operator of (feed-)forward shift \((Eu_l = u_{l + 1})\). They propose a deconvolution procedure (recovery of the input signal) by the least-squares method. It is a certain modification of the procedure published earlier by the second author where the feed-forward (backward) shift operator in equation (1) is replaced by the cyclic forward (backward) shift matrix. This modified procedure is also appropriate for the recovery of input signals of polynomial discrete systems whose description includes input variables of the first degree, for instance in bilinear systems.