Invertibility and inverses of linear time-varying digital filters (Q2732963)

The authors consider a linear time-varying (LTV) digital filter \(G : u \mapsto y \) in the form of the following ARMA equation NEWLINE\[NEWLINE y_k + a_{1, k} y_{k-1} + ... + a_{n_k, k} y_{k-n_k} = b_{0,k} u_k + b_{1,k} u_{k-1} + ... + b_{m_k, k} u_{k-m_k} \tag{1} NEWLINE\]NEWLINE where \( u_k, y_k \in {\mathbb{R}} \) are the input and output of the filter, and \(\;a_{i, k}, b_{j,k} \in {\mathbb{R}} \;\) for \(\;1 \leq i \leq n_k , 0 \leq j \leq m_k \) are time-varying coefficients at each time \(k\) with \(a_{n_k, k}, b_{m_k, k} \neq 0 \). Thus, the LTV filter (1) has time-varying order \( o_k = \max \{ n_k, m_k \} \). A necessary and sufficient condition for the \(d\)-shift invertibility of the LTV filter (1) is developed. A procedure resulting from this condition is presented for computing the \(d\)-shift left and right inverses of a given LTV filter. It is shown that the order of the inverses is also time-varying and that there may exist nonunique solutions for the inverses. Moreover, stability of the given filter and its inverses is discussed and a necessary and sufficient condition for stability of inverse filters is given.