This concept belongs to adaptive signal processing / recursive least squares (RLS) lattice filtering theory, specifically the derivation of order-update recursions for forward and backward linear prediction error vectors. The core mechanism is decomposing the subspace spanned by successively higher-order delayed data vectors into a direct sum of a lower-order subspace and an orthogonal complement, so that the pth-order forward and backward prediction errors (and their exponentially-weighted norm-square quantities, denoted sigma and delta) can be updated to (p+1)th order without recomputing the full least-squares projection from scratch. Unlike stochastic (statistical) lattice filters, which rely on stationarity and expectation-based correlations computed offline, RLS lattice quantities are purely data-dependent and time-varying, requiring both order recursions (across prediction order at fixed time index) and time recursions (across time index at fixed order), with the backward prediction case introducing a boundary condition where insufficient data causes rank deficiency (division-by-zero) in the recursion.
This concept belongs to adaptive signal processing / recursive least squares (RLS) lattice filtering theory, specifically the derivation of order-update recursions for forward and backward linear pre…