A D-step predictor in lattice and ladder form

We use the orthogonalizing property of the two-multiplier linear prediction lattice filter to construct a
d
-step ahead predictor in lattice form. The predictor generates
d
-step forward and backward residuals in a recursive way and possesses most of the interesting properties of the basic one-step prediction lattice filter. An exact solution is presented first assuming a stationary observation process, using orthogonal projections in Hilbert space. Two adaptive implementations are also proposed for the case where the statistics of the signal process are unknown or time varying: a gradient method and a recursive least-squares scheme. Finally, we show how to construct an adaptive
d
-step ahead predictor by adding a ladder part to the
d
-step lattice structure.