Orthonormal lattice filter—a multistage multichannel estimation technique

The orthonormal lattice filter algorithm is a unified recursive technique to estimate the least‐squares coefficients of a broad class of linear models. It consists of a sequence of two recursions. First, the Gram‐Schmidt orthonormalization procedure estimates a set of inner products from multiple signal vectors, for example, multichannel sampled observations. A lattice type network structure, suitable for adaptive applications, implements the Gram‐Schmidt procedure. Second, a recursion introduced in this paper transforms the inner products into model coefficients. In contrast with conventional algorithms, this algorithm does not require the covariance matrix to design the filters but directly filters the observations. We consider a number of special cases of the multichannel model, for example the rational polynomial model, the autoregressive model, and the autoregressive‐moving average model with known and unknown inputs. In addition, we introduce the autoregressive‐orthonormal model for the unknown input case. We also transform the coefficients of the autoregressive‐orthonormal model to estimate the poles and zeros of a signal transfer function and thereby obtain the amplitude and phase spectra of the signal. For each model, we appropriately simplify the general orthonormal lattice filter algorithm and also the filter network structure. It turns out that for the autoregressive model the simplified algorithm is the Itakura‐Saito algorithm which is a direct version of the Levinson recursion and the simplified structure is a lattice type ladder. With the example of a noisy truncated signal, we compare the actual and estimated values of (a) the coefficients of a known‐input autoregressive‐moving average model, and (b) the autoregressive‐orthonormal power density spectrum of the signal.