Next: 5 Nonlinear Filters
Up: 2 Filter Structures supported
Previous: 3 Recursive Filters
Contents
4 Lattice Filters
Figure 2.4:
Block diagram of the lattice predictor.

Lattice structures are widely used in prediction applications. Fig. 2.4 shows
the lattice predictor structure of order M. Stage of the lattice predictor has two inputs
from the previous stage, namely the forward and backward prediction errors
and , respectively, and produces two outputs
and
.
The two outputs are given by the following order update equations

(7) 
The input of the first lattice stage has its forward and backward errors equal to the input signal

(8) 
The coefficient of stage is known as the partial correlation coefficient (PARCOR) or
the reflection coefficient. The set of PARCOR coefficients for an Mstage lattice predictor are related to
the coefficients of the transversal predictor of the same order (see Section 2.4.3). In fact the
lattice and transversal predictors are equivalent. The LevinsonDurbin algorithm is an efficient
procedure to calculate the transversal predictor coefficients from the autocorrelation function
of the input sequence and it also provides the PARCOR coefficients for the corresponding lattice predictor.
The following properties are well known for lattice structures
 The PARCOR coefficients always satisfy the relation
.
 The power of the forward prediction error
and the backward prediction error
of the same stage are equal.
 The backward prediction errors
are uncorrelated
with one another for any input sequence . This property is very important since it shows that
the lattice predictor can be seen as an orthogonal transformation with the input signal samples
as input and the uncorrelated (orthogonal) output as the backward
error from the Mstages.
 The power of the prediction error decreases with increasing lattice order. The error power
decrease is controlled by the PARCOR coefficients according to the relation
, where is the power of the forward or backward prediction error
at stage . This indicates that the closer the value of to unity the higher the
contribution of stage in reducing the prediction error. Usually the first few PARCOR
coefficients have higher magnitude with the magnitude of the coefficients dropping to values
close to zero for later stages.
Figure 2.5:
Block diagram of the joint process estimator.

Although the operation of lattice filters are usually described in the prediction context, the
application of lattice filters is not limited to prediction applications. A traditional adaptive transversal
filter can also be implemented using the lattice structure as shown in Fig. 2.5.
The structure in Fig. 2.5 is known as the joint process estimator since it estimates
a process from another correlated process . The joint process estimator consists of
two separate parts, the lattice predictor part and the linear combiner part. The lattice predictor
part main function is to transform the input signal samples
that
might be well correlated to the uncorrelated backward prediction errors
.
The linear combiner part calculates the equivalent transversal filter output according to the relationship

(9) 
An adaptive joint process estimator adjusts both the PARCOR coefficients
and the linear combiner coefficients
simultaneously. The PARCOR coefficients
are adjusted to minimize the forward and backward prediction error power and the linear combiner
coefficients are adjusted to minimizes the square of error signal
as shown in Fig. 2.5.
Next: 5 Nonlinear Filters
Up: 2 Filter Structures supported
Previous: 3 Recursive Filters
Contents