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4 Lattice Filters



Figure 2.4: Block diagram of the lattice predictor.
\begin{figure}
\begin{center}
\epsfig{file=/home/john/winD/docs/aspt/aspt/figs/lattice.eps,width=\textwidth}
\end{center}
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Lattice structures are widely used in prediction applications. Fig. 2.4 shows the lattice predictor structure of order M. Stage $m+1$ of the lattice predictor has two inputs from the previous stage, namely the forward and backward prediction errors $e_{f_{m}}(n)$ and $e_{b_{m}}(n)$, respectively, and produces two outputs $e_{f_{m+1}}(n)$ and $e_{b_{m+1}}(n)$. The two outputs are given by the following order update equations
\begin{displaymath}
\begin{array}{l l l l l}
e_{f_{m+1}}(n) & = & e_{f_{m}}(n)...
... = & e_{b_{m}}(n-1)& - & k_{m+1} e_{f_{m}}(n).
\end{array}
\end{displaymath} (7)

The input of the first lattice stage has its forward and backward errors equal to the input signal
\begin{displaymath}
e_{f_0}(n) = e_{b_0}(n) = x(n).
\end{displaymath} (8)

The coefficient $k_{m}$ of stage $m$ is known as the partial correlation coefficient (PARCOR) or the reflection coefficient. The set of PARCOR coefficients for an M-stage 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 Levinson-Durbin algorithm is an efficient procedure to calculate the transversal predictor coefficients $\va $ 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



Figure 2.5: Block diagram of the joint process estimator.
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\epsfig{file=/home/john/winD/docs/aspt/aspt/figs/lattice_jpe.eps,width=\textwidth}
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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 $d(n)$ from another correlated process $x(n)$. 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 $x(n), x(n-1), \cdots, x(n-M+1)$ that might be well correlated to the uncorrelated backward prediction errors $e_{b_0}(n), e_{b_1}(n), \cdots, e_{b_M}(n)$. The linear combiner part calculates the equivalent transversal filter output according to the relationship
\begin{displaymath}
y(n) = \sum \limits_{i=1}^{M} c_i \; e_{b_i}(n).
\end{displaymath} (9)

An adaptive joint process estimator adjusts both the PARCOR coefficients $k_i\; i=1,2,\cdots,M; $ and the linear combiner coefficients $c_i;\; i=1,2,\cdots,M$ 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 $e(n) = d(n) - x(n)$ as shown in Fig. 2.5.
next up previous contents
Next: 5 Nonlinear Filters Up: 2 Filter Structures supported Previous: 3 Recursive Filters   Contents