Levinson-Durbin

One of the applications of the Levinson—Durbin formula implemented in this block is the autoregressive (AR) Yule—Walker problem, which consists in modeling an unknown system as an AR process. Such a process can be modeled as the output of an all-pole IIR filter with white Gaussian noise at the input. In the Yule—Walker problem, the usage of the autocorrelation sequence of the signal to obtain an optimal estimate leads to the equation of the above kind, which is most efficiently solved using Levinson—Durbin recursion. In this case, the block input is an autocorrelation sequence, and — A value with zero delay. The output signal on the A port of the block contains the coefficients of the AP process that optimally simulates the system. The coefficients are ordered in descending order of degrees , and the AR process itself has a minimum phase. Prediction error determines the gain for an unknown system:

The output signal on port K contains the corresponding reflection coefficients for the lattice implementation of this IIR filter.

Another common application of the Levinson—Durbin algorithm is linear predictive coding, the purpose of which is to find the coefficients of the moving average (CC) process (or FIR filter) predicting the next signal value based on the current signal sample and a finite number of past samples. In this case, the input of the block is an autocorrelation sequence of the signal, where — a value with zero delay, and the output on port A contains the coefficients of the predictive MA process (in descending order of degree ).

These coefficients solve the following optimization problem:

The output K contains the corresponding reflection coefficients, , for the lattice implementation of this FIR filter.

The algorithm requires operations for each input signal. Therefore, this implementation for large much more efficient than the standard Gaussian elimination method, requiring operations for each signal.