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# levinson

Levinson-Durbin recursion

## Syntax

a = levinson(r)
a = levinson(r,n)
[a,e] levinson(r,n)
[a,e,k] levinson(r,n)

## Description

The Levinson-Durbin recursion is an algorithm for finding an all-pole IIR filter with a prescribed deterministic autocorrelation sequence. It has applications in filter design, coding, and spectral estimation. The filter that levinson produces is minimum phase.

a = levinson(r) finds the coefficients of a length(r)-1 order autoregressive linear process which has r as its autocorrelation sequence. r is a real or complex deterministic autocorrelation sequence. If r is a matrix, levinson finds the coefficients for each column of r and returns them in the rows of a. n=length(r)-1 is the default order of the denominator polynomial A(z); that is, a = [1 a(2) ... a(n+1)]. The filter coefficients are ordered in descending powers of z–1.

$H\left(z\right)=\frac{1}{A\left(z\right)}=\frac{1}{1+a\left(2\right){z}^{-1}+\cdots +a\left(n+1\right){z}^{-n}}$

a = levinson(r,n) returns the coefficients for an autoregressive model of order n.

[a,e] levinson(r,n) returns the prediction error, e, of order n.

[a,e,k] levinson(r,n) returns the reflection coefficients k as a column vector of length n.

 Note   k is computed internally while computing the a coefficients, so returning k simultaneously is more efficient than converting a to k with tf2latc.

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### Algorithms

levinson solves the symmetric Toeplitz system of linear equations

where r = [r(1)  ... r(n+1)] is the input autocorrelation vector, and r(i)* denotes the complex conjugate of r(i). The input r is typically a vector of autocorrelation coefficients where lag 0 is the first element r(1). The algorithm requires O(n2) flops and is thus much more efficient than the MATLAB® \ command for large n. However, the levinson function uses \ for low orders to provide the fastest possible execution.

## References

[1] Ljung, L., System Identification: Theory for the User, Prentice-Hall, 1987, pp. 278-280.