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poly2rc

Convert prediction filter polynomial to reflection coefficients

Syntax

k = poly2rc(a)
[k,r0] = poly2rc(a,efinal)

Description

k = poly2rc(a) converts the prediction filter polynomial a to the reflection coefficients of the corresponding lattice structure. a can be real or complex, and a(1) cannot be 0. If a(1) is not equal to 1, poly2rc normalizes the prediction filter polynomial by a(1). k is a row vector of size length(a)-1.

[k,r0] = poly2rc(a,efinal) returns the zero-lag autocorrelation, r0, based on the final prediction error, efinal.

Examples

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Given a prediction filter polynomial, a, and a final prediction error, efinal, determine the reflection coefficients of the corresponding lattice structure and the zero-lag autocorrelation.

a = [1.0000 0.6149 0.9899 0.0000 0.0031 -0.0082];
efinal = 0.2;
[k,r0] = poly2rc(a,efinal)
k =

    0.3090
    0.9801
    0.0031
    0.0081
   -0.0082


r0 =

    5.6032

Limitations

If abs(k(i)) == 1 for any i, finding the reflection coefficients is an ill-conditioned problem. poly2rc returns some NaNs and provides a warning message in those cases.

More About

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Tips

A simple, fast way to check if a has all of its roots inside the unit circle is to check if each of the elements of k has magnitude less than 1.

stable = all(abs(poly2rc(a))<1)

Algorithms

poly2rc implements this recursive relationship:

k(n)=an(n)an1(m)=an(m)k(n)an(nm)1k(n)2,m=1,2,,n1

This relationship is based on Levinson's recursion [1]. To implement it, poly2rc loops through a in reverse order after discarding its first element. For each loop iteration i, the function:

  1. Sets k(i) equal to a(i)

  2. Applies the second relationship above to elements 1 through i of the vector a.

    a = (a-k(i)*fliplr(a))/(1-k(i)^2);
    

References

[1] Kay, Steven M. Modern Spectral Estimation. Englewood Cliffs, NJ: Prentice-Hall, 1988.

Introduced before R2006a

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