poly2rc

Convert prediction filter polynomial to reflection coefficients

Syntax

k = poly2rc(a)

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

Description

k = poly2rc(a) returns a vector k of lattice-structure reflection coefficients from a vector a of prediction filter coefficients.

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

example

Examples

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Find Reflection Coefficients from Prediction Filter Polynomial

Open Live Script

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)

r0 = 
5.6032

Input Arguments

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`a` — Prediction filter coefficients
vector

Prediction filter coefficients, specified as a vector.

Note

The value of a has the following restrictions:

a(1) cannot be 0.
If a(1) is not equal to 1, the poly2rc function normalizes the prediction filter polynomial by a(1).

Data Types: single | double
Complex Number Support: Yes

`eFinal` — Final prediction error power
`0` (default) | scalar

Final prediction error power, specified as a scalar.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

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`k` — Reflection coefficients
column vector

List of reflection coefficients, returned as a column vector of length p, where p+1 is the number of elements of a.

`r0` — Zero-lag autocorrelation
scalar

Zero-lag autocorrelation, returned as a scalar.

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.

Tips

A simple and quick way to verify if all the roots of a lie inside the unit circle is to check if all the elements of k have magnitude less than 1.

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

Algorithms

The poly2rc function implements the following recursive relationship:

$\begin{matrix} k (n) = a_{n} (n) \\ a_{n - 1} (m) = \frac{a_{n} (m) - k (n) a_{n} (n - m)}{1 - k {(n)}^{2}}, m = 1, 2, \dots, n - 1 \end{matrix}$

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:

Sets k(i) equal to a(i)
Applies the second relationship above to elements 1 through i of the vector a.
```
a = (a-k(i)*fliplr(a))/(1-k(i)^2);
```