poly2rc

R2026b

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.

Examples

Find Reflection Coefficients from Prediction Filter Polynomial

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 = 5×1

    0.3090
    0.9801
    0.0031
    0.0081
   -0.0082

r0 = 
5.6032

Input Arguments

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

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

Assume a vector of polynomial coefficients, a = [1 a₁ ⋯ a_n], where n is the polynomial degree. The poly2rc function finds k = [k₁ k₂ ⋯ k_n] through this recursive relationship:

$$\begin{array}{c}{a}_n{}^{(n)}=a_n\\ k_i=a_i{}^{(i)},i=n,n-1,\cdots ,1\\ a_j{}^{(i-1)}=\frac{a_j{}^{(i)}-k_i{a}_{i-j}{}^{(i)}}{1-k_i{}^2},j=1,2,\cdots ,i-1\wedge i\ge 2\end{array}$$

This relationship is based on Levinson’s recursion [1].

References

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See Incompatibility with MATLAB for Default Dimension Selection (MATLAB Coder).

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Refer to the usage notes and limitations in the C/C++ Code Generation section. The same usage notes and limitations apply to GPU code generation.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The poly2rc function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a

R2026b: Use thread-based environments

You can use poly2rc in thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.