JURYRC Procedure (PV-WAVE Extreme Advantage)

Signal Processing Toolkit User Guide > Reference (G to P) > JURYRC Procedure (PV-WAVE Extreme Advantage)

Synthesizes a Cholesky-factored Toeplitz form from a stable polynomial using the Jury (reflection coefficient) algorithm.

Usage

JURYRC, alpha_in, a, alpha_out, t, c

Input Parameters

alpha_in—A scaling factor (or prediction error variance).

a—An array of polynomial coefficients.

Output Parameters

alpha_out—An array consisting of the elements of the diagonal matrix factor.

t—The upper triangular matrix factor.

c—An array of reflection coefficients.

Keywords

None.

Discussion

Given the coefficients of a stable, monic polynomial:

A(z) = 1 + a1z–1 + ... + anz–n

and a scaling factor (prediction error variance α), JURYRC finds an upper triangular matrix T and a diagonal matrix D that satisfy:

aT–1DT–T = [α, 0, 0, ..., 0]

where the row array a contains the coefficients of the polynomial a = [1, a1, ..., an], and the diagonal matrix is given by D = diag {α(n), α(n – 1), ..., α(0)}. The matrix T–1DT–T = R is Toeplitz. The reflection coefficients of the polynomial are also returned.

This procedure is one part of a suite of functions (JURYRC, LEVCORR, LEVDURB, and TOEPSOL) used to solve Toeplitz linear equations and factorization problems. Given the first row of a symmetric Toeplitz matrix R, the function TOEPSOL is used to solve the equation:

(EQ 1)

where α is chosen so that a(0) = 1 in the array a. JURYRC combined with LEVCORR are, in essence, the inverse of TOEPSOL, in that given the array of polynomial coefficients a and scalar α, EQ 1 is solved for the elements of the first row of R. This inverse operation is accomplished by first finding the elements of the matrices T and D defined above using JURYRC, and then evaluating the product T–1DT–T using LEVCORR.

note	If the zero-order coefficient, a0, of the input polynomial is not equal to 1.0, the value of alpha_in is modified as alpha_in/a0 and the array of polynomial coefficients is modified as a/a0.

Example

This example illustrates the relationship between JURYRC, TOEPSOL, and LEVCORR.

; First order autoregressive autocorrelation sequence used to

; form first row of Toeplitz matrix.

r = [1.0d0, 0.9d0, 0.9d0^2, 0.9d0^3, 0.9d0^4, 0.9d0^5]

PM, TRANSPOSE(r), Title = 'First Row of Original Toeplitz Matrix'

b = [1.0d0, 0.0d0, 0.0d0, 0.0d0, 0.0d0, 0.0d0]

; Normalize the polynomial coefficient array a so a(0) = 1.

; Normalization factor is alpha.

a = TOEPSOL(r, b)

alpha = 1/a(0)

a = a/a(0)

; Compute the Cholesky decomposition (T –1D T –T) from the

; normalized polynomial coefficients contained in the array a

; obtained from TOEPSOL.

JURYRC, alpha, a, alpha_out, t, c

; Compute the first row of the Toeplitz matrix R = T –1D T –T from

; the matrix and array returned from JURYRC.

LEVCORR, r_out, alpha_out, t

PM, TRANSPOSE(r_out), $
Title = 'First Row of Returned Toeplitz Matrix'