Reflection coefficients (RCs) are an alternate way of representing
the information in an AR model.
The RCs can be more convenient and easier to statistically model.
Reflection coefficients (RCs) may be calculated by invertible
AR coefficients or ACF estimates , and therefore we may use
the results of Sections 5.2.5, 5.2.8, or 5.2.8.
Computationally, the best method is to convert from ACF to RC because
the RCs are obtained as a by-product of the Levinson algorithm.
where is the ACF vector,
is the vector of reflection coefficients
augmented by the variance (zero-th lag ACF sample),
ACF to RCs (module_acf2rc.m).
The conversion from ACF to RCs is an invertible
transformation that is characterized by a
Jacobian matrix. The determinant of this
matrix is the J-function of the transformation.
Note that we use and not the AR prediction
This transformation is invertible and is characterized by the
The above is implemented by software/module_acf2rc.m.
AR to RCs(module_ar2rc.m).
You would probably never need to convert from AR coefficients to
RC coefficients because the Levinson algorithm outputs both coefficients as a by-product.
However, for completeness, the algorithm, which makes the conversion in-place is
provided in software/module_ar2rc.m.
The Jacobian for converting from AR to RC coefficients
is obtained by the following algorithm:
where O is the set of odd values of
in the range
where E is the set of even values of
in the same range.