Applications of the Yule-Walker method and algorithm extend well beyond Telecommunication in general let alone Voice Enhancement, which can be considered as a narrow field, in comparison.

yule-walker algorithm
Figure 1: Yule-Walker is used in Voice Enhancement for white noise signal spectrum shaping and transmission channel and audio facility equalization

Nevertheless, the Yule-Walker approach is used in several applications within the Voice Enhancement files. In this note we wish to emphasize applications for the following:

Although other methods can be used in the tasks mentioned, the Yule-Walker method is an approach of choice.

The Yule-Walker method for the filter design is based on recursion and the criterion used for the approximation is based on the least-squares method. More specifically, the Yule-Walker method leads to generation of an IIR filter by fitting a specified frequency response.  It generates the Inverse Fast Fourier Transform (IFFT) of the predefined filter frequency magnitude and solves Yule-Walker equations. As a result of the recursive process IIR filter B and A coefficients are produced.

The resemblance of the IIR amplitude characteristic and predefined target (in the form of specified magnitude of the frequency response) is achieved by using Least Squares criterion. That of course does not mean that the resulting IIF magnitude characteristic passes through the predefined target points. Figure 1 illustrates an example showing the actual relation between the target and the realization of the spectrum shaping filter of order 12.

It is worth mentioning that the Yule-Walker method does not require nor accept the phase information as the input data. Also, the solution provided by Yule-Walker method is not checked against any criteria of generic optimality except for the LS criterion.  After the filter’s coefficient are generated and represented in the target arithmetics, the filter’s instability should be re-examined.

In all three applications of the Yule-Walker approach to the Voice Enhancement are, the task is formulated in a similar fashion: for the predefined power spectral density (or the magnitude of the transfer function) generate an IIR filter of order N (where N is pre-defined as well). Then, the difference between the target and the realization has to be verified and it is not satisfactory, the order must be increased. So the process may require more than one iteration (cf. [1,2]).

The Yule-Walker method goes beyond the applications to generating IIR filters for Telecommunication. It has powerful application in other disciplines such as economic modeling, statistics and in many more. VOCAL’s engineering staff is focused on applications in Telecommunication, Audio and Acoustics. Contact us to discuss details of IIR filter implementations using Yule-Walker approach.

More Information


  1. Optimization-Based Design and Implementation of Multidimensional Zero-Phase IIR Filters, by Dimitry Gorinevsky and Stephen Boyd; in IEEE TRANS. ON CIRC. SYST.—I: REGULAR PAPERS, VOL. 53, NO. 2, Feb. 2006
  2. Review of Digital Filter Design and Implementation Methods for 3-D Sound, by Huopaniemi, J,. and Karjalainen,M. Helsinki Univ. of Techn, 1997.