Block Diagram of AR Software Modules

Figure 10.5 shows the software modules to be discussed and how they relate to the four equivalent representations of the AR process. Note that software/module_ar_clt.m can produce either AR or RC depending on an input flag. There are various means of obtaining each representation. Most important is that one can obtain ACF directly from the raw data using software/module_acf_spax.m, or indirectly from the magnitude-squared DFT (software/module_acf_clt.m). Also, various 1:1 modules are provided to convert between the various representations including software/module_ar2rc.m, software/module_rc2ar.m, software/module_acf2rc.m, software/module_acf2ar.m, and software/module_rlevinson.m. Note that in autoregressive analysis, the prediction error variance (or innovation variance), $\sigma^2$, can be derived if one knows the AR coefficients and the process variance which is the same as the zero-th lag of the ACF, $r_0$. Therefore, the two quantities $\sigma^2$, $r_0$ are interchangeable when grouped with AR or RC coefficients, or the remaining ACF coefficients, from an information standpoint. With both AR and RC, there is a choice of using either prediction error or variance depending on an input flag. Depending on the circumstance, one or the other might be more appropriate. Note that the script software/test_ar.m tests all the modules. The recommended approach is to use software/module_acf_spax.m (or software/module_acf_spa.m if intermediate raw spectrum is needed) followed by the desired conversion module. The recommended feature type is LAR. The software modules that perform the conversions are shown. Relevant sections are software/module_dftmsq.m: Section 4.3, software/module_acf_spa.m: Section 5.2.2, software/module_acf_clt.m: Section 5.2.5, software/module_ar_ml.m: Section 5.2.8, software/module_acf2rc.m: Section 10.4.6, software/module_ar2rc.m: Section 10.4.6, software/module_poly2root.m, software/module_root2poly.m: Section 10.4.7, software/module_bilinear.m: Section 4.1.5.

Figure 10.5: Diagram showing the relationship of five equivalent representations of an autoregressive process, autocorrelation function (ACF), autoregressive coefficients (AR), reflection coefficients (RC), log area ratio (LAR) coefficients and AR polynomial roots. It can be verified that any two paths through the figure that end at the same point will produce the same features and J-function.