Application of SPA to Circular Auto-Correlation Function Analysis

The auto-correlation function (ACF) is widely used in auto-regressive (AR) time-series analysis and spectral estimation [25]. An estimate of the ACF given a length-$ N_t$ time-series can be obtained using the frequency-domain approach by inverse DFT of the raw spectrum. Because the FFT is used, this approach falls under feature extraction methods for circularly stationary processes (See section 9.1). The ``input" data $ {\bf x}$ is the $ N\times 1$ vector of raw spectral values (magnitude-squared DFT), where $ N=N_t/2+1.$ The output ACF feature is $ {\bf z}={\bf A}^\prime {\bf x}$. To compute the $ P$-th order ACF (lags 0 through $ P$), the columns of the $ N\times(P+1)$ matrix $ {\bf A}$ must contain the cosine functions which are the basis functions for the real part of the inverse FFT. To properly compute the ACF, we need to effectively compute the inverse-FFT of the full length-$ N_t$ spectrum (redundant bins duplicated) which requires that we scale the complex bins ( $ i=\{2,3 \ldots N-1\}$) by 2.0 and the real bins ($ i=\{1,N\}$) by 1.0. This un-equal bin scaling can be formalized by the scaling variable $ w_i$, which takes the values of 1 or 2 as indicated. The resulting transformation is

$\displaystyle z_k = \frac{1}{N_t^2} \sum_{i=1}^{N} \; w_i x_i \; \cos\left\{ 2 \pi (i-1) (k-1)/N_t\right\},$ (5.4)

for $ 1\leq k \leq P+1.$ This computes the order-$ P$ circular ACF using the frequency-domain method. In this case, the energy statistic is $ t({\bf x}) = \sum_i \; w_i x_i,$ with un-equal bin scaling, which is a valid norm on $ {\cal P}^N$.

There are two possibilities for computation of the ACF depending on if one is starting with time-series or spectral data. In both cases, we may assume that the time-series is independent Gaussian noise with mean 0 and variance 1. If $ {\bf x}$ is the time-series, then $ p({\bf x}\vert H_0)$ is taken directly from Table 3.1, ``Gaussian" row, and $ p({\bf z}\vert H_0)$ is computed using software/pdf_A_chisq_m.m.

   [lpzH0,ic] = pdf_A_chisq_m(z,A,K,[ic]);
where K is an $ N\times 1$ vector with the degrees of freedom [1,2,2, ... 2,2,1]. See software/module_acf_spax.m for additional details. For testing, use software/module_acf_test.m with TYPE=2.

If $ {\bf x}$ is the raw spectrum, $ p({\bf x}\vert H_0)$ is given by (4.1), with $ {\bf x}$ taking the role of $ {\bf z}$ in the equation, and $ p({\bf z}\vert H_0)$ is computed using software/pdf_A_chisq_m.m. For additional details, see software/module_acf_spa.m. For re-synthesis, use software/module_acf_synth.m. For testing, use software/module_acf_test.m with TYPE=0.

Baggenstoss 2017-05-19