Application of SPA to Circular
AutoCorrelation Function Analysis
The autocorrelation function (ACF) is
widely used in autoregressive (AR) timeseries
analysis and spectral estimation [25].
An estimate of the ACF given a length timeseries
can be obtained using the frequencydomain
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 is the
vector of raw spectral values
(magnitudesquared DFT), where
The output ACF feature is
.
To compute the th order ACF (lags 0 through ),
the columns of the
matrix 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 inverseFFT of the
full length spectrum (redundant bins duplicated) which requires that we
scale the complex bins (
) by 2.0
and the real bins () by 1.0.
This unequal bin scaling can be formalized by the scaling
variable , which takes the values of 1 or 2 as indicated.
The resulting transformation is

(5.4) 
for
This computes the order circular ACF using the frequencydomain method.
In this case, the energy statistic is
with unequal bin scaling, which is
a valid norm on
.
There are two possibilities for computation of the
ACF depending on if one is starting with timeseries
or spectral data.
In both cases, we may assume that the timeseries
is independent Gaussian noise with mean 0 and variance 1.
If is the timeseries, then
is taken directly from Table 3.1, ``Gaussian" row,
and
is computed using software/pdf_A_chisq_m.m.
[lpzH0,ic] = pdf_A_chisq_m(z,A,K,[ic]);
where K is an 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 is the raw spectrum,
is
given by (4.1), with taking the role of
in the equation, and
is computed using software/pdf_A_chisq_m.m.
For additional details, see software/module_acf_spa.m.
For resynthesis, use software/module_acf_synth.m.
For testing, use software/module_acf_test.m with TYPE=0.
Baggenstoss
20170519