The main problem in CLT is finding a positive-valued mean for the floating reference hypothesis, , for which the feature mean is equal to the current feature value. Let be the current ACF feature value. We need to find a positive-valued such that A convenient approach comes from AR analysis, where it is known that the ACF of the AR spectrum is equal to the original ACF . Thus, the AR spectral estimate serves as .
Using the Levinson-Durbin recursion , we may transform the ACF into the AR coefficients , and . The corresponding AR spectrum is written
Under , the elements of are independent and Chi-squared with 1 or 2 degrees of freedom with mean (See Section 16.1.2). Bins are distributed according to
In summary, we take the following steps:
% Nt = FFT size % x is dimension n by 1, where n=Nt/2+1; % A is dimension n by P+1 % k is vector of degrees-of-freedom k = [1;2*ones(n-2,1); 1]; A = cos([0:Nt/2]' *(2*pi/Nt)*[0:P]).*repmat(k,1,P+1)*(1/Nt^2);
[a,e]=levinson(z);The output variable e is the AR innovation variance parameter .
af = fft(a(:),Nt); xz = Nt * e ./ abs(af(1:n)).^2;
lpxHz = expon_dist(x([2:n-1],:),xr([2:n-1],:)) + ... chisq1_dist(x([1 n],:),xr([1 n],:));
format long [A'*xr z]
q = 2./k .* xz.^2;The covariance of under is :
Sz = A'*diag(q)*A;Then, is
ldetSr=log(det(Sz)); lpzHz = -(P+1)/2*log(2*pi) - .5*ldetSz;See software/module_acf_clt.m for additional details. Use software/module_acf_synth.m for inversion (re-synthesis). The module can be tested using software/module_acf_test.m with TYPE=1. We compare this method with other approaches in Section 9.4.10.