## CR bound analysis of circular ARMA parameters pdf_arma_circ.m.

The circular ARMA model lends itself better to analysis. The CR bound for the circular ARMA model parameters

are derived now. We analyze the PDF (9.18) for the CR bound. We apply the results of Section 16.2.2 with

 (9.23)

noting that the ARMA parameters are spectral parameter . Following these results, we need the derivatives of the spectral values with respect to the parameters. The first derivatives are (from eq. 9.22 with replaced by ).

 (9.24)

Thus,

 (9.25)

where

 (9.26)

We can simplify (9.25). For example, we can rewrite as

The in the last term can be ignored since

because is the DFT of an anti-causal sequence. This leaves

 (9.27)

Similarly,

 (9.28)

Using (9.24), (16.9),

 (9.29)

and

 (9.30)

A simple form for can be found. We re-write (9.30) as

 (9.31)

Note that the frequency-domain function

is the DFT of an anti-causal function since both and are causal, so

for giving

 (9.32)

where is the ACF of the -th order AR process with parameters

   rb = rlevinson(b,1);
Ibb = toeplitz(rb(1:Q)) * N;

Similarly,

 (9.33)

which is the same form as (9.30), so we can copy the form of (9.32) to arrive at

 (9.34)

where is the ACF of the -th order AR process with parameters

   ra = rlevinson(a,1);
Iaa = toeplitz(ra(1:P)) * N;

Also,

 (9.35)

We also arrive at

because is zero for negative . Similarly,

(See [25] p. 277 for additional information). A MATLAB implementation for , , is provided below (See software/pdf_arma_circ.m).
>>   A=fft([a(:); zeros(N-P-1,1)]);
>>   B=fft([b(:); zeros(N-Q-1,1)]);
>>   E=exp(-1i*2*pi/N*[0:N-1]' * [0:max(P,Q)]);
>>   Hb = real( E(:,2:Q+1) .* repmat(1./B,1,Q) );
>>   Ha = real( E(:,2:P+1) .* repmat(-1./A,1,P) );
>>   I.a_a = 2 * Ha' * Ha;
>>   I.a_b = 2 * Ha' * Hb;
>>   I.b_b = 2 * Hb' * Hb;


Baggenstoss 2017-05-19