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 anticausal 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 rewrite (9.30) as

(9.31) 
Note that the frequencydomain function
is the DFT of an anticausal 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(NP1,1)]);
>> B=fft([b(:); zeros(NQ1,1)]);
>> E=exp(1i*2*pi/N*[0:N1]' * [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
20170519