5360 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 20, OCTOBER 15, 2015
(15)
Employing the shift invariance property and assuming that the
noise perturbation is small relative to the signal power, we have
(16)
Itiseasytoverifythat
(17)
Let
and be the real and imaginary parts of
, respectively. Then, (16) can be rewritten as
(18)
After some straightforward manipulations, we obtain
(19)
where
, and
.
Adopting the WLS approach, (19) leads to
(20)
where
is the optimal weighting matrix that is defined through
the Gauss-Markov theorem [20], [22], [28], [29]:
(21)
with
(22)
The solution to this unconstrained optimization problem is
(23)
Substituting
into (21) yields
(24)
It is shown in [22], [31], [32] that
1
(25)
where
is the largest singular value of . Substituting
(25) into (24), using
and ignoring the con-
stant term
,wehave
(26)
1
Equation (25) is directly computed from Lemma 1 of [32], albeit it looks a
bit different from that in [32] because
is the singular value of which
is actually a forward-backward averaging matrix. It is easy to follow [32] to
verify that the singular values of
are the same as those of ,
whereas the singular values of
are times the eigenvalues in
. Using these tips, the reader can obtain (25).
TABLE I
2-D U
NITARY PUMA ALGORITHM.
Since is unknown, we can perform the estimation in a re-
laxation manner. It follows from (19) that we can use the least
squares (LS) estimate of
as an initial estimate, i.e.,
(27)
where
for sufficiently high SNR (See
Appendix A). Substituting (27) into (26) leads to
(28)
where
(29)
Recalling that
, the estimate of is then com-
puted as
(30)
The next step is to estimate
.Let and its SVD
be
(31)
where
holds the singular values and
(32)
(33)
Applying the same idea for estimating
and according to
(20)–(30), it is easy to obtain the closed-form expression of
:
(34)
where
(35)
(36)
(37)
(38)
Here,
where and
, and are the real and imaginary parts of
, respectively, and .The
steps for the 2-D unitary PUMA algorithm are summarized in
Table I.
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