4.3 Solution Methods
4.3.1 MLE for Submanifold Defined by Shift-Invariance: Uni-
In this section we consider the submanifold ¯Bdamped; for notational convenience
we drop the subscript and denote this submanifold simply as ¯B. The set ¯VY˜ (641) of
submanifold constraint satisfying tangent vectors is specified in (655) by the matrix Es
and the complex ¯Nb-tuple bn. The matrix Esis a representation of a basis for SY˜. The
¯
Nb-tuple bncorresponds to the tangent vector from ˜Y to the nearest point on ¯B. Together
˜
is known, the ML estimateY ∈ ¯b B is computed from (664), (665) and (663).
Complex Representation of Constraint
In the previous development we treated the submanifold tangent spaces SY as real
vector spaces. This was convenient for analysis purposes, since in the H(θ) model the parameters of interest are themselves real (and potentially odd in number). For the special case of the damped exponential signal model, these spaces may be represented as complex vector spaces of lower dimension. This key feature follows from the form of the derivative matrix that defines the space SY and the related fact that the signal
parameters occur in pairs, so that the real dimension of SY is even.
Lemma 23 (Basis for the subspace SY on ¯Bdamped). The basis matrix Es(653) for the
¯
Bdampedassumption has the form as
Es = h ¯ Es i · ¯Es i (666) where ¯Es isNb× ¯q and ¯q = q2.
Proof: (see appendix C.3)
The columns of ¯Esspan SY as a complex vector space of dimension ¯q = q2. To see
this, let ∆ ∈ SY be arbitrary, where SY is spanned by Es (see (A.161) ). By definition,
this vector has the form ∆ = Y⊥B for some matrix B. Let s be the real q-tuple such
that b = Ess, where b = vec(B). Referring to (666) this can be expressed as
b = Ess = ¯EssR+ i ¯EssI (667)
where the real q-tuple s has been partitioned into two ¯q-tuples. (i.e.; sR and sI are
defined by the first ¯q and last ¯q elements of s, respectively). Define the complex ¯q-tuple ¯ s = sR+ i · sI, so that s = " Re(¯s) Im(¯s) # = " sR sI # . (668)
Using this, (667) becomes
b = Ess = ¯Ess .¯ (669)
The following theorem records the complex form of the derivative matrices Lsand
Lssthat appear in the ML estimate of s using this complex vector space formulation of
the space SY.
Theorem 24. If the basis matrix Es has the form (666) then s = L−1ssLs defined by
(659),(660) is equivalently expressed in complex form as ˆ¯
s ≡ ¯L−1ssL¯s (670)
where ¯Lss≡ ¯EsHΛebbE¯sand ¯Ls ≡ ¯EsHΛebbbn. The realq-tuple s and the complex ¯q-tuple
¯
s are related by (668). Proof: see C.4
As a consequence the ML estimate (663) is given by ˆb = b
n+ ¯Esˆ¯s . (671)
Solution
Elements of ¯Bdamped satisfy a shift invariance condition, defined later, which can
be expressed in terms of a vector valued function r(:) defined on B such that Y ∈ ¯B if and only if r(Y ) = 0. In this case the submanifold membership condition is
expY˜∆ ∈ ¯B ⇐⇒ r(expY˜∆) = 0 (672)
and the set of constraint-satisfying tangent vectors ¯VY˜ (641) is
¯
VY˜ = {∆ ∈ BY˜ such that r(expY˜∆) = 0} .˜ (673)
Using the fact that elements of BY˜ have the general form ∆ = ˜Y⊥B, we express the
constraint as a function of b = vec(B) for a fixed ˜Y⊥
Consider the Taylor series expansion of the constraint function r(b; ˜Y ) r(b; Y ) = r(0; ˜Y ) + ¯ Nb X k=1 ∂r ∂bkb k+ h.o.t. (675)
and for convenience define
h c Him k ≡ − ∂rm ∂bk . (676)
(H is a function of ˜c Y⊥). The first order approximation of the constraint then is
r(b; Y ) ≈ r( ˜Y ) −Hb .c (677)
A critical feature of this approximation is thatH is constructed in such a way thatc
it has a null space null(H) of complex dimension ¯c q = dim S˜
Y, and this null space well
approximates SY˜. Let ˆEs denote the matrix whose columns are a basis for null(H),c
then
col( ˆEs) ≈ col( ¯Es) (678)
where ¯Esis a basis for SY˜.
The set of constraint-satisfying tangent vectors ¯VY˜ (636), (673) is approximated by
¯
VY˜ ≈ {b ∈ C ¯
Nb such that b = min kr( ˜Y ) −Hbkc 2} . (679)
Elements in this set are given by least squares solution of kr( ˜Y ) −Hbkc 2with respect to
b, which is
b = ˆbn+ ˆEss¯ (680)
where
bn= ˆH#r( ˜Y ) (681)
and ¯s is a complex free ¯q-tuple.
Remark: If ˜Y ∈ ¯B then r( ˜Y ) = 0 and therefore ˆbn = 0 (681) so that the solution
(680) simplifies to
Example:1-Dimensional ULA On a N -element uniform line array (ULA) a single damped exponential signal has the form
h = z(0:N −1) (683)
where z = ρ exp(iθ). The parameters θ and ρ are reals and denote the spatial frequency and damping parameter, respectively, of the complex exponential waveform. If J0 and
J1 represent selection matrices that select the elements (0 : N − 2) and (1 : N − 1)
respectively, then J0· h · z = J1· h.
In the case of multiple signals we use subscripts so that hk = z (0:N −1)
k and zk =
ρkexp(iθk). For p signals the signal matrix is H = h
h1 · · · hp i
and has the shift- invariant property
J0HΨ = J1H (684)
where Ψ = diag (z1, · · · , zp).
Let Y be the semi-unitary matrix defined by left singular vector matrix of H so that H = Y T where for T is non-singular. Using this (684) becomes J0Y T Ψ = J1Y T or
equivalently
J0Y · Φ = J1Y (685)
where Φ = T ΨT−1. Evidently
Φ = (J0Y )#J1Y . (686)
By construction, diag (Ψ) = eig(Φ) so that eig(Φ) = h z1· · · zp i
. In the pure har- monic case (ρk = 0) the p-tuple θ is evidently θ =6 (diag(Ψ)) =6 (eig(Φ)).
Because a single damped exponential is specified by two real parameters all sub- manifolds here are of even dimension, that is dim(SY0) = q = 2p. p = ¯q.
Definition: Shift Invariance for Semi-Unitary Matrices: Define the function
where
Φk(Y ) ≡ (JkY )#J0Y (688)
and Jk, J0 ∈ RN (sel)×N are the selection matrices that select the N (sel) elements from
the N elements. A N × p semi-unitary matrix Y is said to be shift invariant with respect to the pair of selection matrices Jkand J0if
Xk(Y ) = 0 . (689)
The relation (689) is often referred to in the literature as the invariance equation [1]. Lemma 25. If Y is shift-invariant with respect to J0 and Jk thenY Q is also shift in-
variant with respect to this same pair. Proof: (appendix C.5)
The following lemma expresses the invariance condition Xk(Y ) = 0 in a reduced
form.
Lemma 26. Let Z = orth(J0Y ), and Z⊥ its orthogonal complement, and define
Rk(Y ) ≡ Z⊥HXk(Y ) . (690)
Then
Rk(Y ) = 0 ⇐⇒ Xk(Y ) = 0 . (691)
Proof: The columns of[ZZ⊥] span the range space of Xkso that
Xk(Y ) = PZXk(Y ) + PZ⊥Xk(Y ) . (692) It follows that Xk(Y ) = 0 ⇐⇒ " ZHX k(Y ) Rk(Y ) # = 0 . (693)
By definition allY ∈ ¯B satisfies Xk(Y ) =0. Furthermore if Y ∈ ¯B then Y Q ∈ ¯B and
therefore ifY satisfies Xk(Y ) = 0 then Xk(Y Q) = 0 for Q unitary. If Rk(Y ) 6= 0 then
Figure 8. Selection Matrices for 2-D Planar Arrays
For the 1-dimensional ULA case a single pair of selection matrices are sufficient to specify ¯B. For M -dimensional array case, M > 1, additional pairs are required and a single constraint Rk(Y ) = 0 is a necessary, but not sufficient, condition for Y ∈ ¯B. The
required set of selection matrices depends on the array geometry (e.g.; rectangular array versus corner clipped array). In what follows we assume a base array defined by the selection matrix J0 and consider shift pairs (J0, J1), (J0, J2), etc. on rectangular arrays
(see figure 8). Defining the vectorized form of the component constraint as rk(Y ) ≡
vec(Rk(Y )), the full constraint takes the form
r(Y ) ≡ r1(Y ) .. . rM¯(Y ) . (694)
Note that each rk(Y ) is a complex ¯Nb- tuple so that r(Y ) is ¯M · ¯Nb × 1. For an appro-
priately chosen a set of ¯M +1 selection matrices the elements of ¯Bdamped satisfy
or equivalently, in terms of the exponential map at ˜Y ,
expY˜∆ ∈ Bdamped ⇐⇒ r(expY˜∆) = 0 . (696)
Approximating the constraint
A key idea in developing the approximation for (694) is incorporating the complex dimension of the submanifold tangent space SY. For p signals on an M -dimensional
array, the complex dimension is dim(SY) = ¯q = M p. Earlier we showed that in the
general case, the set of constraint-satisfying tangent vectors had the form b = bn+ ¯Ess¯
(655) where ¯s was a free complex ¯q-tuple and ¯Eswas an Nb× ¯q complex matrix which
represented a basis for SY.
We next develop an approximation for the submanifold component membership function (690). As a first step in the linearization, we record the expanded form of the function Rk(·) (690) when the argument is Y = expY˜∆. Substituting the expanded˜
form of the exponential map expY˜∆ = ˜˜ Y + ˜∆ + h.o.t into (690) yields
Rk(expY˜∆) ≡ R˜ k( ˜Y ) − M0k( ˜Y ; ∆) − M1k( ˜Y ; ∆) (697)
where
M0k( ˜Y ; ˜∆) = Z⊥H(Jk∆ ˜˜Φk− J0∆)˜ (698)
and
M1k( ˜Y ; ˜∆) = Z⊥HJkY δΦk( ˜Y , ˜∆) + h.o.t. . (699)
In the above we have used
Φk(expY˜∆) = ˜Φk+ δΦk( ˜Y , ∆) (700)
and where ˜Φk ≡ Φk( ˜Y ) and
The tangent vectors of interest are elements of BY˜ < TY˜B¯damped, which have the
general form ˜∆ = ˜Y⊥B. Substituting this form into (698) yields
M0k( ˜Y ; ˜∆) = (Z⊥HJk· ˜Y⊥B · ˜Φk− Z⊥HJ0· ˜Y⊥B) . (702)
Vectorizing (702) and (699) using the vec(ABC) = (Ct⊗ A)b identity yields the vec- torized form for (697) as
rk(expY˜∆) = rk( ˜Y ) − ˜Hkb − Ckvec(δ ˜Φk(b)) + h.o.t (703)
where
˜
Hk = ( ˜Φtk⊗ Z⊥HJkY˜⊥) − (I ⊗ Z⊥HJ0Y˜⊥) (704)
and
Ck = (Ip⊗ Z⊥HJkY )˜ (705)
and where rk(:) ≡ vec(Rk(:)) and δ ˜Φk(b) ≡ δ ˜Φk(Y⊥B). Note that Hk and Ck are
¯
Nb× ¯Nb.
To arrive at the desired form we note that, at high SNR, small kbk, that
kCkvec(δ ˜Φk(b))kF kHkbkF . (706)
Dropping this term and the higher order terms in (703) results in the approximation
rk(expY˜∆) ≈ rk( ˜Y ) − ˜Hkb . (707)
From these component function linearizations (707) we approximate the full constraint (694) as r(Y ) ≈ r( ˜Y ) − ˜H( ˜Y )b (708) where ˜H( ˜Y ) is ˜ H( ˜Y ) = ˜ H1( ˜Y ) .. . ˜ HM¯( ˜Y ) . (709)
The matrix ˜H( ˜Y ) is approximately rank N − ¯q. Motivated by the dimension of the con- straint submanifold tangent space SY with complex dimension ¯q, we replace ˜H( ˜Y ) by
its rank N − ¯q approximation, denotedH. This defines the matrixc H in (677) and (679)c
and using (659) through (671) completes the subspace estimation algorithm, which is summarized in the next section.