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SECOND-ORDER VERSUS FOURTH-ORDER MUSIC ALGORITHMS :
AN ASYMPTOTICAL STATISTICAL ANALYSIS
Eric Moulines, Jean-François Cardoso
Télécom Paris - CNRS URA 820 - GdR TdSI
46 rue Barrault, 75634 Paris Cedex 13, France.
E-mail : [email protected]
ABSTRACT
Direction finding techniques are usually based on the 2ndorder statistics of the received data. In this paper, we propose
a MUSIC-like direction finding algorithm which uses a
matrix-valued statistic based on the contraction of the 4thorder cumulant tensor of the array data (4-2 MUSIC). We then
derive, in a unified framework, the asymptotic covariance of
estimation errors for both 2nd-order and 4th-order based
MUSICs, and show that the 4th-order method can perform
equally well or even better than the 2nd-order method, even
when the noise spatial structure is known.
geometry, see [9]). Another approach is to use a matrix-valued
statistics based on Hybrid Non-Linear Moments : in the ULA
case, an appealling statistic is the cross-covariance matrix
between the signal and its coordinate-wise distortion by a
zero-memory non linearity [7,10].
Given y the complex random vector at the array output, we
propose to estimate :
R̃ = E( y ∗ My y y ∗ ) and R = E(yy ∗ )
and to form the following matrix-valued statistics, depending
on M :
1. INTRODUCTION
Q(M) = R̃ − R M R − R Tr( MR )
Current array processing techniques are based on the
second-order statistics of the received signals. In many
situations, in particular in digital communications, received
signals are non-Gaussian so that they contain valuable
statistical information in their moments of order greater than
two. In these circumstances, it makes sense to develop array
processing techniques that exploit higher-order information
[1-7]. Of particular interest are the higher-order cumulants,
which enjoy the very appealing property of being
asymptotically insensitive to Gaussian noise : it is thus neither
necessary to know, to model, nor to estimate the noise
covariance, as long as the noise is Gaussian, which is a
reasonnable assumption for most communications media [8].
Note that M is an arbitrary hermitian matrix. We demonstrate
in the following that Q (M) is the result of the contraction of
the 4th-order array output cumulant tensor with the matrix M.
This approach offers two distinct advantages over [1,7] : first,
Q(M) provides directly the signal subspace even when signals
are correlated and for any array geometry (at least when M is
definite positive) ; in contrast, the methods in [1,7] are
designed for the ULA case and independent sources. Second,
the "natural" sample estimates of Q (M) are hermitian : no
further processing is needed for eigendecomposition. Finally,
there is an extra degree of freedom in the choice of weighting
matrix M : it can be shown that M acts much like a spatial
beamformer, that can focus on specific look directions.
Here, we consider one particular array processing
problem : high-resolution direction finding. The most popular
methods for estimating the direction-of-arrivals (DOAs) of
signals impinging on an array of sensors are based on the
concept of the signal subspace. This subspace is obtained, in
the standard MUSIC technique, from the eigendecomposition
of the sample covariance matrix.
Besides the derivation of a new method, we present, in a
unified framework, the asymptotic covariance of estimation
errors for both 2nd- and 4th-order MUSIC, and we show that,
in some practical situations, 4th-order based method can
perform equally well or even better than 2nd-order method,
even when the noise spatial structure is known.
Some methods have been recently proposed [3-4] that
exploit the full set of 4th-order cumulants. These approaches
offer some distinctive advantages over 2nd-order MUSIC but
the price to pay is a cumulant estimation cost that grows as m 4
(m being the number of sensors), while covariance estimation
cost grows as m 2 . Hence, it is interesting to consider cumulant
based m×m matrix-valued statistics. Such statistics are, of
course, required to provide consistent estimates of the true
signal subspace. Several approaches have been proposed. The
most natural is to use a m 2 subset (or slice) of higher-order
cumulants : the selection of an appropriate slice depends on
the symmetry of the problem. In the Uniform Linear Array
case (ULA), the 4th-order diagonal cumulant slice is a
reasonable choice [1] (for extensions to arbitrary array
The paper is organized as follows. Section 2 formulates
the problem and establishes model assumptions. Section 3
gives the expression of the contracted 4th-order cumulant
tensor and shows how it can be used to obtain the DOA’s.
Section 4 investigates the asymptotic distribution of the
resulting bearing estimates and compares it, in the single
source case, with the asymptotic distribution of 2-MUSIC.
2. PROBLEM FORMULATION
Assume an array of m sensors, n narrowband far-field
emitters and define a ( θ ) ∈ C m to be the array response for a
narrowband emitter at DOA θ. The array output y ( t ) ∈ C m is
given by
y(t) = As ( t ) + n ( t )
3. ESTIMATION METHODS
where
s ( t ) = [ s 1 (t),..., sn (t) ]T
and
.
.
.
A = [ a ( θ1 ),
, a ( θn ) ]. The signal emitted by the p-th
source, sp (t), is a stationary non-Gaussian complex random
variable. Signals emitted at different times, by the same source
or by different sources, are assumed to be statistically
independent. The additive n (t) is a complex Gaussian noise,
with arbitrary spatial structure. The array output is sampled at
N distinct time instants. Based on these measurements
y ( t ) ,1 ≤ t ≤ N , the problem of interest is to determine the
DOAs θp , 1 ≤ p ≤ n .
As explained in the introduction, we focus here on a class
of algorithms, exploiting only 4th-order cumulants only, while
keeping the same complexity as the standard 2nd-order
methods. The main idea is to contract the AOQ tensor. Let
M ij be a (1,1) hermitian tensor, that is, a linear self-adjoint
operator on C m . The contraction consists in summing over a
pair of indices, which yields
In the following, we assume that signals and noise are
complex circular, which means, omitting the dependence on t :
E ( si si . . . si s ∗i . . . s ∗i ) = 0 p≠q
1
2
p
p +1
p +q
Due to inherent symmetries in symbol constellation, such a
condition is approximately fulfilled by digital communication
signals of PSK type (at least when the signals are
independent). Circularity implies that odd-order statistical
moments of the array output are zero. Therefore, we will
concern ourselves only with the even-order moments, and, in
particular the 4th-order moments. These are given by
µ ij lk = E ( yi y j y k yl )
where y j (resp. y j ) denotes the j-th component of y (resp. y ∗ ).
Under the circularity assumption, 4th-order cumulants are
expressed in terms of 4th- and 2nd-order moments as :
Q ij lk = Cum ( yi , y j , y k , yl )
Q ij lk = µ ij lk − µi k µl j − µi j µl k
The set of four index quantities Q ij lk defines a (2,2)-tensor [9].
This tensor is referred to as the Array Output Quadricovariance
(AOQ). In the same manner, we can also define the source
signal quadricovariance (SSQ) :
K βα δδ = Cum ( s α , s β , s γ , s δ )
We finally drop the explicit dependence of a on θ ; the
concise notation a αi (respectively a− iα ) will now stand for the ith component of the directional vector associated with the
DOA θα , i.e a(θα ) (respectively a ∗ (θα )). Using these
conventions, we directly get, thanks to cumulant multilinearity
and additivity properties
Q ij lk = a αi a− βj a− kγ a δl K βα γδ
whith no noise contribution thanks to the Gaussian hypothesis.
We have used here the standard summation convention :
summation over any index repeated once as a superscript and
once as a subscript is implicit. The range of summation is not
stated explicitely but is implied by the position of the indices.
In principle, both the 2nd-order and the 4th-order
cumulants can be used to estimate the DOAs. However, the
use of the 2nd-order moments requires to know, to model or to
estimate the noise covariance. As long as the noise is
Gaussian, the use of 4th-order cumulants eliminates this need.
jk
l
Q(M) ij ∆
= Qi l Mk
Since Q is a tensor, the contraction has a simple algebraic
interpretation. Let F denote the space of the linear operators
on C m ; the space of (2,2) tensors can be canonically
associated with the space of linear operators on F. The (1,1)
tensor Q(M) resulting from the contraction, is the image by the
AOQ of the operator M (cf appendix). From the above
expressions we get
a αi a− βj ( a− kγ M lk a δl K βα γδ ) = a αi a− βj K βα γδ ( A H M A) δγ
In "index-free" formalism, this last equality can be more
concisely written
Q (M) = A C A H
where C is the contraction of the SSQ tensor on the matrix
A H M A, i.e C = K (A H M A). The last relation is strikingly
similar to the one obtained at 2nd-order with Q (M) in place of
the array output covariance matrix and C in place of the signal
covariance. Since only 4th-order cumulants are used, the
properties of the measurement noise do not appear in the
expression of the exact statistics. Q (M) is rank defective, with
a rank equal to the rank of C, and its range is spanned by the
steering vectors. Hence Q (M) can serve as a basis for standard
DOA estimation methods. From now on, we will concern
ourselves only with MUSIC-type algorithms : a MUSIC
algorithm based on Q (M) will be, for obvious reasons, referred
to as 4-2-MUSIC.
Note that M is a "free" parameter which can be adjusted to
increase DOA performances (see next section). In absence of
any information on DOA and on spatial correlation of the
noise, a neutral choice is : M = I. In presence of known (or
even unknown) coherent interferences, M can be chosen to
"steer nulls" in the direction of the interferers, while ensuring
that signals received from the desired directions pass
unattenuated. Finally, M could be replaced by a consistent
estimate of an appropriate statistic. An interesting choice, the
inverse of signal covariance, is discussed below.
4. PERFORMANCE ANALYSIS
In this section, we sketch the derivation of the statistical
performances of 4-2-MUSIC DOA estimation (details will be
published elsewhere) and compare these performances with
those achieved by 2-MUSIC. In order to avoid index invasion,
we use in the following a few special notations that are defined
in appendix.
We base our analysis on the following continuity theorem :
Let the sample statistic ŜN be a strongly consistent estimate of
N ( ŜN − S ) is asymptotically
S, i.e. ŜN → S (a.s) and √
Gaussian with finite asymptotic covariance
Σ( Ŝ ) ∆
= lim ( N Cov( ŜN − S ) ) < ∞
N→∞
V̂N ∆
= g ( ŜN ) is a strongly consistent estimate of
then
V∆
= g ( S ), provided that g (.) satisfies some regularity
conditions. Moreover,
lim N Cov( V̂N − V ) ) = g ′ ( S ) Σ( Ŝ ) g ′ ( S )H
N→∞
In the above formulation, we have not specified the algebraic
nature of ŜN (and S) because the continuity theorem holds
regardless ofthe nature of ŜN : a scalar, a vector, or even a
linear operator on an Hilbert space. To assess the statistical
perfomances of MUSIC-type methods, we apply the continuity
theorem twice : we first relate the covariance matrix of DOA
estimates θ̂ to the covariance of Π̂ , the estimate of Π which is
the orthogonal projector onto the noise-subspace. Second, we
calculate the covariance of Π̂ from the covariance of the
sample statistics.
The MUSIC estimates are obtained by searching the single
variable localisation function | Π̂ a(θ) | 2 = Tr( Π̂ a (θ)a (θ)∗ )
= Π̂ ∗ a (θ)a (θ)∗ for its minima. The DOA estimates θ̂ are
thus related to the projector onto the sample noise subspace Π̂
by a set of n equations
∗
∂S(
θ̂
)
Π̂ = 0
∂θi
i = 1, . . . , n
H
where S(θ) ∆
= A (θ) A (θ) . This set of n implicit equations
determines θ̂ as a function of Π̂ . The first derivative of this
function with respect to each θi can be shown to be the (1,1)tensor
∂ θ̂
i
∂ Π̂
= − h(θi )−1
∂S(θ)
∗
∂θi
∗ 2
2
where h ( θ ) ∆
= Π ∂ S/∂θ . Provided that Π̂ is a consistent
estimate of Π, it follows directly from the continuity theorem
and the above equation that θ̂ are consistent estimates of the
true parameter θ, with asymptotic covariance given by :
−1
Σ( θ̂i , θ̂ j ) = h ( θi ) h ( θ j
∂S ( θ )
)−1 ∂θi
∗
∂S ( θ )
Σ( Π̂ ) ∂θ j
This expression is a MUSIC "built-in" feature : it is a
consequence of the way the method exploits the information
provided by the sample noise projector. To determine the
asymptotic covariance Σ( Π̂ ), we use the following lemma,
which relates the perturbation of a projector on an invariant
subspace to the perturbation of the operator itself [12].
Lemma : Let B be an hermitian operator and λ0 be an
eigenvalue of B. The orthogonal projector Π( B ) onto the
invariant subspace associated with λ0 is an (infinitely)
differentiable function of B. Its first derivative, denoted Π′(B)
is the (2,2)-hermitian tensor :
Π′( B ) = [ Π , Γ ]2
where Γ is
#
Γ∆
= ( λ0 I − B )
the
pseudo-inverse
of
( λ0 I − B ) :
We now assume, for a fair comparizon with 2-MUSIC, that
the noise is spatially white with power σ. For 2-MUSIC, the
sample noise projector Π̂ is estimated from the
eigendecomposition of the sample covariance matrix R̂. Since
R̂ is a consistent estimator of array output covariance R, Π̂ is a
consistent estimator of Π, which corresponds, in this case, to
the invariant subspace of R associated with the eigenvalue σ.
We can therefore apply directly the above lemma, with
Γ = (σI − R)# . The continuity theorem then yields :
Σ2 ( Π̂ ) = [ Π , Γ ]2 Σ( R̂ ) [ Π , Γ ]2
The last point is to evaluate the asymptotic covariance of the
sample covariance R̂. This result can be directly derived, in the
Gaussian case, from the central moments of the Wishart
distribution. It can be established as well in the non-Gaussian
case, from standard formula on cumulants of ordinary kstatistics [11] : Σ( R̂ ) = Q + R ⊗2 R. Combining these two
results, together with ΠR = RΠ = σΠ and Q (Π)=0, we get :
Σ2 ( Π̂ ) = σ [ Π , Γ R Γ ]2
(1)
The covariance of the DOAs can finally be obtained from the
above expressions. Note that this formula goes beyond
"traditionnal" performance analysis results, since we have
assumed a non-Gaussian model for the signals [15]. It proves
an important result (see also [14]) : the exact joint pdf’s of the
sources has no influence on the performance of 2-MUSIC (the
only quantity of interest is the source covariance matrix).
The covariance of the projector for 4-2-MUSIC is denoted
Σ4−2 (Π) and can be derived in a similar manner ; the sample
projector is a consistent estimator of the orthogonal projector
onto the noise subspace Π, which corresponds to the kernel of
Q (M). The projector perturbation formula can therefore be
applied with Γ=Q (M)# , the pseudo-inverse of Q (M). All the
difficulty lies in the derivation of the covariance of the
contracted quadricovariance [11]. It can be demonstrated that :
Σ4−2 ( Π ) = σ [ Π , Γ L Γ ]2 + [ ΓRΓ , ΠRMRMRΠ ]2
+ [ ΠRMRΓ , ΓRMRΠ ]1
(2)
where L is a (1,1)-tensor depending on the cumulants of the
signals up to order six
jkn m l
L ij = H ilm
Mn Mk
kn l
j
jk m n
l
nk l
m j
jk
m l n
+ Q lm M k M m
n R i + 2Q il M k R m M n + Q il M k M n R m + Q ml M n M k R i
j k m n
k l
k l n m j
+ Ri Rl Mk Rm Mn Rk + Ri Mk Rl Mn Rm
and where the "hexacovariance" (!) H is of course defined by
jkn ∆
j
k
n
H ilm
= Cum(yi ,y ,yl ,y ,ym ,y ).
In its basic version, 4-2-MUSIC uses the identity as a
weighting matrix : M = I and eq. (2) takes the simpler form
Σ4−2 ( Π ) = σ [ Π , Γ L̃ Γ ]2
j
L̃ i
(3)
jab
ja b
ab j
= H iab
+ R ij Q ab
ab + 2 Q ib R a + Q ib R a
jb a
j a b
a b j
+ Q ab R i + R i R b R a + R i R a R b + σ2 R ij
Expression (3) involves the (pseudo)-inverse of the AOQ, Γ,
twice. This suggests that, provided the kurtosis are large
enough, 4-2-MUSIC methods can perform better than the 2MUSIC method, even when the noise cross-spectrum matrix is
known. An identical conclusion is drawn in [5]. This can be
checked analytically in the case of a single source with nonvanishing r-th order cumulant k (r) . Denote k̃ (r) the
standardized r-th order cumulant, i.e. k̃ (r) = k (r) /k r(2)/2 , and ρ the
Signal-to-Noise Ratio (SNR), ρ = k (2) /σ. Let us first choose
M = I. The asymptotic variances for 2-MUSIC and 4-2MUSIC are respectively equal to
Σ2 ( θ̂ ) =
Σ4−2 ( θ̂ ) =
a0
a1
a2
a3
−1
+ρ
1
ρ h (θ)
a + a ρ−1 + a ρ−2 + a ρ−3
3 012
2
ρ h (θ)k̃ (4)
= 2 + 5 k̃ (4) + k̃ (6)
= 6 + 5 k̃ (4)
= m +6
= m +2
For ULAs, the geometrical factor h ( θ ) is shown to be
inversely proportional to the number of sensors m, so that the
asymptotic covariance of 4-2-MUSIC does decrease as m is
increased. To get more quantitative information, let take a
somewhat idealized model of a PSK modulation : assume that
the source is a complex random variable with constant
modulus and a phase uniformly distributed over [ −π , π ]. The
first standardized cumulants of this random variable are
respectively equal to : k̃ (4) = −1, and k̃ (6) = 4. If the SNR is
large enough so that higher order terms can be neglected, it
appears that the performances of 4-2-MUSIC and 2-MUSIC
are identical. The remarkable result is that 4-2-MUSIC
achieves this performance without any information on the
Noise Covariance Matrix (NCM). On the other hand, even at
this large SNR limit, errors in the estimation of the NCM can
severely affect performances of 2-MUSIC.
We now illustrate the benefit that could be drawn from the use
of an appropriate spatial filtering matrix M. Suppose that M is
chosen to be the inverse of the source autocorrelation matrix :
−1/2
in the single source case, M = R −1
a (θ)a (θ)∗ . Of
S = ( k (2) )
course, RS is not known exactly ; however, it can be
demonstrated that the large sample properties of the estimates
are not affected RS is replaced by a consistent (initial) estimate
R̂S , so that the performance analysis remains valid [13]. The
asymptotic covariance is then modified in its higher order
terms which become independent of the number of sensors m :
a 2 = 6 , a 3 = 2. This formula suggests that an appropriate
spatial filtering allows us to significantly decrease the variance
at low SNRs.
CONCLUSION
A new array processing method has been introduced in this
paper based on the contracted quadricovariance of the array
output data. It offers a computational cost comparable to 2ndorder MUSIC while eliminating the need to model additive
Gaussian noise. Performance analysis demonstrates that 4-2MUSIC can yield as good or even better bearing estimates than
2-MUSIC with known noise covariance matrix, provided that
source kurtosis is large enough. These algorithms are of
practical interest for communication signals such as PSK
modulation.
APPENDIX
Let ( ei , i = 1,m ) be an orthonormal basis of E = C m . The
vector space of all linear operators on E is denoted by F :
elements of F are sometimes called "matrices" (in this paper,
matrices are referred to as (1,1)-tensors). We denote A ij the
entries of the matrix associated with an operator A. The
adjoint of A is denoted A H ; its associated matrix is the
transpose-conjugate of the matrix associated with A. F is a
H
Hilbert space, with the inner product A ∗ B ∆
= Tr ( B A ).
Similarly, the vector space G of all linear operators on F can
be identied with the space of (2,2)-tensors.
We now define two "tensor products" : ⊗1 and ⊗2 . They
combine any pair of (1,1)-tensors, A and B in F, to yield a
(2,2)-tensor, according to the defining rules :
( A ⊗1 B ) M
( A ⊗2 B ) M
∆ A B ∗ M = A Tr(MB H )
=
∆ A M BH
=
The first tensor product is a (2,2)-tensor, similar to the
classical vector outer product. The second tensor product
essentially behaves as the Kroenecker product. In index
notation, the coordinates of these tensors on the canonical
basis ( ei e ∗j ) ⊗1 ( ek e ∗l ) are
( A ⊗1 B ) ij lk = A ij (B H ) kl
( A ⊗2 B ) ij lk = A ki (B H ) lj
Finally, we define two symmetrized "tensor products" :
[ A , B ]1,2 ∆
= A ⊗1,2 B + B⊗1,2 A
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