2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Modified Capon Beamformer for High-Resolution Direction-of-Arrival
Estimation
Changnong Liu† , Jingmin Xin†∗ , Yoshifumi Nishio‡ , Nanning Zheng† , and Akira Sano§†

Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, Xi’an 710049, China∗
Xi’an Jiaotong University Suzhou Academy, Suzhou 215123, China‡
Department of Electrical and Electronic Engineering, The University of Tokushima, Tokushima 770-8506, Japan§
Department of System Design Engineering, Keio University, Yokohama 223-8522, Japan
Abstract—In this paper, a modified Capon beamformer (MCB) is proposed for estimating the direction-ofarrivals (DOAs) of multiple noncoherent narrowband signals, where the computationally expensive eigendecomposition is not required. Further the relation between thisnonparametric estimator and the well-known parametricmethod called the multiple signal classification (MUSIC) is
studied, where it is clarified that by increasing the power ofthe inverse of the array covariance matrix, the resolution of
the MCB can be improved significantly compared with thestandard Capon beamformer (SCB) and as better as that ofMUSIC regardless of the signal-to-noise ratio (SNR). The
eﬀectiveness of the proposed MCB and the performance
analysis are verified through numerical examples.
1. Introduction
Direction-of-arrival (DOA) estimation of multiple narrowband signals incoming on an array of sensors is of great
importance in a variety of applications, such as radar, sonar,astronomy, and seismology (e.g., [1]–[2]). The beamforming is one of the oldest ideas for DOA estimation in arrayprocessing [3], and perhaps the most well-known one is theCapon beamformer [4]. The standard Capon beamformer(SCB) has the advantage that any assumption about the statistical properties of the array data is not required andhence it can be used in situations, where the information
about these properties is unavailable (e.g., [2]), but it does
not have the best resolution as the parametric methods suchas the multiple signal classification (MUSIC) [5], which isa relatively simple and eﬃcient subspace-based DOA estimation method and the large sample realization of the maximum likelihood (ML) method in the presence of uncorrelated incident signals [6]. Furthermore, the SCB can be applied to the array with arbitrary geometrical configurations
and is computationally eﬃcient than the (spectral) MUSIC method, where the computationally intensive eigendecomposition is involved besides the one-dimensional (1-D)
searching (cf., [1], [12]), and it may become a tremendous
computational burden in the practice applications of arrayswith large numbers of sensors. However, the DOA estimation performance of the SCB degrades significantly and it isdiﬃcult to distinguish the closely spaced incident signals,

when the signal-to-noise ratio (SNR) is low or the numberof snapshots is small.
The relation between the Capon beamformer and theMUSIC method was studied in [7], where it was shownthat the resolution of the Capon beamformer can be equal
to that of MUSIC when the SNR tends to infinite. In an
attempt to alleviate the limitations of the SCB, some modifications were proposed [8]–[12]. However, there are no
existing techniques for selecting the weighting matrix [12]and the weight parameter [10], while the eigendecomposition is required in [9], and Cholesky decomposition and theselection of LP model order are needed in [11].
Therefore in this paper, we investigate the DOA estimation of multiple noncoherent narrowband signals in acomputationally eﬃcient way and propose a new modified
Capon beamformer (MCB) to overcome the limitation of
the SCB, where the eigendecomposition is avoided. Further the relation between the MCB and the MUSIC method
is clarified explicitly, where it is shown that by increasing
the power of the inverse of the array covariance matrix, theeﬀect of signal components in the spatial spectrum can be
reduced and hence the resolution of the MCB can be improved significantly compared with the SCB and as betteras that of the MUSIC method regardless of the SNR. The
eﬀectiveness of the proposed MCB is verified through numerical examples.
2. Problem Formulation
We consider an array composed of M sensors with theintersensor spacing d and p (p < M) noncoherent narp
rowband signals {sk (n)}k=1
with the wavelength λ incomp
ing from far-filed along the distinct directions {θk }k=1
. The
received array data at the nth snapshot can be expressed asy(n) = A(θ)s(n) + w(n)(1)

where y(n), s(n) and w(n) are the vectors of the received
noisy data, incoming signals and additive noise given by
y(n) , [y1 (n), y2 (n), · · · , y M (n)]T , s(n) , [s1 (n), s2 (n),· · · , s p (n)]T and w(n) , [w1 (n), w2 (n), · · · , w M (n)]T , A(θ)
is the array response matrix given by A(θ) , [a(θ1 ),a(θ2 ), · · · , a(θ p )], while the array response vector a(θk ) isrelated with the array configuration, and ( · )T denotes the- 116 -

transpose. For the uniform linear array (ULA), a(θk ) =[1, e jτ(θk ) , · · · , e j(M−1)τ(θk ) ]T , and τ(θk ) , 2πd sin(θk )/λ.
We make the basic assumptions that the incoming signalp
s {sk (n)}k=1
are temporally complex Gaussian random process with zero-mean and mutually uncorrelated. The adM
ditive noises {wm (n)}m=1
are temporally and spatially complex white Gaussian random process with zero-mean andvariance σ2 and uncorrelated with the incoming signals.
The number of incoming signals p is known or estimatedin advance by using the detection methods (cf., [13]).
Under the basic assumptions, from (1), we easily obtainthe covariance matrix R of the received array data
R , E{y(n)yH (n)} = A(θ)R s AH (θ) + σ2 I M .

(2)

In practice, where the finite snapshots of array data are
available, R is unavailable and usually replaced by its sample estimate R̂ given by
1 ∑y(n)yH (n)
N n=1
N

R̂ =

(3)

where N is the number of snapshots.

m ≥ 1), we can obtain a MCB to design the optimal weightvector w̄ by solving the following problem
min w̄H Rm w̄
w̄

w̄H a(θ) = 1.

subject to(7)

Similarly for the case of finite snapshots, the solutions to
the weight vector and the MCB spatial spectrum are given
byw̄MCB =PMCB (θ) =

R̂

−m

a(θ)

(8)

−m
aH (θ) R̂ a(θ)

1
fMCB (θ)(9)

where fMCB (θ) is the MCB cost function defined by
fMCB (θ) , aH (θ) R̂

−m

a(θ).

(10)

p
Apparently the DOAs {θk }k=1
can be estimated by maximizing the spectrum PMCB (θ) in (9) without the procedure of
eigendecomposition. Note that the proposed MCB in (9)reduces to the SCB in (6) when m = 1, while it turns into the modified Capon method considered in [8] and theimproved Capon method with W = I M suggested in [12]when m = 2.

3. Modified Capon Beamformer for DOA Estimation

4. Mathematical Link between MCB and MUSIC

3.1. Standard Capon Beamformer [4]Here we focus on examining the relationship between
the standard and modified Capon beamformers and thesubspace-based MUSIC method [5].

The SCB [4] is a nonparametric DOA estimator, which selects the weight vector w̄ to maintain a fixed gain at a “lookdirection” while to maximally reject the signals comingfrom other directions by using the following constrained
quadratic problem [4]min w̄H Rw̄
w̄

subject to w̄H a(θ) = 1.

(4)

By exploiting some techniques such as Lagrange optimization method, when the number of snapshots is finite, we
can obtain the weight vector w̄SCB as−1w̄SCB =

R̂ a(θ)−1aH (θ) R̂ a(θ).

(5)

Noting that the presence of additive noise will ensure R̂ ispinvertible, the DOAs {θk }k=1
can be estimated from the locations of the p highest peaks of the Capon spatial spectrum(i.e., array output power) PSCB (θ) given by
PSCB (θ) =1
fSCB (θ)

4.1. MUSIC Method [5]From (2), the eigenvalue decomposition (EVD) of thecovariance matrix R is given by
R = UΣU H = U s Σ s U Hs + Un Σn UnH

where U , [U s , Un ], U s , [u1 , u2 , · · · , u p ], Un ,[u p+1 , u p+2 , · · · , u M ], Σ , diag[Σ s , Σn ], Σ s , diag(λ1 ,λ2 , · · · , λ p ), Σ s , diag(λ p+1 , λ p+2 , · · · , λ M ), and (ui , λi ) is
the ith eigenpair of R with λ1 ≥ · · · ≥ λ p ≥ λ p+1 =· · · = λ M = σ2 , while U s and Un correspond to the signal and noise subspaces, and UU H = I M . By utilizing
the properties that U s and A have the same column space
and the noise subspace Un is their orthogonal complement, i.e., UnH A = O(M−p)×p , when the number of snapshots isfinite, the (spectral) MUSIC method estimates the DOAs
by searching the p highest peaks of the following “spatial
spectrum” given by

(6)

−1

where fSCB (θ) , aH (θ) R̂ a(θ).

(11)

PMUSIC (θ) =1
fMUSIC (θ)(12)

where fMUSIC (θ) is the MUSIC cost function defined byH

3.2. Modified Capon Beamformer
Now by replacing R in the standard Capon constraintfunction in (4) with Rm , where m is a positive integer (i.e.,- 117 -

fMUSIC (θ) , aH (θ)Ûn Ûn a(θ)
M
M
∑
∑
H
2
=
|a (θ)ûi | =fi (θ)
i=p+1

i=p+1

(13)

where fi (θ) , |aH (θ)ûi |2 .
The performance of the MUSIC method has been studied extensively (e.g., [14]), and it is known that the MUSIC method is statistically eﬃcient in cases when either
the number of snapshots or the SNR is suﬃciently large,
but its computational cost is high due to the procedure of
the eigendecomposition. The MUSIC method has higher
resolution than the SCB does mainly because the utilization of the property that the noise subspace is orthogonal tothe columns of A(θ).

Spatial Spectra

310(spectral) MUSIC
Modified Capon with
Eigenvector
SCB
MCB (m=2)
MCB (m=3)
MCB (m=4)

2

Amplitude

10

1

10

0

10

4.2. Relation Between MCB and MUSIC−1

According to (11), the inverse of the array covariancematrix R can be obtainedH−2H
R−1 = UΣ−1 U H = U s Σ−1
s U s + σ Un Un .

(14)

Hence from (14), we easily haveH−2m
R−m = UΣ−m U H = U s Σ−m
Un UnH . (15)s Us + σ

Then when the number of snapshots is finite, the EVD of
H
the sample estimate R̂ in (3) is given by R̂ = ÛΣ̂Û , whereΣ̂ = diag(λ̂1 , λ̂2 , · · · , λ̂ M ), and λ̂1 ≥ · · · > λ̂ p ≥ λ̂ p+1 ≥· · · ≥ λ̂ M > 0, and hence the MCB cost function fMCB (θ) in(10) can be expressed as

10

−80

−60

−40

−20

40

60

80

Figure 1: Spatial spectra of the conventional and modified Capon
beamformers and the MUSIC method for Example 1. (SNR=0dB,
N = 200)

Furthermore by considering the fact that the signal
eigenvalues λ1 , · · · , λ p are larger than the noise eigenvalues−m= 0, it
σ2 , i.e., λ̄−1
i < 1 for i = 1, 2, · · · , p and limm→∞ λ̄i−1
is obvious that the increasing of the power of R (i.e., m)can eﬀectively weaken the proportion associated with thesignal subspace. Hence from (17), we easily have
lim fMCB (θ) = σ−2m lim fMUSIC (θ).

N,m→∞
M
∑
(
)
−m H
H
fMCB (θ) = aH (θ) Û s Σ̂ s Û s +λ̂−mi ûi ûi a(θ)

020Angle (deg)

N,m→∞

(18)

Further when the number of snapshots is suﬃciently large,
we have lim λ̂i = σ2 (i = p + 1, · · · , M), and in the caseAs a result, the MCB spatial spectrum becomes equal tothat of the MUSIC method except for a scaled amplitude,
when the power m and the number of snapshots tend tolarge enough. Therefore by increasing the power of R−1(i.e., m) in (10) to eliminate the influence involved the signal subspace, the DOA estimation performance of the MCB
can be improved significantly, where the procedure of computationally intensive eigendecomposition is not required.

of SNR , ∞ (i.e., σ2 , 0), from (13), (15) and (16), we
easily obtain

5. Numerical Examples

i=p+1
−m

M
∑

H

= aH (θ)Û s Σ̂ s Û s a(θ) +λ̂−mi fi (θ). (16)

i=p+1

N→∞

lim fMCB (θ) = a (θ) lim

(

H

N→∞

N→∞

+
=σ

−2m

(

−m H )
Û s Σ̂ s Û s a(θ)

M (
∑
i=p+1

a

H

lim λ̂−m
i

N→∞

)(

lim fi (θ)

)

N→∞

)

−m
(θ)U s Σ̄ s U Hs a(θ) + lim fMUSIC (θ)
N→∞

(17)
where Σ̄ s , diag(λ̄1 , λ̄2 , · · · , λ̄ p ), and λ̄i , λi /σ2 . Evidently from (16) and (17), we can find that the MCB costfunction (and hence that of the SCB with m = 1) consistsof two terms, where the first one involves the informationof signal subspace, while the latter one corresponds to theMUSIC cost function. Consequently the existence of theterm associated with the signal subspace causes the estimation performance of the Capon method in (6) worse thanthat of the MUSIC method in (12).

Now we evaluate the performance of the proposed MCB
method for estimating DOAs of uncorrelated narrowband
signals with some numerical examples. Two uncorrelated
signals with equal power and distinct angles θ1 = 5◦ andθ2 = 17◦ imping on a ULA composed of M = 6 sensors
spaced half-wavelength apart (i.e., d = λ/2).
Example 1—Improved Resolution through Spatial Spectra Searching: Firstly we examine the spatial spectra of the
SCB [4], the modified Capon beamformer with eigenvector method [9], [3], and the proposed MCB with diﬀerent−1
powers of R̂ (i.e., m). The results are obtained from 100
independent trials and plotted in Fig. 1, where that of the(spectral) MUSIC [5] is also shown for comparison. We
notice that the SCB and the MCB with m = 2 cannot provide evident peaks at the incoming directions, hence they
fail to distinguish this two incident signals. However, byincreasing the power m which can eﬃciently weaken the
proportion associated with the signal subspace in (17), the- 118 -

(a) Estimation Performance versus SNR (θ1)

1

RMSE (deg)10compared with the SCB and as better as that of the MUSIC method regardless of the SNR. The simulation results examined the eﬀectiveness and validity of the proposed
method.

0

10

−1

10

−2

10

−5

0

10
SNR (dB)

15

20

Acknowledgments

25(b) Estimation Performance versus Number of Snapshots (θ1)

2

10
RMSE (deg)5This work was supported in part by the Programme of Introducing Talents of Discipline to Universities under Grant
B13043 and the National Natural Science Foundation ofChina under Grant 61172162.

1

10

0

10

−1

10

1

10

2

10
Number of Snapshots

3

References

10

Figure 2: RMSEs of the estimation of θ̂1 verse (a) SNR and (b)the number of snapshots (dashed line: SCB; solid line with “o”:
MCB with m = 2; dotted line with “×”: MCB with m = 3; solidline: MCB with m = 4; dash-dotted line: (spectral) MUSIC; solidline with “”: root-MUSIC; dotted line: CRB) for Example 2(N = 200) and Example 3 (SNR=0dB).

MCB with m = 3 and m = 4 succeeds in estimating these
two DOAs and its resolution gradually approximate that ofthe (spectral) MUSIC exactly as it is derived in (18) .
Example 2—Performance versus SNR: Now we verify
the performance of the MCB in (16) against SNR, while the
SCB, the (spectral) MUSIC are also included for a comparison. The root-MUSIC [15] and the stochastic Cramer-Rao
lower bound (CRB) [14] are plotted for reference. The results are obtained from 1000 independent trials. The empirical root-mean-square errors (RMSEs) of the estimates θ̂1against SNR are shown in Fig. 2(a). Obviously the MCB
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Example 3—Performance versus Number of Snapshots:
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the increase of m though the number of snapshots is small.
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