IEEE Workshop on Nonlinear Circuit Networks
December 15-16, 2017

Modified Capon Beamforming Method with Array
Interpolation and Spatial Smoothing
Xiao Chen†‡1 , Jingmin Xin† , Yoshifumu Nishio‡2 , Nanning Zheng† , and Akira Sano§3†

Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, Xi’an 710049, China
1
cx2012@stu.xjtu.edu.cn‡
Department of Electrical and Electronic Engineering, Tokushima University, Tokushima 770-8506, Japan2nishio@ee.tokushima-u.ac.jp§
Department of System Design Engineering, Keio University, Yokohama 223-8522, Japan3
sano@sd.keio.ac.jp

Abstract—In this paper, we propose an array interpolation andspatial smoothing based modified Capon beamforming method,where the arbitrary linear array is transformed to a virtual uniform linear array (ULA) by utilizing the interpolation technique,and then the coherency of incident signals can be decorrelatedby employing the spatial smoothing preprocessing. Further byincreasing the power of array covariance matrix, a modified
Capon beamformer is used to estimate the DOAs, where thecomponent corresponding to the signal subspace is suppressed.
We analyze the computational complexity of our method withMUSIC method and illustrate the low complexity of our mehtod.
The proposed method estimates the direction-of-arrivals (DOAs)of coherent narrowband signals in an arbitrary linear arraywith high accuracy and low complexity. The effectiveness of the
proposed method is verified through numerical examples.

I. I NTRODUCTION
In sensor array processing, many DOA estimation methods
were proposed in the literature (see, e.g., [1], [2], [3], [4]and references therein), where the subspace-based methods
and the beamforming techniques are widely used because of
their simplicity. The beamforming is one of the oldest ideas
for DOA estimation, and perhaps the most well-known one
is the Capon beamformer [5], where the DOAs are estimatedby maintaining a constant gain at the incident directions ofdesired signals and minimizing the power contributed by noise
and interference signals. The properties of Capon beamformer
was studied in [6], [7], and the Capon method can be applied
to the array with arbitrary geometrical configurations and has
low computational complexity compared with the subspacebased DOA estimation methods such as the MUSIC [8], where
the computationally tremendous eigendecomposition is not
required. Further, it was clarified that the resolution of theCapon beamformer can be equal to that of MUSIC when the
SNR tends to infinite [9], whereas the performance of Capon
beamformer degrades significantly when the signal to noise
ratio (SNR) is low. Moreover, the array is usually subject togreat uncertainty in practice.
In this paper, we consider the DOA estimation of the coherent narrowband signals impinging on an arbitrary linear array.
In [10], we proposed a new computational efficient modified
Capon beamforming method and in this paper, we analyze
the computational complexity of our mehtod and compared itwith MUSIC method1 . The proposed method overcomes thecommon restriction of ULA geometry and becomes suitable
for more general array geometry than ordinary methods, while
the resolution of DOA estimation is improved to be as goodas that of the subspace-based methods such as the MUSIC,
where the computationally burdensome eigendecomposition is
avoided. Finally the simulation results show that the proposed
method performs well at low SNR.
II. P ROBLEM FORMULATION
We consider an arbitrary linear array composed of M identical and omnidirectional sensors and suppose that p coherent
narrowband signals {si (n)}pi=1 with the center frequency
f0 impinging on the array from far-field along the distinct
directions {θi }pi=1 . The received noisy array data at the nthsnapshot 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 signals, the incident signals, and additive noise respectively given by y(n) , [y1 (n), y2 (n), · · · , yM (n)]T , s(n) ,[s1 (n), s2 (n), · · · , sp (n)]T , and w(n) , [w1 (n), w2 (n), · · · ,wM (n)]T , while A(θ) is the array response matrix given byA(θ) , [a(θ1 ), a(θ2 ), · · · , a(θp )], and (·)T denotes transpose.
Four assumptions need to be made. Firstly, given a setof distinct DOAs {θ1 , θ2 , · · · .θp }, the array response vectors {a(θ1 ), a(θ2 ), · · · , a(θP )} are linearly independent. Secondly, the incident signals {si (n)} are all coherent, which
can be expressed as complex multiples of a common signal s1 (n) as si (n) = βi s1 (n), where βi is a complex
attenuation coefficient of ith signal and the signal s1 (n) is
temporally complex Gaussian random process with zero-mean.
The variance of signal s1 (n) is given by E {s1 (n)s∗1 (t)} =rs δn,t , E {s1 (n)s1 (t)} = 0, ∀n, t where E{·}, (·)∗ , and δn,tdenote the expectation, the complex conjugate, and theKronecker delta respectively. Thirdly, the additive noises{wi (n)} are temporally and spatially complex white Gaussian random process with zero-mean and variance σ 2 , whereE{w(n)wH (t)} = σ 2 I M δn,t , E{w(n)wT (t)} = O M ×M ,-8-

1 The paper includes the contents presented in [10].

while I m , O m×q , and (·)H indicate the m × m identity
matrix, the m × q null matrix, and the Hermitian transpose.
Additionally the additive noises are uncorrelated with theincident signals; Lastly, the number of incident signals p is
known and it satisfies the relation M ≥ 1.5p.
This paper concentrates on estimating the DOAs {θi }pi=1 ofcoherent signals impinging on an arbitrary linear array fromthe noisy array data {y(n)}N
n=1 in a computationally efficientway.
III. P ROPOSED M ETHOD
A. Interpolation Transformation
We design virtual array as uniformly spaced linear array tomake sure the effective implementation of spatial smoothing
preprocessing and polynomial rooting. Firstly we divide the array scanning area into K sectors and implement interpolationtransformation on each sector separately, where the kth sector(1) (2)is defined by the interval [θk , θk ], and we define a set ofangles Θk for each sector with a fixed interval ∆θ [11], [12],then we can get the response matrix Ak of actual array andĀk of virtual ULA associated with the set Θk respectively.
There exists a constant matrix B k which satisfies thatB k Ak = Āk(2)

where the interpolated matrix B k can be estimated as theleast squares solution of (2), and the interpolation error isdefined as ε , (kĀk − B k Ak k)/(kĀk k). According toexperience, we hope that the difference between the positionof the real array and that of the virtual array is as small as
possible [12]. By increasing the number of divided sectors, theinterpolation error decreases, while the computational burden
becomes heavier. Typically we choose the minimum in theset of K, which meet the condition that ε < 10−3 as thenumber of sectors [12]. Then compute B 1 , B 2 , · · · B k in eachsector separately. For notational convenience, hereafter we use
B to represent B 1 , B 2 , · · · B k . From (2), we easily get thecovariance matrix R̄ of the virtual ULA as
H

R̄ = BRB H = ĀRs Ā + σ 2 BB H

(3)

Here the white noise is converted into colored noise, so a
prewhitening is necessary. By partitioning the convariancematrix R as
 p M −p 
R = R11 R12 p
R21 R22 M −p
we get the estimate of noise variance σ 2 as [14]tr{R22 Π}
σ2 =
tr{Π}

(4)

signals imping on the virtual array. Hence the given arbitraryarray is converted into virtual ULA successfully.
B. FBSS Preprocessing
The coherence between the incident signals will lead to therank deficit of covariance matrix R̃ in (6) So we use spatialsmoothing of subarrays to decorrelate incident signals here.
We divide the virtual uniform linear array into L overlapping
subarrays with M0 (M0 ≥ p + 1) sensors in the forward andbackward directions, where L = M − M0 + 1, and sensors{l, l + 1, · · · , l + M0 − 1} or {M − l + 1, M − l, · · · , L − l + 1}
compose the lth forward subarray or backward subarray respectively, where l = 1, 2, · · · , L. From (6), the covariancematrixs of the lth forward subarray and the lth backward
subarray are respectively given by:f2R̃l = A1 D l−1 Rs D −(l−1) AH
1 + σ I M0

(7)2R̃l = A1 D −(M0 +l−2) R∗s D M0 +l−2 AH
1 + σ I M0

(8)

and
b

where A1 is the first M0 rows of response matrix Ā and D
denotes the p × p diagonal matrixD , diag e−j(2πf0 d/c) sin(θ1 ) , · · · , e−j(2πf0 d/c) sin(θp )(9)then we easily obtain the forward spatially smoothed covariance matrix and the backward spatially smoothed covariance
L
L
P
P
b
f
b
f
R̃l and R̃ = L1R̃l .
matrix as R̃ = L1
l=1

l=1

In order to minimize the loss of effective aperture, we
can obtain a interpolated spatially forward-backward smoothed
covariance matrix R̃FB as1 fb
R̃FB =
R̃ + R̃(10)
2
It has been proved that spatially smoothed covariance matrix
R̃FB is nonsingular, when number of subarrays is greater thanor equal to an half of the number of signals, i.e., L ≥ 0.5p
[13].
C. Modified Capon Beamformer
Based on the spatially smoothed covariance matrix R̃FB
in (10), the standard Capon beamformer (SCB) for the DOA
estimation of the coherent signals is given bys.t.
w̄H a(θ) = 1
min w̄H R̃FB w̄
w̄
where the weight vector w̄ is obtained by

(11)

−1

w̄SCB =R̃FB ā(θ)

(6)

(12)
−1
ā(θ)H R̃FB ā(θ)
Then the DOAs of incident signals can be estimated fromthe peaks of spatial spectrum PSCB (θ) or the valleys of costfunction fSCB (θ)
1
1
=
(13)
PSCB (θ) ,−1f
SCB (θ)aH (θ)R̃ a(θ)

By using received data from the real array, we obtain thecovariance matrix R̃ of the virtual array when same incident

By replacing R̃FB in the SCB constraint function with R̃FB ,where m is a positive integer (i.e., m ≥ 1), we can obtain a(5)−1
where Π = I M −p − R21 R†21 , and R†21 = (RH
21 R21 )H·R21 . Clearly by substituting (5) into (3), the ‘prewhitened’
covariance matrix of the virtual array R̃ is given by:
HR̃ , ĀRs Ā + σ 2 I M = R̄ − σ 2 BB H + σ 2 I M

FB

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m

101

MCB to design the optimal weight vector w̄ by solving thefollowing problem

I--FBSS--root--SCBI--FBSS--root--MCB(m=2)
I--FBSS--root--MCB (m=4)
I--FBSS--root--MUSIC
CRB

subject to w̄H a(θ) = 1.
(14)w̄Similarly the solution to the weight vector and the MCB spatial
spectrum are given by−mw̄MCB =R̃FB a(θ)

(15)

−m

aH (θ)R̃FB a(θ)
1
PMCB (θ) =fMCB (θ)
where fMCB (θ) is the MCB cost function defined by−m

fMCB (θ) , aH (θ)R̃FB a(θ).

RMSE (deg)mmin w̄H R̃FB w̄

Apparently the DOAs {θk }pk=1 can be estimated by maximizing the spectrum PMCB (θ) in (16) without the procedure of
eigendecomposition. Noted that the proposed MCB in (16)reduces to the SCB in (13) when m = 1.
Benefiting from the Vandermonde structure of the response
matrix Ā of the virtual array, we use a polynomial rooting
to replace the peaks searching here. By defining a complex
variable z as z = ejω , where ω = 2πf0 d sin θ/c, from (17), we
get the z format cost function and the corresponding relation
between DOA and complex variable z as:
−m

10-1

(16)

10-2
-5

(17)

fMCB (θ) = froot−MCB (z) = āH (z)R̃FB ā(z)

100

(18)

arg(zi )c(19)
2πf0 dwhere ā(θ) = ā(z) = [1, z, · · · , z M −1 ]. Finally, we obtainthe DOAs of incident signals by finding the minimum valuesof froot−MCB (z).
sin θi =IV. C OMPLEXITY A NALYSISThe computational complexity of Capon method is
FCapon = 10N M 2 + Finv (M ) + (8M (M + 1) + 1) F (r),where F (r) is a constant which is only decided by the
angular resolution r, and Finv (M ) represents the complexity of inversion operation of matrix with M dimensions. While, the computation complexity of MUSIC method
is FM U SIC = 10N M 2 + Fevd (M ) + FM DL (M ) +(8 (M − N ) (M + 1) + 1) F (r), where Fevd (M ) represents
the complexity of eigendecomposition operation of matrix
with M dimensions, and FM DL (M ) is the MDL criterion
which has an approximate complexity of O M 2 . Due to thehigh complexity of matrix eigendecomposition and matrix inversion, the complexity of Capon method and MUSIC method
actually is decided by the calculation of eigendecomposition
and inversion of covariance matrix respectively. At the sametime, interpolated transformation has nearly the same influnce
on Interpolated MUSIC method and our proposed method.
If we set M equals to 10, then the average computation of
eigendecomposition operation is 129215 flops, while that ofinversion operation is 10736 flops. We can see the complexityof eigendecomposition is over 10 times than that of inversion,and their difference become even bigger with the increase of
M.

0

5

10

15

20

25

SNR (dB)
Fig. 1. Estimation performance versus SNR

V. N UMERICAL R ESULTS
In this section, we evaluate effectiveness of the proposed
method (I-FBSS-root-MCB) for DOA estimation of coherent
narrowband signals by using an arbitrary linear array, whichis constructed by adding a horizontal shift ∆di to the ith(i = 1, 2, · · · M ) element of a virtual ULA consisting of
M = 9 sensors with element spacing d = λ/2. Here, we
set the horizontal shift vector as∆d = [0, 0, 0.1λ, −0.1λ, 0.05λ,−0.05λ, 0.1λ, −0.05λ, −0.05λ]T .

(20)

Two coherent signals with equal power arrive from angles
θ1 =10o and θ2 = 25o . The number of snapshots is set as
N = 500. The RMSEs of estimates of the DOAs versus SNRs
are shown in Fig. 1.
For comparison, the behavior of the FBSS-based standard
Capon beamformer with interpolation (I-FBSS-root-SCB), the
FBSS-based root-MUSIC with interpolation (I-FBSS-rootMUSIC) [11], and the Cramer-Rao lower bound (CRB) [15]are also carried out. The results shown in Fig. 1 are allbased on 1000 independent trials. The empirical root-meansquare errors (RMSE) in the simulations is calculated by
2
T P
P 
P
(t)
, where T denotes the total
θi − θ̂i
RMSE = T1
t=1 i=1(t)number of trials and θ̂i is the estimate of θi obtained in
the tth trial. In addition, we choose m = {2, 4} for theexperiments.
From the curves in Fig. 1, we can see that all methods
estimate the DOAs of the coherent signals relatively accurately
and the RMSEs decrease significantly with the increase of
SNR, benefiting from the utilizing of interpolation transformand FBSS preprocessing. Meanwhile, the proposed methodcompletely outperforms the I-FBSS-root-SCB method, and its
performance becomes better with the increase of power m.
When m reaches 4, the resolution of proposed method is very
close to the I-FBSS-root-MUSIC method, which is known for
its precision.

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VI. C ONCLUSION
In this paper, we propose an array interpolation and spatialsmoothing based modified Capon beamforming method for
DOA estimation of coherent signals with arbitrary linear
array. The simulation results show that the proposed method
performs as well as the subspace-based method, where thecomputationally burdensome eigendecomposition is avoided.
Meanwhile, the proposed method overcomes the common
restriction of ULA geometry and becomes suitable for moregeneral array geometry than ordinary methods.
ACKNOWLEDGMENT
This work was supported in part by the National Natural
Science Foundation of China under Grant 61671373.
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