IEEE Workshop on Nonlinear Circuit Networks
December 14-15, 2012

A Simple Method for Calibrating Uniform Linear
Array
Xiang Cao #∗1 , Jingmin Xin # , Yoshifumi Nishio ∗2 , Nanning Zheng # , Akira Sano †3#

Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University
Xi’an 710049, China
1∗cao.xiang@stu.xjtu.edu.cn

Department of Electrical and Electronic Engineering, The University of Tokushima
Tokushima 770-8506, Japan2†nishio@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, a simple auto-calibration method
is presented for estimating spatial signature with uncalibrated
uniform linear array. The proposed method is an extension of
Esprit-like method and also extracts two subarrays to construct
an estimator. The difference is that one element of array canbe used several times in one subarray. A closed-form solution is
obtained. Simulation results indicate that our algorithm performs
better than Esprit-like. Furthermore, the simple algorithm could
be used to initialize the multidimensional search method in [1].

I. I NTRODUCTION
In sensor array signal processing, many high-resolution
methods such as ESPRIT [2] have been advanced for estimation of unknown parameters embedded in an array output
model. It is a natural motivation of array calibration, since theperformance of these methods will degrade dramatically when
the ideal array model is damaged by some unknown errors(e.g., mutual coupling and sensor position uncertainties).
Two parametric errors (i.e., statistical and deterministic)
were usually used to extend the ideal array model. This paper
only focus on the deterministic unknown gain and phase errors.
The term deterministic here implies that the unknown error ateach of elements of array is a stable constant during the periodof observation.
Several so-called autocalibration approaches have been developed in literature. Here, the autocalibration indicates thatarray calibration may be accomplished without employing
any dummy elements or transmitters at known direction. In[3], an iterative eigenstructure-based technique of estimating
Direction of Arrival (DOA) in the presence of unknown gain
and phase errors is presented and it can apply to arbitraryarray geometries except Uniform Linear Array (ULA). For
ULA, a phase ambiguity exists between the diagonal error
matrix and the ideal array steering matrix (see [4] and [5]).
Thus, it is impossible to estimate DOA and gain and phaseerrors simultaneously for ULA from array output. A Hermitian
Toeplitz structure of ULA output covariance matrix in theabsence of gain and phase errors is exploited in [6] and [7].
This method takes advantage of the elements equivalence atevery diagonal line to form two equations for estimation ofgain and phase errors. The main drawback of the approach isthat the Toeplitz matrix assumption is only established under
infinite sampling condition. It means that if the number ofsnapshots is fixed, the performance of the algorithm does not
improve after increasing the signal-to-noise ratio (SNR) to acertain point. In addition, the ESPRIT method also can beextended to the case. The idea firstly emerged from [8], andis applied for estimating spatial signature in [9] and [5]. The
Esprit-like method (see [5]) constructs an estimator based on
the maximum overlapping two subarrays. However, it has been
pointed out in [10] that any two subarrays configuration may
be inherently suboptimal. In [1], we proposed a multidimensional search procedure to estimate spatial signature. Unlike
Esprit-like method, the algorithm fully exploits the multiple
invariances inherent ULA even though the array model errors
exist.

In this paper, we consider the spatial signature estimation
problem with ULA in the presence of unknown gain and phaseerrors. The practicality of spatial signature can be found in[9] and [5]. The sensor gain and phase errors are different
from mutual coupling among elements of array and it can not
be affected by other sensors. Furthermore, the ideal steeringmatrix of ULA has a Vandermonde structure. Both of themprovide the possibility of selecting subarrays from uncalibrated
array like ESPRIT. In practice, A simple eigenstructure-basedalgorithm is presented. Two subarrays are firstly extracted
from ULA to construct an estimator. The difference with
Esprit-like method is that one element of ULA can be used
several times in one subarray. Our method firstly estimates therotational DOAs (not absolute DOAs) and error parameters,then obtains the estimation of spatial signature.

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II. DATA MODEL
A uniform linear array with M sensors received narrowband
signals from p far-field sources and the vector response y ∈C M ×1 of the array at time t can be expressed asy(t) = Γ(γ)A(θ)s(t) + n(t),(1)

where s(t) ∈ C p×1 is the vector of incident signals at timet, n(t) ∈ C M ×1 [is the vector of additive noises,] the steeringmatrix A(θ) = a(θ1 ) a(θ2 ) · · · a(θp ) is the ideal
array response, and
]T
[
2π
2π
a(θ) = 1 ej λ d sin(θ) · · · ej λ (M −1)d sin(θ). (2)Here, operator (·)T stands for transpose, θ1 , θ2 , · · · , θp are thedirections-of-arrival of signals, d and λ represent the distance
between two consecutive sensors and the identical wavelength
for all signals, respectively. The diagonal matrix Γ(γ) is givenby[]Γ(γ) = diag γ1 γ1 · · · γM ,(3)and |γi | > 0 denotes the deterministic unknown gain andphase error of sensor i. The vector γ is constructed out ofthe diagonal elements of matrix Γ. For simplicity, Γ(γ) andΓ denote the same matrix.
Two assumptions need to be made. Firstly, signal s(t) is atemporally complex white Gaussian random vector with mean
zero and its covariance matrix Rss has full rank p (assuming
no correlate signals). Secondly, noise n(t) is a temporally andspatially complex white Gaussian random vector with mean
zero and uncorrelated with incident signals. Then, the so-calledsignal subspace E s and noise subspace E n can be easily
obtained from the eigenvalue decomposition of array output
covariance matrix
R = E{y(t)y H (t)} =M∑

λ i ei eH
i ,

(4)

i=1

where E{·} and (·)H denote the statistical expectation andthe complex conjugate transpose, respectively. The eigenvalues
are ordered such as λ1 ≥ λ2 ≥ · · · ≥ λp > λp+1 =· · · = λM . The corresponding signal subspace
E s and]noise
[
e
·
·
· ep
subspace
E
are
given
by
E
=
and
1
n
s
[
]
E n = ep+1 · · · eM .
The focus of this paper is the estimation of Spatial Signaturematrix H = Γ(γ)A(θ) (instead of absolute DOAs) fromthe N snapshots of the array output. One ambiguity for thisproblem may be observed between the unknown signal vectors(t) and H (i.e., Hs(t) = αH · ( α1 s(t)) for an unknown
non-zero scaling α). A reasonable constraint for solving thisscaling ambiguity is to let the first element of diagonal matrix
Γ(γ) be equal to one.

exploiting invariance for ULA even though the gain and phaseof the sensor have not been calibrated.
Since the one-to-one correspondence between rows in H
and elements of array, extracting a subarray from array isequivalent to picking up rows of matrix H. Define a selection
matrix J m×(M −ε) consisting of zeros and ones. Only one
element at every row is equal to one and it corresponds tothe selected sensor of the array. However, the number of one
each column in J m×(M −ε) indicates the number of repetitionsof selected element in subarray.
Then, the spatial signature matrix H can be partitioned into
two subarrays
[
]
Γx Ax
J ΓA =,(5)
Γy Ax Φε
where
J=

[

J1
J2

]

[
=

J m×(M −ε)
0m×ε

0m×ε
J m×(M −ε)

]
,

(6)

and εd denotes the distance between two subarrays. Γx , Γyand Ax may be calculated by the following equations:Γx = diag{J 1 γ}, Γx = diag{J 2 γ}, Ax = J 1 A.

(7)

The displacement diagonal matrix Φ is given by
[ 2π
]
2π
Φ = diag ej λ d sin(θ1 ) · · · ej λ d sin(θp ) .

(8)Here, we give an example of selection matrix. Consider a ULA
with six elements, a possible selection matrix J 9×5 may be
expressed as
1 0 0 0 0
 0 1 0 0 0 
 0 0 1 0 0  0 1 0 0 0 J 9×5 = (9)
 0 0 1 0 0 .
 0 0 0 1 0  0 0 1 0 0 
 0 0 0 1 0 
0 0 0 0 1
The determinant of the matrix Γ(γ) is given bydet(Γ(γ)) =M∏

γi ̸= 0,

(10)

i=1

then the rank of the matrix H is p, i.e., Rank(ΓA) = p.
It is clear that the matrix ΓA spans the same space as thesignal subspace E s . The relationship implies that there exists
a nonsingular p × p matrix T satisfying[]
Ex
J Es =(11)= J ΓAT .
Ey
Combining equation (5) and (11) leads toIII. ESTIMATION ALGORITHM

E x = Γx Ax T , E y = Γy Ax Φ̄T

A simple subspace approach is presented for estimating
spatial signature matrix H in this section. The Vandermondestructure of the ideal array steering matrix A permits optimallywhere diagonal matrix Φ̄ = Φε . Eliminating Ax yields to- 135 -

Γ̄(γ̄)E y = E x Ψ

(12)

(13)

where the m × m diagonal matrix Γ̄(γ̄) = Γ−1y Γx andΨ = T −1 Φ̄T . The vector γ̄ is constructed from the diagonal
elements of matrix
Similar to Esprit-like method in [5], Γ̄(γ̄) and Ψ can be
estimated from the least squares problem as follow

The solution for Ψ is give by()−1Ψ̂ = E H
EHx Exx Γ̄(γ̄)E y .

Esprit−like
Method2 initialized by method1
Method1
Method2 initialized by Esprit−like

0.7
0.6
0.5
RMSE

ˆ , Ψ̂ = arg min Γ̄(γ̄)E − E Ψ 2 .
Γ̄
y
x
F
¯
Γ
,Ψ

0.8

(14)

0.4
0.3
0.2
0.1

(15)

Substituting back to (14) and after some manipulation, theleast squares problem becomes
[
(
)T ]
H
⊥
H
ˆ
γ̄,
ΠE ⊙ E y E yγ̄ = arg min γ̄
(16)
x
γ̄
(
)−1
H
=
I
−
E
E
E
EH
where Π⊥xxx
x . From above equaExtion, an estimate of vector γ̄ may be obtained from theeigenvector associatedwiththe smallest eigenvalue of the()
THmatrix Π⊥. Here we note that constraint may
Ex ⊙ Ey Ey
be added to matrix Ψ and some discussions about this question
can be found in [5]. Consider the example of (9), the two
diagonal matrices Γx and Γy may be expressed asΓx = diag {1, γ 2 , γ 3 , γ 2 , γ 3 , γ 4 , γ 3 , γ 4 , γ 5 }(17)Γy = diag {γ 2 , γ 3 , γ 4 , γ 3 , γ 4 , γ 5 , γ 4 , γ 5 , γ 6 } .

(18)

and

Recall Γ̄(γ̄) = Γ−1y Γx , we can see that once an estimate ofvector γ̄ is obtained from (16), it is convenient to computerthe elements of vector γ. Furthermore, the matrix Ψ can beobtained by Substituting the matrix Γ̄(γ̄) reconstructed fromˆ back to (15).
γ̄The proposed algorithm now is briefly outlined below.
1) Estimate the signal subspace Ê s from the eigendecompositionof the sample covariance matrix R̂ =
∑N
1
H
y(n)y
(n), where N is the finite number ofn=1
N
snapshots.
2) Calculate E x and E y according to (11), then determine
an estimate of vector γ̄ from (16).
3) Calculate the matrix Ψ according to (15), then extract
the rotational DOA θ̂.
4) Construct matrix Γ(γ̂) and A(θ̂), then spatial signaturematrix is estimated as Ĥ = Γ(γ̂)A(θ̂).
IV. EXPERIMENTIn this section, computer simulation were conducted forevaluating the performance of the proposed algorithm. The
results of the methods in [5] and [1] are also included. In
this scenario, a ULA of 9 elements with half of wavelength
element spacing is used. In [1], the ULA is divided into 5
subarrays and the ith subarray is selected by matrix J̃ i =

0
−10

−8

−6

−4

−2

0
SNR(dB)

2

4

6

8

10

Fig. 1. RMSE of the spatial signature matrix H estimation versus SNR.
The number of snapshots N=600[05×(i−1) , I 5×5 , 05×(5−i) ], i = 1, · · · , 5. However, the two
subarrays in this paper are selected by matrices
[
]T
T
T
T
T
J 1 = J̃ 1 J̃ 2 J̃ 3 J̃ 4
(19)
and

[
J2 =

T

J̃ 2

T

J̃ 3

T

J̃ 4

T

J̃ 5

]T
.

(20)

The deterministic gain and phase errors are given by
◦
◦
◦
◦
◦
1, 1.10ej10 , 0.90e−j5 , 1.25ej20 , 0.80e−j9 , 0.96ej15 ,◦◦◦1.18e−j23 , 0.88e−j2 and 0.85ej4 . Two equal-power uncorrelated signals located at 25◦ and 45◦ are considered
and the SNR per element for each source is defined by
SNR = 10 log10 (σs2 /σn2 ), where σs2 and σn2 , respectively,
denote the power of incident signal and that of additive noiseat each sensor. RMSE is used as the performance measure anddefined by
K √
2
1 ∑
RMSE =(21)
Ĥ i − H i ,K i=1
F
where K is the number of trials, Ĥ i and H i are the estimatedspatial signature and the true one at the ith experiment. A totalof 500 trials were performed for each simulation scenario.
In Figure 1, method1 and method2 denote the proposed
algorithm in this paper and in [1], respectively. Note thatmethod1 could be used to initialize method2. The simulation
result shows that our method performs better than Esprit-likeeven at low SNR. It means that the maximum overlapping
subarray is not the optimal choice in the presence of unknown
sensor gain and phase response. In other words, fully exploiting invariance inherent array should be taken into account. In
addition, the behaviour of the iterative scheme (method2) has
been improved significantly when the initial parameter values
were provided by our method.
V. C ONCLUSION
A simple eigenstructure-based algorithm is presented for
spatial signature estimation for ULA with unknown gain/phaseerrors. Our method fully excavates the invariance of the ULA
instead of simply using maximum overlapping subarray in- 136 -

Esprit-like in [5] and give a closed-form solution. How to
select subarrays from ULA, making the algorithm lead to
optimal parameter estimation, is not addressed in this paper.
Some pragmatic discussion about this question can be foundin [10].
ACKNOWLEDGMENT
This work was supported in part by the National Nature
Science Foundation of China under Grant 61172162.
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