Electrical Engineering and Systems Science authors titles recent submissions

Enhanced Array Aperture using Higher Order Statistics for DoA Estimation
Payal Gupta
Centre of Applied Research in Electronics, Indian Institute of Technology, New Delhi 110016, India

Monika Agrawal

arXiv:1711.05923v1 [eess.SP] 16 Nov 2017

Centre of Applied Research in Electronics, Indian Institute of Technology, New Delhi 110016, India

Abstract
Recently, the higher order statistics (HOS) and sparsity based array are most talked about techniques to estimate the
Direction of Arrival (DoA). They not only provide enhanced Degree of Freedom (DoF) to handle underdetermined
cases but also improve the estimation accuracy of the system. To achieve high accuracy and more number of DoF
with limited number of sensors, here we have proposed a method based on the fourth order statistics. The aperture
of virtual array becomes O(16N 4 ) using N physical sensors. Proposed method can be extended to the HOS which
increases the DoF by many folds. Numeric simulation validates these claims that the proposed method increases the
resolution capacity as well as maximize the DoF among all the earlier proposed method.
Keywords: DoA esimation, Array signal processing, Sparse array, Higher order statistics.

1. Introduction

DoA estimation is a classical research problem has
been actively researched since many decades [1]. It has
wide range of applications in the field of communications, radar, radio astronomy, sonar, navigation, tracking of various objects, rescue and other emergency assistance devices etc. [1]-[3]. Traditionally, subspace
based method such as MuSiC, ESPRIT, Root MuSiC
etc. based on the second order statistics, can resolve
only N − 1 sources using N sensors [1]-[3]. To identify
more number of sources than sensors, specialized array
structure i.e. minimum redundancy array (MRA) [10],
minimum hole array (MHA) [11], nested arrays [12],
coprime arrays [13], CaDiS [14] etc. have been suggested. Cross correlation lags are being used to provide
an enhanced virtual array aperture. Total DoF provided
by these structures depend upon the available continuous correlation lags. MRA and MHA [10]-[11] do not
have any closed form expression for array geometry, it
is being designed by searching the best sensor placement for the given number of correlation lags. Other
sparse structure follow some design rules and provide

Email addresses: payal@care.iitd.ac.in (Payal Gupta),
maggarwal@care.iitd.ac.in (Monika Agrawal)
Preprint submitted to Digital signal processing

some relation between number of sensors and number
of continuous lags are collected [19][21]. All these estimated continuous correlation lags are arranged in one
vector, aliasing single snapshot observation of enhanced
virtual array.
Various methods using spatial smoothing [12]-[13],
compressive sensing [14]-[16], vandermonde decomposition [20], covariance augmentation [17]-[18], [29][30] etc. have been used to estimate the DoA. Larger the
number of continuous correlation lags, more is the number of sources that can be discriminated. Many efforts
are being made to design arrays having large contiguous
correlation lags.
Further HOS can also be used not only to enhance
DoF but also to provide better immunity to noise [5]-[6].
A 2qth level nested array structure is demonstrated [12]
which resolve O(N 2q ) sources using N sensors. Nesting
subarrays based structure [24]-[25] have been designed
to increase the array aperture using HOS by many folds.
In this paper we will show that for real valued signals
the number of continuous correlation lags for any given
array structure can be increased by many folds. These
continuous correlation lags increase the available DoF
and hence increases the number of sources which can be

separated for given number of sensors. We have demonstrated the concept with fourth order statistics (FOS) but
August 29, 2017

3. Proposed Algorithm

similarly the results can be extended for HOS to provide
much more number of correlation lags. This conceptualized idea has been authenticate by rigorous numerical
simulations.
This paper is organized as follow. In section II, we
formulate the problem. In section III, we briefly review
of the proposed algorithm. In section IV, discuss the
example to understand the proposed algorithm. In section V, we present the numerical simulation results to
prove the proposed algorithm by examining the MSE
performance with SNR and the number of sources. The
conclusions are in the section VI.

Let us define a vector z(k) ∈ C2N×1 by concatenating
the array output with its conjugate, i.e.
zi (k) = yi (k)

1≤i≤N

zN+i (k) = y∗i (k)

(4)

1≤i≤N

(5)

Substitute the yi (k) from (1) into (5),
D
X

zN+i (k) =

ai (θl )sl (k) + vi (k)

l=1

!∗

i ∈ {1......N} (6)

Using eq (2)
2. Problem Formulation

zN+i (k) =

D
X

(e

jπ(−xi )sin(θl )

)s∗l (k) + v∗i (k)

l=1

Let us assume that an array of N sensors, receives
D narrow band, uncorrelated far field sources from the
direction θ1 , ....., θD w.r.t the normal of the array. The
array output at the nth sensor and at the kth snapshot
can be expressed as
yn (k) =

D
X

an (θl )sl (k)+vn (k)

eq. (7) can be written as
zN+i (k) =

1 ≤ k ≤ K, 0 ≤ n ≤ N−1

D
X

(e

jπ(−xi )sin(θl )

)sl (k) +

!

v∗i (k)

l=1

a(θl )sl (k) + v(k) = As(k) + v(k)

i ∈ {1......N}

(8)
Comparing the (8) with (1), zN+i (k), N + 1 ≤ i ≤
2N can be seen as signal observed at virtual sensor present at (−xi ) location, which is physically not
present. Hence, the complete vector z(k) can be seen

as the signal observed at an array of 2N sensors located
at {−xN , ...., −x1 , x1 , ......, xN }. This array of 2N sensors,
consisting of N physical and N virtual is termed as semi
virtual array and defined as,
Definition 1(Semi Virtual array). Let S be a set of N
sensors locations whose elements denotes as (x1 , ...., xN )
. Then a semi virtual array is formed by placing another
N virtual sensors at [−x1 , −x2 . . . ] i,e Sv is given as,

(1)
(2)

xn denotes the location of nth sensor in term of λ/2, λ is
the wavelength of the received signal, K is the number
of snapshots, sl (k) ∈ R denotes the kth snapshot of lth
source signal. Sources assumed to be real. vn (k) denotes
Gaussian random noise at nth sensor, which is uncorrelated with the source sl (k) ∀ l.


Let y(k) = y1 (k), ....., yN (k) T , where superscript (.)T

denotes matrix transpose operation. Using a(θl ) =
[1, a1 (θl ), ......., aN (θl )]T , y(k) can be written as,
y(k) =

(7)
s∗l (k) = sl (k)

where,

D
X

i ∈ {1......N}

As signal are assumed real, i.e,

l=1

an (θl ) = e jπxn sin(θl )

!

Sv = [w1 , w2 , ...., w2N−1 , w2N ]T = [−xN , .., −x1 , x1 , ..., xN ]T2N×1
(9)
Define,
ãn (θl ) = e jπwn sin(θl )
(10)

∈ C|N|

l=1

(3)

z(k) can be written as mathematically
where, A = [a(θ1 )a(θ2 ), ......., a(θD )] is the N × D array
manifold matrix,
s(k) = [s1 (k), s2 (k), ......., sD (k)]T is the source signal
vector. and
v(k) = [v1 (k), v2 (k), ......., vN (k)]T is the noise vector corresponding to the kth snapshot. Our aim is to estimate

the DoA from the observed signal using the output of
the array.

zn (k) =

D
X

ãn (θl )sl (k) + ṽn (k)

n ∈ {1......2N}

(11)

ã(θl )sl (k) + ṽ(k) = Ã(θ)s(k) + ṽ(k)

(12)

l=1

In vector form
z(k) =

D
X
l=1

2

where, ã(θl ) ∈ C2N×1 denotes the semi virtual steering vector and à = [ã(θ1 ), ......, ã(θD )]2N×D is termed as
semi virtual array manifold matrix, ṽ(k) = [v(k) v∗ (k)].
It can be easily seen that with this virtual array manifold resolvability of the array becomes almost double.
Thereby HOS can further enhance the DoF increasing
the number of resolvable sources.
The fourth order cumulant of z(k) is defined as

the above equation can write as
c p,q,r,m = cum

D
X

ã p (θl1 )sl1 (k),

l1 =1

D
X

D
X

ãq (θl2 )sl2 (k),

l2 =1

ãr (θl3 )sl3 (k),

D
X
l4 =1

l3 =1

!
ãm (θl4 )sl4 (k) +

cum(ṽ p (k), ṽq (k), ṽ(r k), ṽm (k))

p, q, r, m ∈ {1......2N}
!
(16)
The second term of (16) will be vanish due to fourth
c p,q,r,m , cum z p (k), zq (k), zr (k), zm (k) =
order cumulant of Gaussian noise is zero. Assuming
h
i
h
i
E z p (k) zq (k) zr (k) zm (k) − E z p (k)zq (k) E [zr (k)zm (k)] sources are uncorrelated, it follows that
h
i h
i
h
i h
i
−E z p (k)zr (k) E zq (k)zm (k) − E z p (k)zm (k) E zq (k)zr (k)
cum[sl1 (k), sl2 (k), sl3 (k), sl4 (k)] =
(
γl δ(l1 , l2 , l3 , l4 ) l1 = l2 = l3 = l4 = l
1 ≤ p, q, r, m ≤ 2N
(13)
0
otherwise
(17)
4
(
Define cvec a C16N ×1 as,
1 l1 = l2 = l3 = l4
δ(l1 , l2 , l3 , l4 ) =
0
otherwise
h
cvec = c1,1,1,1 ...c1,1,1,2 c1,1,2,1 ....c1,1,2,2N c1,1,2N,1 ....c1,1,2N,2N
where, γl = cum(sl (k), sl (k), sl (k), sl (k)) denotes the
c1,2,2N,1 ....c1,2,2N,2N c1,2N,2N,1 ....c1,2N,2N,2N
fourth order cumulant of the lth source.
i
Using (7), (16) becomes
c2,2N,2N,1 ....c2,2N,2N,2N c2N,2N,2N,1 ....c2N,2N,2N,2N
(14)
!
D
X
Theorem 1: For the uncorrelated sources the vectorã p (θl )ãq (θl )ãr (θl )ãm (θl ) γl
c p,q,r,m =
(18)
ized model of (14) can be expressed as
l=1
p, q, r, m ∈ {1......2N}
cvec = B(θ)p

(15)
The vectorization form of (14) can be written as
16N 4 ×D

where, B(θ) = [b(θ1 ).....b(θD )] ∈ C
,p∈C
represent the 4th order cumulant of the D sources.

D×1

cvec =

D
X

γl b(θl )

(19)

l=1

Proof : From the (13)

where,

c p,q,r,m = cum z p (k), zq (k), zr (k), zm (k)

!

b(θl ) = ã(θl ) ⊗ ã(θl ) ⊗ ã(θl ) ⊗ ã(θl )

1 ≤ p, q, r, m ≤ 2N

(20)

Hence
cvec = B(θ)p

from (11), we get

c p,q,r,m = cum

D
X

ã p (θl1 )sl1 (k)+ṽ p (k)

D
 X

where,B(θ) , [b(θ1 ) b(θ2 )..... b(θD )] and p ∈ CD×1 =
[γ1 γ2 .... γD ] represents the vector of the 4th-order
cumulants (γl , 1 ≤ l ≤ D) of D sources.

ãq (θl2 )sl2 (k)+

l2 =1

l1 =1

••••
One element of vector cvec is given as,

D
D
!
 X
 X
ãm (θl4 )sl4 (k)+ṽm (k)
ãr (θl3 )sl3 (k)+ṽr (k)
ṽq (k)
l3 =1

l4 =1

=

D
X
l=1

e jπ(w p +wq +wr +wm )sin(θl ) γl

!

1 ≤ p, q, r, m ≤ 2N

(21)
which can be seen as the signal observed at a sensor location w p + wq + wr + wm . Hence total cvec corresponds

1 ≤ p, q, r, m ≤ 2N
Due to unorrelatedness between source and noise signal,
3

4. Example and Discussion

to the signal observed at an array whose sensors are located at w p + wq + wr + wm . This gives a notion virtual
array defined as,
Definition 2(Virtual Array) Consider a set Sv be a set
of 2N sensors at (w1 ....w2N ). The virtual Array L consists of sensors located at w p + wq + wr + wm , defined
as,

In this section we showed the enhancement of correlation lags achieved by proposed idea. Table I illustrates
the comparison of different existing structures with proposed method and existing methods. It can be easily
seen that with proposed methods the number of continuous lags becomes almost double. Further if we extend
this idea with HOS much more number of lags can be
obtained.
Let us take an example to elaborate the proposed
method. Consider the sparse array structure [23] with
N = 6 physical sensors, shown in fig. 1(a). In Fig.
1(b), shows the proposed semi virtual array using (12)
and a virtual array shown in fig. 1(c). It can be observe that [24] gives 89 unique lags in which 71 lags are
consecutive lags in range [−35 35] but proposed method
provides 176 unique lags, among them 127 lags are consecutive in range [−63 63]. Table II has enumerated the
no of continous lags achieved by HOS.
Fig. 2 shows the weight function corresponding to
proposed array. The importance of weight function is
to improve the estimate correlation leg, from the fig. 2
approximate all correlation lags frequency is more than
50 so the variance of estimated correlation lag is less.
The first hole pair in the coarray occurs at 64.

L = w p + wq + wr + wm , p, q, r, m ∈ {1, ...., 2N}
Here this virtual Array can have total 16N 4 number of
sensors further it should be noted that all the sensors are
not distinct because the combinations of w p , wq , wr , wm
may not be unique, therefore virtual Array will have
< 16N 4 distinct elements. This virtual array is symmetric about the origin i.e this virtual array has same
number of positive and negative elements. The aperture of the virtual Array is from 4w1 to 4w2N also
4w1 = −4w2N . Let M denotes the maximum continuous number in positive side of the virtual array and −M
to M continuous sensors in this virtual array, i.e. total
effective array aperture is 2M + 1. This 2M + 1 length
continuous length virtual array provides 2M + 1 DoF.
The frequency of the each entry of lag is being defined
as weight function. This help in studying the variance
of estimated lags.
Definition 3:(weight function) The weight function
f (n), counts the periodicity of n in L, defined as

(
)
 w p , wq , wr , wm ∈ Sv
Fn = (w p , wq , wr , wm ) 
w p + wq + wr + wm = n
f (n) = |Fn |

where, |k| is the cardinality of the set k.
Here Fn corresponds to the number of entries in cvec corresponding to the nth virtual sensor location. These entries corresponding to the same lags are combined to
improve the estimate correlation leg. For a good array
weight function should be rectangular [12]-[14] so that
each estimated lag has equal variance
The constructed single snapshot of observation is
used to estimate the DoA of the sources. Single snapshot will give ill conditioned covariance matrix therefore spatial smoothing based approach for the estimation of DoA has been suggested [12] to the use of subspace method. Compressive sensing[14]-[16], covariance matrix augmentation [17]-[18],[29]-[30], vandermonde decomposition [20] and Maximum likelihood
based approaches[26]-[27] etc. can also be used to estimate the DoA from single snapshots. The details of
each of these algorithm are well documented can be
taken from references [12]-[30] and are not discussed
her to avoid repeatability.

Figure 1: (a).NULA array with N=6 sensors (b) Proposed semi virtual
NULA array (c) Proposed virtual ULA

5. Simulation results
In this section, a series of numeric simulation results
are presented to analyze the performance of the proposed approach. We have compared the performance
of proposed method with other three reference methods, i.e. Four Level nested array (FLNA)[23], Three
Level nested array(3LNA)[24] and Two level fourth order Nested array (2L-FONA) [25]. We first compare the
MuSiC spectrum between the FLNA [23] and the proposed method.
4

Table 1: COMPARISON OF NUMBER of LAGS ACHIVED BY DIFFERENT SPARSE ARRAY STRUCTURE AND WITH PROPOSED
METHOD

Comparison of number of lags achieved by different sparse array structure and with proposed method
Total consecutive lags
ArrayStructure

Without Proposed method

With Proposed Method

N=4

21

41

N=6

71

127

N=7

109

169

N=4

23

59

N=6

73

145

N=7

125

141

Number of Sensors

FL-NA [23]

3L-NA[24]

2L- FONA[25]

N=4

49

75

N=6

113

175

N=7

169

259

Table 2: Comparison of number of lags achieved by HOS

Continuous Lags
Order

Array Structure

Number of sensor

FLNA

Proposed method

4th

N=6

71

127

6th

N=8

433

703

8th

N=8

673

1087

Afterwards we conducted Monte-Carlo experiments
to analyze the performance in terms of MSE in DOA estimates as function of input signal-to-noise ratio (SNR)
and the number of sources. The subspace based method
MuSiC and compressive sensing are used to estimate
the DoAs. Other methods [17]-[27] also give similar results but are not shown here to avoid repeatability. We
assumed signal sources power is same and sources are
modeled as real and random Gaussian processes.
5.1. Music Spectra
In this subsection, we compare the MuSiC spectrum
obtained from the FLNA and the proposed approach apply on FLNA. Two FLNA consisting of 6 sensors with
location S = {1 2 3 6 12 24} and 4 sensors with the location of S = {1 2 4 6} is consider in throughout the
simulations. The maximum continuous lags is obtained

L = {−63 − 62......0......62 63} obtained by FLNA with
proposed method for N = 6 sensors. D = 25 far-field
narrowband sources located uniformly between −60◦ to
60◦ impinge on the FLNA for input of SNR is 0 dB, and
the number of snapshots K = 20, 000, the MuSiC spectra are compared in fig. 3 and fig. 4. The MuSiC spectra
for FLNA depicted in fig. 3a and for proposed approach
in fig. 3b. It can be clearly seen that, FLNA has failed
to identify all the sources while proposed method are
approximate resolve all 25 sources with much better accuracy.
Fig. 4 also shows the MuSiC spectrum of FLNA
and proposed method for N = 4 physical sensors. The maximum continuous lags is obtained
L = {−20 − 19......0......19 20} obtained by proposed
method. All the 8 far-field narrowband sources from
the directions {−55 − 39 − 23 − 7 9 25 41 57}, are estimated at 0 dB input SNR and with 20, 000 snapshots. As we can see from the plotted spectra, proposed method MuSiC spectra obtained much sharped
peaks than earlier suggested algorithm. To understand
the behavior of the algorithm in much better form we
illustrate the performance of proposed methods through
Monte Carlo simulations.
5.2. MSE vs SNR
Now, we compared the DoA estimation performance
of different array configuration with proposed method
through Monte Carlo simulations.

5

600

500

weight function

400

300

200

100

X: -84
Y: 4
0
-100

-80

-60

-40

-20

0

20

40

60

80

100

x (element position in virtual array

Figure 2: Weight Function
0

0

-0.5

Normalized MuSiC Spectrum

Normalized MuSiC Spectrum

-0.5

-1

-1.5

-2

-1

-1.5

-2
-2.5

-3
-100

-80

-60

-40

-20

0

20

40

60

80

θ (degree)

100

-2.5
-100

-80

-60

-40

-20

0

20

40

60

80

100

θ (degree)

(a)

(b)

Figure 3: MuSiC Spectrum comparison between two approaches. There are 6 sensors and D=25 sources located uniformly between −60◦ to 60◦ .
The number of snapshots is K = 20,000 and signal-to-noise ratio (SNR) is 0 dB. (a) FLNA in [25]; (b) Proposed approach.

(a)

(b)

Figure 4: MuSiC Spectrum comparison between two approaches for Number of sensors 4 and Number of sources 8. The number of snapshots is K
= 20,000 and signal-to-noise ratio (SNR) is 0 dB. (a) FLNA in [25]; (b) Proposed approach.

We assumed the signal power (σ2 ) of sources is same

in simulation. The SNR is defined as
S NR = 10 log
6

σ2
σ
= 20 log
σn
σ2n

where, σ2n is the variance of noise. The average mean
square error(MSE) of the estimated DoAs, given as
MS E =

60◦ with the separation {91, 54, 33, 26, 22, 20, 17, 15} respectively. Fig. 8 illustrates the effect of number of
sources of FLNA, 2L-FONA and 3LNA array structure
with our proposed algorithm at 20 dB SNR and MSE
increases with sources increases. It can be clearly seen
that the proposed algorithm has less error compared to
the existing methods [23]-[25].

I
D
1 XX ˆ
(θq (i) − θq )2
I i=1 q=1

where, θ̂ is the estimated DoAs for the ith Monte Carlo
trial, i = 1, .......I. Here, we use 100 Monte Carlo trial
for all simulations.
We consider D = 3 sources with the angles
{−60, −5, 50} and D = 8 sources with the angles
{−60, −43, −26, −9, 8, 25, 42, 59, } and compare with the
three array configuration for N = 6 physical sensors with the location of {1 2 3 6 12 24} for FLNA,
{0 1 2 3 21 28} for 2L-FONA and {1 2 3 4 8 30} for
3LNA. The DOA is estimated using the subspace based
MuSiC algorithm. Fig. 8 shows the average MSE of all
DoAs for individual array configuration as a function of
SNR, for 2000 snapshots. The corresponding plots for
D = 3 and D = 8 are given by fig. 5 and fig. 6 for N = 6
sensors. Clearly, all array configuration with proposed
method gives much better MSE. There is huge performance gain of almost 15 − 20 dB.
Figure 8 shows the mean square error of estimating DoA by using compressive sensing method LASSO
with the 0.25 penalty parameter.

0.8
proposed on FLNA
FLNA
2L FONA
proposed on 2L FONA
3L NA
prop on 3L NA

0.6

MSE(dB)(in Degrees)

0.4

0.2

0

-0.2

-0.4

-0.6
-15

-10

-5

0

5

10

15

SNR(dB)

Figure 6: Mean square error (MSE) by MuSiC vs. SNR for 100
Monte Carlo experiments with N = 6 physical sensors with the location of {1 2 3 6 12 24} for FLNA, {0 1 2 3 21 28} for 2L-FONA
and {1 2 3 4 8 30} for 3LNA and D = 8 sources from the direction {−60, −43, −26, −9, 8, 25, 42, 59, }. The number of snapshots is
K = 2000.

0.6
proposed on FLNA
FLNA
2L FONA
proposed on 2L FONA
3L NA
prop on 3L NA

0.4

MSE(dB)(in Degrees)

0.2

0

-0.2

-0.4

-0.6

-0.8
-20

-15

-10

-5

0

5

10

15

20

SNR(dB)

Figure 5: Mean Square Error (MSE) vs. SNR for 100 Monte
Carlo experiments with N = 6 physical sensors with the location of {1 2 3 6 12 24} for FLNA, {0 1 2 3 21 28} for 2L-FONA and
{1 2 3 4 8 30} for 3LNA and D = 3 sources from the direction
{−60, −5, 50} using MuSic algorithm. The number of snapshots is
K = 2000.

Figure 7: Mean square error (MSE) by Compressive sensing vs.
SNR for 1000 Monte Carlo experiments with 6sensors with the
location of {1 2 3 6 12 24} and D = 8 sourcesfrom the direction
{−60, −43, −26, −9, 8, 25, 42, 59, }. The number of snapshots is K =
5000.

5.3. MSE versus Number of sources
In this section we study the impact of variation of
number of sources. Consider N = 4 physical sensors
with the location {1 2 4 6} for FLNA, {0 1 9 12} for 2LFONA and {1 2 3 10} for 3LNA at 2000 snapshots. We
vary sources from D = 2 to D = 9 in the range −60◦ :

6. Conclusion
In this paper we have exploited the property of real
signals to show the separability of the give array struc7

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0.8

0.7

0.6

MSE(dB)(in Degrees)

0.5

0.4

0.3

0.2

0.1
FLNA
proposed on FLNA
2L FONA
proposed on 2L FONA
3L NA
prop on 3L NA

0

-0.1

-0.2
2

3

4

5

6

7

8

9

Number of sources

Figure 8: Mean square error (MSE) by MuSiC vs. Number of sources
for 100 Monte Carlo experiments with 4 sensors at 20 db SNR. The
number of snapshots is K = 2000.

ture can be doubled. It not only increases the DoF
thereby increases the number of signals for which DoA
can be estimated but also decrease the MSE in the estimation. We have shown the concept using forth order
statistics. But much more improvement can be obtained
using further higher order statistics. Numeric simulation proves the efficacy of the proposed algorithm.
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[30] S. Pillai, Y. Bar-Ness, and F. Haber, ”A new approach to array
geometry for improved spatial spectrum estimation,” Proceedings
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Payal Gupta was born in Delhi, India on March
18,1989. She received the B.Tech degree in Electronics and communication engineering from Uttar Pradesh
Technical University, Lucknow, in 2010, and the M.tech
degree in signal processing from Delhi University, India, in 2014.
She is currently pursuing the Ph.D degree in the field
of signal processing at Indian Institute of Technology,
Delhi, India. Her research interests are in sensor array
signal processing, direction of arrival and sparse array
design.
Monika Agrawal can be found at url:
http://care.iitd.ac.in/People/Faculty/
M_Aggarwal.htm

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