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Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom

Nested Arrays: A Novel Approach to ArrayProcessing with Enhanced Degrees of Freedom

Xiangfeng Wang

OSPAC

May 7, 2013


Reference

Reference

Pal Piya, and P. P. Vaidyanathan. Nested arrays: anovel approach to array processing with enhanced degrees offreedom. IEEE Transactions on Signal Processing.58.8(2010): 4167-4181.


Outline

Outline

Motivation and Goal

Co-array Perspective

Nested Array: Optimization

DOA Estimation

Numerical Examples

Conclusion


Motivation and Goal

Motivation and Goal

Antenna arrays

Direction-of-arrival (DOA) estimation

Uniform linear arrays (ULA)

Degree of freedom (DOF) is N − 1 for ULA

Goal: nested arrrays

obtain O(N2) DOF from only O(N) physical sensors



Signal Model

A N element possibly nonuniform linear antenna array

a(θ) ∈ RN×1 be the steering vector

a(θ)i = e j · 2πλ·di ·sin θ

where di denotes the position of the i-th sensor

Assume D narrowband sources from directions{θi , i = 1, 2, · · · ,D} with powers {σ2i , i = 1, 2, · · · ,D}Received signal

x[k] = A · s[k] + n[k]

where A = [a(θ1) a(θ2) · · · a(θD)] denotes the array manifoldmatrix and s[k] = [s1[k] s2[k] · · · sD [k]]T , and n[k] denotestemporally and spatially white, and uncorrelated from thesources.



Difference Co-Array

Rxx = E[xxH

]= ARssA

H + σ2nI

= A

σ21

σ22...σ2D

AH + σ2nI

z = vec(Rxx) = vec

[D∑i=1

σ2i (a(θi )aH(θi ))

]+ σ2n1n

= (A∗ � A) p + σ2n1n



Difference Co-Array

A∗ � A behave like the manifold of a array whose sensorlocations are given by the distinct values in the set{xi − xj , 1 ≤ i , j ≤ N}Difference Co-Array: Let us consider an array of N sensors,with xi denoting the position vector of the i-th sensor. Definethe set

D = {xi − xj}, ∀i , j = 1, 2, ·,N,

where Du denotes the distinct elements of the set D.

The difference co-array is defined as the array which hassensors located at positions given by the set Du.



Difference Co-Array

Weight Function: Define an integer valued functionw : Du 7→ N+ such that w(d) = no. of occurences of d in D,d ∈ Du.

w(0) = N, 1 ≤ w(d) ≤ N − 1,∀d ∈ Du\{0}.w(d) = w(−d),

∑d∈Du ,d6=0 w(d) = N(N − 1)

Cardinality of Du gives the defrees of freedom

DOFmax = N(N − 1) + 1



Difference Co-Array

Summary: If we use second-order statistics, then by exploitingthe degrees of freedom (DOF) of the difference co-array, thereis a possibility that we can get O(N2) degrees of freedomusing only O(N) physical elements

Examples:

an N element ULA ⇒ 2N − 1 elements ULA


Nested Array

Nested Array

Idea: Use a possibly nonuniform array ⇒ its differenceco-array has significantly more degrees of freedom than theoriginal array.

Nested Array: can be generated very easily in a systematicfashion and degrees of freedom of its co-array can be exactlypredict.

Two Level Nested Array

K Levels Nested Array


Nested Array


A concatenation of two ULAs

Inner ULA has N1 elements with spacing d1

Sinner = {md1,m = 1, 2, · · · ,N1}

Outer ULA has N2 elements with spacing d2 = (N1 + 1)d1

Souter = {n(N1 + 1)d1, n = 1, 2, · · · ,N2}

Difference co-array: a ULA with 2N2(N1 + 1)− 1 elements

Sca = {nd1, n = −M, · · · ,M,M = N2(N1 + 1)− 1}

2N2(N1 + 1)− 1 freedoms in the co-array using only N1 + N2

elements


Nested Array


N1 = 3, d1 = d , N2 = 3, d2 = 4d , M = 11


Nested Array


maxN1,N2

2N2(N1 + 1)− 1

s.t. N1 + N2 = N

⇓

N optimal N1, N2 DOF

even N1 = N2 = 12N N2−2

2+ N

odd N1 = N−12

, N2 = N+12

N2−12

+ N

obtain little over half of the maximum


Nested Array


Parameters: K , Ni , i = 1, 2, · · · ,K ∈ N+

S1 = {nd , n = 1, 2, · · · ,N1}

Si = {nd ·i−1∏j=1

(Nj + 1), n = 1, 2, · · · ,Ni}, i = 2, · · · ,K


Nested Array


Degrees of freedom in the corresponding difference co-array

DOFK = 2 {[N2(N1 + 1)− 1] + [(N3 − 1)(N1 + N2 + 1)

+(N1 + 1)] + · · ·+ [(NK − 1)(N1 + N2 + N3 + · · ·· · ·+ NK−1 + 1) + (N1 + N2 + · · ·+ NK−2 + 1)}+ 1

⇓

DOFK = 2

K∑i=1

K∑j=i+1

NiNj + NK − 1

+ 1


Nested Array


maxK∈N+

maxN1,··· ,NK∈N+

DOFK

subject toK∑i=1

Ni = N

Theorem 1. Given a number N of sensors, the optimal number ofnesting levels K and the number of sensors per nesting level aregiven by

K = N − 1,

Ni =

{1, i = 1, 2, · · · ,K − 1,0, i = K .


Nested Array


DOFK = 2

1

2

( K∑i=1

Ni

)2

−K∑i=1

N2i

+ NK

− 1

= N2 −K∑i=1

N2i + 2NK − 1

1 ≤ j ≤ K − 1 ⇒ break Nj into the sum of two smallerintegers Nj1 and Nj2, i.e., Nj = Nj1 + Nj2 ⇒ (K + 1) levels ofnested with {N1,N2, · · · ,Nj−1,Nj1,Nj2,Nj+1, · · · ,NK}sensors ⇒

DOFK+1 = N2 −

∑i 6=j ,i=1

N2i + N2

j1 + N2j1

+ 2NK − 1


Nested Array


∆DOF = DOFK+1 −DOFK = N2j −N2

j1 −N2j2 = 2Nj1Nj2 ≥ 0

breaking up always increases the degrees of freedom

similar analysis for j = K

∆DOF = DOFK+1 − DOFK = 2NK1(NK2 − 1) ≥ 0 �

The corresponding difference array is a nonuniform linear arraywith degrees of freedom given by DOFopt = N(N − 1) + 1which is same as the upper bound.

Structure of the Optimally Nested Array: the optimum nestedarray has sensors located at the positions given by the setSopt = {d , 2d , 4d , 8d , · · · , 2N−1d}The optimally nested array is one with exponential spacing


Application

Spatial Smoothing Based DOA Estimation

Spatial smoothing works only for a ULA and we shall focus onthe two-level nested array or any array whose differenceco-array is a filled ULA.

2-level nested array, N2 sensors in each level

A∗ � A ∈ RN2×D with (N2 − 2)/2 + N distinct rows


Application

Spatial Smoothing Based DOA Estimation

Theorem 2 : The matrix Rss as defined as

Rss = R̂2

where

R̂ =1√

N2

4 + N2

(A11ΛAH11 + σ2nI )

has the same form as the covariance matrix of the signalreceived by alonger ULA consisting of N2/4 + N/2 sensorsand hence by applying MUSIC on Rss, uptp N2/4 + N/2− 1sources can be identified.


Numerical Examples

Environment

6 sensor array (N = 6)

8 narrowband sources (D = 8)

directions of arrival {−60◦,−45◦,−30◦, 0◦, 15◦, 30◦, 45◦, 60◦}noise is assumed to be spatially and temporally white

2 level nested array with 3 sensors in each level

N2/4 + N/2− 1 = 11

SS-method: proposed spatial smoothing based technique

QS-method: KR product based MUSIC in [1] which requiresquasi stationarity


Numerical Examples

MUSIC Spectrum with different numbers of snapshots


Numerical Examples

MUSIC Spectrum with different numbers of snapshots


Numerical Examples

Optimally Nested Array vs 2 Level Nested Array


Numerical Examples

Optimally Nested Array vs 2 Level Nested Array


Conclusion

Conclusion

A novel nested array structure is proposed which can realizesignificantly more degrees of freedom

Optimum nested array structure was found through solving acombinatorial optimization problem

An alternative spatial smoothing based approach tounderdetermined DOA estimation

· · ·Future research ?


Conclusion

References

W.K. Ma, T.H. Hsieh, and C.Y. Chi. DOA estimation ofquasi-stationary signals via Khatri-Rao subspace. ICASSP,April 2009: 2165–2168.

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