Proceedings of 11 th International Conference on Computer and Information Technology (lCCIT 2008)25-27 December, 2008, Khulna, Bangladesh

Direction of Arrival Algorithms for Adaptive Beamforming inNext Generation Wireless Systems

Mohammad Ismat Kadirl., Md. Shadiul Hoque2

., Saiful Islam3

Electronics and Communication Engineering DisciplineKhulna University., Khulna - 920800., Bangladesh

E-mail: ismat_kadir@yahoo.coml.. sadiul_ece04@yahoo.com2

.. sunny_khulna@yahoo.com3

Fig. 1: Beamforming setup with DOA Estimation

received by other sensor elements. Finally, an N-elementbeamforming system is capable of forming up to N beams.

For the beamformer to steer the radiation in a particulardirection and to place the nulls in the interfering directionsthe direction of arrival has to be known beforehand. TheDirection of arrival algorithms does exactly the same; theywork on the signal received at the output of the array andcompute the direction of arrivals of all the incoming signals.Once the angle information is known it is fed into thebeamforming network to compute the complex weightvectors required for beam steering.

3. DIRECION OF ARRIVAL

A. SignalModelLet a uniform linear array be composed of N sensors,

and let it receive M narrow band source signals Sm (t)from , as shown desired users arriving at directions

81" 82 ••••8M :as shown in Figure-2.The array also receives I

narrow band source signals Si (t) from undesired(or

interference)users arriving at directions 81,82.••••••8, . At a

particular instant of time t = 1.,2., ...K ,where K is the total

number of snap shots taken. The desired users signal vector

xs (t) can be defined as

(1)M

xM(t) = La({}m)sm(t)m=1

The performance of next generation wireless can begreatly improved by using adaptive beamformingalgorithms [3], [6].

Beamforming can meet the challenge of increasingspectral efficiency and improving wireless communicationsystem performance by significantly increasing thereception and transmission ranges and reducing theprobability of interception of secure transmission.

Adaptive Beamforming is a technique in which an arrayof antennas is exploited to achieve maximum reception in aspecified direction by estimating the signal arrival from adesired direction (in the presence of noise) while signals ofthe same frequency from other directions are rejected. Thisis achieved by varying the weights of each of the sensors(antennas) used in the array.

Beamforming is a form of spatial filtering used todistinguish the spatial properties between a SOl and thenoise and the interfering signals. Beamforming principlesapply to both the transmission and reception of signals.

Beamforming is accomplished through the use of an arrayof sensors such as antenna, hydrophones and so on. In orderto proceed with the discussion of beamforming, it isimportant to note some basic assumptions. First, a signaloriginating far away from the sensor array can be modeledas a plane wave. Next the signal received by each sensorelement is a time-delayed (phase shift) version of the signal

1. INTRODUCTION

2. BEAMFORMING BACKGROUND

Abstract - Different beamforming algorithms like Side-lobeCancellors, Linearly Constrained Minimum Variance(LCMV), Least Mean Squares (LMS), Recursive LMS, andDirection of Arrival (DOA) exist in literature. Among theDirection of Arrival (DOA) algorithms, MUSIC and ESPRITplay the most important role. These two algorithms wereimplemented and their performances were compared. Thealgorithms were simulated for different signal levels and theDOAs were computed for use in next generation wireless.ESPRIT was found to be a better DOA technique foruncorrelated source used in beamforming.

Keywords - Antenna array, Beamforming, DOA estimation,

Eigenvalues, ESPRIT algorithm, MUSIC algorithm.

1-4244-2136-7/08/$20.00 ©20081EEE 571

and i(t) is the I x 1 undesired (or interference) users sourcewave form vector defined as

We also define the undesired (or interference) users signal

vector X I (t) as

(8)

(7)

where Al is the N x I matrix of the undesired users signal

direction vectors and is given by

F. field sipal IOUIa!

(\\-.vel-a*-.l)

Fig. 2: Geometry of a uniform linear array. (9)

where a(Bm ) is the N x 1 array steering vector which

represents the array response at direction Bm and is given

by

The overall received signal vector XM (t) is given by the

superposition of the desired users signal vector XM (t),

undesired (or interference)users signal vector X I (t) ,and an

N x I vector net) which represents white sensor noise.Hence, x(t)can be written as

(10)X(t) = xM (t) + n(t) + XI (t)where [(.)]T is the transposition operator ,and qJm

represents the electrical phase shift from element to element where net) represents white Gaussian noise. Thealong the array. This can be defined by conventional (forward-only) estimate of the covariance

matrix defined as

'Pm =2TI (1) sin(Bm) (3) R = E{x(t)x H (t)} (11 )

where AM is the N x M matrix of the desired users

signal direction vectors and is given by

where d is the inter-element spacing and A is thewavelength of the received signal. The desired users signal

vector XM (t) of (1) can be written as

(13)

Substituting for x(t) from (10) in (12) yields

where Rss = E{S(t)SH (t)} is an MxM desired users

source waveform covariance matrix; Ru = E{i(t)i H (t)}

is an I x I undesired users source waveform covariancematrix.

Where E{.} represents the ensemble average; and (.) H isthe Hermitian transposition operator. Equation (11) can beapproximated by applying temporal averaging over Ksnapshots (or samples) taken from the signals incident onthe sensor array. This averaging process leads to forming aspatial correlation (or covariance) matrix R given by [8]:

1 K HR = -- L x(k)x (k) (12)

K K=1

(6)

(4)

s(t) = [SI (t), S2 (t) sM (t)]T

AM = [a(B1),a(B2 ) ••••••,a(BM)] (5)

and s(t) is the M x 1 desired users source waveform

vector defined as

572

B. Algorithm for MUSIC estimationConsider an N-element linear array that detects M signals

impinging on it whose directions of arrival need to beknown. From the previous discussion we know that thereceived signals at the output of the array have the followingform.

x(t) = A(B)s(t) +n(t)Or in matrix notation it can be represented as,

C. Algorithm for ESPRIT estimationESPRIT (Estimation of Signal Parameters via Rotational

Invariance Technique) is one of the most efficient androbust methods for OOA estimation. It uses two arrays inthe sense that the second element of each pair is displacedby the same distance in the same direction relative to thefirst element. It is not required to have two separate arraysbut can be realized using a single array by being able toselect a subset ofelements.

x = AS +n (14)Let the array signals received by the two arrays be

denoted by x(t) and y(t) such that

(24)

(23)

_ -1 Arg(A,,,) _() - COS { },m - 1, .... ,M

111 21{~ 0

A is a Kx M matrix; where M is the number of steeringvectors produced by N elements of the array. nx (t) And

ny (t) denotes the noise induced at the elements of the two

arrays. Now, by using the available methods, the numbers ofdirectional sources M, are estimated based on principlessuch as Akaike's information criterion (AIC) and Minimum

description length (MOL). Two matrices U x andUy are

formed which denote the M eigenvectors corresponding tothe largest eigenvalues of the two array correlation matrices

Rxx & Ryy (Array correlation matrices). The eigenvectors

of the following 2M by 2M matrix are obtained and aredenoted by

x(t) = As(t)+nx(t) (21)

Once the eigenvector V is obtained its eigenvalues

Am,m =1, ,M can be computed.

Now the OOA is given by

yet) = ABs(t) +ny (t) (22)

P (8) is

(20)

Let the noise eigenvector be defined as EN such that,

(s - (72 I)EN =0 (18)

Or,

APA*EN=O (19)

Based upon this approach, the pseudospectrumgiven by

P(B) = 1A(B)* ENEN*A(B)

The eigenvectors ofRx must satisfy

RXei =a 2lei (16)

(Rx - (72 I~i =0 (17)

where e is the eigenvector and i varies from 1to N-M.

lAP *AI = IRt - (72 II = 0

Where P = E{SS*}

AP* A =R -a2Ix

When the number of signals M is less than N then AP*A issingular and has a rank less than N. The eigenvalues of canbe found by,

~o is the element separation in terms of wavelength. Othervariations of ESPRIT include beam-space ESPRIT,resolution-enhanced ESPRIT multiple invariance ESPRIT

When the pseudospectrum P (8) is plotted, peaks appear at and higher order ESPRIT.the angles of arrival of the incident signals.

Where S is the signal vector, A is the array propagationvector and n is the noise vector with zero mean andvariance.The covariance matrix is given by,

Rx = E{XX*} = AE{SS*}A *+E{nn*} (15)

=: AP *A + (j2 I

573

4. IMPLEMENTING MUSIC AND ESPRIT

The above algorithms were implemented and thedirections of arrival were computed. It is obvious from theanalysis that MUSIC gives the pseudospectrum from whichwe can compute the direction of arrival as thepseudospectrum peaks. But in case of ESPRIT, we can getthe direct polar plot.

Two sets of amplitudes have been used to be the signals atthe elements of a linear array. Using the algorithms, theDOAs are calculated. The arrays are simulated to have threeelements. There can be the provision for variable number ofelements too. The distance between the elements is A/2 .The amplitudes of three element signals for first case areconsidered to be [1 2 3] and for second case [1.5 2.5 3.5]in voltage units.

Fig. 3: ESPRIT for first signal

•1

.1

Amplitude: {I 2 3J

ESPRtT ALGORITHM4

AmpI/bI.; (1.5 2.5 J.5j

The directions of arrival for different element signals areshown for the first amplitude set in Fig. 5 and those for thesecond set in Fig. 6. The pseudospectrum pee) s for the twocases are plotted in Fig.3 and Fig.4 respectively, from wherethe spectrum peaks can be taken to be the DOAs.

:'1 1I; W~: ~3J

----------f-~---------fh----- --j-------hl---------fl----------

---------.L~--------L--\-----.L---LJ--------LL-------:~ i: I to I. : !~, :1 :\ ~;

!\---:il--\~/,-----:\-~_i~;:/-----·\:~i-I\----fJ.. --\":.,,.!'_ .. ~ _. _. -------~ ---_. ----_. ~ ----_. -----~. ----_. ----~ _. .\""..l. --

• •• I

I •• II •• •

I •• •

I •• I• • I •t • I I

I • I I

Fig. 5: MUSIC for first signal

AllpIiWi . I I • •

11.52.53:sJ : I': : ill : :: : \ : J: :

-----------[,\----------[-1 --------1--------:'(------ .. -1'1----------l :1', :J l : I i.: II: l-~--------+,---------[i --~-------[-------t-\1---------/\- -----_. -~t

II I' :1 -, : 1 I: I II-~t-------t-\-------J----\:-----[-- -- -1----)':-------t-!(-------j1,

I' :', f: \. : l :\ I:, I"

i_\_/-,:::\::-:~:~_,--::_:\,:--J---~;---:--,:\:---:i-:- .\.::/_:,

Fig. 6: MUSIC for second signal

Fig. 4: ESPRIT for second signal

574

Table 1 summarizes the results found in thecorresponding experimentations. We can have acomparative view of the two algorithms for beamformingfrom this result.

Amplitude MUSIC ESPRIT

[1 2 3] 17° 38° 57° 20° 10° 13°

[1.5 2.5 3.5] 15° 38° 55° 15° 19° 10°

Table 1: Comparative DOA

5. CONCLUSION

For first signal in MUSIC Estimation we get the peak ofpower spectrum at 17, 38, 57 degrees and for second signalat 15,38, 55 degrees respectively. This gives an idea aboutthe process of DOA estimation. The element angles exhibitgreater variation, although in case of signal amplitudevariation, the angles of peak power spectrum does not differthat much. In case of ESPRIT estimation we get 20, 10, 13degrees and 15, 19, 10 degrees for the first and the secondsignals for the three elements respectively. No doubt, thevariations are not that much pronounced and the estimatedangles for different elements remains to be almost same.

So, the ESPRIT DOA algorithm can be treated as a morerobust and faster estimation technique as compared toMUSIC. The computation is also less complexcomparatively. However, ESPRIT has the disadvantage ofnot being capable of handling correlated sources. Finally,this accurate direction will help to obtain betterbeamforming which is very essential for next generationwireless systems.

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