International Journal of Emerging Technology and Advanced Engineering

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 10, Oct 2017)

224

Tracking Algorithm for Directive Arrays

A. V. L. Narayana Rao

1

, Dharma Raj Cheruku

2

1

Dept. of ECE, SSNCET, GIT, Ongole, AP, India. 2_{Dept. of ECE GITAM University Visakhapatnam, AP, India.}

Abstract—Elliptical arrays (EA) for direction of arrival (DOA) with directional elements is studied in this paper. Majority of previous works assume isotropic antenna elements for work demonstratration. Micro strip patch antennas approximating the optimal theoretical gain pattern are designed to compare the resulting DOA estimation accuracy with a EA using dipole antenna elements. Simulation results show improved DOA estimation accuracy and robustness using micro strip patch antennas as opposed to conventional elopements. It is shown that the bandwidth of the EA for DOA estimation is limited by the broadband characteristics of the directional antenna elements and not the electrical size of the array. DOA estimating algorithms consider Omni directional elements in general. This paper deals with MUSIC Algorithm modified for directive antenna. Results show the improvement in directivity.

Keywords—Patched antenna, smart antenna, kalman filter, Tracking algorithm, Modified MUSIC, DOA.

I. INTRODUCTION

SMART antennas have been a subject of research interest for several decades, motivated initially by military applications. Over the past 10–15 years however, this topic has received widespread interest due in large part to the proliferation of mobile communications devices. Smart antennas, or adaptive arrays, have several advantages over traditional arrays including increased coverage, improved robustness to multipath, increased system capacity, and resistance to signal interception and interference [1]. The primary features of smart antennas are the ability to determine an incoming signal’s direction of arrival (DOA) and the ability to control the radiation pattern (beam forming). This paper focuses on DOA estimation and the effect of the antenna element characteristic son the achievable accuracy [2]–[5] for a sample of recent work on DOA estimation). In general, for DOA estimation, separate receiver channels (RF front-end and digitizer) are required for each antenna element, and additional receiver channels can significant ly increase size, weight, power, and cost. Therefore, it is desirable to maximize performance witha minimum number of antenna elements.

The vast majority of the literature on smart antennas for DOA estimation and beam forming assume either isotropic antenna elements or, equivalently, unidirectional elements with the analysis limited to the plane where the antenna gain is equal in all directions [6,8]. While the literature on the individual topics of directional antennas and smart antennas is vast, there is a limited number of published papers that discuss the use of directional antennas as the elements of a smart antenna array for DOA estimation.

Smart antenna array for DOA estimation. Switched antenna UCAs have been used for direction finding. In the case of [9], omnidirectional monopole antennas were used whereas in directive patch antennas were used on the flat surfaces of a semi dodecahedron.

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In this section, a design that approximates the pattern found to have the lowest DOA estimation error in the previous section will be discussed and its performance will be compared to λ/2dipole antennas for a 4-element EA.

II. MICRO STRIP PATCH ANTENNA

[image:2.612.352.537.132.272.2]

Micro strip patch antenna was designed by using an optimization algorithm where the goal was to minimize the difference between the simulated far-field radiation pattern and the theoretical pattern with e = 0.4,CST STUDIO. The resulting optimized dimensions of the micro strip antenna was as follows: Wsub = _=2 = 148:0 mm, Lsub = _=2 = 148:0 mm, W = 106:8 mm, L = 96:5 mm,W50 = 9:8 mm, Wslot = 2:4 mm, Lslot = 21:7 mm,t = 3:175 mm, _r = 2:2 (duroid 5880 substrate).The design used a relatively small substrate in order to realize arrays with radii as low as e =0.4. The use of an inset feed (Wslot _ Lslot) is a common technique to improve the input impedance match of micro strip antennas. The radiation pattern of the theoretical directional element with e = 0.4 is compared to the gain pattern of a micro strip patch obtained from EM simulation. in Fig. 3. Also evident from the gain pattern is the backside radiation which is present since the ground plane is not infinite in extent. The difference in peak gain between the theoretical pattern and the micro strip patch is approximately 0.7 dB. Frequency of operation is 2.69GHz,Main lobe magnitude is 3.99dB,Main lobe direction315.0deg,beam width 57.4deg, Design of antenna is illustrated by Figs1,2.

Fig. 1 Patch Design

[image:2.612.401.496.307.415.2]

Fig.2 Layout Of The Patched Array

Fig.3 Beam Pattern Of Patched Directive Array

2.1 Elliptical Array Of Directional Elements:

Fig.4 Geometry Of Ellipse In Xy- Plane

In=amplitude of excitation

αn=phase of the n th element

θ=elevation angle from z axis.

øn=Azimuth angle measured from x axis for n th element.

[image:2.612.61.277.477.590.2]

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A smart antenna receiver system used to describe a Elliptical array for DOA estimation is shown in Fig. 4. In this figure, a 8-element Array with eccentricity (e) and inter-element distance d the array output is given

Y(k)=wHx(k) (1)

where k is the sample index,

wH represents the Hermitian transpose,

w is a complex weight vector =[w1 w2 : : : wN]T ), and

the received signal at the antenna elements is given by:

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=A.s(k)+n(k) (3)

a,b=semi major and minor axises respectively

.

e=eccentricity of elliptical array and is 0.5. N-element array response vector is given by Equation (2), A is the array manifold. The noise is represented by n(k) in equation (3).Steering vector represents the relative phases of the received signals at the antenna elements. Obviously, in the case of isotropic antennas or antennas that are Omni directional in _, g0 = g1 = : : : = gN�1 = 1,which means

that all DOA information is derived from the relative phase differences between the elements. In contrast, with directional elements, the DOA estimation algorithms can exploit the gain variation at different incident angles as well as phase differences to increase accuracy and robustness to ambiguous DOA estimates. The bandwidth of a DOA system is defined in this work to be the frequency range over which the accuracy of the DOA estimates meets the desired application’s requirements. Is assumed that the N antenna elements in the Elliptical array identical and have a maximum gain in the direction that is radially outward from the center of the array,

a( (4)

Where R= r-sinө(xcosөnax+ysinөn ay) depends on

ecentricity.Equation (4) shows the directional vector of elliptical array with directive elements.

III. MODIFIED GAIN KALMAN FILTER

The KFA (Kalman filter Algorithm) can be applied as an estimator of the state of a dynamic system described by the linear difference equation

𝒙𝑘=𝐀 𝑘−1𝒙 𝑘−1+𝐁 𝑘−1𝒖 𝑘−1+𝒘 𝑘−1, (5)

where the matrix 𝐀𝑘−1 = relationship between the state of the system , 𝒙𝑘−1 = column vector time k-1, 𝒙𝑘 at time

k= state vector.𝐁 𝑘−1= matrix relating control input vector 𝒖 𝑘−1 to the state vector.

𝒘𝑘−1 = process noise. Measurement vector 𝒛𝑘 (equation (6))is related to the true state of the system by the equation

𝒛𝑘=𝐇𝑘𝒙𝑘+𝒗𝑘, (6)

where the vector 𝒗𝑘 = measurement noise

𝐇𝑘 = the observation matrix, which makes it possible to have more parameters/dimensions in our estimate than we are actually getting from our measurement inputs.

𝒘 𝑘−1(process noise) and the 𝒗𝑘(measurement noise) are

assumed to be independent of each other and normal distributed with zero mean.

From these two noise vectors the system noise covariance matrix 𝐐𝑘−1=E(𝒘𝑘−1𝒘𝑘−1T) (where E denotes the expected value) and the measurement noise covariance matrix 𝐑𝑘=E(𝒗𝑘𝒗𝑘T), are formed. Where column vector

𝒗𝑘 and row vector is 𝒗𝑘T_{. Just like the method of least}

squares, the KFA estimation of the system is statistically optimal with respect to a quadratic function of the estimate error, minΣ(𝑠𝑘−𝑠 𝑘)2𝑁𝑘=1,

where 𝑠𝑘 is the true state of some system and 𝑠 𝑘 is the estimate of the same. It is equivalent to minimizing the trace of the error covariance matrix 𝐏 𝑘∣𝑘 of the estimate 𝒙 𝑘∣𝑘,

𝐏𝑘∣𝑘=cov (𝒙𝒌−𝒙 𝑘∣𝑘,−𝒙 𝑘∣𝑘)=E[(𝒙𝒌−𝒙 𝑘∣𝑘)(𝒙𝑘−𝒙 𝑘∣𝑘)T]. (7) The estimate vector 𝒙 𝑘∣𝑘 is referred to as the a posteriori

estimate since it is an estimate of the system at time k calculated including the measurement 𝒛𝑘. This differs from the a priori estimate, denoted 𝒙 𝑘∣𝑘−1, which is an estimate of the state at time k made only with the available measurements up to k-1, but not including the latest measurement 𝐳k. This notation is also used, having the same meaning as above, to separate the error covariance matrices 𝐏𝑘∣𝑘 given by equation (7)and 𝐏𝑘∣𝑘−1, as well as separating the error vector of the estimate 𝜺𝑘∣𝑘=𝒙𝑘−𝒙 𝑘∣𝑘

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3.1 Minimizingtheerrorcovariance

The equation of the error covariance of the estimate

𝐏𝑘∣𝑘=E[(𝒙𝑘−𝒙 𝑘)(𝒙𝑘−𝒙 𝑘)T], together with KFA stating that

𝒙 𝑘∣𝑘=𝒙 𝑘∣𝑘−1+𝐊𝑘(𝒛𝑘−𝐇𝑘𝒙 𝑘∣𝑘−1). Using these two expressions

and the fact that the measurement 𝒛𝑘 is the true state of the system, affected by the observation matrix and corrupted by the measurement noise so 𝒛𝑘=𝐇𝑘𝒙𝑘+𝒗𝑘, they get

𝐏𝑘∣𝑘=E{[(𝐈−𝐊𝑘𝐇𝑘)(𝒙𝑘−𝒙 𝑘∣𝑘−1)−𝐊𝑘𝒗𝑘][(𝐈−𝐊𝑘𝐇𝑘)(𝒙𝑘−𝒙 𝑘∣𝑘−1)−

𝐊𝑘𝒗𝑘]T}.

By the definitions of the error covariance matrix 𝐏𝑘∣𝑘−1 and the measurement covariance matrix 𝐑𝑘 this can be written as 𝐏𝑘∣𝑘=(𝐈−𝐊𝑘𝐇𝑘)𝐏𝑘∣𝑘−1(𝐈−𝐊𝑘𝐇𝑘)T+𝐊𝑘𝐑𝑘𝐊𝑘T_{, which}

after expansion becomes 𝐏𝑘∣𝑘=𝐏𝑘∣𝑘−1−𝐊𝑘𝐇𝑘𝐏𝑘∣𝑘−1− 𝐏 𝑘∣𝑘−1 𝐇𝑘T_𝐊𝑘T₊_𝐊𝑘₍_{𝐇𝑘𝐏 𝑘∣𝑘−1}_𝐇𝑘T₊_𝐑𝑘₎_𝐊𝑘T_{. (8)}

The mean squared error is then minimized taking the trace of 𝐏𝑘∣𝑘 (the sum of the elements on the main diagonal), differentiating equation (8) with respect to 𝐊𝑘, and setting the result equal to zero, so we have that

∂Tr(𝐏𝑘∣𝑘)/∂𝐊𝑘=−2(𝐇𝑘𝐏𝑘∣𝑘−1)T+2𝐊𝑘(𝐇𝑘𝐏 𝑘∣𝑘−1𝐇𝑘T₊_𝐑𝑘_)=0.

Hence 𝐊𝑘=𝐏 𝑘∣𝑘−1 𝐇𝑘T(𝐇𝑘𝐏 𝑘∣𝑘−1 𝐇𝑘T+𝐑𝑘)−1, which is

kalman gain for minimum error in co variance.

3.2.Reduced Order Kalman Filter

Trying to coping with the nonlinear nature of the models of the real world. The solution they came up with was to linearize their equations around the estimate and then applying the KFA. This is essentially the extended type of Kalman filter.

If one or both of the equations describing the state transition and the observation matrix are nonlinear, a solution might be to use the EKF. An example hereof is if the state transition of KFA is a nonlinear function described by the equation

𝒙 𝑘∣𝑘−1=𝒇(𝒙 𝑘−1∣𝑘−1), instead of the original 𝒙

𝑘∣𝑘−1=𝐀 𝑘−1𝒙 𝑘−1∣𝑘−1.

This will also affect the rest of the KFA in such a way that 𝐏 𝑘∣𝑘−1=𝐀 𝑘−1 𝐏 𝑘−1∣𝑘−1𝐀 𝑘−1T+𝐐 𝑘−1, where 𝐀 𝑘−1 will be defined as the Jacobian matrix of the

function

(𝒙 𝑘−1∣𝑘−1), 𝐀𝑘−1=𝜕𝒇𝜕𝒙│𝒙 𝑘−1∣𝑘−1.

So 𝐀𝑘−1 will thus be the function (𝒙 𝑘−1∣𝑘−1)

differentiated with respect to 𝒙 and then evaluated at the point 𝒙 𝑘−1∣𝑘−1. Fig.5 shows the flow chart for modified kalman filter

Fig.5 Flow Chart Of Proposed Modified Gain Kalman Filter

3.3 Matematical Model Of Peoposed Algorithm

In this approach an adaptive array system is considered in conjunction with constraints on a weight vector given as If Rxx is a covariance matrix of received signals plus

thermal noise, the output power of the system will be equation(9)and (10)

P=WHRXXW ( 9)

and if the system is subjected to a constraint

||W||=WHW=1 (10)

(This is Euclidean norm constraint, any non-zero value can be chosen).

The so called cost function of the system can be written as equation(11)

H(W)=WHRxxW+λ(1-WTW) (11)

The aim is to minimize this function after incorporating the constraint with the undetermined Lagrange multiplier λ and then differentiating the function on a complex weight vector by using (complex gradient operator)

∆W*H(W)= RxxW-λW (12)

and then putting it equal to zero leads to

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It can be seen that the stationary value occurs when W is an eigenvectors of Rxx. When W is one of the normalized eigenvectors with corresponding eigen value λi. It can also

be seen that lowest eigen value is corresponding to the minimum output power which are relating to the channel thermal noise powers. In general if there are N elements and D source in the environment at the frequency of interest, when [N>D] there will be [N-D] eigen values corresponding to the receiver thermal noise powers.

Now the array gain in the constraint C direction (i.e. spatial vector in a look direction)with the weight vector W is CWT , where C is a set of element gain in the direction of (өi).

So the normalized gain will be ‖CWT‖⁄(‖C‖‖W‖)and this gain has a maximum value of unity when W=C*

For each look direction we can expandc∗ in terms of eigen values is equation (14)

(14) But now we restrict W (equation (15)to be a component

of c∗which is made up of only the eigenvectorswhich give receiver noise at the array output.

= (15)

Where i<N-D corresponds to the noise level eigenvectors. In this case normalized gain is generally slightly less than unity. However in the signals directions all the noise eigenvectors produce zero array gain, and so will any linear combination of them.

An Alternative(proposed) Approach for DOA Estimation

The DF function given by equation (16) plotted is in fact the reciprocal normalized gain, give peaks in the signal direction, and becomes with this choice of.

(16) The proposed approach can be summarized as shown in

the flowing diagram Fig. (6)

λ⁄2Dipole Elements:

Six λ⁄2dipoles array elements with 0.5 λ inter element spacing is assumed with a single source come from (100°).Fig. (6) Shows that the MUSIC DF interpreted three sources from [0° ,100° ,180° ]rather than the actual single source come from 0°,100°. This means that the MUSIC DF experienced false reading from angles [0°,180°] due to the use of λ⁄2dipole elements. The false reading is due to the existence of true nulls from [0°,180°] in the original pattern of half wave length dipole element as shown in figure (7).

The MUSIC DF function is mainly depending on creating orthogonal nulls in the final pattern of the arraying the direction of received sources and then interprets these nulls plus the nulls in the original pattern of half wave dipole (the elements of array) by the formula given by Equation(17) as a peaks in the arrival directions of these sources which leads to this wrong results

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Fig.6 Flow Chart Of Proposed Algorithm

IV. RESULTS AND CONCLUSION

The proposed approach with the equations (16),(17) is tested under the same assumption given. Fig.(10),(11)&(12) shows that the final DF pattern exhibit a direction to a single source from 200_{,60° and nothing from (0°,}

180°) as the case of MUSIC.

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sources come from angels (0°, 180°) which means that the alternative approach is overcoming the problem raised in MUSIC when it is used with a practical halfwave dipole array elements.

Angle of arrival (AOA) estimation based on MUSIC algorithm with isotropic and λ⁄2dipole array elements is investigated and the results show the following. With isotropic

array elements the MUSIC DF can

simultaneously estimate the directions to the single and multiple sources with a good accuracy even for those sources which are placed at angles [0°or 180°].When λ⁄2 dipole is used as an array element, it has been found that the MUSIC DF system exhibit a false

direction reading from angles [0°, 180°]

because of these directions are coincide with the nulls in the element pattern.

FIG.7 DOA by MUSIC (20,60 DEG)

FIG.8 DOA BY MUSIC_GAIN (20,60 DEG)

FIG.9 DOA by MODIFIED_MUSIC (20, 60 DEG)

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FIG.11 MUSIC GAIN_EKF(MUSEK) AT30MHz

FIG 12 MUSIC GAIN_EKF(MUSEK) AT 5GHz

REFERENCES

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[2] Z. Ye and C. Liu, ―2-D DOA estimation in the presence of mutual coupling,‖ IEEE Transactions on Antennas and Propagation, vol. 56,no. 10, pp. 3150–3158, October 2008.

[3] E. Ben-Ari and J. Remez, ―Performance verification of a multimodalinterferometric DOA-estimation antenna,‖ IEEE Antennas and WirelessPropagation Letters, vol. 10, pp. 1076–1080, 2011.

[4] S. Hwang, S. Burintramart, T. Sarkar, and S. Best, ―Direction of arrival(DOA) estimation using electrically small tuned dipole antennas,‖ IEEETransactions on Antennas and Propagation, vol. 54, no. 11, pp. 3292–3301, November 2006.

[5] D. E. N. Davies and M. S. A. S. Rizk, ―A broadband experimental null-steering antenna system for mobile communications,‖ Radio and Electronic Engineer, vol. 48, no. 10, pp. 509–517, October 1978. [6] J. R. F. Guy and D. E. N. Davies, ―UHF circular arrays incorporating open-loop null steering for communications,‖ IEE Proc. on Communications, Radar and Signal Processing, vol. 130, no. 1, pp. 67–77, February1983.

[7] J. Liu, l. Li, and W. Huazhi, ―Investigation of different types of array structures for smart antennas,‖ in Int. Conf. on Microwave and Millimeter Wave Technology (ICMMT), 2008, pp. 1160–1163. [8] M. R. Islam and I. A. H. Adam, ―Perfomance study of direction

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[9] B. Wu, ―Realization and simulation of DOA estimation using MUSIC algorithm with uniform circular arrays,‖ The 4th Asia-Pacific Conf. on Environmental Electromagnetic, pp. 908–912, 2006.