# Non Unuiform Phased array Beamforming with Covariance Based Method

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2 on Unuiform Phased array Beamforming with Covariance Based Method II. COVAVIACE BASED METHOD FORMULATIO Assume that we have a collection of transmitter antennas which are located at known coordinates x i = x 1,i, x,i, x 3i = (x, y, z) in some spherical coordinate along the z-axis as shown in Fig. 1. In the presented study and in all of the examples and formulas of current paper, it is assumed that these transmitter antennas are along the z-axis. It is assume that each transmitter antenna is driven by a specific signal on the carrier frequency of f c or with wavelength of λ and complex envelope ofs i t, i = 1,,. At a specific point in space with distance of r and direction of k(θ, ϕ) from the transmitter antenna, each radiated signal s i (t) gives rise to a "signal" the far field at radius r, with complex envelope given by y i t, r, θ, ϕ = 1 4πr s i t r c π ej λ x i T k(θ,ϕ) (1) Where, in this equation, k is a unit vector in the (θ, ϕ)direction. Fig. 1. T/R modules and spherical coordinate system At the far field, these signals add linearly and the radiated powers P i add linearly as well. At this point assume that the i-th element location is on the z-axis at coordinate z i. The signal at position (r, θ, ϕ) resulting from all of the transmitted signals at far field will be: = y t, r, θ, φ = 1 4πr i=1 i=1 y i t, r, θ, φ s i t r πz i c ej λ sin θ () The power density of the entire signals then given by = 1 4πr k=1 l=1 P y r, θ, ϕ < s k t s l t > e j π z k z l λ And it is known that the complex signal cross-correlation is defined by sin θ (3) R kl =< s k t s l t > (4) With defining the direction vector as below a θ = e j πz 1 λ The normalized power density P(θ, ϕ) of signals, in (W/ster), would be: sin θ,..., e j πz λ sin θ T (5) P θ, φ = 1 4π k=1 l=1 j π z R kl e λ k z l sin (θ) (6) 38 P a g e

3 on Unuiform Phased array Beamforming with Covariance Based Method Recognizing that (6) is quadratic form in the Hermitian matrix R which is the cross-correlation matrix of signals, this can be written compactly as P θ, ϕ = 1 4π a θ Ra θ (7) This normalized power density P(θ, ϕ) is exactly the beampattern which is desirable to find [8, 9]. In the following some examples of beampatterns produce from such a cross-correlation matrix has been shown. Fig. shows the beampattern produced by signal cross-correlation matrix of (8), (9) and (10) respectively. It is noticeable that these figures are beam-patterns of 10-element uniform linear array (ULA) with half-wavelength spacing (8) (9) Beampattern of (8) Beampattern of (9) Beampattern of (10) Fig.. Beampattern respect to (7). The blue one is corresponds to cross-correlation matrix of (8), The black one is corresponds to cross-correlation matrix of (9) and the red one is corresponds to cross-correlation matrix of (10) In general case the elements of the signal cross-correlation matrix are complex values except the diagonal elements that are real. This general case is related to MIMO radars but in the case of phased array radar, all the transmitter signals are correlated with each other and then absolute value of all the elements in R, are equal to 1(blue one at Fig. ). III. USIG COVARIACE MATRIX TO OPTIMIZE BEAMPATTER In this section we introduce our cost function. Let us assume that there are ( < t ) targets of interest which t is total number of transmitter elements. Without loss of generality, they are assume at locations θ k k=1. Then the cumulated power of the probing signals at the target locations is given by Where Anlge (rad) k=1 a θ k Ra(θ k ) tr RB (11) B = a θ k a (θ k ) k=1 max In [10] authors have defined a cost function of tr(rb) with some constraints on correlation R matric of R. In this equation B is an estimate of B which means that some prior information is available on target locations. In our case as we are working on phased array radars, all the elements of cross correlation matric are equal to 1, then we cannot solve that problem respect to R. As a result we will form this equation respect to a vector, and consequently we use of direct estimation of θ or a, instead of B. Then our cost function will be the form below (1) 39 P a g e

4 on Unuiform Phased array Beamforming with Covariance Based Method max tr RB a R = allonematric B = a θ k a θ k k=1 subjectto a = 1 (13) In many cases we wish to focus our transmitter energy to one desire direction, tracking applications are such cases. To simplicity and without loss of generality we will consider the case which focusing the total energy at θ = 0 is of our interest. In other words it is assume that there is only one target which is placed at θ = 0 angle. In such a case our optimization problem will convert to: max tr RB a R = allonematric B = a θ a θ subjectto a = 1 (14) We will solve this optimization problem with some numerical and iterative method and show the result at the next section. There is one important point at this solution, regardless of other exist algorithm which optimum the placement of the phased array antennas, that place the antennas at a class of specific places of multiple of half wave length, this algorithm has no such a constraint. IV. UMERICAL EXAMPLES In this section we will show the results by some numerical examples. We suppose that our antenna has 7 elements as it stated before without loss of generality we suppose that our transmitter's elements are placed along the z-axis and the radiation pattern has been measured at distance of 1m far from the transmitter along the x-axis and in a range between -1m and +1m along the z-axis. Fig. 3 shows the results of such geometry for conventional phased arrays and our arrangement of antennas placements Half wave spacing Modified spacing Distance (m) Fig. 3 comparison between uniform and non uniform linear array beampattern In this figure blue plot is beam-pattern of conventional phased array transmitter that is obtained from equation (7) which has 7 elements in it's transmitter antenna with half wave length spacing between each antennas: Z = λ 3,, 1, 0, 1,, 3 The red plot in Fig. 3 Shows the beampattern which result from our algorithm that is obtained from two iterations. This result is for below spacing between the antennas elements: Z = λ 3.7,.3, 1, 0, 1,., 3.7 As it seen from this figure with modifying of spacing between elements result from covariance based method we achieve to lower sidelobe level and better transmitter beam-pattern. As we continue the presented algorithm iterations we will achieve more convenient results. To show the usefulness of this algorithm we wish to consider another example. In this example we assume that we have a planar array with 7*7 elements along the z-axis and y-axis. 40 P a g e

5 y y on Unuiform Phased array Beamforming with Covariance Based Method Just like the latter example, for conventional phased array we use half wave length spacing between transmitter's elements in both z and y axis as shown in Fig. 4 and for modified phase array we use of spacing from presented algorithm like Fig x Fig. 4uniform planar array x Fig. 5non uniform planar array Resulting beam-patterns are shown in Fig. 6 and Fig 7. other parameters like distance to measuring the power is the same as latter example. As it is seen from this figure, the resulting beam-pattern from our algorithm with modified antenna placement has lower side-lobe level and sharper peak respect to conventional array with half wave spacing. Fig. 6 uniform planar array beampattern 41 P a g e

6 on Unuiform Phased array Beamforming with Covariance Based Method Fig. 7non uniform planar array beampattern V. COCLUSIO In this paper it presented that with a new approach to defining of beampattern of a phased array antenna based on cross-correlation matrix of transmitter signals, we can have a new vision of optimizing beampattern to steer a desire beampattern energy at a specific direction by optimizing placements of transmitter's antenna elements, also it showed that with this approach we can achieve a pattern that has lower side-lobe level and has sharper peak at desire direction respect to conventional phased arrays with half wave length spacing. REFERECES [1] M. Stojanovic and Z. Zvonar. Adaptive Spatial/Temporal Multiuser Receivers for Time-Varying Channels with Severe Isi. In Proc. of Coq! On Signals and S-ystems, Princeton, March [] B. Ottersten, M. Viberg, P. Stoica, and A. ehorai. Exact and large sample ML, techniques for parameter estimation and detection in array processing. In Haykin, Litva, and Shepherd, editors, RadarArray Processing, pages Springer-Verlag, Berlin, 1993 [3] S. Shamsunder and G. Giannakis. Signal Selective Localization of onganssian ICyclostationary Sources. IEEE Trans. SP, 4: , Oct [4] C.P. Mathews and M. I). Zoltowski. Eigenstructure Techniques for -D Angle Estimation with Unifom Circular Arrays. IEEE Trans. on SP, 4: , Sept [5] J. Li and P. Stoica. Efficient Parameter Estimation of Partially Polarized Electromagnetic waves. IEEE Trans. on SP, 4( 11):3 I , ov [6] H. rim and J.G. Proakis. Smoothed Eigenspace-Based Paramerter Estimation. Auromaticu, Special Issue on Statistical Signal Processing and Control, Jan [7] J.H. Cozzens and M.J. Sousa. Source Enumeration in a Conelated Signal Environment. IEEE Trans. on Acoust., Speech and Signal Proc., 4(): , Feb [8] D. R. Fuhrmann and G. S. Antonio, Transmit beamforming for MIMO radar systems using partial signal correlation, in Proc. 38th IEEE Asilomar Conf. Signals, Syst., Comput., ov. 004, pp [9] D. R. Fuhrmann and G. S. Antonio, Transmit beamforming for MIMO radar systems using signal cross-correlation, IEEE Trans. Aerosp. Electron. Syst., vol. 44, pp , Jan [10] P. Stoica, J. Li, and Y. Xie, On probing signal design for MIMO radar, IEEE Trans. Signal Process., vol. 55, no. 8, pp , Aug P a g e

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