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A Novel Method of Beamforming to Improve the Space Diversity

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A Novel Method of Beamforming to Improve the Space Diversity A Novel Method of Beamforming to Improve the Space Diversity Marco Antonio Acevedo Mosqueda, Emmanuel Martínez Zavala, María Elena Acevedo Mosqueda, and Oleksiy Pogrebnyak Abstract —At present, the systems of mobile communications The method to increase secondary lobes is described in demand major capacity in the services, therefore optimum space Section IV and finally, Section V presents different diversity is needed. In this paper, we propose a new method of simulations to obtain a radiation pattern by increasing the beam-forming to improve the space diversity and achieve magnitude of side lobes, which is comparable to the radiation frequency reuse. Our proposal consists in solving a system of pattern of Dolph-Chebyshev which has a polynomial order of linear equations from an 8-elements linear antenna array that increases the size of the side lobes. This method is compared with m = 8 and a maximum ratio of 4 dB side lobes. the algorithm of Dolph-Chebyshev. Our method shows competitive results in increasing the space diversity. II. ARRAY FACTOR FOR LINEAR WEIGHTED UNIFORM SYMMETRICAL ARRAYS Index Terms —Beamforming, Space diversity, Linear antenna An antenna array is a set of simple antennas which are array, Dolph-Chebyshev. connected under certain conditions, and they are usually equal and oriented in the same direction. They are arranged in a I. INTRODUCTION specific physical layout and relatively close with respect to each other. The combination of all these parameters provides a N RECENT DECADES , the interest in the antenna array has unique radiation pattern with desirable features such as width grown significantly due to it plays an important role in of the main lobe, the location of zero and maximum angles, mobile communications systems. Now these systems I among others [11]. demand more services due to in the constant increase in the Figure 1 shows the linear weighted uniform symmetrical number of users, and as a possible solution to this problem array used in this work. several studies have focused on a new mechanism known as division multiple access space (Space Division Multiple Access, SDMA) [1–7] which provides the ability to reuse resources (frequency bands, slots time, codes or any combination) simultaneously within the same service area. The goal of this paper is to propose a method to optimize the space diversity of a linear antenna array. A new technique of solving linear equations of a linear array of 8 elements to increase the magnitude of the side lobes and improve space diversity is proposed. This method is compared with the array factor of Dolph-Chebyshev [8] which is used to improve space diversity and it is considered important because it has the Fig 1. Linear weighted uniform symmetrical array. largest amplitude of side lobes around the main lobe side. In Section II the method for designing a weighted and In Figure 1, d is the distance between adjacent elements, θ symmetrical array factor (AF) [9] is described. The array has a is the angle of measurement and are the respective weights total of elements equal to 8 and an equidistant separation for each element of the linear array. between the elements of d= λ/2. Section III will show the The array factor is obtained by adding the result of the algorithm for calculating the factor of Dolph-Chebyshev array weighing of each element such that: [8], with a Chebyshev polynomial order m = 8 and a side lobe level R = 15 dB [10]. The radiation pattern of Section II is used for a first comparison with the pattern of Dolph-Chebyshev radiation in … (1) Section III. … , where k . We can express the above formula in terms of Manuscript received April 17, 2012. Manuscript accepted November 20, 2012. the cosine function using Euler's identity so that: M.A. Acevedo, E. Martinez, and M.E. Acevedo are with the Department of Telecommunications Section of Postgraduate Studies and Research, School of Mechanical and Electrical Engineering Zacatenco, DF, Mexico (e- ) mail: [email protected], [email protected], [email protected]). (2) O. Pogrebnyak is with the Center for Computing Research, National c s "2$ % 1'(, Polytechnic Institute (CIC-IPN), DF, México (e-mail: [email protected]). * pp. 93–98 93 Polibits (46) 2012 Marco Antonio Acevedo Mosqueda, Emmanuel Martínez Zavala, María Elena Acevedo Mosqueda, Oleksiy Pogrebnyak where , note that the array factor is maximum if For an 8-elements array factor, the positive and negative the angle' +,-.$/. angles where maximums and zeros appear are shown in In this/ paper 0 we use the following array factor for a Table I. number of elements equal to 8 with an equidistant distance between elements of d = λ/2. TABLE I. POSITIVE AND NEGATIVE ANGLES FOR MAXIMUM AND NULL . 12 14 Maximum Nulls 3 o o 1 (3) ±22.02 ±14.47 5 6 o o 1 ±38.68 ±30.00 89 7 : ±61.04 o ±48.59 o 14 12 o o ; ) ±119.8 ±90.00 Figure 2 shows the radiation pattern of the array factor used in this article. III. DOLPH -CHEBYSHEV ARRAY FACTOR The method Dolph-Chebyshev Array was initially investigated by Dolph and it is based on the studies by Chebyshev. Dolph in 1946 proposed a method to calculate the array factor; this method is based on the approximation to the structure of the linear array by means of the Chebyshev polynomials of order m [9]. 1) The Chebyshev polynomials (ChP) The ChP of order m are defined as: Fig 2. Radiation pattern of linear uniform symmetric array of 8 elements and equidistant distance of d=λ/2. T %1 c sh T 1 LXYYZ-[RUR U \ 1 (8) Null or zero angles of the array factor is obtained when: STU V c s "T 1 LXYYZ- U( % 1 \ U \ 1 c s ["T 1 LXYYZ-[U( U ] 1 Chebyshev polynomials can also be generated using the 3C (4) following recurrent formula: /<==8 sin @ AB D % EFG, $ 1,2,3,… where is element array, δ is the difference of the electrical (9) phase between two adjacent elements. For real angles sin θ is I ^ ^ ^3 less than or equal to one, therefore, the argument of the S U 2US U %S U. We find the first ChP by evaluating equation (9), and the equation must be less than or equal to one, this means that first eight polynomials are: only a finite set of values satisfy the equality. The angles of the side lobes are the greatest when: ( ) = 0 zt 1 () = 1 zzt (10) ()zt = 2z 2 −1 3JC (5) 2 () = 3 − 8 3 zt 4z 3z / sin @ AB D % EFG $ 1, 2, 3, … ()zt = 8z 4 − 8z 2 +1 The width of the main beam is defined as the angular 4 ()zt =16 z 5 − 20 z 3 + 5z aperture of the main lobe measured at a given constant power 5 ()zt = 32 z 6 − 48 z 4 +18 z 2 −1 level and it is obtained when: 6 ()zt = 64 z 7 −112 z 5 + 56 z 3 − 7z 7 (6) B3.;)3 It can be observed that if the module of z is greater than or /B sin @ A D % EFG, equal to 1 then the ChP are directly related to the cosine where are half-power points and they are function. normalized/J L$, / to a . The bandwidth of medium power is determined as follows: 0.707 2) Properties of Chebyshev polynomials We must take into account some of the properties of (7) Chebyshev polynomials: NOPQ R/J %/R Polibits (46) 2012 94 A Novel Method of Beamforming to Improve the Space Diversity a) All polynomials of any m order will pass through the point b) Within1,1. the range , the polynomials have values in [-1, 1] . %1 \ U \ 1 c) All null values occur within the range . d) The maximum and minimum values within %1 \the U range \ 1 z ∈ [-1, 1] have values and respectively. e) The higher the order of1 the polynomial,%1 the steeper the slope of The zeros of Chebyshev RUR ] 1 polynomials are calculated as follows: (2m −1)⋅π cos , (11) 2N Fig. 4. Radiation Patterns from Linear Array and Dolph- T 0, 1, … , I % 1 Chebyshev. In Figure 4 we can observe that the width of the main lobe 3) The designing of the Dolph-Chebyshev Array Factor is the same in both radiation patterns. The amplitude of the first side lobe of the radiation pattern from the linear array is The approximation of the linear array of antennas through greater than the amplitude of the first side lobe of the radiation the ChP is accomplished by the proposal of the number of pattern from Dolph-Chebyshev method.00000000000000000. elements in the array, in this case we proposed N = 8. The level of the amplitude of the second and third , (12) secondary lobes of the radiation pattern from Dolph- Chebyshev are greater than the second and third side lobes of where m is the Torder I %of 1 Chebyshev polynomial. the radiation pattern from linear array. The first null in both Chebyshev polynomials are identical to equation (2), if we radiation patterns is located at the same angle, but the second write them as follows: and third nulls from Dolph-Chebyshev are shifted with respect (13) to those from the linear array. D _* ` a This comparison suggests that there is not a significant where S U , = ChebyshevL YZ- coefficients.2$ % 1 ' 1b difference between the two radiation patterns. The level YZ- / of side lobesL in dB and the rectangular shape can be obtained by: IV. METHOD FOR INCREASING SIDE LOBES For increasing the side lobes the weighing of the antenna lm d k d d 3k array has to be chosen so that a null value will be placed in c Ref gh R , c 20iZj c , c 10 , 1n where is array factor of side lobes, is the level of the calculated directions and a maximum value will be placed in main lobe =8 and R is the level of the side lobescd with respect to the angles of the side lobes and leaving the peak of the main the main lobe.
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