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Space-Borne Multibeam Array Pattern Synthesis for Increasing Capacity.

1. INTRODUCTION

In SatCom system, the contradiction between high gain and wide coverage of antenna can be resolved by space-borne multibeam array effectively, which has been a typical payload in global SatCom system [1, 2]. One of the key technologies in space-borne multibeam array antenna is the power pattern synthesis technology, which requires not only broad main-lobe beamwidth, but also low side-lobe level (SLL). It may be quite difficult for the classic analytic method, such as Dolph-Chebyshev, to achieve a satisfactory multibeam pattern, especially when the numerous elements are used.

Since genetic algorithm (GA) was applied to pattern synthesis by Johnson and Rahmat-Samii in 1994 [3], evolutionary algorithm has been widely used in the large-scale array pattern synthesis problem. 16 shaped beams were synthesized via real-code GA (RCGA) algorithm in space-borne hexagonal transmitter array with 61 elements [4, 5], and Recioui employed PSO to suppress SLL in linear array [6]. Recent advances in intelligent pattern synthesis focus on merging different kinds of intelligent algorithm, such as hybrid differential evolution (DE), and artificial bee colony (ABC) synthesis method has been applied to reconfigurable array synthesis [7]. Another trend in this field is the synthesis method based on convex optimization, which has remarkable global convergence properties and fast solving speed via interior-point algorithm. Wang et al. and Fuchs et al. proposed sequential convex optimization algorithm to implement arbitrary array pattern synthesis [8, 9].

However, a major problem of the above-mentioned synthesis method is considering only the array radiation performance improvement, no matter whether antenna pattern enhances the whole communication system performance or not. User capacity in multibeam SatCom system is a key performance metric, and almost no objective of space-borne array pattern synthesis method is to maximize the user capacity in the current SatCom system. Gilhousen et al. firstly analyzed the capacity of multibeam SatCom system using CDMA, but the analysis did not consider the inter-cell interference caused by side-lobe of beam pattern [10]. Fu et al. established a comprehensive capacity analysis model of multibeam CDMA-based SatCom and showed the effect of imperfect power control; however, the basis of their analysis relied on a non-shaped tapered-aperture antenna pattern, such that the accuracy of capacity analysis would decline [11, 12]. In [13], seven-beam pattern with "iso-flux" coverage was synthesized by GA, which is performed for standard hexagonal array (SHA), and capacity was analyzed in the 7-beam CDMA-based SatCom system with the existence of power control error (PCE), but this study made no attempt to optimize beam pattern for capacity increase.

This paper focuses on the pattern synthesis of receiving SHA with 19 elements, which is aimed at increasing user capacity under the background of multibeam SatCom system using CDMA. The outline for this paper is listed below. Section 2 shows the gain pattern model of SHA, and then develops capacity analysis model in practical shaped 7-beam SatCom system using CDMA. In Section 3, on the basis of the initial multibeam pattern given by RCGA, SA-based PSO for increasing capacity pattern synthesis algorithm is described in detail. Section 4 provides the numerical simulation results. Finally, some meaningful conclusions are presented in Section 5.

2. SYSTEM DESCRIPTION

2.1. SHA Antenna Pattern and Seven-Beam Coverage

The geometry of receiving SHA antenna with 19 elements is depicted in Figure 1. Element space is 0.545[lambda], where [lambda] is the wavelength of the L-band sinusoidal wave. The array response can be written as:

S([theta], [phi]) = [19.summation over (i=1)] [I.sub.i] exp {j[k.sub.0]([x.sub.i]u + [y.sub.i]v) + [[alpha].sub.i]} [F.sub.i]([theta], [phi]) (1)

where [theta] and [phi] are the elevation and azimuth angles in the spherical coordinate, and [I.sub.i] and [[alpha].sub.i] are amplitude and phase excitation of element i in Figure 1. [x.sub.i] and [y.sub.i] are x- and y-coordinate positions of element i. [k.sub.0] = 2[pi]/[lambda] is the wavenumber, and u = sin [theta] cos [phi], v = sin [theta] sin [phi]. In (1), [F.sub.i]([lambda], [phi]) is the radiation pattern of individual element i, which is a dual-fed and dual-layer stacked patch antenna. Figure 2 shows three actual patterns of elements 2, 8 and 15 at [phi] = 90[degrees], measured in spherical near-field anechoic chamber (Satimo) [14]. The normalized gain of SHA is given as follows:

[mathematical expression not reproducible] (2)

where [eta] [approximately equal to] 0.9 is radiation efficiency.

We employ the uplink seven-beam gain coverage mentioned in [13]. The ideal beam coverage is shown in Figure 3. The inner coverage contains center beam 1, whose half-power beamwidth (HPBW) of [lambda] is required for greater than [+ or -] 20[degrees]. Outer coverage has six beams, the [lambda]-HPBW of beam 1/6/7 is 25[degrees] ~ 55[degrees], and the left beam's 9-HPBW is -55[degrees] ~ -25[degrees]. The desired seven-beam patterns are generated under the above requirements, and RCGA-based pattern synthesis method is adopted, which minimizes the mean-squared error between synthesized pattern and the desired pattern (discuss in Section 3). Figure 4 depicts the optimized seven-beam gain contours in uv space, and the dashed circle means the coverage requirement of [+ or -] 55[degrees].

2.2. The Interference Model of Seven-Beam CDMA-Based SatCom System

The interference of the desired user in the single satellite system using CDMA is from: a) the multiple access interference from other users in the same beam cell; b) interference from the users of adjacent beam in the same satellite; c) co-channel interference from other satellite systems. This paper does not take intra-system interference into account. It is assumed that users are distributed uniformly in all seven beams shown in Figure 3, and the user of interest is in the center beam. [I.sub.S0] and [I.sub.S1] represent the interferences of categories a) and b), respectively. The total interference [I.sub.S] of user of interest is the sum of [I.sub.S0] and [I.sub.S1].

Power control is employed to reduce the multiple-access interference in CDMA SatCom System, in order to improve user capacity. The round trip time (RTT) of LEO SatCom is greater than 10 ms, so open-loop power control is used generally [15]. If imperfect open power control is taken into account, [I.sub.S0] is expressed as follows:

[mathematical expression not reproducible] (3)

where [P.sub.0] is the nominal user's terminal power into space-borne receiver with perfect power control. [gamma].sub.jk] is the kth user's open-loop PCE in the jth beam cell, and ([j.sub.0], [k.sub.0]) [not member of] S0 means the user in center beam except the users of interest.

Beam cell is essentially formed via HPBW area of beam pattern. Due to the roll-off of main-lobe and SLL, multibeam cells are not absolutely separated. So, [I.sub.S1] is written as follows:

[mathematical expression not reproducible] (4)

where (j, k) [member of] [S.sub.1] means all the users in outer six-beams, and 2 [less than or equal to] j [less than or equal to] 7. [[beta].sup.2.sub.jk] is the antenna gain pattern discrimination factor, which is given as:

[[beta].sup.2.sub.jk] = [G.sub.1] ([[theta].sub.jk], [[phi].sub.jk])/[G.sub.j] ([[theta].sub.jk], [[phi].sub.jk]) (5)

where [G.sub.j] ([[theta].sub.jk], [[phi].sub.jk]) is the k th user's pattern gain of the j th cell, and the practical meaning is power compensation that keeps the same power in seven-beam coverage. [G.sub.1] ([[theta].sub.jk], [[phi].sub.jk]) is the kth user's gain of the center beam cell, which reflects the amount of the isolation between different beams. Formula (5) establishes an important relationship between the multibeam pattern and SatCom system, which is the basis of the following discussion.

2.3. Capacity Analysis

With the increase of users in SatCom system, the interference rises to the user of interest. When the desired user's bit energy-to-noise radio is below the threshold, the communication will be broken off. Wherefore, the outage probability of desired user is defined as:

[P.sub.out] = Pr {EINR [less than or equal to] EIN [R.sub.req]} (6)

where EINR can be expressed as:

EINR = B/[R.sub.b] x [[gamma].sub.0][P.sub.0]/I + [eta] (7)

where [[gamma].sub.0] is the PCE of the desired user, B the channel bandwidth, [R.sub.b] the information bit rate, and [eta] the noise power. After some simple transformations, the outage probability can be given by:

[mathematical expression not reproducible] (8)

where [[eta].sub.0] is noise power spectral density, and [[eta].sub.0] = k[T.sub.s], where k is the Boltzmann constant and [T.sub.s] the equivalent system noise temperature of SHA antenna. [E.sub.b]/[[eta].sub.0] in Equation (8) can be written as:

[mathematical expression not reproducible] (9)

where [P.sub.t] and [G.sub.t] are the transmitting power and antenna gain of the user terminal, respectively. [G.sub.r] is the receiving gain of space-borne SHA antenna, d the distance between user terminal and satellite, [L.sub.a] the axial ratio loss of SHA antenna, and [N.sub.F] the noise figure of RF-front system.

The total normalized interference [I.sub.s]/[P.sub.0] is expressed as:

[mathematical expression not reproducible] (10)

where [[gamma].sub.jk] is lognormal random variable [15], and [mathematical expression not reproducible], where [delta].sub.jk] is a zero-mean Gaussian random variable in nepers. The mean and variance of [I.sub.s]/[P.sub.0] are given by:

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

where [g.sub.1] and [g.sub.2] are shown as:

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

In (13) and (14), A is shadow probability factor, and A is usually 0.3 in LEO [18]. [[sigma].sub.s] and [[sigma].sub.us] are the standard deviations of PCE for shadowed users and unshadowed users in dBs, respectively. h = ln(10)/20, which converts nepers into dBs.

When the desired user's PCE [[gamma].sub.0] is fixed, the conditional outage probability is:

[P.sub.out] | [delta] = 1/2 er fc ([??] - [[mu].sub.I]/[square root of 2[[sigma].sup.2.sub.I]) (15)

where [??] is expressed as:

[??] = B/[R.sub.b] [[e.sup.[delta]]/EIN [R.sub.reg] - 1/[E.sub.b]/[[eta].sub.0]] (16)

because the desired user may be in shadowed or unshadowed state, the unconditional outage probability is given by:

[mathematical expression not reproducible] (17)

where f ([[sigma].sub.s]) and f ([[sigma].sub.us]) are the Gaussian distributions of [sigma] for shadowed and unshadowed states, respectively. Formula (17) contains only one variable: user capacity N, which can be obtained when [P.sub.out] is fixed.

3. PATTERN SYNTHESIS FOR INCREASING CAPACITY 3.1. Initial Element Excitations

The initial element excitation is acquired from RCGA-based pattern synthesis method, whose fitness function is conveyed as follows:

[mathematical expression not reproducible] (18)

where [N.sub.[theta]] and [N.sub.[psi]] are the numbers of samples in the elevation plane and azimuth plane. [G.sub.d]([[theta].sub.m], [[phi].sub.n]) is the desired beam pattern. [G.sub.d1] ([[theta].sub.m], [[phi].sub.n]) and [G.sub.d2] ([[theta].sub.m], [[phi].sub.n]) are the desired patterns of center beam and beam 2 and given by:

[mathematical expression not reproducible] (19)

The excitation of each element's amplitude and phase is selected as gene, and the float value of excitation is acted as gene encoding. The structure of this chromosome is defined as: ([I.sub.1], [I.sub.2], ..., [I.sub.19], [[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.19]). The parameters of RCGA-based pattern synthesis are shown as follows: a) the population size is 200, and the maximum number of generations is 250. b) the generation gap factor is 0.96, and re-insect method is applied that can replace least-fit parents. c) intermediate crossover is used, and the crossover probability is 0.9. d) the mutation probability is 0.15.

3.2. Simulated Annealing-Based PSO Pattern Synthesis Algorithm

PSO is an effective population-based intelligence algorithm inspired by the social behavior of bird flocking or fish schooling. In PSO, each particle searches the best positions in multi-dimensional space according to its self-cognitive and social experience. Clerc and Kennedy proposed inertial weight and constriction factor to adjust the velocity of the particle for obtaining better convergence [16].

We select the chromosome defined in RCGA as the position vector in our PSO algorithm, and in the (n + 1)th iteration the position of ith particle is given by:

[x.sub.i](n + 1) = [x.sub.i](n) + [v.sub.i](n + 1) (20)

where [x.sub.i](n) is the position vector of nth iteration, and [v.sub.i](n + 1) is velocity vector in the (n + 1)th iteration, which determines the search direction and can be expressed as:

[v.sub.ij] (n + 1) = [chi] [[v.sub.ij] (n) + [c.sub.1][r.sub.1]j (n) ([p.sub.ij] (n) - [x.sub.ij] (n)) + [c.sub.2][r.sub.2]j (n) ([[??].sub.gj] (n) - [x.sub.ij] (n))] (21)

where [v.sub.ij](n) is the dth component of velocity vector of ith particle, and j = 1, 2, ..., 38. [c.sub.1] and [c.sub.2] are cognitive and social parameters, respectively, and they are always positive constants, which control the impact of the local-best position [p.sub.ij] (n) and global-best position [[??].sub.gj] (n). Both [r.sub.1j] (n) and [r.sub.2j] (n) are random numbers distributed uniformly between 0 and 1. [chi] is constriction factor, and [chi] = 2/[absolute value of 2-C-[square root of ([C.sup.2]-4C)], where C = [c.sub.1] + [c.sub.2], and the general requirements C > 4 can obtain better convergence.

Although constriction factor can improve the global convergence of PSO, the immature convergence problem cannot be avoided. A PSO based on SA pattern synthesis algorithm is applied in this paper. SA algorithm imitates the temperature falling process of heated solid, and it is a quite effective method to avoid trapping into local optimum [17]. The thoughts of the proposed algorithm is that [[??].sub.gj] (n) in (21) is updated in probability under the control of SA. In this algorithm, the normalized receiving probability is calculated for every particle, which is expressed as follows:

[mathematical expression not reproducible] (22)

where [f.sub.i] = f ([X.sub.i](n)), [f.sub.g] = f ([[??].sub.g](n)). T = T(n) is annealing temperature parameter, which decreases with increasing time. According to the receiving probability, the global best particle is selected by roulette wheel strategy, and the selection probability P[b.sub.r] is defined as:

P[b.sub.r] = [P.sub.r]/[r.summation over (i=1)] [P.sub.i] (23)

where [P.sub.r] is the receiving probability of the rth particle and [p.sub.bet] a random number between 0 and 1. When P[b.sub.r] [greater than or equal to] [p.sub.bet], the position of rth particle is chosen as global best position in the current iteration.
Procedure: SA-based PSO for Increasing Capacity (SAPSOIC) pattern
synthesis algorithm

Step 1: Set the number of particle [N.sub.p] and the maximal
        iteration number M. Initialize the parameter [c.sub.1] and
        [c.sub.2], and the annealing constant [[lambda].sub.SA].
        Set the limit of user per beam [[N.sub.l], [N.sub.u]].
Step 2: Set [x.sub.i](0) = w(0) + [DELTA][w.sub.i], where
        [DELTA][w.sub.i] = [[[DELTA][I.sup.0.sub.1x19],
        [DELTA][[alpha].sup.0.sub.1x19]].sup.T],
        [DELTA][I.sup.0.sub.1x19] and
        [DELTA][[alpha].sup.0.sub.1x19] are randomized uniformly
        vector in [-2, 2], and [0, 2[pi]], i = 1, 2, ..., [N.sub.p].
        w(0) is the initial excitations from RCGA.
Step 3: for m = [N.sub.l] to [N.sub.u]
          Calculate [P.sub.out], set [f.sub.i] = [P.sub.out], and
            evaluate [f.sub.g] = min = ([f.sub.i], 1, 2, ...,
            [N.sub.p]), set the best position [p.sub.g];
         Set the initial annealing temperature T = -[f.sub.g]/ln (5);
         for t = 1 to M
           Evaluate receiving probability [P.sub.i] of every particle;
           for r = 1 to [N.sub.p]
             Evaluate every particle's selection probability
             P[b.sub.r]; [p.sub.bet] = rand(0, 1);
             if [p.sub.bet] [less than or equal to] P[b.sub.r]
               [p.sub.g] = [x.sub.r](t);
               break;
               end if
           end for
           Update the velocity vector [v.sub.i](n + 1) and, position
             vector [x.sub.i](n + 1), i = 1, 2, ..., [N.sub.p];
           Evaluate the new fitness of every particle, and, update
             [p.sub.g] and [p.sub.i], i = 1, 2, ..., [N.sub.p];
           Update T = [[lambda].sub.SA]T;
         end for
         Randomize the new excitations [DELTA][w.sub.i], and set
           [x.sub.i](0) = [p.sub.g] + [DELTA][w.sub.i], i = 1, 2, ...,
           [N.sub.p];
Step 4: end for
end procedure


Capacity N is given in terms of outage probability, and it is quite difficult to derive the analytic expression of capacity. However, outage probability is risen as capacity is increased, and if outage probability is minimized for a given capacity, then in the same requirements of outage probability, e.g., [P.sub.out] = 10%, communication system has more considerable user capacity. So, the optimized model of SA-based PSO pattern synthesis is given as:

[mathematical expression not reproducible] (24)

where [N.sub.u] is the current user number per beam, which will be increased with the algorithm iterations. The pseudocode of SA-based PSO for Increasing Capacity (SAPSOIC) pattern synthesis algorithm is given.

4. NUMERICAL SIMULATION

The geometric formula is given in [13], which is about the distance r from user's position in beam coverage to the sub-satellite point and the elevation [theta] of SHA antenna, and the azimuth of user in beam is the same as [phi] of antenna. By randomizing r and [phi] repetitively, users are distributed uniformly in the overall beam coverage, and then the initial pattern gain of user is immediately obtained according to the excitations of RCGA synthesis. Finally, the initial outage probability can be calculated by means of Monte-Carlo method. In every iteration of our proposed algorithm, the fitness function, i.e., outage probability, is evaluated by repeating the same process mentioned above.

4.1. Input Parameter

The main SatCom system parameter is listed below: satellite orbit altitude H = 900 km; bit rate [R.sub.b] = 9.6 kbps; channel bandwidth B = 7 MHz; user terminal [P.sub.t][G.sub.t] = 5 dBw; axial ratio loss [L.sub.a] = 3 dB; noise figure [N.sub.F] = 3 dB; system noise temperature [T.sub.s] = 500 K. Additionally, according to [15], the typical standard deviation [[sigma].sub.s] of PCE for shadowed user is 2 dB ~ 4 dB, and the [[sigma].sub.us] value of PCE for unshadowed user is 1 dB ~ 2 dB. The required threshold EIN[R.sub.req] is 5.5 dB.

4.2. Numerical Results

4.2.1. The Result of SAPSOIC Multibeam Pattern Synthesis Method

In Figures 5 and 6, 3-D gain pattern and contour in uv-coordinate of SAPSOIC synthesis are depicted, respectively. Compared with Figure 4, the gain is more focused in the required coverage area ([+ or -] 55[degrees]) after SAPSOIC pattern synthesis, and the optimized pattern gain at the edge of beam coverage is decreased, but the gain in only a fraction of outskirts area is dropped to 9 dB. Moreover, the area where gain is greater than 12 dB is enlarged in center beam contour of Figure 6, and meanwhile, the beam gain is increased in overlap area (between center beam and outer six beams).

Figure 7 shows sectional gain patterns of beam 1, beam 4 and beam 7 at [phi] = 0[degrees] before SAPSOIC synthesis optimization, and the optimized result is depicted in Figure 8. It is apparent that the first pattern null of optimized center beam moves to -55[degrees] and 55[degrees], and the maximal main-lobe gain of center beam is increased by about 3 dB. Moreover, the roll-off of these three beams' main-lobe is much steeper than non-optimized case. Both the optimized outer beam 4 (plotted as a dashed line with "*" marker) and beam 7 (plotted as a dashed line) patterns have moving trends toward the middle. The sectional beam pattern comparisons among beam 1 and beam 3 and beam 6 at [phi] = 120[degrees] are described in Figures 9 and 10, and Figures 11 and 12 illustrate the non-optimized and optimized sectional beam patterns of beam 1, beam 2 and beam 5 at [phi] = 210[degrees] and give the same results.

According to the above comparison, it appears clearly that the optimized pattern achieves a "max-gain-flux" seven-beam coverage. Generally, the "iso-flux" beam coverage is a final objective pursued by many traditional pattern synthesis methods of space-borne multibeam antenna [19,20], as depicted in Figure 7, where center-beam gain is less than outer six-beams gain. However, in terms of capacity increase, it is strongly suggested that "max-gain-flux" beam coverage has the competitive advantage over the "iso-flux" beam coverage in multibeam SatCom system using CDMA.

4.2.2. Capacity Analysis Result

Figures 13 and 14 compare the outage probability affected by PCE for unshadowed and shadowed users in two kinds of beam coverage, respectively. The values of PCE parameter, [[sigma].sub.s] and [[sigma].sub.us], are 2.5 dB in Figure 13 and 1 dB in Figure 14. As shown in these two figures, user capacity is reduced obviously with [[sigma].sub.us] or [[sigma].sub.s] increases. Particularly, user capacity is improved more than one time higher in the "max-gain-flux" coverage synthesized by SAPSOIC than that of non-resynthesis "iso-flux" coverage. As shown in Figure 13, when [P.sub.out] = 10%, the user number per beam is only 5 at [[sigma].sub.us] = 1 dB (depicted as solid line) and 7 at [[sigma].sub.us] = 0.5 dB in the original seven-beam coverage; however, the user numbers per beam cell are nearly 11 and 15 in the same condition after SAPSOIC synthesis. In Figure 14, user capacity is about 3 when [P.sub.out] = 10%, [[sigma].sub.s] = 3.5 dB in the non-resynthesis coverage, but capacity is increased to 8 in the synthesized "max-gain-flux" coverage.

5. DISCUSSION

According to the results of Section 4, user capacity is improved obviously under the "max-gain-flux" beam coverage; however, the quality of coverage in the outskirts area is degraded slightly, compared with the original seven-beam pattern. The decrease of gain at the edge of beam coverage is acceptable in LEO SatCom system, due to the high-speed moving of LEO satellite. It is assumed that the satellite moves in near-circular orbit, and the moving velocity v of this satellite is expressed as: v = [square root of (GM/([R.sub.e] + H))], where G is gravity constant, M the mass of earth, and [R.sub.e] the radius of earth. So v [approximately equal to] 6.49 km/s. The maximal spending time of one user from [theta] = 55[degrees] to [theta] = 54[degrees] is about 13 s, without considering earth rotation. The time in poor coverage is short, and the beam gain of user at the edge of coverage formerly can be improved quickly.

6. CONCLUSION

This paper uses a standard hexagonal array antenna combined with SA-based PSO to increase user capacity for CDMA-based SatCom system. We hunt an important relationship between user capacity and multibeam pattern, and propose a novel object function of array pattern synthesis, which is communication outage probability instead of the antenna metric function. The most interesting finding of this paper is that user capacity is doubled at least in the optimized beam coverage. For increasing capacity in the multibeam SatCom system using CDMA, the space-borne array pattern should have the following features: 1) the beam pattern needs to drop rapidly in the overlap area of multiple beam; 2) the "max-gain-flux" multibeam coverage can improve capacity compared with the traditional "iso-flux" coverage, at the expense of the gain decrease in some outskirts area. Although the degradation of coverage is acceptable, the multi-objective evolutionary algorithm (MOEA) is considered to seek a tradeoff between the quality of coverage and the maximal user capacity in the future.

ACKNOWLEDGMENT

The work was supported by the Innovation Foundation of Chinese Academy of Sciences (No. CXJJ-11-S107) and the Natural Science Foundation of Shanghai City (11ZR1435000).

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Haiwei Song (1,2),*, Guang Liang (1,2), Wenbin Gong (2), and Jinpei Yu (1,2)

Received 6 January 2014, Accepted 25 February 2014, Scheduled 12 March 2014

* Corresponding author: Haiwei Song (shw.phd@hotmail.com).

(1) Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Science, China. (2) Shanghai Engineering Center for Microsatellites, China.

Caption: Figure 1. The geometry of SHA antenna.

Caption: Figure 2. Measured element pattern at [phi] = 0[degrees].

Caption: Figure 3. The desired seven-beam pattern coverage of receiving SHA antenna.

Caption: Figure 4. The contour of RCGA-based synthesis seven-beam pattern antenna.

Caption: Figure 5. 3D seven-beam gain pattern of SAPSOIC synthesis.

Caption: Figure 6. The contour of SAPSOIC-based synthesis seven-beam pattern.

Caption: Figure 7. The sectional gain patterns of the original three beams at [phi] = 0[degrees].

Caption: Figure 8. The sectional gain patterns of the optimized three beams at [phi] = 0[degrees].

Caption: Figure 9. The sectional gain patterns of the original three beams at [phi] = 120[degrees].

Caption: Figure 10. The sectional gain patterns of the optimized three beams at [phi] = 120[degrees].

Caption: Figure 11. The sectional gain patterns of the original three beams at [phi] = 210[degrees].

Caption: Figure 12. The sectional gain patterns of the optimized three beams at [phi] = 210[degrees].

Caption: Figure 13. User capacity affected by [[delta].sub.us] and beam pattern ([[delta].sub.s] = 2.5 dB, EIN[R.sub.req] = 5.5 dB).

Caption: Figure 14. User capacity affected by [[delta].sub.s] and beam pattern ([[delta].sub.us] = 1 dB, EIN[R.sub.req] = 5.5 dB).
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Author:Song, Haiwei; Liang, Guang; Gong, Wenbin; Yu, Jinpei
Publication:Progress In Electromagnetics Research
Date:Feb 1, 2014
Words:4948
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