A Method for Designing Fixed Multi-Beam Antenna Arrays in WCDMA Systems Afif Osseiran∗1,2 and Andrew Logothetis1 Ericsson Research, 16480 Stockholm, Sweden, 2 Royal Institute of Technology (KTH), Stockholm, Sweden ∗ Tel: +46858532670 Fax: +4687575720 E-mail:
[email protected]
Abstract— In this paper, Simulated Annealing is used for designing a fixed multi-beam antenna array (AA) in a multi-user radio WCDMA system. The method can be used dynamically in a real system and/or in a radio planning tool. The proposed method is applied on a linear array by designing the Antenna Element (AE) pattern, the number of fixed beams, the AE spacing, and the AA transmit weights. The proposed method leads to a substantial BS power reduction which can be used to increase the system capacity or data throughput in a WCDMA system.
I. I NTRODUCTION Conventional wireless systems make use of omni-directional or sectorized antenna systems. The major drawback of such antenna systems is that electro-magnetic energy, intended for a particular user located in a certain direction, is radiated unnecessarily in every direction within the entire cell (i.e. sector). One way to limit this source of interference and direct the energy to the desired user, is to introduce smart antennas. Smart antennas can be divided into two major groups depending on the level of intelligence: fixed multi-beam (also called switched beam) systems and steered beam (also called adaptive array) systems. Fixed multi-beam systems consist of a finite number of fixed beams with predefined beam patterns and fix pointing directions. In the uplink (UL) the antenna elements are used to collect all incoming energy resulting in increased antenna gain, while in the downlink (DL) the beam with the largest UL received power will be used for transmission to the mobile of interest. The second type of antenna system is the more advanced steered beam solution. That is, the shape of the beams may or may not be fixed but the pointing directions are steered towards the mobile of interest both in UL and DL directions within the cell. Besides requiring stringent uplink and downlink calibration (coherency requirement), the steered beam solution shows only a marginal system gain [1] relative to a fixed multi-beam system in WCDMA. The two major impairments that limit the full potential offered by a fixed-beam Antenna Array (AA) system are (see Fig. 1) : 1) Straddling loss: which represents the loss of the antenna gain of the fixed multi-beam system compared to the steered beam system for users who are not located in the beam direction, 2) Side-lobes: which naturally arise in antenna arrays that must be suppressed as much as possible since they generate interference to users not located in the main beam. In this paper, AA parameters are optimally designed to minimize the total BS power required to guarantee an ac-
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Fig. 1.
Straddling loss and side lobes of a three fixed-beam antenna array.
ceptable Quality of Service (QoS) to all users within a cell of a WCDMA system. The AA parameters that are optimized are: Antenna Element (AE) pattern (azimuth 3dB beam-width), number of fixed beams, AE spacing, and AA transmit weights. Simulated Annealing (SA) is the method used to perform the optimization. The method can be used dynamically (e.g online) in a real network system and/or (off line) in a classical radio planning tool. It in interesting to note that the proposed beam-shape optimization method can be used to optimize the uplink system capacity, where for instance the AA parameters can be designed in order to minimize the noise rise. II. S IMULATED A NNEALING SA [2], [3] is a widely known simulation optimization technique. Its strength remains in the fact that it renders possible to compute the global maxima of a multi-variable nonlinear function and avoid being trapped in local maxima, whereas other optimization methods tend to get trapped in local maxima. SA methods are based on an analogy with thermodynamics and the way that liquids freeze and crystallize. In short, SA consists of finding a good solution to an optimization problem by trying random variations of the current solution. In SA the constraints are expressed through
the minimization of a loss function or cost function. Starting from an initial configuration (which corresponds to a value of the cost function), a sequence of iterations is generated. Each iteration consists of 1) random selection of a configuration from the neighborhood of the current configuration and, 2) the calculation of the corresponding change in cost function. The neighborhood is defined as a transition from one configuration into another by provoking a small perturbation. The transition is unconditionally accepted if the cost is lowered (corresponds to a negative cost function). On the other hand, a higher cost is accepted as the new solution with a probability which decreases as the computation proceeds. Toward the end of the iteration, the probability of accepting a worse solution (higher cost function), is nearly zero, and thus the solution converges to a global maximum or at least proceeds away from local maxima depending on the formulation of the cost function. The application of SA to optimize linear sparse arrays was first suggested by [4], [5] for acoustic imaging applications. The optimizations in [4], [5] were limited to a single beam and thus they are inappropriate for optimizing fixed multi-beam systems in a multi-user dynamic system. III. S YSTEM S ETUP AND A LGORITHM Let N denote the number of elements of the AA, θ3dB is the 3dB azimuth beam-width of each AE, d is a 1 × N vector representing the coordinates of the AEs, b is the number of beams, and finally W is the N × b BeamForming Network (BFN) matrix (e.g. the kth column of W has unit norm and represents the weights of the kth beam). The simulation consisted of a large number of snapshots where for each snapshot, a number of users are generated randomly in a cell. Taking into account the fast fading, the log-normal fading and assuming the COST 231-Walfisch-Ikegami path loss model, the BS power is computed per snapshot. Some of the relevant system deployment parameters are summarized in Table I. The aim of the proposed method is to optimize a fixed multibeam system such that the BS transmit power is minimized given that all served users are guaranteed an acceptable QoS. Hence the SA algorithm aims simply at minimizing the total consumed power by exciting some of the linear array parameters: AE pattern (azimuth 3dB beam-width), number of fixed beams, AE spacing and AA transmit weights. Parameter Cell radius [m] Path Loss Max BS output power [W] Average Orthogonality factor Spreading Factor base station height STD of the Log Normal fading Noise power
Value 1000 COST 231-Walfisch-Ikegami 20 0.64 128 50 8 7dB
TABLE I S YSTEM AND S IMULATIONS PARAMETERS .
A. The Algorithm A flowchart of the SA optimization algorithm is shown is Fig 2. Before describing the SA algorithm, let us define the following: • •
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λ is the wave length. Pi (b, θ, d, W ) is the total BS power of the ith snapshot corresponding the linear array parameterized by (b, θ, d, W ). IN ×N is the N × N identity matrix. fft(X) is the discrete Fourier transform of a matrix X. randn() sample of a normally distributed random variable with mean 0 and variance 1. crandn() sample of a complex circulant symmetric normally distributed randompvariable (i.e. crandn() = (randn() + i ∗ randn())/ (2)). rand() sample of a uniformly distributed random variable between 0 and 1. randint(1, N ) generates a random integer between 1 and N . Pb , Pd , Pθ and PW are respectively the probability of updating the number of beams, AE spacing, AE 3dB beam-width, and BFN matrix. αd , αθ and αW denotes the standard deviation of the perturbation for the AE spacing, the AE beam-width, and the transmit weight, respectively. τk is the cooling schedule of the SA (limk→∞ τk = 0).
First, the antenna parameters are initialized. Then for each iteration k, a random number is generated by the random generator rand(). This number is compared to Pd . If it is greater than Pd then the array spacing is kept as it is, otherwise the AE spacing is slightly modified. Once all AE spacing have been perturbed, then the AE beam-width, the AA transmit weight and the number of beams are perturbed in the same manner. Once these parameters are perturbed, the total BS e d, eW f ), of the ith snapshot corresponding the power, Pi (eb, θ, e d, eW f ), is computed. newly perturbed array parameters (eb, θ, Then the power difference ∆Pi between the newly computed BS power and one from the previous SA iteration is computed for all snapshots. The cost function ∆P , is defined as the 95th percentile of the power difference for all snapshots (i.e P r{∆Pi ≥ ∆P } = 0.95). Once the decision to update the array parameters is done, then the iteration index k is increased by one. This process will be repeated until k reaches the value K (i.e. the desired number of simulated annealing iterations). IV. R ESULTS Several aspects of the proposed schemes have been evaluated and tested via simulations. The solution is illustrated by two examples. In the first example, the improvement and robustness of the proposed method is shown in case of weight optimization, only i.e Pb = 0, Pd = 0 and Pθ = 0. Whereas the second example demonstrates the effectiveness of the proposed method by jointly optimizing the number of beams and the beam shape, i.e. Pd = 0 and Pθ = 0.
k: = 1, θ := π / 2
b := N ,
W := fft ( I N × N ), d := [0 λ / 2
P := ∞
( N − 1)λ / 2]
~ d =d ~ θ =θ ~ b =b ~ W =W
n = randint(1, N ) ~ d (n) = d (n) + α d randn()
yes
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yes
k = k +1
rand() < Pd
~ d =d ~ θ =θ ~ b=b ~ W =W
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STOP
~
θ = θ + αθ randn()
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yes ~ n = randint (1, N ) m = rand int(1, b ) ~ ~ ~ ~ W (n, m) = W (n, m) + α W crandn () W (:, m) = W (:, m) / W (:, m)
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rand() < Pb 2
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∆P = 95th percentile of ∆Pi no
randn() < 0
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~ b = b +1
~ ~ ~ ~ ∆Pi = Pi (b ,θ , d ,W ) − Pi (b,θ , d , W ), ∀i
~ insert new column in W
~ b = b −1
no
~ randomly delete one column in W
(a) Fig. 2.
Algorithm Description for designing a Fixed-Beam Antenna Array.
A. Antenna Weight Optimization
B. Number of Beams & Antenna Weight Optimization
In this example, the downlink power consumption was reduced by exciting the transmit antenna weights when the traffic load was high. A conventional three column AA was assumed. The conventional fixed multi-beam antenna pattern is depicted in dashed line in Figure 3. By applying the proposed method, the DL power was reduced by up to 50% compared to a Butler BFN Matrix (BFNM). The optimized beam weights are shown in bold in Figure 3. It can be observed that the interference from other beams is minimized at the cross over regions, in contrast to the conventional solution where the nulls are located near the maximum of the beam lobes, the nulls in the optimized case are closely located at the cross over regions. Note that the side lobe levels were also suppressed by approximately 4 to 5dB, and finally the straddling loss was slightly reduced.
In order to overcome the straddling loss problem it may be necessary to increase the number of beams while keeping the number of antenna columns fixed. On the other hand, introducing additional beams will require extra power resources for the complementary common pilot signals, since each fixed beam is associated with a unique pilot signal used for pilot assisted channel estimation at the mobile. Therefore there is a trade-off between increasing the number of beams and the total power allocated to the common pilots and data signals. In Figure 4, a four multi-beam antenna system generated by an AA of four columns served as a starting point to optimize the BS consumed power. In a fashion similar to the previous example, the number of beams and the antenna weight were exited until the simulated annealing converged to the global maximum or at least avoided to be locked in a local maximum.
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Fig. 3. Weight optimization of a three fixed beams antenna system (optimized beams are in bold lines whereas original ones in dotted lines).
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V. C ONCLUSIONS Simulated Annealing is used for designing a fixed multibeam antenna linear array in a multi-user dynamic WCDMA simulator. The objective is to minimize the total BS power required to guarantee an acceptable quality of service to all users within a cell. The advantages of the proposed method may be summarized as follows: • Straddling Loss Mitigation: by optimizing the number of fixed beams and their patterns. • Interference Management: since side-lobes are suppressed and the nulls are steered towards the cross over regions. • Resource Optimization: the BS power was reduced substantially which translates to system capacity increase or higher system throughput or lower hardware cost. System simulations showed that the proposed method yields up to 50% reduction in the transmitted BS power compared to a conventional Butler beamforming network matrix. ACKNOWLEDGEMENT The authors would like to thank Dr. Sverker Magnusson from Ericsson and the reviewers for their helpful comments. R EFERENCES [1] A. Osseiran et al., “Downlink Capacity Comparison between Different Smart Antenna Concepts in a Mixed Service WCDMA System,” in Proceedings IEEE Vehicular Technology Conference, Fall, vol. 3, Atlantic City, USA, 2001, pp. 1528–1532. [2] S. Ross, Simulation, 2nd ed., ser. Statistical modeling and decision science. Academic Press, 1997.
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The optimized solution provided a DL power gain of nearly 50% compared to a Butler BFNM shown in Figure 4(a). As it turns out, the optimal number of beams is six. Furthermore, the nulls are located closer to the cross over regions whereas in conventional Butler BFNM the peaks of the side lobes are located at the cross over regions (see Figure 4(b)). Finally, note that the straddling loss was almost eliminated.
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(b) Optimized. Fig. 4.
A conventional and optimized 4 column antenna array systems.
[3] S. Kirpatrick, C. Gelatt, and M. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, no. 4598, pp. 671–680, May 1983. [4] V. Murino, A. Trucco, and C. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Processing, vol. 44, pp. 119–123, Jan. 1996. [5] A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE Journal of oceanic engineering, vol. 24, no. 3, July 1999.
