**4.1. Cell to switch assignment in cellular networks using barebones PSO**

Cell assignment is an important issue in the area of resource management in cellular networks. The problem is an NP-hard one and requires efficient search techniques for its solution in realtime. We briefly present an example case of solving this problem using the barebones PSO [9]. The effective assignment of cells to switches in order to minimize the cost of network deployment is a challenging issue in cellular networking.

The cell-to-switch assignment (CSA) problem consists of optimally assigning cells to network switches while respecting certain constraints such as the call volume of each cell and the switches capacity [63]. The objective of the optimization is the reduction of implementation and operational costs. Usually, the cost function considers the cost of linking cells to switches (cabling cost) and the cost of handoff between different cells (handoff cost). The problem is an NP-hard one with exponential complexity and cannot be solved analytically in real size networks. Other evolutionary techniques like tabu search [64], ACO [65] and GAs [66] have been used in the literature for solving this problem. The problem formulation is given below.

We consider *n* unique and distinct cells in a given service area and *m* switches with known location and traffic parameters. The objective is the optimum assignment of cells to switches in order to minimize the total cost that comprises the handoff and cabling costs.

In single-homing CSA, each cell belongs to one cluster and it is assigned to one switch at a time. In this case, the objective function to be minimized is [63]:

$$\sum\_{l=1}^{n} \sum\_{j=1}^{m} \mathbf{c}\_{lk} \mathbf{x}\_{lk} + \sum\_{l=1}^{n} \sum\_{j=1 \atop j \neq l}^{n} h\_{ij} \left( 1 - y\_{ij} \right), \quad k = 1, \dots, m \tag{16}$$

where *cik* is the cabling cost per time unit between cell *i* and switch *k*, *xik* is a parameter that takes the value one when cell *i* is assigned to switch *k* (otherwise, *xik*=0) and *hij* is the cost per time unit for the handoffs that occur between cells *i* and *j*. The first term in (16) gives the total cabling cost and the second one is the total handoff cost per time unit among cells. Therefore, *yij* is defined as

$$\mathbf{x}\_{ij} = \sum\_{k=1}^{m} \mathbf{x}\_{ik} \mathbf{x}\_{jk} \quad \text{i.e.} \ j = 1, \dots, n \tag{17}$$

where *yij* is one when cells *i* and *j* are connected to the same switch, otherwise it is zero. The product *xikxjk* in (17) defines the variable

$$\mathbf{x}\_{\text{ijk}} \mathbf{w}\_{\text{ik}} \mathbf{x}\_{\text{jk}} \quad \mathbf{i}\_{\text{}} \mathbf{j} = \mathbf{1}, \dots, n \text{ and } k = \mathbf{1}, \dots, m \tag{18}$$

that is zero unless cells *i* and *j* are connected to switch *k*. In this case, it is one.

Cell assignment is subject to further constraints. The call handling capacity of each switch should not be violated at any time, i.e.

**4.1. Cell to switch assignment in cellular networks using barebones PSO**

deployment is a challenging issue in cellular networking.

10 Contemporary Issues in Wireless Communications

Cell assignment is an important issue in the area of resource management in cellular networks. The problem is an NP-hard one and requires efficient search techniques for its solution in realtime. We briefly present an example case of solving this problem using the barebones PSO [9]. The effective assignment of cells to switches in order to minimize the cost of network

The cell-to-switch assignment (CSA) problem consists of optimally assigning cells to network switches while respecting certain constraints such as the call volume of each cell and the switches capacity [63]. The objective of the optimization is the reduction of implementation and operational costs. Usually, the cost function considers the cost of linking cells to switches (cabling cost) and the cost of handoff between different cells (handoff cost). The problem is an NP-hard one with exponential complexity and cannot be solved analytically in real size networks. Other evolutionary techniques like tabu search [64], ACO [65] and GAs [66] have been used in the literature for solving this problem. The problem formulation is given below.

We consider *n* unique and distinct cells in a given service area and *m* switches with known location and traffic parameters. The objective is the optimum assignment of cells to switches

In single-homing CSA, each cell belongs to one cluster and it is assigned to one switch at a

( )

where *cik* is the cabling cost per time unit between cell *i* and switch *k*, *xik* is a parameter that takes the value one when cell *i* is assigned to switch *k* (otherwise, *xik*=0) and *hij* is the cost per time unit for the handoffs that occur between cells *i* and *j*. The first term in (16) gives the total cabling cost and the second one is the total handoff cost per time unit among cells. Therefore,

, , 1,...,

where *yij* is one when cells *i* and *j* are connected to the same switch, otherwise it is zero. The

*y xx ij n*

*cx h y k m*

1 , 1,...,

åå åå + -= (16)

= = å (17)

= , 1,..., and 1,..., *ijk ik jk z xx ij n k m* = = (18)

in order to minimize the total cost that comprises the handoff and cabling costs.

*ik ik ij ij*

*j i*

1

that is zero unless cells *i* and *j* are connected to switch *k*. In this case, it is one.

=

*m ij ik jk k*

¹

time. In this case, the objective function to be minimized is [63]:

*nm nn*

11 11

*ij ij*

*yij* is defined as

product *xikxjk* in (17) defines the variable

== ==

$$\sum\_{k=1}^{n} \lambda\_i \mathbf{x}\_{ik} \le M\_{k'} \quad \text{i} = 1, \dots, n \tag{19}$$

where *λ<sup>i</sup>* is the number of calls that cell *i* handles per unit time and *Mk* is the call handling capacity of switch *k*. Also, each cell is assigned only to one switch, i.e.

$$\sum\_{k=1}^{m} \mathbf{x}\_{ik} = \mathbf{1}, \quad i = \mathbf{1}, \ldots, m \tag{20}$$

The optimization problem defined by (16) and subject to (17)-(20), can be converted [63] to an integer programming one by replacing (18) with the

$$\begin{cases} 0 \le \mathbf{z}\_{j|k} \le \mathbf{x}\_{j|k'} \mathbf{x}\_{ik} \\ \mathbf{z}\_{jk} \ge \mathbf{x}\_{ik} + \mathbf{x}\_{jk} - \mathbf{1}\_{\prime} \end{cases} \quad \text{i.e.,} \\ \mathbf{j} = \mathbf{1}, \dots, n \text{ and } k = \mathbf{1}, \dots, m \tag{21}$$

We compare BB and BBExp PSO with ACO [65] and binary PSO (BPSO) [67]. We run 100 independent trials for each algorithm. The statistical results are presented and compared In the proposed barebones PSO variants, the only parameter we set was the swarm size. In the examples presented here, this was set to 5 particles. The ACO parameters were the same as in [65]. For the BPSO, we set the learning factors *c*1 and *c*2 equal to two. Systems with varied number of cells and switches that range from 15 to 200 and from 2 to 7, respectively, were considered.

The percentage of successfully obtained solutions as a function of the number of cells and switches indicates the solution accuracy of the algorithms. Figure 1 shows the results of the application of BB PSO, BBExp PSO, BPSO and ACO in a single-homing system for swarm size equal to 5 particles. In the first case, BB PSO outperforms the other methods in systems with small complexity but as the complexity increases (*n*/*m*=150/6 and 200/7) BBExp gives the best results. As it was expected, solution accuracy decreases with system complexity. In general, BB PSO outperforms the other methods for small *n* and *m*. However, its performance degrades with system complexity; in this case, BBExp gives better results. In any case, at least one of BB and BBExp PSO is better than BPSO and ACO. We have also evaluated the different algorithms in terms of the computational time required for the derivation of the previous results. Figure 2 shows small differences in results between the four algorithms. In all cases, barebones PSO algorithms are slightly faster. BBExp outperforms BB as system complexity increases. The computational cost of the methods grows exponentially with *n*/*m* and increases with the swarm size. More details about this problem can be found in [9].

**Figure 1.** Successful solutions vs cells/switch

**Figure 2.** Computational time vs cells/switch chart

#### **4.2. A PTS technique based on ACO and ABC for PAPR reduction of OFDM signals**

A major drawback of OFDM signals is the high value of peak to average power ratio (PAPR). Partial transmit sequences (PTS) [68], is a popular PAPR reduction method with good PAPR reduction performance. However, PTS requires an exhaustive search in order to find the optimal phase factors. Thus, the search complexity is high. Several methods have been published in the literature for PAPR reduction using PTS with low search complexity [10, 69, 70]. The problem description is given below.

In an OFDM system, the high-rate data steam is split into N low-rate data streams that are simultaneously transmitted using N subcarriers. The discrete-time signal of such a system is given by

$$s\_k = \frac{1}{\sqrt{N}} \sum\_{n=0}^{N-1} S\_n e^{\frac{j2\pi nk}{LN}} \quad k = 0, 1, \ldots, LN - 1 \tag{22}$$

where *L* is the oversampling factor, *S* = *S*0, *S*1, ..., *SN* <sup>−</sup><sup>1</sup> *<sup>T</sup>* is the input signal block. Each symbol is modulated by either phase-shift keying (PSK) or quadrature amplitude modulation (QAM).

The PAPR of the signal in (22) is defined as the ratio of the maximum to average power and is expressed in dB as

$$PAPR(s) = 10\log\_{1i} \frac{\max\_{0 \le i < l \le \dotsb} \left| s\_i \right|^2}{E\left[ \left| s\_i \right|^2 \right]} \tag{23}$$

where *E* . is the expected value operation.

**Figure 1.** Successful solutions vs cells/switch

12 Contemporary Issues in Wireless Communications

**Figure 2.** Computational time vs cells/switch chart

70]. The problem description is given below.

**4.2. A PTS technique based on ACO and ABC for PAPR reduction of OFDM signals**

A major drawback of OFDM signals is the high value of peak to average power ratio (PAPR). Partial transmit sequences (PTS) [68], is a popular PAPR reduction method with good PAPR reduction performance. However, PTS requires an exhaustive search in order to find the optimal phase factors. Thus, the search complexity is high. Several methods have been published in the literature for PAPR reduction using PTS with low search complexity [10, 69,

In the PTS approach the *S* input data OFDM block is partitioned into *M* disjointed subblocks represented by the vector *S<sup>m</sup> m*=1, 2, ..., *M* −1 and oversampled by inserting (*L* −1)*N* zeros. Then the PTS process is expressed as

$$\mathbf{S} = \sum\_{\star=1}^{M} \mathbf{S}\_{\star} \tag{24}$$

Next, the subblocks are converted to time domain using *LN* point inverse fast fourier transform (IFFT). The representation of the OFDM block in time domain is expressed by

$$\mathbf{s} = IFFT\left\{\sum\_{n=1}^{M} \mathbf{S}\_{n}\right\} = \sum\_{n=1}^{M} IFFT\left\{\mathbf{S}\_{n}\right\} = \sum\_{n=1}^{M} \mathbf{s}\_{n} \tag{25}$$

The PTS objective is to produce a weighted combination of the M subblocks using *b* = *b*1, *b*2, ..., *bM <sup>T</sup>* complex phase factors to minimize PAPR. The transmitted signal in time domain after this combination is given by

$$\mathbf{s}'(\mathbf{b}) = \sum\_{\star=\star}^{M} \mathbf{b}\_{\star} \mathbf{s}\_{\star} \tag{26}$$

In order to reduce the search complexity the phase factor possible values are limited to a finite set. The set of allowable phase factors is

$$\mathcal{A} = \left\{ e^{\frac{\beta \hbar \nu}{\hbar}} \, \middle| \, \mu = 0, 1, \dots, W - 1 \right\} \tag{27}$$

where *W* is the number of allowed phase factors. Therefore in case of *M* subblocks and *W* phase factors the total number of possible combinations is *W <sup>M</sup>* . In order to reduce the search complexity we usually set fixed one phase factor.

The optimization goal of the PTS scheme is to find the optimum phase combination for minimum PAPR. Thus, the objective function can be expressed as

Minimize

$$F(b) = 10 \log\_{10} \frac{\max\_{b \le b \le N-1} \left| s'(b) \right|^2}{E\left[ \left| s'(b) \right|^{1^\*} \right]} \tag{28}$$

subject to

$$\Phi \in \left\{ e^{\left. \phi\_{\dots} \right|} \right\}^{\text{M}} \text{ where } \phi\_{\dots} \in \left\{ \frac{2\pi l}{W} \middle| l = 0, 1, \dots, W - 1 \right\} \tag{29}$$

We have evaluated objective function above using evolutionary algorithms and methods found in the literature. We have used two main measurement criteria namely the comple‐ mentary cumulative distribution function (CCDF) and the computational complexity. In all our simulations, 10E5 random OFDM blocks are generated. The transmitted signal is over‐ sampled by a factor L=4. We consider 16-QAM modulation with N=256 sub-carriers which are divided into M=16 random subblocks. The phase factors for W=2 are selected. We consider the first phase factor to be fixed so the total number of unknown phase factors is M-1.

The control parameters in all simulations are given below. In the PSO algorithm *c*1 and *c*2 are set equal to 2.05 while the inertia weight is linearly decreased starting from 0.9 to 0.4. For ACO the initial pheromone value *τ*0 is set to 1.0e-6, the pheromonone update constant Q is set to 20, the exploration constant *q*0 is set to 1, the global pheromone decay rate *ρ<sup>g</sup>* is 0.9, the local pheromone decay rate *ρ<sup>l</sup>* is 0.5, the pheromone sensitivity *α* is 1, and the visibility sensitivity is *β* is 5.

Figure 3 presents the comparison between the CCDF by different PTS reduction techniques. For Pr(*PAPR* <sup>&</sup>gt;*PAPR*0) =10−<sup>3</sup> the PAPR of the original OFDM transmitted signal is 10.84dB. For all evolutionary algorithms, the population size NP is set to 30 and the maximum number of

In order to reduce the search complexity the phase factor possible values are limited to a finite

0,1,..., 1

where *W* is the number of allowed phase factors. Therefore in case of *M* subblocks and *W* phase factors the total number of possible combinations is *W <sup>M</sup>* . In order to reduce the search

The optimization goal of the PTS scheme is to find the optimum phase combination for

0 1 10 2 max ( )

*E* ££ - ¢

where 0,1,..., 1 *<sup>m</sup>*

p Î Î= f

We have evaluated objective function above using evolutionary algorithms and methods found in the literature. We have used two main measurement criteria namely the comple‐ mentary cumulative distribution function (CCDF) and the computational complexity. In all our simulations, 10E5 random OFDM blocks are generated. The transmitted signal is over‐ sampled by a factor L=4. We consider 16-QAM modulation with N=256 sub-carriers which are divided into M=16 random subblocks. The phase factors for W=2 are selected. We consider the

The control parameters in all simulations are given below. In the PSO algorithm *c*1 and *c*2 are set equal to 2.05 while the inertia weight is linearly decreased starting from 0.9 to 0.4. For ACO the initial pheromone value *τ*0 is set to 1.0e-6, the pheromonone update constant Q is set to 20, the exploration constant *q*0 is set to 1, the global pheromone decay rate *ρ<sup>g</sup>* is 0.9, the local pheromone decay rate *ρ<sup>l</sup>* is 0.5, the pheromone sensitivity *α* is 1, and the visibility sensitivity

Figure 3 presents the comparison between the CCDF by different PTS reduction techniques.

all evolutionary algorithms, the population size NP is set to 30 and the maximum number of

( ) *k LN*

 ì ü í ý

î þ *<sup>b</sup>* (29)

the PAPR of the original OFDM transmitted signal is 10.84dB. For

*s b*

é ù ¢ ë û

*s b*

2

î þ (27)

(28)

2

*j n <sup>W</sup> Aen W* p

= = ì ü í ý

set. The set of allowable phase factors is

14 Contemporary Issues in Wireless Communications

complexity we usually set fixed one phase factor.

Minimize

subject to

is *β* is 5.

For Pr(*PAPR* <sup>&</sup>gt;*PAPR*0) =10−<sup>3</sup>

minimum PAPR. Thus, the objective function can be expressed as

*F*

*<sup>M</sup> <sup>j</sup>*

f *b*

( ) 10 log

=

{ } <sup>2</sup>

*m <sup>l</sup> <sup>e</sup> l W W*

first phase factor to be fixed so the total number of unknown phase factors is M-1.

**Figure 3.** PARP reduction performance comparison of the BBO-PTS algorithms with other PTS schemes for NP=30, G=30.

generations G is set to 30. Thus, the computational complexity of this case is *NP* × *G* =900. The computational complexity of the exhaustive search is *W <sup>M</sup>* <sup>−</sup><sup>1</sup> =32768 while the PAPR for this case is 5.86dB. The PAPR by the iterative flipping algorithm for PTS (IPTS) [69] is 7.55dB with search complexity (M-1)W=30. The PAPR by the gradient descent method (GD) [71] with search complexity *CM* <sup>−</sup><sup>1</sup> *<sup>r</sup> W <sup>r</sup> I* =*C*<sup>15</sup> <sup>2</sup> 2<sup>2</sup> 3=1260 is 6.96dB. The PAPR by ABC [10], PSO [72], and ACO [6], is 7.01dB, 7.13dB, and 6.52dB, respectively. Table 1 holds the comparison of the search complexity among the different methods for *CCDF* =10−<sup>3</sup> , NP=30, and G=30. It is obvious that ACO presents the better performance among the other methods with the same search com‐ plexity.


**Table 1.** Comparison of computational complexity for CCDF=1e-3 among different PTS Schemes

#### **4.3. Dual-band microwave filter design using SADE**

Microwave filters are among the important components of a modern wireless communication system. Several papers exist in the literature that address the filter design problem [73]. Open Loop Ring Resonator (OLRR) filters, which consist of two uniform microstrip lines and pairs of open loops between them, are widely used as the building block in several multiband bandpass filter design cases [74]. In [74], two pairs of folded OLRRs operating at two passbands are proposed to produce dual-band response.

A dual band OLRR filter is shown in Figure 4. The frequency response of such a filter depends on the filter dimensions and spacings between microstrip lines [74, 75]. The design parameters for this case are the ones shown in Figure 4, (*W*1, *W*2, *L* <sup>1</sup>, *L* 2, *L* 3, *L* 4, *L* <sup>5</sup>, *S*1, *S*2, *S*3, *G*), all expressed in mm.

Such a filter design problem can be defined by the minimization of |*S*<sup>11</sup> | in the passband frequency range. This design problem is therefore defined by the minimization of the objective function:

$$\mathsf{F}(\overline{\mathsf{x}}) = 20 \log \left\{ \max \left| \mathbb{S}\_{\mathsf{u}}(\overline{\mathsf{x}}, f) \right|, \ f \in \mathsf{S}\_{\mathsf{p}} \right\} \tag{30}$$

where *x*¯ is the vector of filter geometry parameters and *Sp* is the set of distinct frequencies in the desired passband frequency ranges.

The filter is designed for operation in two WiMax (IEEE 802.16) frequency bands. These are the 3.5GHz and the 5.8GHz frequency bands. For this case, we set *Sp* ={3.55, 3.6, 5.75, 5.8}. Figure 5 shows the simulated frequency response of this design. The simulated current distribution for the 3.6GHz and 5.8GHz frequencies is presented in Figure 6, where the resonating ring in each case is clearly seen. In the first passband between 3.508 and 3.809 GHz, the filter has a return loss less than 10dB and insertion loss greater than 0.5dB. In the second passband between 5.744 and 6.121 GHz the results also show a return loss less than 10dB and insertion loss greater than 0.5dB. The rejection band (between 4.236 and 5.367 GHz) has an insertion loss less than 20dB. In the first passband the return loss is less than 29dB at both 3.533 GHz and 3.759 GHz. In the second passband the return loss is less than 22dB at 5.794GHz and less than 28dB at 6.07GHz.

**Figure 4.** Dual-band filter geometry

Evolutionary Algorithms for Wireless Communications — A Review of the State-of-the art http://dx.doi.org/10.5772/59147 17

**Figure 5.** Simulated frequency response of the dual-band filter.

Figure 5. Simulated frequency response of the dual‐band filter.

Figure 6. Current distribution simulations for dual‐band filter at (a) 3.5GHz and (b) 5.8GHz **Figure 6.** Current distribution simulations for dual-band filter at (a) 3.5GHz and (b) 5.8GHz

#### A brief survey of different evolutionary algorithms and their application to different problems in wireless communications has been presented. It must be pointed out that several evolutionary algorithms exist in the literature. **5. Conclusion**

iterations.

using such algorithms.

**5. Conclusion**

Loop Ring Resonator (OLRR) filters, which consist of two uniform microstrip lines and pairs of open loops between them, are widely used as the building block in several multiband bandpass filter design cases [74]. In [74], two pairs of folded OLRRs operating at two passbands

A dual band OLRR filter is shown in Figure 4. The frequency response of such a filter depends on the filter dimensions and spacings between microstrip lines [74, 75]. The design parameters for this case are the ones shown in Figure 4, (*W*1, *W*2, *L* 1, *L* 2, *L* <sup>3</sup>, *L* 4, *L* 5, *S*1, *S*2, *S*3, *G*), all

Such a filter design problem can be defined by the minimization of |*S*<sup>11</sup> | in the passband frequency range. This design problem is therefore defined by the minimization of the objective

where *x*¯ is the vector of filter geometry parameters and *Sp* is the set of distinct frequencies in

The filter is designed for operation in two WiMax (IEEE 802.16) frequency bands. These are the 3.5GHz and the 5.8GHz frequency bands. For this case, we set *Sp* ={3.55, 3.6, 5.75, 5.8}. Figure 5 shows the simulated frequency response of this design. The simulated current distribution for the 3.6GHz and 5.8GHz frequencies is presented in Figure 6, where the resonating ring in each case is clearly seen. In the first passband between 3.508 and 3.809 GHz, the filter has a return loss less than 10dB and insertion loss greater than 0.5dB. In the second passband between 5.744 and 6.121 GHz the results also show a return loss less than 10dB and insertion loss greater than 0.5dB. The rejection band (between 4.236 and 5.367 GHz) has an insertion loss less than 20dB. In the first passband the return loss is less than 29dB at both 3.533 GHz and 3.759 GHz. In the second passband the return loss is less than 22dB at 5.794GHz and

F( ) 20 log max ( , ) , *x* = { *S xf f S* <sup>11</sup> Î *<sup>p</sup>*} (30)

are proposed to produce dual-band response.

16 Contemporary Issues in Wireless Communications

the desired passband frequency ranges.

less than 28dB at 6.07GHz.

**Figure 4.** Dual-band filter geometry

expressed in mm.

function:

one has to consider the problem characteristics. Another key issue is the selection of the algorithm control parameters, which is also in most cases problem‐dependent. One may also use at first the control parameters for these algorithms that commonly perform well regardless of the characteristics of the problem to be solved. The example of the CSA problem in cellular networks showed the better performance of BB and BBExp compared to BPSO and ACO in terms of successfully obtained solutions and execution time. In the PTS optimization problem ACO outperformed ABC and PSO. The selection of the SADE technique for microwave filter design has lead to a successful filter design which exhibits low loss in the passbands and high isolation between the passbands. The DE algorithms are also robust optimizers. In classical DE algorithms, the selection of the appropriate strategy for trial vector generation and control parameters requires additional computational time using a trial‐and‐error search procedure. Therefore, it is not always an easy task to fine‐tune the control parameters and strategy given also that commonly the appropriate control parameters and strategy selection are problem dependent. The SaDE advantage though, is the fact that no additional time for solving a A brief survey of different evolutionary algorithms and their application to different problems in wireless communications has been presented. It must be pointed out that several evolu‐ tionary algorithms exist in the literature. GAs and SI algorithms are among those most commonly used. In order to select, the best algorithm for every problem one has to consider the problem characteristics. Another key issue is the selection of the algorithm control parameters, which is also in most cases problem-dependent. One may also use at first thecontrol parameters for these algorithms that commonly perform well regardless of the characteristics of the problem to be solved. The example of the CSA problem in cellular

given problem is required. SaDE requires only the adjustment of two parameters: the population size and the number of

The practical examples subject to several constraints presented in this chapter show the applicability and the efficiency of

GAs and SI algorithms are among those most commonly used. In order to select, the best algorithm for every problem

networks showed the better performance of BB and BBExp compared to BPSO and ACO in terms of successfully obtained solutions and execution time. In the PTS optimization problem ACO outperformed ABC and PSO.

The selection of the SADE technique for microwave filter design has lead to a successful filter design which exhibits low loss in the passbands and high isolation between the passbands. The DE algorithms are also robust optimizers. In classical DE algorithms, the selection of the appropriate strategy for trial vector generation and control parameters requires additional computational time using a trial-and-error search procedure. Therefore, it is not always an easy task to fine-tune the control parameters and strategy given also that commonly the appropriate control parameters and strategy selection are problem dependent. The SaDE advantage though, is the fact that no additional time for solving a given problem is required. SaDE requires only the adjustment of two parameters: the population size and the number of iterations.

The practical examples subject to several constraints presented in this chapter show the applicability and the efficiency of using such algorithms.
