*3.1.6. Maximum number of iterations (IMAX)*

The strength of SQ method lies in its ability to statistically deliver a true global minimum. However, as mentioned, one of the drawbacks of this method is its slow delivery of the ideal solution once the global minimum area is located. As no indication of time required for reaching the optimal solution is available, the search process is typically suspended after the allowed search time is spent or the cost function has reached the desired level of precision. *Standard Values Values based on Empirical*

0, *<sup>i</sup>* <sup>=</sup> 2, ..., 20 *<sup>R</sup>*<sup>0</sup> <sup>=</sup> 0.0005

<sup>2</sup> *Qsuf f* <sup>=</sup> 0.005

*Ressymb* = 0.00001 *NVCPL* = 30 *Rescoef f* = 0.00001 *α* = 0.995

*<sup>i</sup>* <sup>=</sup> 1..20; *<sup>ξ</sup><sup>i</sup>* <sup>∈</sup> �−1, 1� *IMAX* <sup>=</sup> <sup>16000</sup>

*Coef*0(*i*) = 1, *<sup>i</sup>* <sup>=</sup> <sup>1</sup>

*Coefnew*(*i*) = *ξiR*<sup>0</sup> + *Coefact*(*i*)

*<sup>p</sup>*(Δ*Q*) = *<sup>e</sup>*<sup>−</sup> <sup>Δ</sup>*<sup>Q</sup>*

*<sup>k</sup>* <sup>=</sup> <sup>−</sup> <sup>3</sup>*σ*Δ*<sup>Q</sup> ln*(*pmax* ) *TL* = *<sup>α</sup>TL*−<sup>1</sup> *T*<sup>0</sup> = 1, *α* < 1, *L* = 1, 2, 3... *pmax* = 50% *pmin* = 0.1%

*<sup>i</sup>*=<sup>1</sup> (*x*1*<sup>i</sup>* − *x*ˆ1*i*)

*kT*

**Figure 10.** Ten instances of adopted SQ search algorithm under moderate coupling

at 4000 iterations preserving the decoupling precision.

function solutions are kept in the same orders of magnitude throughout the entire search process. The small differences between the temporary cost function progress and stable final solution precision confirms that the SQ parameters have been selected adequately. The search successfully avoids local minimum traps and follows the correct search path. Furthermore, since BER performance enters saturation for *Q* < 0.005 the presented search can be stopped

Even though the obtained cost function value is similar for all ten instances of conducted SQ search, the presented decoupling coefficients numerically offer completely different solutions. Two different sets of 20 decoupling coefficients, obtained as a result of two execution of classical SQ algorithm under the same coupling conditions, are presented in Figure 11. The difference between the cost functions in this case is almost zero as surface cost functions are

*Q* =

 <sup>1</sup> *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup>*

**Table 2.** Set of SQ search parameters after empirical analysis

*Decoupling Analysis*

Simulated Quenching for Cancellation of Non-Linear Multi-Antennas Coupling 145

**Figure 9.** *Q* in dependence of the number of iterations under two different coupling levels

In Figure 9, *Q* is depicted as a function of the number of iterations under two different coupling conditions. Both *Q* functions experience three different segments which correspond to surface wandering at the beginning, almost downhill movement in the middle, and error floor at the end. In both cases error floor is reached after around 20000 iterations and it would make sense to use this number as maximum number of permitted iterations. Still, if we take into account that surface with Q level smaller than 0.005 produce gratifying decoupling results, in this cases the search can be stopped after 4000 and 12000 iterations respectively. The process of establishing the satisfying level of cost function (*Qsuf f* ) is explained in [1] where relation between BER and the corresponding quadratic mean square cost function level is analyzed. In the end we propose to set maximum number of iterations to 16000 and use it in combination with satisfying cost function level to control the search execution.

#### **3.2. SQ example**

The SQ search is conducted under moderate coupling conditions of [-10,-12,-10] dB using the SQ search parameters adopted to nonlinear decoupling problem gathered inside Table 2. The table consist of two columns with standard values based on theoretical assumptions occupying the left column, and values based on empirical decoupling analysis in the right column.

The behavior of adopted SQ search algorithm is shown in Figure 10, where cost function is presented in dependence of number of iterations for 10 different SQ search executions. The image shows that the search path in first 500 iterations is similar in all ten instance. The curves then start to spread, and are grouped again at the end. Nevertheless, the cost



**Table 2.** Set of SQ search parameters after empirical analysis

The strength of SQ method lies in its ability to statistically deliver a true global minimum. However, as mentioned, one of the drawbacks of this method is its slow delivery of the ideal solution once the global minimum area is located. As no indication of time required for reaching the optimal solution is available, the search process is typically suspended after the allowed search time is spent or the cost function has reached the desired level of precision.

**Figure 9.** *Q* in dependence of the number of iterations under two different coupling levels

combination with satisfying cost function level to control the search execution.

**3.2. SQ example**

column.

In Figure 9, *Q* is depicted as a function of the number of iterations under two different coupling conditions. Both *Q* functions experience three different segments which correspond to surface wandering at the beginning, almost downhill movement in the middle, and error floor at the end. In both cases error floor is reached after around 20000 iterations and it would make sense to use this number as maximum number of permitted iterations. Still, if we take into account that surface with Q level smaller than 0.005 produce gratifying decoupling results, in this cases the search can be stopped after 4000 and 12000 iterations respectively. The process of establishing the satisfying level of cost function (*Qsuf f* ) is explained in [1] where relation between BER and the corresponding quadratic mean square cost function level is analyzed. In the end we propose to set maximum number of iterations to 16000 and use it in

The SQ search is conducted under moderate coupling conditions of [-10,-12,-10] dB using the SQ search parameters adopted to nonlinear decoupling problem gathered inside Table 2. The table consist of two columns with standard values based on theoretical assumptions occupying the left column, and values based on empirical decoupling analysis in the right

The behavior of adopted SQ search algorithm is shown in Figure 10, where cost function is presented in dependence of number of iterations for 10 different SQ search executions. The image shows that the search path in first 500 iterations is similar in all ten instance. The curves then start to spread, and are grouped again at the end. Nevertheless, the cost

*3.1.6. Maximum number of iterations (IMAX)*

**Figure 10.** Ten instances of adopted SQ search algorithm under moderate coupling

function solutions are kept in the same orders of magnitude throughout the entire search process. The small differences between the temporary cost function progress and stable final solution precision confirms that the SQ parameters have been selected adequately. The search successfully avoids local minimum traps and follows the correct search path. Furthermore, since BER performance enters saturation for *Q* < 0.005 the presented search can be stopped at 4000 iterations preserving the decoupling precision.

Even though the obtained cost function value is similar for all ten instances of conducted SQ search, the presented decoupling coefficients numerically offer completely different solutions. Two different sets of 20 decoupling coefficients, obtained as a result of two execution of classical SQ algorithm under the same coupling conditions, are presented in Figure 11. The difference between the cost functions in this case is almost zero as surface cost functions are


**4. Improved Fast Simulate Quenching (IFSQ)**

method:

• annealing schedule

• probability transition function • maximum allowed displacement.

The proposed Improved Fast SQ (IFSQ) approach applies changes to three vital parts of SQ

Simulated Quenching for Cancellation of Non-Linear Multi-Antennas Coupling 147

The IFSQ implements the annealing schedule function according to Boltzman function while the probability transition function corresponds to Caucy distribution. The novel part of IFSQ approach is also to keep the maximum allowed displacement factor a function of temperature. All enhanced SA methods labeled as fast are based on steeper annealing schedule function when compared to traditional SA approach. Previously described SQ approach is based on the exponential annealing schedule function where the temperature level is decreased by a fixed factor during the entire search process. However, it makes sense to decrease the temperature more rapidly at the beginning when wandering is desirable, and than reduce the temperature carefully towards the end when SQ turns into a greedy search algorithm. This is precisely the effect achieved with the following annealing schedule function used in IFSQ approach:

*TL* <sup>=</sup> *<sup>T</sup><sup>α</sup>*

initial temperature level of SQ search algorithm:

*αT*<sup>0</sup> = *<sup>T</sup><sup>α</sup>*

*ln*(*L* + 1)

*TSQ*

where *L* is the actual temperature level and *T<sup>α</sup>* is the temperature constant. The numerical value of *T<sup>α</sup>* is actually irrelevant to performance of IFSQ search. In real physical process this factor controls the initial temperature and as such influences the probability of uphill movement during the search since probability is directly related to temperature with Boltzmann constant *kB*. As decoupling problem is not related directly to the physical energy *kB* is replaced with problem specific constant *kc* which is adjusted according to the initial temperature level. In order to easily compare the SQ to IFSQ search *T<sup>α</sup>* is chosen to equal the

> <sup>1</sup> <sup>=</sup> *<sup>T</sup>IFSQ* 1

with *T*<sup>0</sup> = 1 and *α* = 0.995 which corresponds to the temperature reduction factor of the adjusted SQ search. As a result, the initial temperature level of IFSQ method is set to *T<sup>α</sup>* = 0.69.

The difference between SQ and IFSQ annealing schedules based on adjusted parameters is seen in Figure 13. As can be seen the temperature level of the SQ algorithm is kept high during the first ten levels. At this time, IFSQ temperature is decreased to less then 30% of the initial temperature and shortly afterwards starts to decrease slowly, while SQ temperature decrease now gets steeper. At the end, IFSQ keeps temperature almost constant allowing the thorough search of actual neighbor space. On the other hand, SQ search turns into a greedy algorithm without any uphill excursions allowing only perturbations which lead to better solution. In short, when compared to SQ method IFSQ annealing schedule is steeper at the beginning and

; *L* = 1, 2, 3.. (17)

*ln*(2) <sup>⇒</sup> *<sup>T</sup><sup>α</sup>* <sup>=</sup> *<sup>α</sup>ln*(2) (18)

**Figure 11.** Two sets of decoupling coefficients for [-10,-12,-10]dB coupling environment

**Figure 12.** Performance of SQ based decoupling module under moderate coupling

*Qa* = 0.0009 and *Qb* = 0.0008 respectively. This points out that the constructed inverse coupling surfaces are similar. However, they are based on different building material as for example *a*<sup>7</sup> is very close to zero in first solution while in second one is *a*<sup>7</sup> = 0.015. On the other hand *a*<sup>5</sup> and *b*<sup>3</sup> show opposite behavior, as they present higher importance in second than in first solution. Furthermore, the amplitudes of decoupling coefficients don't exhibit the tendency of reduction as the exponential order is increased. For example, *a*<sup>8</sup> *a*<sup>3</sup> or *b*<sup>7</sup> *b*<sup>3</sup> indicating that the surface simplification based on simple reduction of the order of polynomials might not achieve the same decoupling efficiency.

The system performance of decoupling module based on the obtained coefficients traces almost the same curve inside *Eb*/*N*<sup>0</sup> -BER graph as shown in Figure 12. The entire description of system used for transmission simulations is found in [1]. The upper curve on the image presents the system behavior under coupling of [-10,-12,-10] dB and without decoupling module. The lowest curve shows the theoretical system performance without any distortion and only with additive white Gaussian noise (AWGN). The two curves in the middle show that the system performance with the decoupling module is very close to the theoretical system performance This confirms that the cost function and inverse surface model are both selected adequately and that SQ search produces accurate results.

### **4. Improved Fast Simulate Quenching (IFSQ)**

The proposed Improved Fast SQ (IFSQ) approach applies changes to three vital parts of SQ method:

• annealing schedule

16 Will-be-set-by-IN-TECH

**Figure 11.** Two sets of decoupling coefficients for [-10,-12,-10]dB coupling environment

**Figure 12.** Performance of SQ based decoupling module under moderate coupling

polynomials might not achieve the same decoupling efficiency.

selected adequately and that SQ search produces accurate results.

*Qa* = 0.0009 and *Qb* = 0.0008 respectively. This points out that the constructed inverse coupling surfaces are similar. However, they are based on different building material as for example *a*<sup>7</sup> is very close to zero in first solution while in second one is *a*<sup>7</sup> = 0.015. On the other hand *a*<sup>5</sup> and *b*<sup>3</sup> show opposite behavior, as they present higher importance in second than in first solution. Furthermore, the amplitudes of decoupling coefficients don't exhibit the tendency of reduction as the exponential order is increased. For example, *a*<sup>8</sup> *a*<sup>3</sup> or *b*<sup>7</sup> *b*<sup>3</sup> indicating that the surface simplification based on simple reduction of the order of

The system performance of decoupling module based on the obtained coefficients traces almost the same curve inside *Eb*/*N*<sup>0</sup> -BER graph as shown in Figure 12. The entire description of system used for transmission simulations is found in [1]. The upper curve on the image presents the system behavior under coupling of [-10,-12,-10] dB and without decoupling module. The lowest curve shows the theoretical system performance without any distortion and only with additive white Gaussian noise (AWGN). The two curves in the middle show that the system performance with the decoupling module is very close to the theoretical system performance This confirms that the cost function and inverse surface model are both


The IFSQ implements the annealing schedule function according to Boltzman function while the probability transition function corresponds to Caucy distribution. The novel part of IFSQ approach is also to keep the maximum allowed displacement factor a function of temperature.

All enhanced SA methods labeled as fast are based on steeper annealing schedule function when compared to traditional SA approach. Previously described SQ approach is based on the exponential annealing schedule function where the temperature level is decreased by a fixed factor during the entire search process. However, it makes sense to decrease the temperature more rapidly at the beginning when wandering is desirable, and than reduce the temperature carefully towards the end when SQ turns into a greedy search algorithm. This is precisely the effect achieved with the following annealing schedule function used in IFSQ approach:

$$T\_L = \frac{T\_\aleph}{\ln(L+1)};\ L = 1,2,3... \tag{17}$$

where *L* is the actual temperature level and *T<sup>α</sup>* is the temperature constant. The numerical value of *T<sup>α</sup>* is actually irrelevant to performance of IFSQ search. In real physical process this factor controls the initial temperature and as such influences the probability of uphill movement during the search since probability is directly related to temperature with Boltzmann constant *kB*. As decoupling problem is not related directly to the physical energy *kB* is replaced with problem specific constant *kc* which is adjusted according to the initial temperature level. In order to easily compare the SQ to IFSQ search *T<sup>α</sup>* is chosen to equal the initial temperature level of SQ search algorithm:

$$aT\_0 = \begin{array}{c} T\_1^{SQ} \\ \frac{T\_0}{\ln(2)} \end{array} \implies \begin{array}{c} T\_1^{IFSQ} \\ \Rightarrow \end{array} \tag{18}$$

with *T*<sup>0</sup> = 1 and *α* = 0.995 which corresponds to the temperature reduction factor of the adjusted SQ search. As a result, the initial temperature level of IFSQ method is set to *T<sup>α</sup>* = 0.69.

The difference between SQ and IFSQ annealing schedules based on adjusted parameters is seen in Figure 13. As can be seen the temperature level of the SQ algorithm is kept high during the first ten levels. At this time, IFSQ temperature is decreased to less then 30% of the initial temperature and shortly afterwards starts to decrease slowly, while SQ temperature decrease now gets steeper. At the end, IFSQ keeps temperature almost constant allowing the thorough search of actual neighbor space. On the other hand, SQ search turns into a greedy algorithm without any uphill excursions allowing only perturbations which lead to better solution. In short, when compared to SQ method IFSQ annealing schedule is steeper at the beginning and

**Figure 13.** Comparison of standard SQ and IFSQ annealing schedule functions

kept at higher level towards the end of the search which allows higher convergence speed at start up, and better analysis of close neighborhood area at lower temperature levels.

The second important difference between SQ and IFSQ approach is the choice of the probability of uphill movement. The SQ method is based on Boltzmann probability density function with the drawback in its thin tail, which makes the configurations with substantial cost function difference rather unreachable under small number of iterations. It was noted in [17] that Cauchy distribution, which has a fatter body than Boltzmann distribution, permits easier access to larger set of candidates without any significant algorithm convergence loss. The authors define modified Cauchy distribution, which is also used in IFSQ, as:

$$p(\Delta Q) = \frac{T\_L}{\left(\Delta Q^2 + k\_c T\_L^2\right)^{(D+1)/2}}\tag{19}$$

**Figure 14.** Transition probability functions of SQ and IFSQ approach

search advances. The dependence is made according to the:

loss.

obtained.

As expected the probability function based on modified Cauchy distribution is fatter than the one based on Boltzmann function allowing uphill movements with more ease. The advantage of IFSQ probability transition function lies inside area which covers Δ*Q* from 0.1 to 0.6. With Δ*Q* = 0.2, for example, transition probability of IFSQ marks approximately 65%, while SQ is limited to approximately 32%. B including this middle class of Δ*Q* values more often into the search path, search robustness is increased but convergence speed doesn't suffer substantial

The depth of exploration depends not only on the transition probability function but also on the annealing schedule and maximum displacement factor *R*0. The annealing schedule function of IFSQ is very steep at the beginning, suppressing the initial deep exploration rather fast and keeping the rest of the search concentrated on small energy differences around the actual point. Thus, so called deep exploration which easily permits uphill movements is available only at the beginning. Still, these initial search opportunities can not be exploited if *R*<sup>0</sup> factor is small as only small part of the potential search space can be reached for the analysis. Small maximum displacement factor is adequate choice for the advanced search status when global minimum area is well located, and extense search is required in order to find the best solution. By keeping *R*<sup>0</sup> constant during the entire search process, as in SQ approach, two opposite search requirement can not be met and consequently *R*<sup>0</sup> is chosen as their compromise. This is changed in IFSQ approach maximum displacement factor is set to high value covering whole search space at the beginning, and it is gradually decreased as the

where *L* stands for the corresponding temperature level, *δ* is the search space reduction factor, and *r* the initial maximum allowed displacement. The numerical values of *δ* and *r* are determined through empirical analysis. Additionally, since IFSQ approach is not oriented only on precision, the maximum number of iterations is not limited to probability, but the search is stopped if in five consecutive temperature levels no cost function improvement is

*R*0(*L*) = *δR*0(*L* − 1), *R*0(0) = *r*, *L* = 1, 2, ... (22)

Simulated Quenching for Cancellation of Non-Linear Multi-Antennas Coupling 149

where *kc* relates *TL* to Δ*Q* and *D* defines the degrees of freedom. According to eq. 9 parameter *D* is equal to 20 while modified Cauchy distribution constant *kc* is chosen to ensure maximum probability of transition (*pmax*) at first temperature level *T*<sup>1</sup> to be satisfied for 99% of all possible cost function differences. Since Δ*Q* follows the Gaussian distribution 99% of all possible numerical values of Δ*Q* matches three standard deviations *σ*Δ*<sup>Q</sup>* of the corresponding distribution. Hence, *pmax* can be expressed as:

$$p\_{\text{max}} = \frac{T\_1}{\left(\left(3\sigma\_{\Delta Q}\right)^2 + k\_c T\_1^2\right)^{(D+1)/2}}\tag{20}$$

By extracting *kc* the following statement is obtained:

$$k\_{\varepsilon} = \sqrt{\frac{1}{T\_1^2} \left( \sqrt{\frac{T\_1}{p\_{\max}}} - \left( 3\sigma\_{\Delta Q} \right)^2} \right)}\tag{21}$$

The difference between these two probability transition approaches are seen easily in Figure 14 where transition probability functions of SQ and IFSQ approach are presented.

**Figure 14.** Transition probability functions of SQ and IFSQ approach

18 Will-be-set-by-IN-TECH

kept at higher level towards the end of the search which allows higher convergence speed at

The second important difference between SQ and IFSQ approach is the choice of the probability of uphill movement. The SQ method is based on Boltzmann probability density function with the drawback in its thin tail, which makes the configurations with substantial cost function difference rather unreachable under small number of iterations. It was noted in [17] that Cauchy distribution, which has a fatter body than Boltzmann distribution, permits easier access to larger set of candidates without any significant algorithm convergence loss.

Δ*Q*<sup>2</sup> + *kcT*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *kcT*<sup>2</sup> 1

The difference between these two probability transition approaches are seen easily in Figure

− 3*σ*Δ*<sup>Q</sup>* 2 

where *kc* relates *TL* to Δ*Q* and *D* defines the degrees of freedom. According to eq. 9 parameter *D* is equal to 20 while modified Cauchy distribution constant *kc* is chosen to ensure maximum probability of transition (*pmax*) at first temperature level *T*<sup>1</sup> to be satisfied for 99% of all possible cost function differences. Since Δ*Q* follows the Gaussian distribution 99% of all possible numerical values of Δ*Q* matches three standard deviations *σ*Δ*<sup>Q</sup>* of the corresponding

*L*

(*D*+1)/2 (19)

(*D*+1)/2 (20)

(21)

start up, and better analysis of close neighborhood area at lower temperature levels.

The authors define modified Cauchy distribution, which is also used in IFSQ, as:

distribution. Hence, *pmax* can be expressed as:

By extracting *kc* the following statement is obtained:

*kc* = 1 *T*2 1

*<sup>p</sup>*(Δ*Q*) = *TL* 

*pmax* <sup>=</sup> *<sup>T</sup>*<sup>1</sup> 3*σ*Δ*<sup>Q</sup>*

> (*D*+1)/2 *T*1 *pmax*

14 where transition probability functions of SQ and IFSQ approach are presented.

**Figure 13.** Comparison of standard SQ and IFSQ annealing schedule functions

As expected the probability function based on modified Cauchy distribution is fatter than the one based on Boltzmann function allowing uphill movements with more ease. The advantage of IFSQ probability transition function lies inside area which covers Δ*Q* from 0.1 to 0.6. With Δ*Q* = 0.2, for example, transition probability of IFSQ marks approximately 65%, while SQ is limited to approximately 32%. B including this middle class of Δ*Q* values more often into the search path, search robustness is increased but convergence speed doesn't suffer substantial loss.

The depth of exploration depends not only on the transition probability function but also on the annealing schedule and maximum displacement factor *R*0. The annealing schedule function of IFSQ is very steep at the beginning, suppressing the initial deep exploration rather fast and keeping the rest of the search concentrated on small energy differences around the actual point. Thus, so called deep exploration which easily permits uphill movements is available only at the beginning. Still, these initial search opportunities can not be exploited if *R*<sup>0</sup> factor is small as only small part of the potential search space can be reached for the analysis. Small maximum displacement factor is adequate choice for the advanced search status when global minimum area is well located, and extense search is required in order to find the best solution. By keeping *R*<sup>0</sup> constant during the entire search process, as in SQ approach, two opposite search requirement can not be met and consequently *R*<sup>0</sup> is chosen as their compromise. This is changed in IFSQ approach maximum displacement factor is set to high value covering whole search space at the beginning, and it is gradually decreased as the search advances. The dependence is made according to the:

$$R\_0(L) = \delta R\_0(L-1), R\_0(0) = r, L = 1, 2, \dots \tag{22}$$

where *L* stands for the corresponding temperature level, *δ* is the search space reduction factor, and *r* the initial maximum allowed displacement. The numerical values of *δ* and *r* are determined through empirical analysis. Additionally, since IFSQ approach is not oriented only on precision, the maximum number of iterations is not limited to probability, but the search is stopped if in five consecutive temperature levels no cost function improvement is obtained.
