*4.1.1. Search space reduction factor and initial maximum allowed displacement*

Since probability transition function in IFSQ presents high values for all energy differences, it can be stated that IFSQ favors the wide search space exploration during the entire search process. However, 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 deep exploration rather fast and keeping the rest of the search concentrated on small energy differences around the actual point.

**Figure 15.** IFSQ search based on different sets of (*r*, *δ*) factors under strong coupling

If the starting search space is large, and the reduction factor very close to one, the search will progress rather slowly. In this case the probability of worse movement decreases and even enters the downhill search but the search space is kept large all the time. As a consequence five consecutive temperature levels without the reduction of cost function occur far before the global minimum area is located. This is confirmed in Figure 15, where the precision of IFSQ search based on different sets of (*r*, *δ*) factors under strong coupling is presented. As is seen in the right image side, when the starting search space is set to *r*<sup>0</sup> = 0.1 or *r*<sup>0</sup> = 0.3 the algorithm presents poor precision for all space reduction factors except for *δ* = 0.5. Still, with reduction factor of 50% the search is easily drown into saturation as clear tendency path can hardly be established at such fast pace.

If the starting search space is too small, the search progress cannot move far from the starting point and the obtained solution results depend largely on the coupling strength. Under strong coupling the starting search space corresponding to *r*<sup>0</sup> = 10−<sup>5</sup> and *r*<sup>0</sup> = 10−<sup>4</sup> represents the selection of narrow starting search space. The obtained cost function precision is generally low, but as the reduction factor gets closer to one, it is increased in both cases. This is logical as if the small search space is additionally reduced at fast pace the algorithm is easily left without any new candidates and no progress can be made.

Consequently, two starting search spaces, in the middle of the image, built around *r*<sup>0</sup> = 0.001 and *r*<sup>0</sup> = 0.01 offer the adequate selection. In fact the best precision under strong coupling is reached with the following four pair of factors: (0.001, 0.99),(0.001, 0.999),(0.01, 0.90) and (0.01, 0.95) . The pairs are in accordance with the theoretical expectations as they ratify that in vicinity of adequate set of parameters smaller search space requires lower reduction factor to obtain the same precision level.

**Figure 16.** IFSQ search based on different sets of (*r*, *δ*) factors under moderate coupling

The same analysis is repeated for system under moderate coupling of [13,-12,-10] dB in Figure 16. The behavior of algorithm is similar and the potentially adequate set of parameters is found in the central part of the image. However, in this case the best values are moved slightly to the left since the coupling strength is lower and the the ideal solution is located closer to the initial search point. Comparing the results obtained at two different coupling levels, the only pair of parameters appearing in both is *r*<sup>0</sup> = 0.001 and *δ* = 0.99 . Hence, these values are used as adequate parameters for IFSQ search adopted to nonlinear decoupling problem.

#### *4.1.2. Number of visited points per level*

20 Will-be-set-by-IN-TECH

The parameters describing IFSQ approach can be divided into the ones inherited from the SQ method and the ones developed for the decoupling solution which correspond to three new

The inherited SQ factors are set to the numerical values obtained in previous SQ analysis while three new IFSQ factors have to be set empirically (*r*, *δ*, *NVCPL*). When determining the correct level of the unknown parameters, the parameters that are still not analyzed are over-dimensioned which guarantees the correct functioning of IFSQ search at the cost of execution time. The first set of simulations focuses on adequate pair of parameters for maximum allowed displacement definition (*r*, *δ*). Afterwards, the obtained pair is used for analyzing the appropriate number of visited candidates per temperature level (*NVCPL*).

Since probability transition function in IFSQ presents high values for all energy differences, it can be stated that IFSQ favors the wide search space exploration during the entire search process. However, 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 deep exploration rather fast and keeping the rest of the search concentrated on small energy

**4.1. Tuning of IFSQ parameters**

differences around the actual point.

established at such fast pace.

IFSQ parameters that have to be set empirically (*r*, *δ*, *NVCPL*).

*4.1.1. Search space reduction factor and initial maximum allowed displacement*

**Figure 15.** IFSQ search based on different sets of (*r*, *δ*) factors under strong coupling

If the starting search space is large, and the reduction factor very close to one, the search will progress rather slowly. In this case the probability of worse movement decreases and even enters the downhill search but the search space is kept large all the time. As a consequence five consecutive temperature levels without the reduction of cost function occur far before the global minimum area is located. This is confirmed in Figure 15, where the precision of IFSQ search based on different sets of (*r*, *δ*) factors under strong coupling is presented. As is seen in the right image side, when the starting search space is set to *r*<sup>0</sup> = 0.1 or *r*<sup>0</sup> = 0.3 the algorithm presents poor precision for all space reduction factors except for *δ* = 0.5. Still, with reduction factor of 50% the search is easily drown into saturation as clear tendency path can hardly be

The number of visited candidates per temperature level defines the number of energy configurations analyzed between two temperature changes. If this number is large, the search space is analyzed in detail which produces stable results but consumes a lot of the processing time. Nevertheless, if the number of visited configurations is small, the search progress is made based on unreliable data which leads to poor cost function results.

The performance of IFSQ search method based on different *NVCPL* number under strong coupling conditions of [-8,-6,-6] dB is depicted in Figure 17. As expected, when *NVCPL* number is small, like for *NVCPL* = 5 or *NVCPL* = 3, the search is easily drown into a blind ally.

*Inherited SQ Values New IFSQ Functions*

 1 *T*2 1

<sup>2</sup> *r* = 0.001

*ln*(*L*+1) *T<sup>α</sup>* = 0.69, *L* = 1, 2, 3...

Simulated Quenching for Cancellation of Non-Linear Multi-Antennas Coupling 153

(Δ*Q*<sup>2</sup>+*kcT*<sup>2</sup> *L* ) (*D*+1)/2

*pmax* <sup>−</sup>

3*σ*Δ*<sup>Q</sup>* 2 

(*D*+1)/2 *<sup>T</sup>*<sup>1</sup>

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

*δ* = 0.99

*Qsuf f* = 0.005

*Ressymb* <sup>=</sup> 0.00001 *TL* <sup>=</sup> *<sup>T</sup><sup>α</sup>*

*Rescoef f* <sup>=</sup> 0.00001 *<sup>p</sup>*(Δ*Q*) = *TL*

*pmax* = 50% *NVCPL* = 10

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

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

*Q* =

reduce of the processing load gain.

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

*Coefnew*(*i*) = *ξiR*<sup>0</sup> + *Coefact*(*i*) *i* = 1..20; *ξ<sup>i</sup>* ∈ �−1, 1�

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

**Table 3.** Theoretic set of IFSQ search parameters after empirical analysis

**Figure 18.** Ten instances of adjusted IFSQ search under moderate coupling

BER enhancement for *Q*<0.005. Since IFSQ is not oriented only on precision of search can be stopped when the cost function reaches *Q* limit. This occurs on average for 1500 number of iterations as shown in Figure 19 where the corresponding cost function value after 1500 iteration is presented for all ten search instances. The average of all ten executions is located at *Q*¯ = 0.0056 which indicates that the search under moderate coupling can be suspended when 1500 iterations are surpassed. When compared with 4000 iterations that is required to SQ approach to reach *Q*=0.005, this presents the reduction of processing load of approximately 37%. When compared to the SQ approach which is not suspended at some cost function limit, the processing gain is even higher as it is reduced from approximately 16000 iterations to 1500 corresponding to reduction of more than 90%. Eventually, if after 1500 iterations the quality of the result is far from *Q*=0.005 the search can easily be repeated with the consequent slight

**Figure 17.** Cost function of IFSQ search with different *NVCPL* factor under [-8,-6,-6] dB coupling

Local minimum area is located well, but as all temperature levels are spent fast, no uphill movements are available, and no additional progress can be made despite the additional number of iterations.

On the other hand, if number of visited candidates is large and the temperature reduction function is not smooth enough the search can get stacked while in wandering phase and can only offer local minimum as final solution. Eventually the best precision under strong coupling is reached with *NVCPL* = 10 and *NVCPL* = 15 .

When the analysis is centered on moderate coupling the behavior of IFSQ method changes very little and the adequate selection of visited number of candidates per level lies between 5 and 15. Since strong coupling conditions discard *NVCPL* = 5 as too small, and moderate coupling exhibits slightly better results with *NVCPL* = 10 than with *NVCPL* = 15, the adequate parameter adjusted to the nonlinear coupling problem is set to *NVCPL* = 10.
