**4.2. IFSQ example**

The parameters of IFSQ algorithm are adjusted to nonlinear coupling problem through empirical analysis based on different set of simulations. The adopted set of IFSQ search parameters is presented in Table 3. The table consists of two columns with inherited SQ values in the left column, and new IFSQ functions and parameters values based on empirical analysis in the right one.

The behavior of IFSQ algorithm applied to coupling of [-10,-12,-12]dB is shown in Figure18, where cost function is presented in dependence of number of iterations for 10 instances of IFSQ search under same operating conditions.

Unlike SQ search which produced 10 closely related curves, the IFSQ curves are spread over wider area as the search progresses due to the higher influence of probability since precision is sacrificed in favor of smaller number of iterations.

Furthermore, the IFSQ cost function is also based on quadratic mean square function for which it has been demonstrated in [1] that smaller *Q* is not transformed into any significant


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

22 Will-be-set-by-IN-TECH

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

coupling is reached with *NVCPL* = 10 and *NVCPL* = 15 .

IFSQ search under same operating conditions.

is sacrificed in favor of smaller number of iterations.

number of iterations.

**4.2. IFSQ example**

in the right one.

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

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

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

The parameters of IFSQ algorithm are adjusted to nonlinear coupling problem through empirical analysis based on different set of simulations. The adopted set of IFSQ search parameters is presented in Table 3. The table consists of two columns with inherited SQ values in the left column, and new IFSQ functions and parameters values based on empirical analysis

The behavior of IFSQ algorithm applied to coupling of [-10,-12,-12]dB is shown in Figure18, where cost function is presented in dependence of number of iterations for 10 instances of

Unlike SQ search which produced 10 closely related curves, the IFSQ curves are spread over wider area as the search progresses due to the higher influence of probability since precision

Furthermore, the IFSQ cost function is also based on quadratic mean square function for which it has been demonstrated in [1] that smaller *Q* is not transformed into any significant

parameter adjusted to the nonlinear coupling problem is set to *NVCPL* = 10.

**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 reduce of the processing load gain.

**5. Conclusion**

behavior and excellent decoupling results.

The effect of coupling are minimized with the software module positioned at the output of analog-to-digital converter and before any further digital signal processing. Inside the module, signal reconstruction based on the approximation of inverse nonlinear coupling is achieved. The generation of inverse coupling function is based on fitting surface approach which can be divided into two steps: calibration process and point-to-point real-time decoupling. Inside calibration process inverse nonlinear coupling surface is build based on function formed with 20 degrees of freedom. Inspired by SA class of approaches, two SQ

Simulated Quenching for Cancellation of Non-Linear Multi-Antennas Coupling 155

Even though classical SQ method can not be used under real time conditions it provides the upper limit of the decoupling precision module and is used for tuning the required system parameters like for example calibration data size. It is intended only for proof of concept and precision reference to other two decoupling search approaches. Eventually actually confirms the adequate selection of the surface model and proves the concept of decoupling software module. The performance of decoupling module based on the SQ search method shows stable

The proposed Improved Fast SQ search method is included in decoupling module as the fast version of the previously developed SQ search. It follows the same search logic as the SQ algorithm with the difference in the annealing schedule which is much steeper, the initial search space is wider and the probability of uphill movement is higher at the beginning. This way the search is carried out more thoroughly in the initial stage, allowing stepper search progress sooner than in the original SQ approach. Furthermore, since the search space is reduced according to the exponential function, the maximum allowed displacement is reduced with temperature, and as consequence the search concentrates more on small space around the global minimum area. The performance of IFSQ decoupling module, based on 1500 iterations, shows good decoupling results, with slightly lower BER level when compared to the standard SQ search. Nevertheless, the required number of iterations is reduced by more than 90% which justifies the precision loss. The approach requires less processing power than

algorithms are proposed for locating the optimal inverse function parameters.

SQ, and as such might be used in mobile phones inside the decoupling module.

on the physical coupling source nor signal constellation.

Igor Arambasic, Javier Casajus Quiros and Ivana Raos *ETSI Telecomunicacion, Universidad Politecnica de Madrid, Spain*

**Acknowledgments**

**Author details**

With the introduction of normalization, the search domain is well defined between �−1, 1�, and the search depth is easily controlled with symbol resolution factor. This proposal allows the adaptation of implemented search methods to general decoupling problem offering at the same time constant precision and efficient optimization of search process with no restrictions

This work has been carried out in the frame of Spanish MCIN project TEC2009-14219-C03-01.

**Figure 19.** The precision of IFSQ search after 1500 iterations

**Figure 20.** Performance of IFSQ based decoupling module under moderate coupling

The reduction of permitted number of iterations to 1500 is verified in Figure 20 where the performance of IFSQ decoupling module under moderate coupling conditions is presented. The first curve on the image presents the behavior of system without the decoupling module under coupling of [-10,-12,-10] dB. The lowest curve shows the theoretical system performance with only AWGN. Two curves in the middle present two decoupling module instances based on IFSQ approach stopped after 1500 iterations. Actually, the coefficients used for generating this two curves correspond to the seventh and tenth IFSQ execution depicted in Figure 19. Regardless of the numerical difference of cost functions, the performance of both decoupling modules is very similar. Since the search is abandoned when cost function reaches sufficient *Q* value defined with *Qsuf f* = 0.005 the performance of decoupling module does not follow the theoretical transmission curve as close as is the SQ approach.
