**2.1. Non-linear coupling model**

Coupling is commonly defined as undesirable transfer of energy between physical mediums (like metallic wires or optical fibers) or circuit segments. Inside multi-antennas mobile terminal the data received at the antennas is affected on its path both by direct distortion imperfections and coupling as modeled in Figure 1.

**Figure 1.** Simplified direct distortion and coupling

In most practical cases the reasonable assumption for the coupling functions is to model them as polynomials and limit them to third order nonlinear behavior which is consistent with the analog electronic elements and follows the nonlinear behavior experienced in many radio frequency (RF) front ends [1]. Hence, if no direct distortion is present, the input of two AD converters can be described mathematically as:

$$\begin{aligned} y\_1 &= \mathbf{x}\_1 + (c\_{11}\mathbf{x}\_2 + c\_{12}\mathbf{x}\_2^2 + c\_{13}\mathbf{x}\_2^3) \\ y\_2 &= \mathbf{x}\_2 + (c\_{21}\mathbf{x}\_1 + c\_{22}\mathbf{x}\_1^2 + c\_{23}\mathbf{x}\_1^3) \end{aligned} \tag{1}$$

*P*10, *Pi P*<sup>20</sup> *P*<sup>30</sup> Binary 2.00 6.00 20.00 QPSK 2.00 6.00 20.00 8-PSK 2.00 6.00 20.00 16QAM 2.00 7.92 39.20 64QAM 2.00 7.92 39.20

Simulated Quenching for Cancellation of Non-Linear Multi-Antennas Coupling 133

**Table 1.** The unit power of three interference orders for different constellations

**Figure 2.** Coupling strength classification depending on the lost system performance

levels as they are more demanding cases.

**2.2. Coupling cancellation module**

coupling function.

that almost completely destroys the received signal and whose system performance almost shows flat line. Moderate coupling level follows the theoretical curve, but the system performance exhibits slow improvement, while the weak coupling system follows closely the ideal theoretical curve but the system performance is still visibly deteriorated. In the following, search for the decoupling function is mainly made for strong and medium coupling

The coupling cancellation is performed with nonlinear software decoupling module located at the output of analog-to-digital (AD) converter as shown in Fig. 3. The advantage of this position is that it deals directly with sampled physical data avoiding any dependency on specific signal type. Inside the module, cancellation is achieved by determining the inverse

where *x*<sup>1</sup> and *x*<sup>2</sup> are the signals received on the corresponding antennas, and *y*1, *y*<sup>2</sup> the signals at the input of AD converter. Parameters *cij* corresponding to the coupling of the *j th* order (*j* = 1..3) experienced on the *i th* (*i* = 1, 2) input of AD converter signals. These elements describe energy absorption/dissipation that the signals are experiencing when passing through RF front-end. Hence, the direct coupling function is either described mathematically with three parameters (*c*1, *c*2, *c*3) or as [*C*1, *C*2, *C*3] in [dB] describing the physical process of coupling power. The relation between interference amplitude in the coupling function and the interference power at physical level is analyzed in [1] and described as:

$$C\_{ij} = 10 \log \frac{c\_{ij}^2 P\_{j0}}{P\_i} [dB] \quad \Leftrightarrow \quad c\_{ij}^2 = \frac{P\_i}{P\_{j0}} 10^{C\_{ij}/10} \tag{2}$$

with *Pi* the power of the received signal at input *i*, *Pj*<sup>0</sup> normalized power of the interference of the *j th* order and *Cij* the coupling parameter expressed in [dB] for the *i th* antenna and *j th* coupling order. In Table 1 the unit power of three interference orders for different constellations is given. These numbers can be used for simulation purposes when the power of transmitted signal is normalized to 1.

The influence of coupling, depending on the system performance, can be divided in 3 levels: strong, moderate and weak. The strict definition of these three levels can not be made as it depends on the constellation type. However, the borders can graphically be seen in Figure 2 where the presented curve shows theoretical performance of the corresponding system without coupling. Thus when referred to strong coupling, we are referring to coupling


**Table 1.** The unit power of three interference orders for different constellations

2 Will-be-set-by-IN-TECH

In most practical cases the reasonable assumption for the coupling functions is to model them as polynomials and limit them to third order nonlinear behavior which is consistent with the analog electronic elements and follows the nonlinear behavior experienced in many radio frequency (RF) front ends [1]. Hence, if no direct distortion is present, the input of two AD

where *x*<sup>1</sup> and *x*<sup>2</sup> are the signals received on the corresponding antennas, and *y*1, *y*<sup>2</sup> the signals

energy absorption/dissipation that the signals are experiencing when passing through RF front-end. Hence, the direct coupling function is either described mathematically with three parameters (*c*1, *c*2, *c*3) or as [*C*1, *C*2, *C*3] in [dB] describing the physical process of coupling power. The relation between interference amplitude in the coupling function and the

[*dB*] <sup>⇔</sup> *<sup>c</sup>*<sup>2</sup>

with *Pi* the power of the received signal at input *i*, *Pj*<sup>0</sup> normalized power of the interference

*th* coupling order. In Table 1 the unit power of three interference orders for different constellations is given. These numbers can be used for simulation purposes when the power

The influence of coupling, depending on the system performance, can be divided in 3 levels: strong, moderate and weak. The strict definition of these three levels can not be made as it depends on the constellation type. However, the borders can graphically be seen in Figure 2 where the presented curve shows theoretical performance of the corresponding system without coupling. Thus when referred to strong coupling, we are referring to coupling

*th* order and *Cij* the coupling parameter expressed in [dB] for the *i*

<sup>2</sup> <sup>+</sup> *<sup>c</sup>*13*x*<sup>3</sup> 2)

<sup>1</sup> <sup>+</sup> *<sup>c</sup>*23*x*<sup>3</sup>

*th* (*i* = 1, 2) input of AD converter signals. These elements describe

*ij* <sup>=</sup> *Pi Pj*<sup>0</sup> <sup>1</sup>) (1)

10*Cij*/10 (2)

*th* order (*j* =

*th* antenna and

*y*<sup>1</sup> = *x*<sup>1</sup> + (*c*11*x*<sup>2</sup> + *c*12*x*<sup>2</sup>

*y*<sup>2</sup> = *x*<sup>2</sup> + (*c*21*x*<sup>1</sup> + *c*22*x*<sup>2</sup>

at the input of AD converter. Parameters *cij* corresponding to the coupling of the *j*

interference power at physical level is analyzed in [1] and described as:

*c*2 *ijPj*<sup>0</sup> *Pi*

*Cij* = 10*log*

of transmitted signal is normalized to 1.

**Figure 1.** Simplified direct distortion and coupling

converters can be described mathematically as:

1..3) experienced on the *i*

of the *j*

*j*

**Figure 2.** Coupling strength classification depending on the lost system performance

that almost completely destroys the received signal and whose system performance almost shows flat line. Moderate coupling level follows the theoretical curve, but the system performance exhibits slow improvement, while the weak coupling system follows closely the ideal theoretical curve but the system performance is still visibly deteriorated. In the following, search for the decoupling function is mainly made for strong and medium coupling levels as they are more demanding cases.
