## Correlation Coefficient

The *correlation coefficient* is the relationship between the incoming
signals at the antenna ports in an array. Mutual coupling in array systems degrades the
performance of the array. The correlation coefficient between antennas is used as a
performance metric in multiple-input multiple-output (MIMO) systems to quantify the system
performance and efficiency of the antennas. By using the correlation coefficient, a MIMO
system designer is able to understand the level of coupling that exists between the antenna
ports in the system. To minimize the mutual coupling would imply to reduce the correlation
coefficient between the pairs of ports. Antenna designers use two approaches to the calculate
correlation coefficient: the far-field Radiation pattern and S-parameters.

### Far-Field Radiation Pattern

The correlation coefficient of a two antenna array system is:

$${\rho}_{e}=\frac{{\left|{\displaystyle \underset{4\pi}{\int}{\displaystyle \int [\overrightarrow{{F}_{1}}(\theta ,\varphi )\u2022\overrightarrow{{F}_{2}}(\theta ,\varphi )]d\Omega}}\right|}^{2}}{{\displaystyle \underset{4\pi}{\int}{\displaystyle \int {\left|\overrightarrow{{F}_{1}}(\theta ,\varphi )\right|}^{2}d\Omega {\displaystyle \underset{4\pi}{\int}{\displaystyle \int {\left|\overrightarrow{{F}_{2}}(\theta ,\varphi )\right|}^{2}d\Omega}}}}}$$

where $$\overrightarrow{{F}_{i}}(\theta ,\varphi )$$ is the radiation pattern of the antenna system when port
*i* is excited. Computing the correlation coefficient using this formula,
requires the radiation pattern of the antenna. This approach is hard and time
consuming.

### S-Parameter Characterization

Antenna Toolbox™ uses the S-parameter characterization to calculate correlation between antenna elements in an array. This approach is simpler than the far-field approach because the S-parameter calculation does not use the radiation patterns of the antennas. Correlation coefficient is calculated using S-parameter by using:

$${\rho}_{e}=\frac{{\left|{S}_{11}^{*}{S}_{12}+{S}_{21}^{*}{S}_{22}\right|}^{2}}{(1-({\left|{S}_{11}\right|}^{2}+{\left|{S}_{21}\right|}^{2}))(1-({\left|{S}_{22}\right|}^{2}+{\left|{S}_{12}\right|}^{2}))}$$

The advantages of this method are quick analysis and broadband
correlation results. However, this approach assumes that the antennas are lossless and that
incoming waves are uniformly distributed. To calculate and plot the correlation between
antennas in an array, use the `correlation`

function in Antenna Toolbox™.

## References

[1] Blanch, S. Romeu, J. and Corbella, I. *Exact Representation of antenna
system diversity performance from input parameter description*

[2] Stutzman, W.L. Thiele, G.A. *Antenna Theory and Design*, 3rd
Edition. New York: Wiley, 2013, p. 307.