# Mode-Stirred Chamber Sample Selection Technique Applied to Antenna Correlation Coefficient

Authors: Paul Hallbjörner; Juan D. Sánchez-Heredia; Antonio M. Martínez-González
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

Mode-stirred chambers provide an excellent tool for antenna characterization. Recent research has focused on making the mode-stirred chamber technique more versatile. One result of these efforts is the sample selection technique, by which a subset of data with specific properties is extracted from a measured set of raw data. This paper presents how to apply the sample selection technique to measurements of the correlation coefficient in dual antenna measurements, for example, for diversity or MIMO applications. Theoretical background and useful mathematical relations are presented. The application is verified through measurements.


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## Body

## 1. Introduction

Mode-stirred chambers (MSCs) have since about a decade come to widespread use for antenna characterization and tests of wireless terminals [1–5]. Lately, much research has focused on how to make MSCs more flexible and versatile. One example is the sample selection technique [6–8]. It has been demonstrated in previous work to be applicable to the amplitude distribution of a measured signal. This paper presents an extension of the sample selection technique, in which it is applied to the correlation coefficient between the received signals from two antennas. The fundamental principle of the sample selection technique is the same as in previous work; it is merely applied to another parameter. Applying it to the correlation coefficient improves the usefulness of the sample selection technique in tests of diversity and/or MIMO antennas, where the correlation coefficient is critical. The aim of the presented work is to study the efficiency and accuracy of the proposed technique and demonstrate its practical usefulness.


## 2. Sample Selection Applied to Correlation Coefficient

In an MSC measurement, a sequence of samples is recorded. The sample set has a certain statistical distribution, described by a probability density function (PDF). The correlation coefficient is calculated from dual channel measurements of the signals from a pair of antennas. Dual channel data can be described by a bivariate (2D) PDF, according to [9, 10](1)f(x1,x2)=x1x2(1-ρ2)s12s22exp⁡⁡(-12(1-ρ2)⋅(x12s12+x22s22))⋅I0(ρx1x2(1-ρ2)s1s2),
where I0 is the modified Bessel function of the first kind and of order zero. The parameters s1 and s2 in (1) define the amplitudes on the two channels, respectively, and ρ is the complex correlation coefficient between the data on the two channels, that is, ρ=|ρc|. There can be an imbalance in the initial data, which can be accounted for in the PDF by letting s1≠s2.Sample selection is here applied for the purpose of achieving a different correlation coefficient than the one at hand for the whole sequence. This means that for one and the same antenna pair, different scenarios can be extracted in subsets of the initial data, something which can be useful in various performance analyses and tests. The sample selection technique can also be used to change the channel imbalance, but this is also possible by simply weighting the data on one of the channels. In this paper, it is assumed that any imbalance in the initial data should be maintained after sample selection.For illustration, two examples of PDFs according to (1) are plotted in Figure 1. In each plot, the channels are balanced (s1=s2), and different values of ρ are applied.PDF for 2D Rayleigh distributed data,ρ=0 (a) and ρ=0.9 (b).
(a)(b)


### 2.1. Accuracy in the Result

The accuracy with which the target correlation coefficient can be achieved is investigated through numerical simulations using Matlab. The initial data consists ofNi samples distributed according to the PDF fi(x). From these, Nt samples are selected which have a specific target PDF denoted by ft(x). The individual branch distributions are first verified, by generating two Rayleigh distributed sequences of initial data with Ni=50000, s1i=1, s2i=2, and ρi=0.7. The selection process is carried out with maintained branch amplitudes in the target data, that is, s1t=1 and s2t=2, and with target correlation ρt=0.2. Figures 2(a) and 2(b) show the histogram PDF of the simulated data after sample selection, together with a solid curve marking the theoretical Rayleigh PDF.(a) PDF of the amplitudes on branch 1 of simulated sequence after sample selection, together with theoretical Rayleigh curve fors=1. (b) PDF of the amplitudes on branch 2 of simulated sequence after sample selection, together with theoretical Rayleigh curve for s=2.
(a)(b)Following this, the accuracy of the achieved correlation coefficient is investigated by performing sample selection on 500 realizations and plotting the histogram PDF of the achieved correlation coefficients. This is done for two values ofρi (0 and 0.7) and four values of ρt (0.3, 0.5, 0.7, and 0.9), thus in total eight cases. Each case uses Ni=10000, s1i=1, and s2t=2. For comparison, in each case, sequences of length Nt and with target correlation coefficient ρt are also generated directly, that is, without any sample selection process. Due to the finite length of the sequences, these will also show some spread in the correlation coefficient. Figures 3(a) and 3(b) show PDFs of the achieved correlation coefficients, with histograms for the sample selection results, and solid lines for the directly generated data. The numbers next to the peaks are the Nt values. Apparently, the accuracy depends on ρi and ρt. Comparing the sample selection results with the directly generated data shows that the selection process itself does not increase the uncertainty by any significant amount, even though a small shift in the distribution is seen in Figure 3(a) for ρt=0.3 and ρt=0.9. The small shift, as well as some irregularities in the histograms, can be attributed to fluctuations due to the limited number of samples.(a) PDFs forρi=0 and ρt=0.3, 0.5, 0.7, and 0.9.Ni=10000 in all cases. Numbers next to the peaks are Nt values. Histograms are the results of simulations with 500 realizations. Solid lines show spreads due to finite sequence length for the Nt values at hand. (b) PDFs for ρi=0.7 and ρt=0.3, 0.5, 0.7, and 0.9. Ni=10000 in all cases. Numbers next to the peaks are Ntvalues. Histograms are the results of simulations with 500 realizations. Solid lines show spreads due to finite sequence length for the Ntvalues at hand.
(a)(b)VaryingNi shows that the spread of the achieved correlation coefficient follows the relation(2)σρ∝1Nt.
This means that the proportion of remaining samples is critical for the accuracy.


### 2.2. Proportion of Remaining Samples

The proportion of remaining samples (Nt/Ni) in the proposed technique is limited by a theoretical relation [11], which for the general case is(3)q=min⁡⁡(fi(x)ft(x)).
Substituting (1) in (3), it is found after some manipulations that q is a function of ρi and ρt, and the expression depends on whether the correlation coefficient is reduced or increased in the sample selection process. For a reduction in correlation coefficient, it is(4)q-=1-ρi21-ρt2,
whereas for an increase in correlation coefficient, it is(5)q+=(1+ρi)(1-ρt)(1-ρi)(1+ρt).
Equations (4) and (5) are illustrated in Figure 4, as functions of ρt for three different values of ρi.Figure 4
Proportion of remaining samples as a function of target correlation coefficientρt, for three different values of initial correlation coefficient ρi.


## 2.1. Accuracy in the Result

The accuracy with which the target correlation coefficient can be achieved is investigated through numerical simulations using Matlab. The initial data consists ofNi samples distributed according to the PDF fi(x). From these, Nt samples are selected which have a specific target PDF denoted by ft(x). The individual branch distributions are first verified, by generating two Rayleigh distributed sequences of initial data with Ni=50000, s1i=1, s2i=2, and ρi=0.7. The selection process is carried out with maintained branch amplitudes in the target data, that is, s1t=1 and s2t=2, and with target correlation ρt=0.2. Figures 2(a) and 2(b) show the histogram PDF of the simulated data after sample selection, together with a solid curve marking the theoretical Rayleigh PDF.(a) PDF of the amplitudes on branch 1 of simulated sequence after sample selection, together with theoretical Rayleigh curve fors=1. (b) PDF of the amplitudes on branch 2 of simulated sequence after sample selection, together with theoretical Rayleigh curve for s=2.
(a)(b)Following this, the accuracy of the achieved correlation coefficient is investigated by performing sample selection on 500 realizations and plotting the histogram PDF of the achieved correlation coefficients. This is done for two values ofρi (0 and 0.7) and four values of ρt (0.3, 0.5, 0.7, and 0.9), thus in total eight cases. Each case uses Ni=10000, s1i=1, and s2t=2. For comparison, in each case, sequences of length Nt and with target correlation coefficient ρt are also generated directly, that is, without any sample selection process. Due to the finite length of the sequences, these will also show some spread in the correlation coefficient. Figures 3(a) and 3(b) show PDFs of the achieved correlation coefficients, with histograms for the sample selection results, and solid lines for the directly generated data. The numbers next to the peaks are the Nt values. Apparently, the accuracy depends on ρi and ρt. Comparing the sample selection results with the directly generated data shows that the selection process itself does not increase the uncertainty by any significant amount, even though a small shift in the distribution is seen in Figure 3(a) for ρt=0.3 and ρt=0.9. The small shift, as well as some irregularities in the histograms, can be attributed to fluctuations due to the limited number of samples.(a) PDFs forρi=0 and ρt=0.3, 0.5, 0.7, and 0.9.Ni=10000 in all cases. Numbers next to the peaks are Nt values. Histograms are the results of simulations with 500 realizations. Solid lines show spreads due to finite sequence length for the Nt values at hand. (b) PDFs for ρi=0.7 and ρt=0.3, 0.5, 0.7, and 0.9. Ni=10000 in all cases. Numbers next to the peaks are Ntvalues. Histograms are the results of simulations with 500 realizations. Solid lines show spreads due to finite sequence length for the Ntvalues at hand.
(a)(b)VaryingNi shows that the spread of the achieved correlation coefficient follows the relation(2)σρ∝1Nt.
This means that the proportion of remaining samples is critical for the accuracy.


## 2.2. Proportion of Remaining Samples

The proportion of remaining samples (Nt/Ni) in the proposed technique is limited by a theoretical relation [11], which for the general case is(3)q=min⁡⁡(fi(x)ft(x)).
Substituting (1) in (3), it is found after some manipulations that q is a function of ρi and ρt, and the expression depends on whether the correlation coefficient is reduced or increased in the sample selection process. For a reduction in correlation coefficient, it is(4)q-=1-ρi21-ρt2,
whereas for an increase in correlation coefficient, it is(5)q+=(1+ρi)(1-ρt)(1-ρi)(1+ρt).
Equations (4) and (5) are illustrated in Figure 4, as functions of ρt for three different values of ρi.Figure 4
Proportion of remaining samples as a function of target correlation coefficientρt, for three different values of initial correlation coefficient ρi.


## 3. Measurement Example

Measurements are performed in a mode-stirred chamber, on a pair of mobile terminal antennas. The measurement frequency is 2600 MHz. The data is recorded with a network analyzer, as two simultaneous sequences of transmission coefficients, each of 1601 samples in length. These sequences constitute the initial data. A small imbalance of 1 dB (amplitude ratio 1.13) and a correlation coefficient of 0.05 are found in the initial data.The sample selection technique is then applied for the purpose of increasing the correlation coefficient to 0.9, while maintaining the same imbalance. According to (5), q=0.058, which means that it should be possible to have 93 samples in the output sequences. In practice, for relatively short sequences such as these, different realizations will give slightly different values of Nt due to statistical fluctuations. Results of two realizations are presented in Figures 5 and 6. Figure 5 shows the resulting sequences on the two channels, with Nt=88 and ρt=0.9 achieved. Figure 6 shows the sample vectors, that is, (x1,x2) pairs, of a resulting sequence. These figures show that the vectors are correlated and that the imbalance, marked with a solid line in Figure 6, is maintained after sample selection.Figure 5
Remaining sequences after selection, withρt=0.9. Solid line is channel 1 and broken line is channel 2.Figure 6
Remaining samples after selection, withρt=0.9.


## 4. Conclusion

It is demonstrated how the sample selection technique can be applied to the correlation coefficient between the signals from a pair of antennas. The initial sequences can have any correlation coefficient in the interval(0,1), as well as any mean power imbalance. The sample selection technique can achieve any correlation coefficient within the interval (0,1) in the result. Simulations show that using sample selection does not reduce the accuracy in the result, compared to data generated directly with the desired correlation coefficient. Expressions for the maximum proportion of remaining samples are presented. The proposed technique is demonstrated on measured dual channel data from a mode-stirred chamber.



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*Source: 986514-2012-03-28.xml*