Technische Universität Ilmenau
Fakultät für Elektrotechnik und Informationstechnik
Institut für Informationstechnik
Fachgebiet Elektronische Messtechnik

Dissertation zur Erlangung des
akademischen Grades Doktor-Ingenieur (Dr.-Ing)

Limitations of Experimental Channel Characterisation

Markus Landmann

 

Table of Contents

Title ... i

Abstract ... iii

Kurzfassung ... v

Acknowledgements ... vii

Contents ... ix

List of Figures ... xv

List of Tables ... xix

1 Introduction ... 1

1.1 Channel Modelling and Experimental Channel Characterisation ... 1
1.2 Drawbacks of the Experimental Channel Characterisation and Motivation ... 4
1.3 Overview and Contributions ... 5

2 Channel Measurement ... 9

2.1 Channel Measurement Techniques ... 9
2.2 MIMO Channel Sounding Measurement Technique ... 12
2.3 Detailed Configuration of the Applied MIMO Sounder Systems ... 13
    2.3.1 Tx/Rx Synchronisation in Remote Operation ... 16
    2.3.2 Back-to-Back Calibration ... 16
    2.3.3 Receiver Sensitivity ... 18
    2.3.4 Phase Noise ... 20
    2.3.5 Arrangement of External Amplifiers in the RF Signal Path ... 22
2.4 Antenna Arrays ... 26

3 Antenna Array Data Model ... 33

3.1 Broadband Model of a Single Antenna Element ... 34
3.2 Narrow Band Model of the Measured Radiation Pattern ... 36
3.3 The Effective Aperture Distribution Function ... 38
    3.3.1 The Idea behind the EADF ... 38
    3.3.2 Construction of the 2D Periodic Radiation Pattern ... 38
    3.3.3 EADF calculated from the 2D Periodic Radiation Pattern ... 40
    3.3.4 Analytic Expression for Radiation Patterns and Derivatives of an Antenna Array ... 41
    3.3.5 Model Error Dependent on the Number of Relevant Samples Used for the EADF ... 43
3.4 Performance and Accuracy Comparison Between Different Interpolation Methods ... 44
3.5 Minimum Angular Sampling Grid for Antenna Array Radiation Patterns ... 45

4 Antenna Array Calibration ... 51

4.1 Obtaining the Antenna Array Radiation Patterns from Measurement ... 51
    4.1.1 Measurement System Setup ... 52
    4.1.2 2D Antenna Positioning System ... 52
    4.1.3 Back-to-Back Calibration of the Measurement System ... 53
    4.1.4 Calibration of the Dual Polarised Reference Horn Antenna ... 55
    4.1.5 Calibration of Channel Measurement Antenna Arrays ... 57
4.2 Estimation of EADFs from Measured Radiation Patterns ... 58
    4.2.1 Data Model of the Measured Radiation Patterns Including a Collective Phase Change ... 58
    4.2.2 EADF Estimation Algorithm ... 60
    4.2.3 Algorithm performance dependent on SNR and number of elements ... 62
    4.2.4 Measurement Example ... 62
    4.2.5 Conclusion on EADF Estimation Algorithm ... 64
4.3 Measured Radiation Patterns and EADFs of Different Array Types ... 65
    4.3.1 “Single Polarised” Antenna Array UCAx1x1x16 ... 65
    4.3.2 Polarimetric Antenna Array SPUCPAx2x4x24 ... 68
    4.3.3 Polarimetric Antenna Array PULPAx2x1x8 ... 68

5 Channel Modelling and Parameter Estimation ... 73

5.1 Radio Channel Model ... 73
    5.1.1 The Specular Path Model (SC) ... 74
    5.1.2 The Dense Multipath Model (DMC) ... 77
5.2 Maximum Likelihood Parameter Estimator RIMAX ... 78
    5.2.1 Formulation of the Estimation Problem ... 79
    5.2.2 Global Search for New Paths ... 81
    5.2.3 Local Search and Discussion of the Algorithmt’s Convergence ... 84
    5.2.4 Estimation example ... 87
5.3 Discussion about the Limitations of the Estimator and its Model ... 87

6 Performance Evaluation of Practical Antenna Arrays ... 89

6.1 CRLB Based Evaluation Framework for Practical Antenna Arrays ... 89
    6.1.1 Example: Comparison of a Theoretical and Measured CUBA ... 91
6.2 Verification of the Antenna Array Performance Evaluation Framework ... 94
    6.2.1 Single Path Scenario ... 95
    6.2.2 Coherent two path scenario ... 96
6.3 Conclusion Chapter 6 ... 100

7 Consequences of Modelling Errors in Channel Parameter Estimation ... 101

7.1 Analysis Procedure and Definition of Basic Parameters Used in this Chapter ... 101
7.2 Antenna Array Related Model Mismatch ... 103
    7.2.1 Systematic Error Related to the Quality of the Calibration Measurement and to the Narrow Band Model ... 103
        7.2.1.1 Accuracy of the Narrow Band Model derived from Anechoic Chamber Measurements (Angular Domain) ... 105
        7.2.1.2 Consequences of distorted Radiation Patterns on the Calculated EADFs ... 110
        7.2.1.3 Simplified Reflection Model of the Positioning System and Distorted EADFs ... 115
        7.2.1.4 Systematic Error of the Estimation Result Caused by Distorted EADFs ... 118
        7.2.1.5 Concluding Remarks on Systematic Error Related to the Quality of the Calibration Measurement and to the Narrow Band Model ... 120
    7.2.2 Systematic Error Due to Incomplete Data Models ... 121
        7.2.2.1 Effect of Ignoring Elevation Characteristics ... 123
        7.2.2.2 Ignoring Polarisation Characteristic ... 131
        7.2.2.3 Consequences of the “Plane Wave Assumption” ... 135
        7.2.2.4 Concluding Remarks on Systematic Error Due to Incomplete Data Models ... 145
7.3 System Related Consequences ... 146
    7.3.1 Consequence of Phase Noise on the DoD/DoA Estimation ... 146
        7.3.1.1 Long Term Phase Drift ... 146
        7.3.1.2 Phase Noise ... 150
        7.3.1.3 Estimation of Artefacts as Consequence of Phase Noise ... 155
    7.3.2 Consequence of an Unsuitable Calibrated External LNA ... 157
    7.3.3 Concluding Remarks on System Related Consequences ... 160
7.4 Conclusions Chapter 7 and Array Error Chart ... 160

8 Overall Limitations of Experimental Channel Characterisation ... 163

8.1 Definition of Metrics ... 163
    8.1.1 Antenna Independent Metrics ... 164
    8.1.2 Antenna Dependent Metrics ... 166
        8.1.2.1 Relative Error of the MIMO Channel Diversity Metrics ... 168
8.2 Error Analysis Based on Ray-tracing ... 169
    8.2.1 Description of the 3D Ray-tracer ... 169
    8.2.2 Ray-tracing Based Analysis Procedure ... 171
    8.2.3 Consequences of an Overall Model Accuracy Lower than the Maximum SNR in the CIR ... 174
        8.2.3.1 Relevance of the Estimated DMC ... 175
        8.2.3.2 Angular Power Spectrum of the SC ... 175
        8.2.3.3 ECM Mismatch of the SC ... 176
        8.2.3.4 MIMO Capacity Error ... 178
        8.2.3.5 NPCG Error ... 179
    8.2.4 Consequences of an Overall Model Accuracy Higher than the Maximum SNR in the CIR ... 180
        8.2.4.1 Relevance of the Estimated DMC ... 181
        8.2.4.2 Angular Power Spectrum of the SC ... 181
        8.2.4.3 ECM Mismatch of the SC ... 184
        8.2.4.4 MIMO Capacity Error ... 184
        8.2.4.5 NPCG Error ... 185
8.3 Error Analysis Based on Measurements ... 186
    8.3.1 Description of the Measurement ... 186
    8.3.2 Measurement Based Analysis Procedure ... 188
    8.3.3 Consequences of an Overall Model Accuracy Lower or Higher than the SNR in the CIR ... 191
        8.3.3.1 Relevance of the Estimated DMC ... 191
        8.3.3.2 MIMO Capacity Error ... 192
8.4 Conclusion Chapter 8 ... 195

9 Conclusions and Future Prospects ... 197

Appendix A Channel Measurement ... 201

A.1 Estimation of the Phase Noise Properties of the MIMO Sounders Used ... 201
A.2 Correction for the Switched Reference Attenuator ... 202

Appendix B Antenna Array Data Model ... 205

B.1 Efficient Antenna Array Data Format ... 205
    B.1.1 Compressed EADF for Single Antenna Element ... 205
    B.1.2 Efficient Matrix Notation for Joint Description of All Antenna Elements ... 207

Appendix C A General Characterisation of the Antenna Arrays Used ... 211

C.1 Definition of Mean Antenna Array Element Parameters ... 211
    C.1.1 Maximum Gain and Mean Gain of All Antennas ... 211
    C.1.2 Mean XPD of All Antennas of the Array ... 212
    C.1.3 3 dB beam width in azimuth and co-elevation ... 212
C.2 Overview of All Arrays ... 212

Appendix D Performance Comparison of Practical Antenna Arrays ... 215

D.1 Settings for the Analysis ... 216
D.2 Constant SNR ... 217
D.3 Constant transmit power and receiver sensitivity ... 218
    D.3.1 Single path scenario ... 220
    D.3.2 Coherent two path scenario ... 223
        D.3.2.1 Settings and Parameter Definition ... 223
        D.3.2.2 Azimuth Results for Constant Co-Elevation ... 227
        D.3.2.3 Co-elevation Results for Constant Azimuth ... 229

Appendix E Glossary of Notations, Operators, Matrices, Symbols, and Acronyms ... 233

E.1 Notations ... 233
E.2 Mathematical operators ... 233
E.3 Special matrices ... 234
    E.3.1 Fourier matrix ... 234
    E.3.2 Noise matrix or vector ... 234
    E.3.3 Reflection and Selection matrices ... 234
E.4 List of Frequently Used Symbols ... 236
E.5 Acronyms ... 238

Bibliography ... 241

Theses ... 253

 

 

1. Introduction

The design of future mobile radio networks (i.e., beyond third generation (3G)) requires research towards new air interfaces which are characterised by highest bandwidth efficiency and unprecedented flexibility. It is commonly understood that radio systems equipped with multiple antennas at both the Mobile Station (MS) and the Base Station (BS) have a huge potential to increase the bit-rates of wireless links. This is possible thanks to a simultaneous transmission of multiple data streams [35]. This multi-antenna technique is called Multiple-Input Multiple-Output (MIMO) and can optimally exploit the spatial diversity of the multiple propagation paths existing in a rich scattering environment. Conceptually, the multipath propagation of the radio channel gives rise to different spatio-temporal signatures for the different transmit data streams, which permits a receiver equipped with multiple antennas to separate those data streams that are otherwise not orthogonal in any of the conventional communication signal dimensions, i.e., in time, frequency, or code. Keeping this in mind, it is not really surprising that the performance of a MIMO system will strongly depend on the radio channel conditions. A key question for a system design and implementation is therefore, whether it is possible to find practically feasible schemes which are sufficiently robust for this task. Another related issue is determining what specific features are required for a practical MIMO system to work reliably under a wealth of various propagation conditions.

1.1 State of the Art Channel Modelling and Experimental Channel Characterisation

The thorough investigation of the multidimensional wave propagation mechanisms is a prerequisite for understanding the spatial and temporal structure of the channel transfer function, and thus, for optimum design and realistic performance evaluation of multiple antenna systems. There are many ways to simulate the input-output behaviour of the radio channel. Basically, channel modelling activities can be divided into deterministic and stochastic approaches.

Examples of deterministic modelling are Finite-Difference Time-Domain (FDTD) and ray-tracing. Ray-tracing is a physically motivated approach and is based on electromagnetic wave propagation analysis (e.g. Uniform Geometrical Theory of Diffraction (UTD)) and uses ray-optical models [36–38]. These models can be very exact and can also describe the time variant channel in certain propagation environments [39, 40]. In case of ray-tracing or ray-launching, a detailed database (i.e., a precise description of the environment) and high computation times are required. To balance the complexity of ray-tracing, stochastic modelling approaches for diffuse scattering are proposed in [41–45].

There are also completely statistical models trying to reproduce the input/output behaviour in a statistical sense by formal assumptions of correlation coefficients and distributions resulting at the transmit antenna and receive antenna ports disregarding the geometrical distribution of the reflectors. A disadvantage of these non-geometric models is that they are inherently specific for a certain antenna characteristic.

For antenna independent modelling (which allows antenna de-embedding and embedding), geometry based channel models are a must [46]. Hereby, the position and the distribution of the scattering areas are generated according to statistical assumptions. In the European forum for cooperative scientific research COST 259 [47], COST 273 [48], and in standardisation bodies (e.g. 3GPP [49], WINNER [50]) Geometry-Based Stochastic Channel Models (GBSCM) were extensively discussed. These models are parametrised based on measurements in typical mobile radio environments. Since the geometry of the propagation environment is taken into account, the influence of the characteristic of the antenna system, such as inter element coupling and polarisation dependent radiation patterns can be included in GBSCM based system simulations. Hereby, the directions of the transmitted and received waves need to be considered ([51, 52], Double Directional Channel Model).

Since the complexity of wave reflection, scattering, diffraction, etc. in real propagation environments can never be completely reproduced by electromagnetic simulation and due to the strong simplifications of the statistical approaches, all models have to be verified and/or parametrised by propagation measurements. So called real-time channel sounders [53, 54] with huge memory capacity and flexibility in using various antenna arrays are employed for such measurements. Directional channel models can be deduced directly from measurements in real propagation environments estimating the geometric parameters of the paths from the recorded data. Given a ray-optical path model, the parameters normally used to model a propagation path are Direction of Arrival (DoA) at the receiver array, Direction of Departure (DoD) at the transmitter array, Time Delay of Arrival (TDoA), Doppler shift, and the complex polarimetric path weight matrix. Figure 1.1 highlights the double directional structure of the multipath channel. Specifically, a double directional measurement, which includes joint DoD/DoA estimation, allows the separation of the directional and polarisation dependent influence of the antennas from the measurements which is a prerequisite of antenna independent channel characterisation. This approach, which has the potential to characterise the radio channel in an antenna independent fashion based on measurements, will be called Experimental Channel Characterisation.

A channel modelling approach that is directly based on the results of Experimental Channel Characterisation is proposed in [1]. This so called Measurement Based Parametric Channel Modelling (MBPCM) is essentially a two-step procedure with an Experimental Channel Characterisation step and a follow-up synthesis step. With the estimated path parameters from Experimental Channel Characterisation an antenna independent description of the radio channel is given. The synthesis step gives us the flexibility to generate realistic MIMO channel transfer functions employing arbitrary application antenna arrays based on the estimated path parameters. However, the estimated path parameters are only valid in a limited area around the original antenna position (during measurement). To cope with this problem the MBPCM approach is extended in [55] and additionally to the path parameters (described before), the reflection points of each path are estimated from the results of Experimental Channel Characterisation. With the knowledge of the reflection points it is possible to extend the valid area around the original antenna position for the syntheses step of the MBPCM approach.

Note that ray-tracing, GBSCM, and MBPCM are mainly verified or parametrised via Experimental Channel Characterisation. Consequently, the accuracy of these modelling approaches are limited to the accuracy of the Experimental Channel Characterisation itself.

Fig. 1.1: Double directional DoD/DoA structure of a multipath radio channel. [only available in download-version]

The radio channel responses can, in general, be observed only within a limited aperture volume which is defined by the array size, frequency bandwidth and temporal observation window. This strictly limits the achievable resolution and accuracy with respect to the angular, delay, and Doppler domain, respectively, when classical non-parametric estimation algorithms are applied. Therefore, high-resolution parameter estimation algorithms have to be envisaged to enhance the resolution by fitting an appropriate data model of the radio channel and measurement system to the measured data. Mainly two different methods are discussed in literature [56]: subspace methods like Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) [57, 58], Root-Multiple Signal Classification (MUSIC) [59] and Maximum Likelihood (ML) methods like Space Alternating Generalized Expectation Maximisation (SAGE) [60, 61]. In these cases the radio channel is modelled by a number of specular propagation paths (Specular Components (SC)) that are described by the parameters DoA, DoD, TDoA, Doppler shift, and the complex polarimetric path weights, which are independent from the antennas used during the measurement. The resolution of these parameters is limited by the measurement Signal to Noise Ratio (SNR), antenna and device imperfections, calibration quality, and the limited validity of the model of the measurement system. The angular resolution performance mainly depends on the antenna array architecture and its manufacturing quality, which includes low electromagnetic coupling, high electrical and mechanical stability. In this context, also the definition of the data model is crucial. It has to be accurate enough to represent the wave propagation and the influence of the measurement system. On the other hand, the radio channel model must not be excessively detailed since the amount of information gathered by the MIMO sounder is always limited and may not be enough to estimate all model parameters precisely. A proper choice of the model structure and order can dramatically reduce the algorithmic complexity and enhance the accuracy and resolution as well as the reliability of the results. It was observed that with the specular paths only 20% to 80% of the received channel sounding signal power can be described. To solve this, the ML algorithm RIMAX [2] using a hybrid radio channel model was proposed. The word hybrid is used to indicate that the radio channel is described by a superposition of SC that result from specular-like reflection and Dense Multipath Components (DMC). Up to now (see [62]) it is assumed that only SC which could not be estimated (unresolved SC) and distributed diffuse scattering are contributing to the DMC. The DMC can be described by a few parameters that essentially parameterise an exponential decaying function in the delay domain. Other causes for the DMC such as model errors related to the measurement system and practical antenna arrays were not considered up to the present.

1.2 Drawbacks of the Experimental Channel Characterisation and Motivation

The accuracy and limits of high-resolution parameter estimation techniques are commonly evaluated theoretically by computing the Cram´er-Rao Lower Bound (CRLB), which defines the fundamental limitations on the achievable parameter variance. In most publications (e.g. in [60, 63, 64]), highresolution parameter estimation techniques are verified based on the CRLBs, simulated antenna arrays, and synthetic radio channels. It is assumed that white Gaussian measurement noise is the only error source. Furthermore, it is assumed that the model of the antenna arrays and the radio channel data model matches reality exactly. Based on this verification, most of the parameter estimation techniques that are well known from theory are applied in practice in a straightforward manner without further considerations (e.g. in [57, 60, 65–71]). Partially unsustainable assumptions are intentionally or unintentionally made with respect to the measurement systems and parameter estimation models applied. For the measurement systems, e.g., the impact of phase noise or unproper calibrated systems are often ignored. For the data models, a common approach is to use incomplete data models (e.g. ignoring full polarisation characteristic and/or elevational characteristic of antenna elements) or to ignore the finite accuracy of the antenna array calibration measurements. Nevertheless, the results of parameter estimation with unsustainable assumptions are used for further processing such as clustering algorithms, channel capacity calculations and parametrisation of GBSCMs. Contrary to the above mentioned publications only a few publications are found that at least try to avoid unsustainable assumptions while estimating the radio channel parameters [3, 4, 72, 73] or try to evaluate practical antenna arrays based on measurements [74–76]. From the above discussion and from a significant experience with measurement analysis [1–3, 5–20] we observed that the reliability, the accuracy, and the limitations of Experimental Channel Characterisation using high-resolution parameter estimation techniques are not clearly defined. This work deals in detail with the limitations of Experimental Channel Characterisation. Methods for accurate and efficient modelling and performance evaluation of practical antenna arrays are proposed. The entire processing chain is analysed in terms of possible error sources. To this end, measurement system impairments, antenna array calibration errors as well as limitations of the radio channel model applied by the high-resolution parameter estimation procedures are investigated. The resulting estimation quality limits are analysed in detail by simulations and measurements. Finally, the impact of parametrising the geometry based channel modelling approach MBPCM with the results of Experimental Channel Characterisation is shown.

The major contribution of the work is a clear answer to five crucial questions:

1. What are the resolution limits of practical antenna arrays?

2. How can we obtain reliable estimates of the radio channel parameters from propagation measurements?

3. What are the fundamental limitations of Experimental Channel Characterisation?

4. Do the DMC result only from unresolved SC and distributed diffuse scattering, strictly speaking are they related to the real propagation conditions only?

5. What is the consequence when parametrising geometry based channel modelling approaches in particular MBPCM with the results of Experimental Channel Characterisation?

The following section illustrates the outline of the work.

1.3 Overview and Contributions

Chapter 2 reviews the most commonly used measurement techniques to characterise a radio channel. Furthermore, it briefly introduces the functionality of the MIMO channel sounding systems applied in the measurements discussed in this thesis. The major sources of error when using such measurement systems are emphasised. Measurement setups are proposed for accurate calibration of the applied measurement systems. System parameters such as phase noise, phase drift, and receiver sensitivity are obtained from measurements. These system parameters are especially relevant in terms of reliability and accuracy of the measurement data and define the limits of the Experimental Channel Characterisation. The purpose and some basic design considerations of high-resolution antenna arrays are discussed. The practical antenna arrays used in this work are presented.

Chapter 3 deals in detail with the efficient and the accurate modelling of measured polarimetric antenna array radiation patterns, which is a prerequisite for an antenna independent description of the radio channel and performance evaluation of practical antenna arrays. A novel and analytic description of measured antenna array radiation patterns by means of the Effective Aperture Distribution Function (EADF) is proposed. As opposed to other methods, the derivatives of the radiation patterns dependent on the DoD/DoA can be easily calculated analytically based on the EADFs. It is shown that the proposed EADF outperforms the Spline interpolation method as well as the vector spherical harmonics (Vector Spherical Harmonics (VSH), [21]) in terms of calculation time by achieving the same interpolation accuracy. The EADF’s low computational complexity, the analytic description of the radiation patterns and its derivatives, and the performance advantage is already exploited in several applications such as the IST-WINNER Channel Model [22], IlmProp (geometry-based Multi-User MIMO Channel Modelling tool) [55, 77]) and the RIMAX parameter estimation framework [53].

An angular sampling grid with a minimum number of samples for the efficient calculation of the radiation patterns of practical antenna arrays is proposed since it is often required for Experimental Channel Characterisation. This grid is based on the idea of the EADF and fulfils the Nyquist theorem in the angular domain.

Parts of the material dealt in this chapter have been published in [13, 21, 23–28].

Chapter 4 proposes a full polarimetric two dimensional (2D) array calibration procedure for the accurate measurement of the radiation patterns of practical antenna arrays. The procedure involves the calibration of the entire measurement setup by means of the MIMO channel sounder, the dual polarised reference horn antenna, and all devices in the Radio Frequency (RF) signal path that do not correspond to the antenna array under test.

Due to the employment of a MIMO channel sounder, we have to cope with a varying phase offset during the 2D antenna array calibration measurement time of several hours (with a Network Analyser (NWA) it would be several days!). These phase offsets prohibit the direct derivation of the EADF model from measurements. A novel gradient based ML estimation algorithm is proposed to correct the measured radiation patterns for the collective phase offset, which consequently allows the accurate derivation of the EADF model.

Applying both the array calibration procedure and the correction for the phase offset the radiation patterns and their EADFs for three different array types are discussed. With this discussion the necessity of accurate full-polarimetric 2D array calibration for Experimental Channel Characterisation and channel modelling is emphasised.

The authors’s contribution related to some of the topics of this chapter has been published in [29].

Chapter 5 reviews the radio channel model applied for the gradient based ML parameter estimation framework RIMAX. The theory of this framework was developed in cooperation with Andreas Richter [62]. In this work the practical implementation of the RIMAX for arbitrary practical antenna arrays and measurement environments was carried out. For the RIMAX implementation, a deep knowledge in estimation theory and handling of measurement data was required.

The radio channel model of the RIMAX comprises two components, the specular (SC) and the dense multipath components (DMC). The SC result mainly from specular-like reflections and the DMC are assumed as unresolved SC and distributed diffuse scattering. As opposed to other parameter estimation algorithms this hybrid model preserves the balance between the maximum information we can gather from measurements and the number of estimated parameters. Further advantages of the RIMAX e.g. the convergence also in the presence of closely spaced propagation paths and the internal reliability check of the estimated paths during the iterative estimation process are highlighted.

However, the drawback of most parameter estimation frameworks is that the impact of error sources other than additive Gaussian measurement noise are neglected. The fundamental limitations of the data model used for parameter estimation with respect to the practical measurement system and practical antenna arrays is briefly introduced.

Own contributions related to the topics of this chapter have been published in [2, 8, 9, 13, 30, 31].

Chapter 6 deals in detail with a powerful framework for the performance evaluation of practical antenna arrays with respect to their angular resolution limits. The framework is based on the calculation of the CRLB of the parameters of the SC. The method can provide information about the DoD/DoA resolution limits of multiple propagation paths and its parameter variances. As opposed to other publications the framework is applicable to any practical antenna array. The advantage of the EADF model using the measured radiation patterns including all imperfections of the antenna array is hereby exploited. Time consuming performance evaluation measurements other than the antenna array calibration become unnecessary with the proposed framework.

 

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