Validation of the Decomposition Method for Fast MIMO Over-the-Air Measurements

Over-the-air (OTA) throughput tests of wireless Multiple-Input Multiple-Output (MIMO) devices are an important tool for network operators and manufacturers. The user equipment (UE) is placed in an anechoic chamber and a random fading process is emulated by a base-station emulator (BSE).The antenna characteristic of theUE is taken into account by sampling the sphere around the UE with the BSE test antenna at a large number of positions. For low-variance throughput results, long measurement intervals over many fading realizations are required, leading to long and expensive measurement periods in an anechoic chamber. To speed up the OTA test, we analyze the Decomposition Method (DM). The DM splits the throughput measurement into two parts: (1) a receiver algorithm performance tests taking the fading process into account and (2) an antenna performance test without fading process emulation. Both results are combined into a single throughput estimate.The DM allows for a measurement time reduction of more than one order of magnitude. We provide an analytic and numerical analysis as well as measurements. Our detailed results show the validity of the DM in all practical settings.


I. Introduction
W ITH the introduction of MIMO and receiver diversity fourth and fifth generation (4G, 5G) wireless devices allow for large gains in throughput performance. These gains are highly dependent on the performance of the receive-antenna system [1] and the receiving algorithm [2] and the propagation environment [2]. The devices can change the behavior of the antenna systems, e.g. by using beamforming mechanisms [3]. They also can adapt software algorithms to suit the environment they are currently used in.
Wireless equipment manufacturers as well as network providers are pushing to have performance tests of the devices available at hand. Network providers want to recommend the user equipment (UE) with the best performance to their customers, manufacturers want to be able to compare the quality of their own UE to the one of the competitor. These comparisons should include the antenna systems, the analog frontends, digital receiving algorithms and baseband processing. One of the methods proposed by 3GPP, a yet very promising one -the Decomposition Method (DM), will be investigated further in this paper. The authors' scientific contributions described in this paper are: • Analytical and numerical validation of the DM • Validation of the method by measurement • Detailed analysis of error sources and error bounds Several other methods exist for performance testing of MIMO enabled UEs [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. A good overview is provided in [23]. Nevertheless, this paper focuses only on the DM, believing that this method can provide good and reliable results with limited effort, knowing there are limits of the method. Still, all other existing test methods also have strong sides and weaknesses too. These other performance test methods will not be treated in this paper. The main advantage of the DM is that it delivers good and reliable results with limited measurement device effort and that the calibration effort of the setup is limited too.
II. Accelerated Testing of the UE by employing the Decomposition Method In a perfect test scenario, the performance of a certain UE will be tested almost like it would be in the real world. The antenna performance and the algorithmic performance of the UE will be tested together in a single step (Fig. 1, 'real world'). The influence of the fading channel and the antennas performance are tested in one step. The test antennas have to be placed in many points to cover the whole test sphere (Fig. 2, Fig. 5(a)). In every single point the test antennas are placed, the performance of the entire UE is tested. To get representative figures of merit of the UEs performance on the propagation channel, lots of different channel realizations need to be examined. This results in a long measurement time, which is a big disadvantage in nowadays labs. Splitting up the ideal 'real world' scenario into two parts: a conducted and a radiated test setup for the downlink case of a 2 × 2 MIMO system [24] To accelerate the tests, the idea is to measure the time consuming algorithmic performance tests only once ( Fig.  1, 'conducted'). The antenna performance tests still have to be done in many positions, but if there is no fading channel, the wireless channel is static (Fig. 1, 'radiated'; Fig. 2) and the measurements can be done very fast, as no stochastic channel behavior has to be examined. Later on the results of the antenna performance tests ('radiated' measurements) and the algorithmic performance tests ('conducted' measurements) are combined in a way that they are almost congruent with the 'perfect' test method ('real world' scenario), which tests all in one step. Later on the results will be combined to one single figure of merit. This two step-approach is called Decomposition Method (DM), as it decomposes a single measurement into different single steps. The DM [24], [25] maps the 'real world' test scenario (Fig. 1) in a conducted and a radiated setup. The 'real world' scenario consists of a transmission chain from the base station, its antenna system the wireless propagation channel to the UE receive (RX) antennas. This scenario is split into two different measurement setups, which speeds up the measurement by my more than an order of magnitude. Both measurement results are used as a basis for calculating the throughput vs. downlink power of the UE. The radiated measurements require an anechoic chamber with two moveable test antennas (Fig.  2, Fig. 5(a)).

A. Radiated and Conducted Setup
The conducted setup (Fig. 1, 'conducted') consists of the Base Station Emulator (BSE) connected to channel emulator, which is linked to the UE by wired connections. The conducted setup is employed for two measurements: • Using a fading channel ('channel' measurement). The channel emulator emulates the wireless propagation  [27] channel. It alters the signals transmitted by the BSE in a way a physical wireless propagation channel would do it, including multi-path propagation (frequency dependent channels) and Doppler effects. • Using an identity matrix as channel matrix -no fading is applied ('baseline' measurement). This measurement can be seen as if the UE would directly be connected to the BSE by wires. The results of the 'baseline' measurement will deliver the lowest possible downlink power values for a certain data throughput rate. These two measurements employing the conducted setup can be done in normal laboratory environment and do not require an anechoic chamber. The radiated setup (Fig. 1, 'radiated') employs the Two-Channel Method for the radiated over-the-air (OTA) tests [26]. It employs an anechoic chamber with two test antennas that are positionable in a Θ-plane, a turntable for UE placement that is turnable in the Φ-plane, as depicted in Fig. 2. The UE is placed in the center of the anechoic chamber (quiet zone). The test antennas always have a constant distance to the UE and they are fed with signals from the BSE. The radiated setup is static and does not contain any fading, it is used for antenna performance tests. The two test antennas generate the EM field impinging on the UE for the radiated tests. The polarizations of the test antennas can be switched independently between vertical (Θ) and horizontal (Φ) polarization. The 'radiated' measurement is done with the radiated setup. The name Two-Channel Method originates from the two downlink channels towards the UE enabled by the two test antennas (Fig. 2).

B. The Split of the Measurements
The DM requires three performance test results of the UE: two using the 'conducted' and one employing the  'radiated' setup, as depicted in Fig. 3: 1) 'Baseline' measurement: conducted measurement without fading. The user equipment is connected to the base station emulator by two coaxial cables, the channel matrix H ch = I. The UE exhibits the best performance, it is only limited by the intrinsic UE noise. The result of the 'baseline' measurement is the downlink power P bl (y) required for a certain relative throughput y, see Fig. 4. 1SP12_1e Rohde & Schwarz Two-Channel-Method 23

UE coordinate system
The following figure is a sketch of the test site with the UE attached to a phantom head. The coordinate system of the UE, xyz, is shown as well as the angles ! and . ! ranges from 0° (top) to 180° (bottom). = 0° is in the orientation of the +x axis, increasing to 90° along the +y axis.

Set-up with antennas in a vertical plane
An approach for a set-up for antennas in a vertical plane is using moving arms on which the test antennas are attached.
For the azimuth angles, the UE is positioned on a turn table in the center axis of the chamber.
(a) Rotation planes of test antenna and device under test in the anechoic chamber [26] MIMO OTA System Implementation 1SP12_1e Rohde & Schwarz Two-Channel-Method 28 Two quad-ridged horn antennas are utilized as test antennas ANT DL1, ANT DL2. Each antenna is capable to create or orthogonal components of linearly polarized field through the dedicated and antenna connectors. The test antennas provide the downlink connection to the UE. The circularly polarized communication antenna ANT UL provides the uplink connection to the BSE. The antenna ANT UL is integrated inside the azimuth positioner. All five antenna ports are connected with RF cabling to the external instrumentation through RF connectors (DL1 , DL1 , DL2 , DL2 , UL) mounted on the access panel (AP).

External system components
The Wideband Radio Communication Tester R&S CMW500 is utilized as a base station emulator with LTE signaling functionality. In a configuration with two RF frontends it can carry two independent LTE downlink data streams x1 and x2 when set to spatial diversity mode. Another RF port receives the LTE uplink signal. 2) 'Channel' measurement: conducted measurement including fading. The signal is dynamically faded, Doppler shifts and spreads are applied. The ability of the device to cope with these circumstances without the UE antenna system influence is tested. The devices performance will be degraded in comparison to the 'baseline' test, resulting in higher needed downlink power for a certain throughput. The conducted faded result is P ch (y), where the subscript ch reflects the 'channel' measurement ( Fig. 4). 3) 'Antenna' measurement: radiated measurement without fading. The Two-Channel Method [26] is used to check the performance of the device antennas (including the receiving algorithms without its fading behavior). The radiated coupling between the test antennas and the UE is exhibited in Fig. 5(a). The user has to choose a proper set of test antenna and UE positions, so called constellations. These constellations also include the test antenna polarizations, that are used during the measurements [28]. The result of the measurement is P ant (y), where the subscript ant reflects the 'antenna' measurement ( Fig. 4).

C. Combining the DM Results to One Figure Of Merit
To get a single figure of merit, the results gathered using the DM ('baseline' result, 'channel' result and 'antenna' result) have to be combined: measurement results of the DM and how to combine them [24] is shown graphically in Fig. 4. Eq. (2) explains itself by graphical summation and subtraction of the results 10 logP dc (y) = 10 [( log P ant (y) − log P bl (y) ) + ( log P ch (y) − log P bl (y) ) + log P bl (y) ].
(1) which leads to 10 log P dc (y) = 10 log P ant (y) + 10 log P ch (y) − 10 log P bl (y). In a linear domain, the result P dc (y) is computed by P dc (y) = P ant (y) P ch (y) P bl (y) .
The DM result P dc (y) is ideally the same as the 'real world' scenario result P oa (y) with the deviation d being as small as possible: 10log(d) = 10log(P dc (y)) − 10log(P oa (y)) .

D. OTA Measurement Time Reduction using the DM
The 'overall' scenario ( Fig. 1, 'real world') with testing fading and radiated performance at once, is the ideal setup for measuring the UE, delivering exact results. Using the 'overall' setup for a measurement of N = 128 constellations, takes the time t overall . Measuring the UE's behavior on the channel with block fading takes the time t bf_ch , whether radiated for a single constellation or conducted. For the results of the static radiated 'antenna' results, it takes the time t ant,st . An example with reasonable time values from practical measurements is given: In comparison, if the measurements are split into conducted and radiated tests, the complete test time needed is t dc This is an acceleration of the measurements of a factor a ≈ 36! Therefore measuring the 'overall' setup is impractical, and the DM with its splitted measurements shows its strengths. The author uses the 'overall' results only for comparison to the DM results.
In the test, data payload is organized in subframes, e. g. for the mobile communication standard LTE. The times t dc and t overall were measured with a data payload of 400 sub-frames for the static measurements and 20000 subframes for the faded measurements per measurement or constellation.
If the fading measurements require more sub-frames, the acceleration rate is even bigger than 36. This acceleration also reduces the anechoic chamber occupancy. The high side of the DM is that faded measurements have to be done only once, resulting in the mentioned acceleration of test speed.

III. Signal Model
The validation of the DM is the goal of this paper. We validate the hypothesis that radiated fading measurements can be split in conducted and radiated measurements and combined afterwards into a single figure of merit. The complete transmission chain is depicted in Fig. 6. We can express the received signal vector for a single flat-fading subcarrier as where s contains the encoded random transmit symbols, n ∼ CN {0, σ N I} denotes additive white symmetric complex Gaussian noise and H ant is the antenna matrix, describing the angle dependent complex antenna gain. We assume linear processing employing whether an ICD/ZF (inverse channel detector / zero forcing) detector with C = H P or Minimum Mean Square Error (MMSE) detector at the receiver side with C = (H H H + σ 2 N I) −1 H H , where P is the pseudoinverse and H is the hermitian transpose. A general channel matrix is denoted by and σ 2 N is the noise power density. The symbol estimates are demapped and decoded to the RX data stream, as depicted in Fig. 6. For uncorrelated channels H α = H β = I and H = H ch .

A. Matrix Models for the Transmission Channels
In general, transmission channels with frequency flat fading have been applied. This results in 2×2 channel matrices, where each channel element h ij simply represents the complex transmission gain. The authors employ an identity matrix as channel matrix for the 'baseline' result All channel power gains are average channel power gains. The channel power gain is represented by the expectation value of the squared Frobenius norm of the channel matrix, which is denoted by the .
The average power gain of the identity matrix is The average power gain of the Rayleigh fading channel has to be smaller than 2, which reflects the fact that the passive transmission channel has less power gain than the identity channel matrix. The identity channel matrix is physically represented by a wired connection between BSE and UE. The authors set the average power gain of the Rayleigh fading channel to ν where the condition ν < 2 has to be fulfilled.  not necessarily need to be round numbers. The condition of ν < 2 gives credit to the fact that a transmission over a general channel matrix H ch has less energy than a transmission over coaxial cables, represented by H ch = I. The second moment for a complex gaussian distribution [31] is where σ is the standard deviation of the complex Gaussian distributed channel parameters h ch,i,j and the expectation value of the squared Frobenius norm of H ch is where N TX is the number of transmit antennas, and N RX the number of receive antennas. With the standard deviation σ = 1, the introduced scaling factor is All simulations in this paper employ H ch including the scaling factor A = 0.2263 and a complex Gaussian distributed channel with σ = 1.

1) Kronecker Channel Model:
The general correlation matrix of a MIMO channel is defined as where vec(H) stacks the matrix elements in a vector. In this work we use the simplified correlation based Kronecker channel model [32,33] where the receiver and transmitter correlation matrix [33] is denoted by and respectively. The off-diagonal elements α and β define the cross coupling. We use the following parameterization of α and β as proposed in [32,34]: • High correlation: α = 0.9, β = 0.9 Random correlated channel realizations are computed according to where H w denotes a MIMO matrix with independent and identically distributed (i.i.d.) complex Gaussian entries CN {0, 1} with zero mean and variance one.

2) Antenna Matrices:
We assume no coupling between the base station antennas. The DUT antenna coupling is denoted by H ant . It is calculated using the complex E-field pattern [26]: where E denotes the complex E-field pattern of the receiving antenna 1 or 2. We denote the polarizations of the test antennas with {p, q}. These polarizations {p, q} can be in the Φ-or in the Θ-plane.
Matrix H ant is shown in Tab. I for three exemplary antennas with different condition numbers κ log , namely for fully decoupled antennas (κ = 0 dB) to strongly coupled antennas (κ = 40 dB). The antenna matrix H ant is normalized to a squared Frobenius Norm (SQFN) of 2 which corresponds to the power gain of an identity matrix, shown in (12). The reason is that results can be compared easily, if the power gain is the same for all H ant . The antenna condition number in linear terms is the ratio of the largest to the smallest eigenvalue of the antenna matrix H ant (Ω 1 , Ω 2 , p, q) (24). The bigger κ lin is, the worse is the matrix conditioning. For visualization purposes the author defines κ log = 20 log 10 (κ lin ).
The identity antenna matrix corresponds to fully decoupled antennas, with κ log = 0 dB. Increasing coupling between the antennas provide antenna matrices with higher condition numbers.

B. LTE as a Show Case for MIMO Enabled Communication Standards
The wireless mobile communication standard Long Term Evolution (LTE) is a common use case for the application of the DM. Therefore main elements of the standard are used for the numerical validation.
1) Modeling of the Physical Layer (PHY) for Downlink / Transmitter Side: Figure 7 shows a block diagram of an LTE transmitter [35]. User payload is segmented into blocks, then channel coding with rate matching is applied. LTE employs turbo coding for channel coding, code blocks are concatenated to create codewords. The user data is scrambled, which is an operation where the user data is convolved with a pseudo random sequence resulting in the scrambled bit stream. In the modulation mapper the bit stream is mapped to a certain complex in phase / quadrature phase (I/Q) modulation scheme, like 2) Channel Estimation Errors: Another part of the investigations is to have a closer look on channel estimation errors. In considerations of receiver types like the ICD/ZF or the MMSE, it is assumed that the transmission channel is perfectly known. The following investigations also consider imperfect knowledge of the channel taking into account the length of the pilot signals that are used for channel estimation. Mengali et al proposed a method to compute the variance of the estimation error for QPSK. The error follows a normal distribution with a first moment equaling zero and variance [36] where L is the length of the pilot sequence and P/(BN 0 ) is the symbol energy to the noise energy (SNR). The channel estimation in LTE is executed over 19 OFDM subcarriers and 10 subframes, each containing 2 slots. Every slot contains one symbol for estimation. This results in L = 19 subcarriers × 10 subframes × 2 slots × 1 symbol = 380 symbols (27) per estimation [3], [34]. With (26) and (27), the channel estimation error variance σ 2 est_ err can be calculated as Equation (26) shows the dependency of the estimation error on the symbol energy to noise energy P/N 0 . The equation is valid for flat fading and time invariant channels.

IV. DM Validation -MIMO Channel Capacity Approach A. Validation procedure
As an additional step, the authors employ the MIMO channel capacity expression, therefore a generic view can be achieved in the investigations. This approach imposes the use of a receiver of general type working close to or at the Shannon bound. For the investigations a 2 × 2 MIMO transmission system with two uncorrelated data streams and symbol energy of P/N T per stream is chosen. The symbol energy is equally distributed over the two transmission paths. The signal covariance matrix of the symbols is a diagonal matrix in this case [37], where N T is the number of transmit antennas. The sum of the main diagonal elements of R ss is the symbol energy [37] trace (R ss ) = P .
Using these assumptions, the rate of the MIMO channel simplifies to [37]   where N R is the number of receiving antennas, H is the channel matrix, the trace operator denotes the sum of the main diagonal elements of a matrix, I N R the identity matrix with dimension N R × N R , E is the expected value and det is the determinant, respectively. For the sake of simplicity, the validations in this section use the absolute throughput rate C, and not the relative throughput rate y as in sec. II-C. In both cases the validity of the Decomposition Method can be shown. Equation (31) can also be expressed in terms of the eigenvalues of HH H employing the Eigendecomposition reducing to where Λ contains the non-negative Eigenvalues on its main diagonal [37], [39].
Using the Eigenvalue matrix Λ and its elements λ i , the expected (ergodic) rate can also be written as [39] We use (31) to calculate the ergodic rate for the three DM scenarios and the overall scenario numerically. The channel matrix is modified according to the different steps of the DM as follows: The required transmit energy P is a function of the channel rate C (for fixed noise power spectral density N 0 ) for a given set of channel matrices, providing P oa (C), P ant (C), P ch (C), and P bl (C). Inserting these results in (2) and (4) we can compute the deviation d. Figure 8 exhibits the absolute throughput rate C versus P/N 0 for different antenna correlation condition numbers (κ = 10 dB). In theory, the channel rate C does not have a limit, as long as P/N 0 is rising. In practical systems the throughput will always saturate at the maximum data rate related to the chosen modulation and coding scheme of a wireless transmission. Figure 8 also shows that the DM result does not deviate at κ = 0 dB. The deviation d, as defined in (4), is increasing with rising κ and is also depending on the P/N 0 . The deviation d versus the channel capacity C is depicted in Fig. 9. The plots show d over C for κ ∈ {0 dB, 10 dB, 20 dB, 30 dB, 40 dB}. For κ log = 0 dB, the deviation d is zero, the DM works perfectly. For rising κ, d is increasing and the maximum of d shifts towards bigger P/N 0 and decreases slower than at lower values of κ. Nevertheless, even at a very bad conditioned antenna matrix with κ = 40 dB, the maximum deviation d is below 1.72 dB.

B. Conclusion on the Channel Capacity Approach
The authors validate the DM by using the MIMO channel capacity formula. MIMO channel capacity curves are simulated and overlayed using the DM. This section compares the 'overall' results to the 'DM' results by showing the occuring deviations d. For the given antenna condition numbers the DM works within the error margin given in Fig. 9. This error margin of d lies within the error margin given by measurement uncertainties in practical OTA measurements [40]. Dozens of measurement uncertainty sources are listed in [41], [42]: mismatch of transmitter chain, insertion loss of transmitter chain, influence of the probe antenna, uncertainty of BSE power level, statistical uncertainty of throughput measurement, fading flatness  within the LTE band, uncertainty of the network analyzer, instability of cable connections just to name a few.

V. DM Validation -Numerical Analysis Approach A. Introduction
To validate the DM from another perspective, the authors choose the numerical link level simulation of a wireless MIMO transmission system. The DM is intended for testing general UEs employing MIMO antenna technology. An important point is to investigate the validity of the DM for the use with LTE devices. It will be a main application of the method. As the number of employed signal processing elements for LTE is high, this paper focuses on the most important blocks of the transmission in the PHY layer for the simulation and numerical validation. The authors again realize a transmission using the block diagrams in Fig. 6 for the numerical simulation of the transmission. They discuss the results of this transmission system and present the validation of the DM under the given conditions.

B. Concept of the Numerical Simulations
As LTE is a complex transmission protocol containing a lot of different functionality blocks, it is necessary to focus on the blocks that might be influential on the DM. All investigations are done with flat fading wireless transmission channels. If problems show up already with the simple model of flat fading channels, all scenarios containing more complex transmission models including tapped delay channel models and OFDM schemes will also have a problem. A wireless system employing an OFDM scheme, splits the complete channel bandwidth of the frequency dependent channel in many narrow-band channels that are assumed to be of flat fading nature. In the following investigations, the authors deal with one of these small flat fading OFDM channels. This has the consequence, that it is not necessary to simulate all the OFDM channels. All steps in the DLSCH block (Fig. 7) are implemented, as well as the modulation mapper and the MIMO processing blocks. The elements in the Physical Downlink Shared Channel (PDSCH) like scrambling, RE mapping and OFDM processing have been omitted, as the authors expect minor influence on the results when employing frequency flat transmission channels.

C. Simulation Setup Configuration and Figures of Merit
The parameters for the simulation setup and the necessary figures of merit (FOM) are introduced in this section, as well as how they can be calculated. The block diagram of the transmission systems in Fig. 6 eases the understanding of the parameters and the FOMs.

Transmission Data Block Size and Subframe Error Rate (SFER):
The transmitted data streams have a length of 2 8 bits per realization of the channel matrix H ant . The transmitted (TX) bit stream is compared to the received bit stream (RX). This is done with a test window size of 2 4 bits, so-called subframes (SF). If the RX subframe completely matches the TX subframe, it is recognized as correct. If there are one or more bits wrong, the complete subframe is rated to be incorrect. The mathematical formulation of the Subframe Error Rate (SFER) is SFER = number of correctly received SF number of all transmitted SF , it ranges from 1 (no subframe (SF) transmitted correctly) to 0 (all SFs transmitted correctly).
It ranges from 0 (no SF transferred correctly) to 1 (all SFs transferred correctly).

Downlink Power, Bandwidth and Noise
The downlink power used for the simulation ranges from -130 dBm to -50 dBm. It is always related to a subcarrier bandwidth of B = 15 kHz. The noise energy is N 0 = −174 dBm + 10 log 10 (B) + N f = −174 dBm +10 · log 10 (15 kHz) + 5.2 dB = −127 dBm, (37) where N f = 5.2 dB is the noise figure of the UE employed for the measurements in this paper; white noise is assumed.

Simulation Parameters and Number of Realizations:
The numerical simulations employ a variation of settings: • Detector type: ICD/ZF or MMSE • Ideal or non-ideal channel estimation • DLSCH processing enabled / disabled • Low, medium or high correlation matrices H α and H β • Receive antenna condition number κ The authors implemented the transmission chain following the block diagram in Fig. 6 for the simulations with DLSCH processing without DLSCH processing. A realization is a certain transmission system instance employing one set of stochastic parameters, like channel parameters or channel estimation errors. The authors executed all simulations with 5000 realizations of the channel matrix H ch . They averaged the throughput of all realizations linearly for each downlink power value throughout this paper. This averaged throughput is displayed in the result throughput vs. downlink power curves. Every realization employs a new set of transmission data blocks. Figure 10 exhibits an example for a set of simulation result curves with relative throughput over downlink power. The mentioned result figure employs an antenna condition number of κ = 40 dB and other different transmission parameters, as mentioned in the caption of the figures. The examplarily taken graph shows all three necessary curves to calculate the DM result: the 'baseline', 'channel' and 'antenna' curves. The DM result and the 'overall' result are also depicted. If the DM works well within the given parameters, the DM result and the 'overall' result overlap, as it is the case in Fig. 10. The quantitative measure how well the DM works, is the deviation d. It shows the difference between the actual value (DM result) and the ideal value ('overall' result), as depicted in (4). The authors simulated the different transmission scenarios for a wide range of parameters. All downlink power vs. RTP show the same 'baseline' result curve, which exhibits the best performance a transmission system can reach. The higher the performance, a certain throughput can be received with less downlink power. The BSE is directly connected to the receiver:

1) Relative Throughput vs. Downlink Power Curves:
The slope of the 'channel' curve is less steep in comparison to the 'baseline' curve, because of the complex normal distribution of the channel parameters in a general H ch matrix. In the case that the receiving antenna matrix is unity and uncorrelated (κ = 0 dB), the 'antenna' result overlaps with the 'baseline' result. The 'antenna' simulations employ H β H ch H α = I. The overlapping of the 'overall' curve and the DM curve is the ideal case, in this case the DM works perfectly. As the antenna elements starts to be correlated (condition number κ of the antenna matrix H ant > 0 dB), the receiver performance is degraded. This degradation leads to a result curve (the 'antenna' result curve) that is a right shifted version of the 'baseline' result curve, as shown in Fig. 10. This right-shift means that the receiver needs more signal downlink power to reach the same throughput as with an uncorrelated channel. The reason for this behavior is that the receiver cancels the influence of the channel, but it also enhances the existing intrinsic noise, generated by the RF frontend of the UE. The more the receiving antennas are coupled, the worse is the conditioning of the antenna matrix (indicated by κ). Looking at Fig. 10, the antenna matrix condition number κ = 40 dB. For a relative throughput of 0.7, the 'baseline' setup needs a downlink power of −117 dBm, while the 'antenna' setup needs −83 dBm, which is a difference of 34 dB. So to say, the degradation through the correlation of the receiving antennas results in a degraded performance of the whole transmission system. The influence of the different simulation parameters will be discussed in the following section.   Figure 11. Deviation d versus antenna condition number for DLSCH processing on, non-ideal channel estimation; low, medium and high channel correlation; ICD/ZF and MMSE receiver

2) Influence of the Transmission System Parameters on the Deviation d:
The authors show the influence of the simulation parameters on the deviation d. These results are exhibited in Fig. 11 -Fig. 16 and the authors will discuss them elaborately. The different simulation parameters are • DLSCH processing on/off • Ideal/non-ideal channel estimation • Low, medium and highly correlated channel parameters These figures show the investigation of the deviation d versus the antenna condition number κ for this set of parameters. They are arranged in different ways to highlight certain ways of behavior. Certain curves are exemplarily taken to explain a certain way of behavior. Some facts are mentioned twice, to ease understandability. Please note that the depicted deviations d are maximum values that will not be reached in practice, when a real antenna system is employed. Influence of the Condition Number κ: Generally speaking, the absolute deviation |d| rises with rising antenna condition number κ. Figure 11 is taken exemplarily to show this behavior, which appears in all simulations. The absolute deviation starts to grow rapidly and settles more or less at κ = 20 dB. The employed simulation parameters for the deviation results in Fig. 11 are: DLSCH processing on, non-ideal channel estimation; low, medium and high channel correlation; ICD/ZF and MMSE receiver. It is interesting to see that the DM works almost perfectly for low correlation and the ICD/ZF receiver. In comparison, the MMSE receiver causes an approximate maximum deviation d = −1 dB at low correlation. With rising correlation, both receiver types, MMSE and ICD/ZF, cause bigger absolute deviation |d|. The value of |d| is greater for MMSE than for ICD/ZF in general.
Influence of the Channel Correlation: Figure 11 also points out the differences of the different channel correlations low, medium and high. DLSCH processing is applied and the channel estimation is non-ideal. The DM works almost perfect for ICD/ZF receivers at low correlations. In contrast, the receiver Type MMSE delivers mentionable deviations already at low channel correlations. As soon as channel correlation becomes medium or high, the absolute deviation |d| increases. The deviation |d| is always bigger for the MMSE receiver than for the ICD/ZF receiver. Figure 12 shows the same trends but with higher absolute deviation |d| (no DLSCH processing is applied). Influence of DLSCH processing on / DLSCH processing off: Figure 16 depicts the differences of the deviation d, between DLSCH processing being turned on and off for ideal channel estimation. The channel correlations are again low, medium and high. In contrast, Fig. 14 exhibits the results for DLSCH processing being turned on and off for non-ideal channel estimations. The main message of both graphs is that if DLSCH processing is turned on, the abs. deviation |d| becomes smaller for all three correlation types and two receiver types (MMSE and ICD/ZF), in comparison to DLSCH processing turned off. This is true for all channel correlation types: low, medium and high. This is the reason why the DM is very suitable for LTE performance tests, as LTE always uses DLSCH processing. The influence of the channel estimation on the results of d is very small. Summarizing, the following facts can be pointed out: • The DM suits well its purpose, the performance testing of MIMO enabled wireless UEs, especially for low correlation channels, it delivers small deviations d for both receiver types: ICD/ZF and MMSE. • In general, if the antenna condition number κ is rising, the absolute deviation |d| rises. • If channels correlations are medium or high, the absolute deviation |d| is bigger than for low correlation channels. • The DM delivers better results with the ICD/ZF receiver in comparison to the MMSE receiver for all channel correlations. • The influence of the channel estimation error is negligible in the case of an LTE pilot sequence length as a reference example. • If a transmission system employs DLSCH processing, the DM performs better as without it. This behavior makes it very suitable for performance tests with LTE, as it always uses DLSCH processing. • Rising deviation occurs for antennas with very poor behavior, but manufacturers will strongly avoid such poor designs as it results in a significantly reduced MIMO system performance.
All diagrams with deviation d versus antenna condition number κ show maximum errors, which will not occure in practice. The reason is that the condition number of the antenna matrix H ant is different for every constellation. Therefore the authors execute investigations with a practical antenna set and the mean deviationd in the following chapter.

3) Investigating the Deviation d Employing LTE Reference Antennas:
The deviation d versus condition number κ graphs tend to overestimate practically arising deviations,  , with a printed circuit board carrying a closed metal cavity for the UE, two printed antennas and connection cables [43] antenna elements with coaxial cables, the internal antennas are deactivated. A picture of the reference antenna is exhibited in Fig. 17. It is available in three different performance types: performance The performance naming refers to the coupling of the antenna elements in the different reference antenna versions. This coupling is expressed with condition number κ, which is different for every constellation. A constellation contains test antenna positions and polarizations, as well as UE placement. The authors use the 128 constellations [28] for statistical investigations on the logarithmic antenna condition numbers κ. Every single constellation delivers a certain κ, which can be calculated employing the E-field antenna pattern as shown in Sec. III-A2. Figure 18 shows the distribution of κ for the 128 constellations for all three antenna types. Also the mean logarithmic value of the condition numberκ is shown in the graphs, where N is the number of constellations (e.g. N = 128), κ i is the logarithmic antenna condition number of a constellation i. The reference antennas behave as they are named. Going from 'Good' to 'Bad',κ rises:   The mean logarithmic deviation is where N again is the total number of constellations, and d i is the deviation at the constellation i. The DM can be validated for all three types of antennas within a |d| of 4.69 dB. This maximum of the averaged values ofd lies below the peak value of |d|, as shown in Fig. 15. In the case a UE employs LTE, DLSCH processing is always applied. In this case the absolute mean logarithmic deviation |d| will always be below 3.57 dB for all antenna condition numbers. If LTE UEs are tested only with low correlation of channels, the method works almost perfectly with |d| below 0.29 dB. Finally, it can be pointed out that: • The mean deviation |d| is always below the maximum of the absolute deviations |d|. In the case of ICD/ZF receiver, low correlated channel, DLSCH processing and non-ideal channel measurement even for the 'Bad' antenna it will be only 0.1 dB. • In the practical case of rather well conditioned antenna matrices and DLSCH processing applied, the method works very well. • Comparing the receiver types, the MMSE receiver causes higher mean deviationsd than the ICD/ZF receiver. • The investigations exhibit that the mean deviation d decreases when going from antenna type 'Bad' to 'Good'. • If a very poor antenna systems is measured, there is a possibility of calibrating the system, as the deviation d is static. • The DM is an innovative method to deliver fast and reliable performance test results.

VI. Validation of the DM by OTA Measurements of a UE A. Measurements
The authors conducted measurements with a commercial LTE device, namely the Sony Xperia smart-phone, for a round-robin campaign for radio area network group 4 (RAN4) of 3GPP. They tested the UE in an anechoic chamber using the 'Good' antenna, employing the setup in Fig. 2 at 128 constellations [28]. The authors measured the 'DM' results and the 'overall' results of the Sony Xperia smart-phone with different transmission channel models than in the simulation to calculate the deviation d. Figure 19 shows the experimental results of the distribution of the deviation d. The authors determined the deviation d for the different transmission channel models for all 128 constellations. The results of d for the different channel models are color coded in the diagram. The measurement setup employs a full LTE transmission containing tapped delay channel models, OFDM and resource block allocation [7]. These tapped delay models were used: Urban Micro (Umi), Urban Micro (Umi) with all tap correlations α = 0 and Urban Macro (Uma). These models are proposed for OTA testing of UEs by 3rd Generation Partnership Project (3GPP) [44]. In comparison to the measurements, the simulations in Sec. V use a simplified setup. Nevertheless, a comparison is valid, as the DM is validated by simulation for a single OFDM subcarriers that has a bandwidth of 15 kHz, in the LTE case. This narrow bandwidth allows a treatment of the channel as frequency flat fading ones. The DM is validated for the smallest element, the single OFDM channel. Therefore it is also valid for the complete fading channel, which consists of many small OFDM carriers.   Figure 19 points out that the deviation d between the measured 'overall' and DM results are distributed around a mean values from −0.16 dB to 0.34 dB for all employed channel models, which is very low regarding all other uncertainty sources [41], [42]. The approximately Gaussian shaped distribution of the deviation d is resulting from the mentioned measurement uncertainties.

B. Comparison of Measurements to the Simulations
The measurements in Sec. VI-A were conducted with a slightly different set of parameters as proposed by 3GPP [32], [34]. The authors present another set of simulations, as close as possible to the measurements. The channel models employ tapped delay models, they were used in the measurements. The taps have different correlation values α, while β is always zero. To be able to use the different correlations of the taps for the simplified model, the authors employ a linear power related averaging of α. This averaging results in these values of α: • Channel model Uma: α = 0.1172 (light correlation) • Channel model Umi with (α = 0): the mean value also is α = 0 (all taps are zero; uncorrelated) • Channel model Umi: α = 0.9474 (strong correlation) Results are summarized in Tab. III. The mean deviation d is given for the measurements and the closest possible simulations for several channel models. The mean deviationd is very close to zero. So to say, all validations with theoretical results, simulation results and measurement results (Tab. II), point in the same direction: the validity of the DM. The mean value of the deviationd is within the given limits for all employed channel models.

VII. Conclusion
This paper covers performance testing of Multiple-Input Multiple-Output (MIMO) enabled wireless user equipment. It deals with Over-The-Air (OTA) performance tests for complete MIMO enabled User Equipment (UE), including antennas, front-end and algorithmic performance. The authors investigate a very promising method, the Decomposition Method (DM). It does not compare the DM to other test procedures. The best way to test a UE would be to check the algorithmic and the antenna performance at the same time. Unfortunately, this is very slow. To speed up the measurement procedure, the Decomposition Method splits the performance tests into two parts: a static test for the antenna performance and a fading test for the algorithmic performance. The results from the separated measurements are combined and deliver accurate and repeatable results. By applying this split, the measurements can be speeded up by a factor of 36 in comparison to the combined measurements. This paper introduces the Decomposition Method and deals with the validation of the split of the measurements. The authors validate the Decomposition Method by theoretical investigations, by numerical simulation as well as by measurement. As many MIMO enabled devices also use Orthogonal Frequency Division Multiplexing (OFDM), the single orthogonal frequency bands have small bandwidths. The paper exploits this fact by simplifying the employed models from generic tapped delay channel models to frequencyflat response channels, without loosing validity. The authors validate the method by theoretical means using the MIMO channel capacity. They also validate it by numerical means, simulating a complete transmission system with Kronecker channel models with frequency flat fading. They show that the applicability of the Decomposition Method is given for most use cases. Low correlated channels deliver an absolute deviation of the Decomposition Method result lower than 0.7 dB. To round up the picture, the validity of the method is shown by measurement results for the Sony Xperia smart-phone. The mean deviationd is within the interval {-0.16 dB to 0.34 dB}.