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Notations
In the following discussions, a bold notation in small letters indicates a vector and bold notation with capital letters indicates a matrix. On some occasions, calligraphic font with capital letters is also used for matrices. Unless otherwise specified, vector refers to a column vector. The operation « diag » has an interpretation identical to that in Matlab. When applied to a vector, it produces a diagonal matrix with the diagonal elements formed out of the same vector. When the operation is performed on a diagonal matrix, the result is a column vector formed out of the diagonal elements of this matrix. E(¢) is the expectation operator. › refers to the Kronecker product. In the following text, the notation jAj refers to the determinant of the matrix A. ln refers to natural logarithm. C N(„,C) refers to a circularly complex Gaussian distribution with mean „ and covariance C. Tx may denote transmit/transmitter/transmission and Rx may denote receive/receiver/reception. The list of abbreviations has also been separately tabulated.
Organization of the thesis
Chapter 2 contains the entire theoretical research on reciprocity calibration for MaMIMO. First, the reciprocity model is introduced clearly demarcating the propagation channel which is recip-rocal and the non-reciprocal radio frequency (RF) chain. We then explain various terminology, namely, UE aided calibration and Internal calibration, Coherent and non-coherent calibration, etc. This is followed by some of the relevant state of the art (SoA). We derive the CRB for these algorithms under a more general setting that allows for antenna grouping. Our results apply to both coherent and non-coherent calibration techniques. We also present new optimal algorithms for transmit antenna array calibration.
Chapters 3,4 deal with the topic of sum rate maximization for a point to point MIMO transmission under Doppler. Chapter 3 considers the problem of precoder design for an HST scenario under complete CSIT. To make the analysis tractable, the channel variation is assumed to be linear as has been done previously in the literature and it is shown that this approximation is justified for our problem scenario. With this setting, we observe that the problem of precoder design under ICI is similar to that of a MIMO Interfering Broadcast Channel (IBC) precoding design and hence tap into existing solutions in that space. The design is extended to account for the presence of Excess CP (ExCP) wherein optimal window parameters are also derived to take advantage of the ExCP. The same problem is treated under a more realistic setting of partial CSIT in Chapter 4. Chapter 5 continues to analyze the problem of partial CSIT that was started in Chapter 4. However, the problem formulation in Chapter 5 considers a general setting of MIMO IBC and analyses the possibility of approximating the expected weighted sum rate (EWSR) metric with the expected-signal-expected-interference-WSR (ESEI-WSR) metric. Finally, in Chapter 6 we perform an experimental validation on the Eurecom MaMIMO testbed that exploits some of the theory that has been discussed so far. The first experiment performs precoding for a TDD MaMIMO scenario using DL channel estimated via reciprocity calibration. The next experiment focuses on multi-user multi-cell precoding under partial CSIT while taking into account the end-to-end channel non-reciprocity. A third experiment explores the use of multiple antennas to mitigate self-interference for a Full Duplex (FD) scenario.
The final chapter provides the conclusions. At the end of every chapter, the list of contributions and associated publications are provided. However, for easy reference, the final chapter also lists out the contributions from our research chapter-wise. In this thesis, if any particular result is not the work of the author, an explicit reference to the source is provided.
Introduction
In this chapter, we present our work on reciprocity calibration for a MaMIMO scenario. Consider a MaMIMO BS with MA antennas talking to a user equipment (UE) having MB number of antennas. To exploit the advantages of Massive MIMO, it is key to have channel state information at the transmitter (CSIT). The channel of interest here is of dimension MB £ MA. In the DL, estimation of the channel from each BS antenna requires a dedicated pilot to be transmitted from that antenna. As the number of antennas is massive, this demands a huge number of pilot transmissions resulting in severe loss of overall system throughput. At the same time, in the uplink (UL), a single pilot is sufficient to determine the channel from one UE antenna to all the BS antennas. Thus, in a MaMIMO scenario, it is more attractive to perform UL channel estimation. Hence, there is a lot of interest in deriving the DL channel estimates from the UL channel estimates. TDD Massive MIMO (MaMIMO) is an ideal candidate for this due to the common assumption of channel reciprocity. Simply put, this would imply that the estimated channel in the UL can be directly used as DL channel. However, the overall digital channel is not reciprocal. Fig. 2.1 shows a detailed picture of the components forming the overall digital channel between a pair of radios A and another pair of radios B. It is clear that the RF components are not reciprocal as the UL and DL signals traverse completely different paths beyond the switch.
In Fig. 2.1, C refers to the propagation channel which is reciprocal. The (i, j)th entry of C corresponds to the propagation channel between the antennas i and j. Hence, all diagonal elements of C are undefined. The overall DL and UL channels observed in the digital domain are noted by HA!B and HB!A. In the frequency domain, over a narrow frequency band, they can be represented by:
Matrices TA, RA, TB, RB model the response of the transmit and receive RF front-ends and are called the absolute calibration factors. The diagonal elements in these matrices represent the linear effects attributable to the impairments in the transmitter and receiver parts of the RF front-end respectively, whereas the off-diagonal elements correspond to RF cross-talk and antenna mutual coupling. Thus, the DL channel HA!B may be derived from the UL channel HB!A as follows.
F is called a relative calibration factor as it is obtained as a ratio of the absolute calibration factors. It is important to note that for the purpose of DL channel estimation, there is no need to estimate the absolute calibration factors. Instead, we need only the relative calibration factors. We would also like to point out here that for the purpose of precoding, we only need the knowledge of HA!B up to a complex gain factor. This can be seen from the fact that the final Tx output power is determined by the Tx power constraints and that a common phase factor on all the Tx antennas does not have any impact on the precoding. In turn, hence, FA, FB also only need be determined up to a complex scale factor.
A TDD reciprocity based MIMO system normally has two phases for its function. First, during the initialization of the system or the training phase, the reciprocity calibration process is activated, which consists in estimating FA and FB. Then during the data transmission phase, these calibration coefficients are used together with instantaneous measured UL channel ˆ HB!A
to estimate the CSIT HA!B, based on which advanced precoding algorithms can be performed. Since the calibration coefficients stay stable during quite a long time [37] (in the order of hours), the calibration process doesn’t have to be done very frequently.
Co-located MaMIMO refers to a single BS with a massive number of antennas that are co-located. In contrast, in distributed MaMIMO the antennas are spread out over the cell. It was shown in [41] that a distributed MaMIMO can achieve higher performance compared to co-located MaMIMO.
UE aided calibration and Internal calibration
There are two main approaches to reciprocity calibration based on whether or not a UE is involved in its determination.
1. In UE aided calibration, explicit channel feedback from a UE during the calibration phase is used to estimate the calibration parameters. Hence, during a training phase, explicit pilots are exchanged between the BS and UE over-the-air. Based on these pilots, the UE feeds back its estimate of the channel to the BS which together with its estimate of the UL channel derives the calibration parameters.
2. A second approach is to estimate only the FA up to a scale factor and not estimate the FB at all. Of course, if all the UEs have just one antenna each, FB is just a complex scalar and need not be estimated. In the general case of UEs with multiple receive antennas, the existing literature [32, 33] justifies this approach. Such an approach is called internal calibration or self-calibration where the calibration is performed entirely between the antennas of the BS. An important advantage of this kind of calibration is that it ensures tight clock and frequency synchronization amongst the antennas that are being calibrated in the case of co-located MaMIMO. The self-calibration may be performed over-the-air (OTA) or via additional hardware circuitry specifically for calibration. The OTA approach is a more popular method today and our research is focused on this topic.
Coherent and Non-coherent calibration scheme
The calibration parameters of the antenna may be considered to be constant in the order of several hours. However, the variation of the physical propagation channel is typically much faster.
This leads to two ways of approaching the estimation of the relative calibration parameters. We could complete the entire estimation of these parameters in a short time span where the propagation channel stays a constant. Such a time duration would be called a coherent time slot. When the estimation happens within one coherent time slot, it is called a coherent estimation scheme. Alternatively, the problem may well be formulated over several different coherent time slots (during which the calibration parameters themselves are assumed constant), and in this case, it is called non-coherent estimation. This is also illustrated in Figure 2.2.
Key assumptions
1. We discuss the estimation of the reciprocity calibration factors over a narrow frequency band where the calibration factors are assumed to be a constant. In the case of wideband signals, we assume a multi-carrier system like OFDM and the estimation happens on a per sub-carrier basis.
2. It is assumed that the impact of the Tx and Rx chains may be modeled as a linear scaling factor over a narrow frequency band. This has been validated in several real implementa-tions like [37], [35]. This can also be seen later in Chapter 6 as shown in Figure 6.6 where the performance with calibration matches that with the ideal DL channel.
3. The calibration matrix F is diagonal. This was validated experimentally in [20] where the off-diagonal elements of F were found to be less than 30dB compared to the diagonal elements. Note, however, that this does not necessarily imply no mutual coupling or cross-talk. If MA, MB represent reciprocal non-diagonal matrices that encapsulate the mutual Hence, by treating the mutual coupling and cross-talk as part of the propagation channel, we get back the diagonal calibration factors! Hence, the diagonal F could be a result of either no mutual coupling or just reciprocal mutual coupling. The existing literature that uses mutual coupling to perform reciprocity calibration implicitly assumes it to be reciprocal. Only [45] considers mutual coupling as non-reciprocal. However, the same authors treat it as reciprocal in their next work [46]. In summary, however, what is important in our research is the diagonal nature of F that is justified by the numerous experimental validations based on that assumption. In fact, our own experiments reported in Section 6.1.1 are in agreement with this assumption.
State of the Art
In this section, we discuss the estimation schemes available in the literature.
Argos
Argos was the first published Massive MIMO prototype supporting 64 antennas simultaneously serving 15 terminals [37]. This work introduced the internal calibration procedure where the calibration procedure is done exclusively at the BS without involving the UE. Transmission and reception RF chain asymmetries are modeled by scalar coefficients while RF crosstalk or mutual coupling are ignored resulting in a diagonal F. The idea is to use one of the BS antennas as a reference and derive the relative calibration factors of the rest of the BS antennas relative to this reference antenna. As mentioned before, the relative calibration factor F can only be estimated up to a scale factor. As the matrix F is assumed to be diagonal, only the diagonal elements f are relevant, where f ˘ diag(F). The relative calibration factors of the individual antennas are denoted as fi. Let ti and ri represent corresponding Tx and Rx absolute calibration parameters for the radio i, such that f ˘ ti . Then, transmission from antenna i. Note that 0 is considered to be the index for the reference antenna. The calibration procedure in the Argos system thus uses a bidirectional transmission between the reference antenna and other antennas to derive their relative calibration coefficients. Assuming the pilots to be unity, the calibration parameters are estimated as follows:
However, this method is sensitive to the position of the reference antenna which can result in significant channel amplitude difference for antennas close to the reference antenna and those far away.
Rogalin Method
The Rogalin method [35] can be regarded as an extension of the Argos calibration. It was primarily intended for a distributed Massive MIMO system but can be equally applied to a co-located MIMO scenario. In this method, no more reference antenna is defined and the calibration is performed among different antenna element pairs.
The principle of the Rogalin method is as follows. Consider the set of all transmissions between antennas i and j that have happened in the same coherence time, and assume pilots as unity.
The Argos method can be viewed as a special case every antenna (with indices 2 till M) forms a transmission pair only with the reference antenna (element 1) at the center.
By generalizing the Argos method to an LS formulation, Rogalin method gets rid of the need for a reference antenna. By involving bi-directional transmission between any radio element pairs, it outperforms the Argos method which relies only on the transmission between the reference antenna and other antenna elements. In a distributed MaMIMO setting, this work also proposes hierarchical calibration, i.e. grouping radio elements into different clusters and performing intra-cluster and inter-cluster calibration separately.
Avalanche
The typical estimation methods mentioned in 2.2.1 and 2.2.2 need M channel uses to complete the estimation of the calibration parameters. In other words, the time to estimate the calibration parameters is linear in the number of antennas. Avalanche [31] is a fast recursive coherent calibration method that can perform the estimation in O( M ) channel uses. The algorithm successively uses already calibrated parts of the antenna array to calibrate uncalibrated radios which, once calibrated, are merged into the calibrated array. At a given point during calibration, assume M0 radios have already been calibrated using L channel uses. Refer to this as a reference set 0 whose calibration factor is f0. With these calibrated radios, a new set of radios M1 will be calibrated. During the calibration process of the reference set, another antenna j belonging to the set M1 would have received L transmissions from this reference set. The L length vector thus observed at the antenna j would be, (2.13) yj ˘ PT T0C0! j r j ¯n0! j.
Here, PT represents the transmit pilots and n0! j refers to the vector of noise observed at antenna j. Next, consider a single pilot (unity pilots) transmission from all the new M1 radios to be calibrated. This results in an observation vector of length M0 at the reference antennas.
Then, it is proved in this paper that if M1 • L, the calibration factors for antenna j in this new set of radios is given by, Here, Y ˘ [y1 ,y2 , . . . ,yM1 ]. A key drawback of this algorithm is that of error propagation as it uses previously estimated calibration values to estimate new ones. Note, however, that at any time instant t, the new number of antennas that can transmit would be max(t ¡ 1,1). Hence, the maximum number of antennas that can be transmitted with L channel uses would be ip˘2 max(i ¡ 1,1) ˘ ¡2 . Thus, the overall estimation for M antennas may be performed in O( M ) channel uses. More details on this method are provided in the simulation section 2.8.
Method in [43]
In this work, a penalized ML based algorithm is proposed to estimate the calibration parameters. The overall received signal may be expressed as Here, the (i, j)th entry of the matrix Y corresponds to the received signal at antenna j from antenna i. Hence, the diagonal entries of this matrix are undefined. N corresponds to the matrix of noise observed and again has diagonal entries undefined. Denoting f as the diagonal elements of F, a penalized ML is then formulated as, where † is an arbitrary parameter chosen to control the convergence of the algorithm. The algorithm proceeds by alternately optimizing bf and Hcas follows.
Group calibration System Model
Here we present a more general system model [? ] that allows the grouping of multiple antennas during transmission. This model falls back to the single antenna transmission scenario when each grouping has only one antenna.
Here, as shown in 2.3, the M antennas are partitioned into G groups with Mi antennas each. Each group Ai transmits pilots Pi for Li channel uses. Let Yi! j be the received signal at antenna j upon transmission of pilot Pi from antenna i. Then for every pair of transmission between antennas i and j (bi-directional Tx).
N i! j represents the noise seen at antenna j when antenna i is transmitting. Equation (2.19) also shows the dimensions of the matrices involved for clarity. It is important to note that the channel is assumed to be constant during this bi-directional Tx. Eliminating the propagation channel Ci! j.
where Fi ˘ R ¡iT Ti and Fj ˘ R¡jT Tj are the calibration matrices for groups i and j. Using the vec operator and its properties, equation (2.20) may be rewritten as Here, we have used the property that for any matrices X1,X2, and X3, where › denotes the Kronecker product. Ni j ˘ vec(Pi Fi Nj!i ¡Ni! jFj Pj). In addition, as the matrices, Fi are diagonal, all the columns corresponding to the zero entries of the calibration matrices can be eliminated. Hence, (2.21) may be further rewritten as, where ⁄ denotes the Khatri-Rao product [22] (or column-wise Kronecker product
Least Squares Solution
Collecting all these bi-directional transmissions, we arrive at a least-squares formulation to solve for the relative calibration factors f.
in order to exclude the trivial solution bf ˘ 0 in (2.24). The constraint on bf may depend on the true parameters f. As we shall see further this constraint needs to be complex valued (which represents two real constraints). Typical choices for the constraint are 1) Norm plus phase constraint (NPC):
If we choose the vector g ˘ f and c ˘ jjfjj2, then the Im{.} part of equation (2.28) corresponds to (2.27). The most popular linear constraint is the First Coefficient Constraint (FCC), which is (2.28) with g ˘ e1, c ˘ 1.
Fast Calibration
Here, we consider a coherent calibration scheme and derive parameters for the group calibration that result in the estimation of the relative calibration factors f using the minimum number of channel uses. Consider a scheme where every antenna group Ai transmits pilots Pi in a round robin fashion. Once all the subgroups have transmitted, we will get the following structure by stacking the individual equations. Consider the following sequence
• When group 2 transmits, we can formulate L2L1 number of equations.
• When group 3 transmits, we can formulate L3L1 ¯ L3L2 number of equations.
• When group i transmits, we can formulate Pij¡˘11 Li L j number of equations.
• Thus, the total number of equations after all the groups transmit is P G P i 1 i˘1 j¡˘1 Li L j.
This process continues until group G finishes its transmission. During this process of transmission by the G antenna groups, we can start forming equations as indicated, that can be solved recursively for subsets of unknown calibration parameters, or we can wait until all equations are formed to solve the problem jointly. Finally, stacking equations (2.23) for all 1 • i ˙ j •
Table of contents :
1 Introduction
1.1 Notations
1.2 Organization of the thesis
2 Reciprocity Calibration for Massive MIMO
2.1 Introduction
2.1.1 UE aided calibration and Internal calibration
2.1.2 Coherent and Non-coherent calibration scheme
2.1.3 Key assumptions
2.2 State of the Art
2.2.1 Argos
2.2.2 Rogalin Method
2.2.3 Avalanche
2.2.4 Method in [43]
2.3 Group calibration System Model
2.3.1 Least Squares Solution
2.3.2 Fast Calibration
2.3.3 Non-coherent estimation
2.4 Cramer Rao Bound
2.5 Optimal Algorithms
2.5.1 Alternating Maximum Likelihood (AML)
2.5.2 Variational Bayes approach
2.6 Maximum likelihood vs. least squares
2.7 Analysis of least squares methods
2.8 Simulation Results
2.8.1 Comparison of grouping based schemes
2.8.2 Comparison of single antenna transmission schemes
2.9 Summary of Contributions
3 Precoder design under Doppler – Full CSIT
3.1 Introduction
3.1.1 Key Assumption
3.2 System Model
3.3 Precoder Design
3.3.1 Covariance matrix update
3.3.2 Power allocation across the subcarriers
3.3.3 Optimization of window parameters – Gradient descent
3.3.4 Overall Algorithm and Convergence
3.4 Simulation Results
3.5 Summary of Contributions
4 Precoder design under Doppler – partial CSIT
4.1 System Model
4.2 Large MIMO asymptotics
4.2.1 Precoder Design
4.2.2 Simulation Results
4.3 Summary of Contributions
5 Analysis of the Gap between EWSR and ESEI-WSR
5.1 MIMO IBC Signal Model
5.2 EWSR
5.3 MaMIMO limit and ESEI-WSR
5.4 EWSR to ESEI-WSR gap Analysis
5.4.1 Monotonicity of gap with SNR
5.4.2 Second-Order Taylor Series Expansion of EWSR
5.4.3 MISO correlated channel
5.4.4 MIMO zero mean i.i.d channel
5.5 Actual EWSR Gap
5.6 Simulation Results
5.7 Conclusion
5.8 Summary of Contributions
6 Experimental results
6.1 Downlink channel estimation via RCMM.
6.1.1 Massive MIMO testbed
6.1.2 Frame structure
6.1.3 Results
6.2 MU-MIMO precoding for a 2 BS, 2 UE scenario
6.2.1 Channel Estimation
6.2.2 EWSR Lower Bound: EWSMSE
6.2.3 Dual DL precoder
6.2.4 Results
6.3 Multiple antenna based self-interference cancellation for Full Duplex
6.3.1 USRP based testbed
6.3.2 Frame structure
6.3.3 Results
6.4 Summary of Contributions
7 Conclusion
7.1 Contributions
7.1.1 Chapter 2
7.1.2 Chapter 3
7.1.3 Chapter 4
7.1.4 Chapter 5
7.1.5 Chapter 6
