# I/Q Imbalance Aware Widely-Linear Receiver for Uplink Multi-Cell Massive MIMO Systems

## Abstract

In-phase/quadrature-phase (I/Q) imbalance is one of the most important hardware impairments in communication systems. It arises in the analogue parts of direct conversion radio frequency (RF) transceivers and can cause severe performance losses. In this paper, I/Q imbalance (IQI) aware widely-linear (WL) channel estimation and data detection schemes for uplink multi-cell massive multiple-input multiple-output (MIMO) systems are proposed. The resulting receiver is a WL extension of the minimum mean square error (MMSE) receiver and jointly mitigates multi-user interference and IQI by processing the real and the imaginary parts of the received signal separately. The IQI arising at both the base station (BS) and the user terminals (UTs) is then taken into account. The considered channel state information (CSI) acquisition model includes the effects of both estimation errors and pilot contamination, which is caused by the reuse of the same training sequences in neighboring cells. We apply results from random matrix theory to derive analytical expressions for the achievable sum rates of the proposed IQI aware and conventional IQI unaware receivers. Our simulation results show that the performance of the proposed IQI aware WLMMSE receiver in a system with IQI is close to that of the MMSE receiver in an ideal system without IQI. Moreover, our results for the sum rate of the IQI unaware MMSE receiver reveal that the performance loss due to IQI can be large and, if left unattended, does not vanish for large numbers of BS antennas.

## 1Introduction

-input multiple-output (MIMO) techniques have become a central part of modern communication systems such as Long Term Evolution (LTE) and WiMAX. With MIMO technology, high throughput and transmission reliability can be achieved. An emerging research field in wireless communications are so-called massive MIMO systems [1], [2]. Massive MIMO systems employ a large number of antennas, e.g. one hundred antennas or more at the base station (BS), and can achieve very high spectral and energy efficiencies [3]. Moreover, in massive MIMO systems, the transmit power of the BS and the user terminals (UTs) can be decreased by increasing the number of antennas at the BS [3]. These and other desirable features render massive MIMO a promising technology for future wireless communication systems. In this paper, we consider the uplink of a multi-cell massive MIMO system. In an uplink single-cell massive MIMO system, which embodies a multiple access channel (MAC), successive interference cancellation (SIC) is the optimal detection scheme [4]. Nevertheless, since SIC is not feasible in most practical systems due to its high computational complexity, linear detectors such as matched filter (MF) and minimum mean square error (MMSE) detectors are often preferred as they provide a good trade-off between performance and complexity [2], [3].

However, hardware (H/W) impairments, which exist in all practical systems, can severely degrade the performance of linear detectors. One of the most important H/W impairments in digital communication systems is in-phase/quadrature-phase (I/Q) imbalance, which arises in direct conversion transceivers [5]. In such systems, the real and imaginary parts of the received RF signal are mixed with the high-frequency carrier signal and its phase-shifted version, respectively, to produce the baseband signal. Ideally, the phase difference between the carrier signal and its phase-shifted version is exactly and the mixers for the real and the imaginary part have the same amplitude gain. However, in practical systems, both amplitude and phase mismatches between the real and imaginary parts occur, which leads to an I/Q imbalance (IQI) in each antenna branch at the base station and in the RF chain of each UT, and impair the received data vectors [6], [7]. Thus, for reliable detection, besides the mitigation of multi-user interference, IQI compensation is necessary as well. One approach to overcome the negative effects of IQI is to measure and compensate the individual IQIs in each antenna branch. However, this solution becomes very costly in massive MIMO systems, where the BS may be equipped with several hundred RF chains. Another approach, which we consider in this paper, is joint data detection and IQI mitigation. In this case, the equivalent channel, which comprises the actual channel and the IQI, is estimated at the BS. For this estimation, we exploit the received training sequence and the channel statistics which also include the effect of IQI. The detection matrix is then constructed based on this equivalent channel estimate and can be used for joint data detection and IQI mitigation. To the best of the authors’ knowledge, the problem of joint channel estimation, data detection, and IQI mitigation for uplink multi-cell massive MIMO systems has not been investigated yet.

Recently, a WLMMSE beamformer for systems with IQI has been proposed in [8]. Here, the authors propose a beamforming scheme, where a multi-antenna receiver suffering from IQI detects signals coming from a specific direction while suppressing signals arriving from other directions. Another related work is [9], where the authors propose a detection scheme for uplink single-cell multi-user-MIMO (MU-MIMO) systems impaired by IQI. In [10], the authors model the residual H/W impairments, which remain after compensation, in massive MIMO systems as an additive Gaussian impairment and derive a capacity bound. Moreover, in our recent work [11], we have proposed an IQI aware precoder for downlink single-cell massive MIMO systems assuming perfect CSI and IQI present only at the BS. Another recent work is [12], where the authors consider the uplink of a single-cell massive MIMO system with IQI present only at the BS. In this paper, we propose a widely-linear MMSE (WLMMSE) receiver for CSI acquisition and data detection in an uplink multi-cell massive MIMO system, where both the BS and the UTs are impaired by IQI and both the received data and training signals for channel estimation are affected by IQI. WL filtering was introduced in [13] and is used to estimate complex signals by filtering the real and imaginary parts separately. Several works have employed WL filtering in single-user MIMO systems, cf. [14], [15]. WL processing results in a higher performance than strictly linear processing, if rotationally variant signals are involved, which is the case when IQI is present. This motivates the use of WL filtering for channel estimation and data detection at the BS of uplink multi-cell massive MIMO systems suffering from IQI.

In contrast to [8], [9], and [11], where a single-cell system with perfect CSI was considered, in this paper, a more sophisticated system model is adopted, which includes the effects of multi-cell interference and CSI imperfection originating from pilot contamination, channel estimation errors, and IQI. Furthermore, contrary to [12], where a single-cell system with ideal UTs was considered, our system model includes multi-cell interference and takes IQI at both the UTs and the BSs into account. Moreover, as opposed to [10], where the residual H/W impairments were modelled by an additional additive Gaussian noise term and a non-augmented system model was employed, we adopt an augmented system model, which is essential for the analysis and mitigation of IQI. In addition, we use results from random matrix theory and provide analytical results for the sum rate performance of the proposed IQI aware (IQA) and conventional IQI unaware (IQU) receivers.

This paper is organized as follows. In Section 2, the system model is presented. In Section 3, the conventional IQU-MMSE receiver is investigated and an analytical expression for its sum rate in the presence of IQI is derived. In Section 4, we introduce the proposed IQA-WLMMSE receiver, and present an analytical expression for the corresponding sum rate. Numerical results are provided in Section 5, and conclusions are drawn in Section 6.

Notation:

Boldface lower and upper case letters represent column vectors and matrices, respectively. is a block diagonal matrix with matrices on its main diagonal. denotes the identity matrix and , , and stand for the th row, the th column, and the element in the th row and the th column of matrix , respectively. denotes the complex conjugate and , , , , and are the determinant, Frobenius norm, trace, transpose, and Hermitian transpose of a matrix, respectively. and denote the real and imaginary parts of a complex number, respectively. stands for the expectation operator and denotes a circular symmetric complex Gaussian distribution with mean vector and covariance matrix . Moreover, stands for almost sure convergence.

## 2System Model

In this paper, we consider the uplink of a multi-cell massive MIMO system with universal frequency reuse. The number of cells is denoted by , and in each cell, single-antenna UTs simultaneously transmit data to a BS with antennas. and are assumed to be very large with their ratio being constant. Furthermore, we assume a block fading channel. The channel matrix between the UTs in the th cell and the BS in the th cell is denoted by . Here, is the channel vector between the th UT in the th cell and the BS in the th cell, where and represents the channel covariance matrix. Since the detection schemes considered in this paper, i.e., IQU-MMSE and IQA-WLMMSE detection, have fundamentally different structures, we adopt two different representations for the system model, namely a complex-valued and a real-valued representation, which are presented in the following subsections.

### 2.1Complex-Valued Representation

In this subsection, the complex representation of the system model, which is used for the IQU-MMSE detector, is introduced. The transmitted data symbols of the UTs in the th cell are stacked into a vector, which is denoted by , where is the data symbol transmitted by the th UT in the th cell. The received signal at the th BS can be modeled as

where denotes the uplink transmit signal-to-noise ratio (SNR), and and with and representing the IQI at the th UT in the th cell. and denote the phase and amplitude imbalances at the corresponding UT, respectively. The IQI at the th BS is modelled by diagonal matrices and , where and with and being the phase and amplitude imbalances at the th antenna branch of the th BS, respectively. If IQI is absent, we have , and , . represents the complex additive white Gaussian noise (AWGN) at the th BS.

### 2.2Augmented Real-Valued Representation

Since IQI affects the real and imaginary parts of a signal differently, IQA channel estimation and data detection should allow for processing the real and imaginary parts of received signals individually. Hence, we use an augmented representation for the IQA system model, where the real and imaginary parts of the signals are stacked together. More precisely, the real and imaginary parts of the independent and identically distributed (i.i.d.) zero-mean complex Gaussian transmit data symbols of the UTs in the th cell are stacked into the augmented vector , where and contain the real and imaginary parts of the complex transmit data symbols in the th cell, respectively, and . The augmented data vector received at the th BS, , from all UTs in the cells can be expressed as

where and are the real and the imaginary parts of the received signal in (Equation 1), respectively. Here, the augmented real-valued channel matrix is given by with and being the real and imaginary parts of the complex-valued channel matrix , respectively. The augmented real-valued vector contains the real and imaginary parts of the complex-valued AWGN vector at the th BS, respectively. models the IQI at the th BS. Here, is a permuted version of , where , represents the IQI of the th RF branch of the th BS and is given by [6]

The elements of permutation matrix are defined as

This permutation is required, since the stacked received data vector contains the in-phase and quadrature-phase components in its upper and lower parts, respectively. Moreover, in (Equation 2), denotes the stacked IQIs of the UTs in the th cell, where the permutation matrix is similarly defined as and is obtained by replacing with and with in (Equation 4). Furthermore, , where is the IQI matrix of the th UT in the th cell, which can be expressed as

## 3IQI Unaware MMSE Receiver

In this section, as a performance benchmark, the sum rate of a conventional IQU-MMSE receiver comprising an IQU-MMSE channel estimator and an IQU-MMSE data detector is investigated in the presence of IQI. For IQU estimation and detection, we adopt the conventional MMSE estimator and detector, respectively, which are not designed for IQI mitigation.

### 3.1Channel Estimation

In this subsection, channel estimation for an IQU system in the presence of IQI is presented. For channel estimation, at the beginning of every coherence interval, training sequences are transmitted by all UTs to their serving BS. Due to the limited length of the coherence interval, there are not enough orthogonal training sequences for all UTs in all cells. Hence, UTs with the same index in different cells use the same training sequence [16]. This leads to a corrupted channel estimate and this effect is known as pilot contamination in the massive MIMO literature [16]. Since we consider full pilot reuse, when pilot contamination is present, UTs having the same index in different cells employ the same training sequence , where is the length of the training sequence. The received training signal at each BS is multiplied by the original transmitted training sequence to eliminate the interference caused by other UTs. Considering (Equation 1), and the orthonormality of the training sequences, i.e., , , we have the following expression for the received training sequence of the th UT in the th cell

where is the received training signal at the th BS. Here, is the transmit training signal-to-noise ratio (SNR) and , where is the AWGN at the th BS during the training period. In this paper, for IQU channel estimation, we assume that the estimator tries to estimate as the desired channel between the th UT in the th cell and the th BS. Since the IQU estimator does not process the real and imaginary parts of the received training sequence separately, it can consider only one component, i.e., of the equivalent channel vector. We note that with the conventional complex-valued system model, which is assumed for the IQU-MMSE estimator, it is not possible to fully model the equivalent channel vector, which comprises both the actual channel and IQI. Taking this into account and considering as the observation, the MMSE channel estimate can be expressed as [17]

where is the cross-correlation matrix of the desired channel estimate and the observation, and given by

The auto-correlation matrix of the received signal in (Equation 6) can be expressed as

Now, we substitute (Equation 8) and (Equation 9) into (Equation 7) and obtain the following expression for IQU-MMSE estimation of the th UT’s channel vector

where deterministic matrix is given by

If pilot contamination is absent, (Equation 10) reduces to

where the deterministic matrix is equal to if we set and in (Equation 11).

### 3.2Data Detection

In this subsection, we investigate IQU-MMSE data detection. The IQU-MMSE detector adopted here is the conventional single-cell MMSE detector. The IQU-MMSE detection vector for the th UT at the th BS is given by

where the th column of the estimated channel matrix is and given in (Equation 10) and (Equation 12) for the cases with and without pilot contamination, respectively. Thus, the detected signal corresponding to the th UT in the th cell at the output of the IQU-MMSE detector of the th BS can be expressed as

### 3.3Asymptotic Sum Rate Analysis

The performance metric considered in this paper is the ergodic sum rate, which is a commonly used metric for performance evaluation of wireless communication systems. For the th cell, the ergodic sum rate is given by

where the expectation is taken with respect to the channel realizations. is the signal-to-noise-plus-interference ratio (SINR) for the th UT in the th cell at the th BS and given by

where , , and are the useful signal power, interference power, and noise power for the th UT in the th cell, respectively. In this paper, using results from random matrix theory, first, an analytical expression for , the asymptotic value of for large numbers of antennas , is derived. Then, using , the asymptotic sum rate is calculated as

In the following Theorem, we provide an analytical expression for the asymptotic SINR of the IQU-MMSE detector.

### 3.4Asymptotic Sum Rate Analysis for Single-Cell Case

Due to the very general setting considered in Theorem ?, the obtained analytical expression for the asymptotic SINR of the IQU-MMSE detector is quite involved. Nevertheless, using these analytical results for performance evaluation is still much more convenient than performing lengthy Monte-Carlo simulations. However, to get some insight for system design, in this subsection, we provide analytical results for the simplified single-cell case with i.i.d. channel vectors and perfect CSI. In particular, we investigate the impact of the IQI at the BS and at the UTs separately to determine whether the IQI at the BS or at UTs is more harmful.

Remark ? reveals that for a fixed number of users , the SINR increases with increasing number of BS antennas .

From ( ?), it can be seen that the SINR loss of the IQU-MMSE detector compared to the ideal case without IQI does not vanish even in the asymptotic scenario where the number of the BS antennas is much larger than the number of the UTs. This motivates the need for a receiver, which mitigates both multi-user interference and IQI, cf. Section 4.

From Corollary ?, we observe that, in an uplink massive MIMO system with IQI only at the UTs, for the SINR does not increase with increasing number of BS antennas. Moreover, by comparing ( ?) and ( ?) we conclude that assuming identical values for the amplitude and phase imbalances at the BS and the UTs, and , the SINR of a massive MIMO system with IQI only at the BS is higher than that of a massive MIMO system with IQI only at the UTs. Thus, if not compensated, IQI at the UTs is more harmful than IQI at the BS.

## 4I/Q Imbalance Aware WLMMSE Receiver

In this section, the proposed IQA-WLMMSE receiver, which comprises an IQA-WLMMSE channel estimator and an IQA-WLMMSE data detector, is presented. The IQA-WLMMSE channel estimator and detector process the real and imaginary parts of the received signals separately. The widely-linear filtering is necessary for mitigation of the IQI, since the real and imaginary parts of the signals are affected differently by the IQI.

### 4.1Channel Estimation

The proposed IQA-WLMMSE channel estimation scheme is the widely-linear extension of the strictly-linear IQU-MMSE channel estimator introduced in Section ? and performs joint IQI compensation and MMSE channel estimation. Since we assume full pilot reuse, UTs in different cells with the same UT index use the same training sequence resulting in the same augmented training sequence in the augmented system model. In order to mitigate the interference from other UTs, the received training signal, , is multiplied by the augmented training sequence . Considering (Equation 2), and taking into account the orthonormality of the pilot sequences, the training signal of the th UT received at the th BS is given by

with , where is the augmented AWGN matrix at the th BS. In (Equation 15), represents the augmented channel between the th UT in the th cell and the th BS and is defined as

where and is the square root of the augmented channel covariance matrix between the th UT in the th cell and the th BS. Here, is given by

The proposed WLMMSE channel estimator estimates the equivalent augmented channel matrix of the th UT, which is given by , where , , and represent the IQI at the th BS, the actual augmented channel of the th UT, and the IQI at the th UT, respectively. Note that for the IQA-WLMMSE detector, both and are required and are used for detection of the real and imaginary parts of the data signal, respectively, c.f. Section 4.2. The proposed WLMMSE channel estimate is given by

where is the auto-correlation matrix of the received training sequence and is given by

Here, is obtained as

where and the last step involved Lemma ? and straightforward mathematical operations. In (Equation 16), is the cross-correlation between the received training sequence and the desired channel estimate, and is given by

Substituting (Equation 18) and (Equation 17) into (Equation 16) leads to the following expression for the IQA-WLMMSE estimate of the channel between the th UT in the th cell and the th BS

where the IQA-WLMMSE channel estimator can be expressed as

For the case without pilot contamination, the IQA-WLMMSE channel estimates are given in (Equation 19) after setting and by (Equation 19) and (Equation 20).

### 4.2Data Detection

The proposed IQA-WLMMSE data detector is the widely-linear extension of the conventional MMSE detector considered in Section , and employs the estimate of the equivalent channel, , which comprises the actual channel, the IQI at the BS, and the IQI at the UT. The IQA-WLMMSE detector includes the filter vectors for the real and imaginary parts of the signal of the th UT at the th BS and is given by

where and are the th and the th columns of the estimated augmented channel matrix , respectively, and are given in (Equation 19). Hence, the decision variable at the output of the IQA-WLMMSE detector at the th BS corresponding to the th UT can be expressed as

### 4.3Asymptotic Sum Rate Analysis

The ergodic sum rate of the IQA-WLMMSE receiver is given by

where the expectation is taken with respect to channel realizations. Here, is the SINR of the th UT in the th cell at the th BS and is defined as

where , , and are the useful signal power, interference power, and noise power of the th UT at the th BS, respectively. Using again results from random matrix theory, we will show that the asymptotic sum rate of the IQA-WLMMSE detector can be expressed as

where the asymptotic SINR expression is provided in the following theorem. In Section 5, we show that the derived asymptotic sum rate accurately predicts the ergodic sum rate, which is obtained through lengthy Monte-Carlo simulations.

### 4.4Asymptotic Sum Rate Analysis for the Single-Cell Case

Although the provided asymptotic sum rate expression is easy to evaluate numerically, since Theorem ? considers a very general case, it does not offer much insight for system design. Hence, in order to get some insight regarding the influence of IQI on the performance of uplink massive MIMO systems employing the IQA-WLMMSE receiver, similar to the analysis for the conventional IQU-MMSE receiver, we consider the simpler single-cell case with perfect CSI, and i.i.d. channels for all UTs. In particular, in order to investigate the influence of the IQI on the performance, the cases with IQI present only at the BS and only at UTs are analyzed separately and their asymptotic SINRs and the corresponding improvements compared to the conventional IQU-MMSE receivers are evaluated in the following Corollaries.

For the system described in Corollary ? and identical IQI at all BS antenna branches, i.e., , the asymptotic SINR is given by

Substituting into (Equation 24) leads to for an ideal system without IQI. From (Equation 24), we observe that with increasing number of BS antennas, the sum rate of the proposed IQA-WLMMSE receiver increases too. In particular, it can be shown that for , which is valid for typical IQI values, the asymptotic SINR in (Equation 24) is smaller than but very close to , i.e., the sum rate of the ideal system without IQI. In addition, comparing the SINR of the IQA-WLMMSE receiver with IQI present only at the BS given in (Equation 24) and the corresponding SINR of the conventional IQU-MMSE receiver given in ( ?), the following asymptotic SINR loss can be obtained

From (Equation 25), it can be observed that for systems with IQI only at the BS, the SINR loss increases with increasing SNR and increasing number of UTs.

Substituting typical values for and into ( ?), it can be observed that similar to the system with IQI only at the BS, the asymptotic sum rate of the system with IQI only at the UTs increases with increasing number of BS antennas, and is smaller than but almost identical to the sum rate of an ideal system without IQI for and . Moreover, considering ( ?) and ( ?), we obtain the following asymptotic SINR loss of the conventional IQU-MMSE receiver compared to the IQA-WLMMSE receiver in a system, where the IQI is only present at the UTs

From (Equation 26), we observe that, if IQI is present only at the UTs, the SINR loss of the conventional IQU-MMSE receiver compared to the IQA-WLMMSE receiver increases with increasing number of BS antennas, . Substituting typical values for and , it can also be observed that the SINR loss of the conventional IQU-MMSE receiver compared to IQA-WLMMSE receiver in systems with IQI only at the UTs is slightly larger than the corresponding loss in systems, where the IQI is present only at the BS. We validate this observation in Section 5 for multi-cell systems and more general settings.

## 5Numerical Results

In order to evaluate the performance of the proposed IQA-WLMMSE receiver and to validate our analytical results, Monte-Carlo simulations have been performed. Here, we assume a system consisting of seven hexagonal cells with a normalized cell radius of one. Without loss of generality, we further assume that the central cell is the target cell. In each cell, there is a BS in the cell center and there are UTs, which are uniformly distributed on a circle with a radius of 2/3. The channel model used here comprises path-loss, antenna correlation, and Rayleigh fading. Moreover, we assume that the BS employs a uniform linear array (ULA) and adopt the ULA channel correlation model used in [18]. In particular, we have , where is the distance between the th UT in the th cell and the th BS, and and are an all-zero matrix and the number of dimensions of the antenna’s physical model, respectively. We adopt , where the steering vector is defined as with being the th angle of arrival (AoA), and and being the antenna spacing and the wavelength, respectively [18].

In Figure 1, the ergodic sum rates of the IQA-WLMMSE receiver, the IQU-MMSE receiver, the MMSE receiver in the absence of IQI, and the MMSE receiver in an ideal system with perfect CSI are depicted. Here, we assume and , and we further assume that the IQI is present at both the BS and the UTs. Moreover, except for the perfect CSI case, full pilot contamination is assumed. The number of UTs is set to and the amplitude and phase mismatches at the UTs and the different antenna branches of the BSs are randomly and uniformly distributed in the range of , respectively. As can be observed from Figure 1, even small amplitude and phase mismatches lead to a high sum rate loss of the IQU-MMSE receiver compared to the ideal system without IQI. As expected from the analysis of the simplified single-cell channel model in Section 3.4, the rate loss associated with IQI does not vanish even if the number of BS antennas is much larger than the number of UTs. For example, for , the rate loss of the IQU-MMSE receiver compared to the system without IQI is approximately . Furthermore, as expected from the analysis of the simplified single-cell channel model in Section 4.4, the proposed IQA-WLMMSE receiver achieves a substantially higher sum rate than the IQU-MMSE receiver and closely approaches the sum rate of the MMSE receiver in an ideal system without IQI. In Figure 1, we also present analytical results for the asymptotic sum rates of the IQU-MMSE receiver and the IQA-WLMMSE receiver given in (Equation 14) and (Equation 23), respectively. For large , a perfect match between analytical and simulation results is observed for all receivers. Nevertheless, even for small numbers of BS antennas, the match between asymptotic and simulation results is good.

In Figure 2, we investigate whether IQI at the UTs or IQI at the BSs is more harmful. To do so, we compare the sum rate performance of systems with IQI only at the BS (BS-IQI), IQI only at the UTs (UT-IQI), and IQI at both BSs and UTs (BSUT-IQI). For clarity of presentations, only analytical results are shown in Figure 2. However, all results were verified by simulations. Here, we consider a system without pilot contamination but with channel estimation errors, and and . The amplitude and phase mismatches are generated in the same manner as for Figure 1. From Figure 2, we observe that if the IQU-MMSE receiver is employed, the system with IQI at both UTs and BSs yields the lowest sum rate, as expected. Furthermore, the system with IQI only at the BSs achieves a higher sum rate than the system with IQI only at the UTs. We note that this effect could also be observed in Section 3.4, where analytical expressions for the asymptotic SINRs in the simplified single-cell system were derived. In fact, the sum rate of the system with IQI both at the BSs and the UTs approaches the sum rate of the system with IQI only the UTs for large numbers of BS antennas. A similar behavior can be observed for the sum rate performance of the IQA-WLMMSE receiver. Again, the system with IQI only at the BSs achieves the highest sum rate followed by the system with IQI only at the UTs and the system with IQI both at the UTs and the BSs. We note that this behavior supports the results in [10], where the authors claim that in the asymptotic regime, where the number of BS antennas is very large, H/W imperfections at the UTs are more harmful than those at the BS.

In Figure 3, the influence of the amplitude mismatch on the sum rate of the conventional IQU-MMSE and the proposed IQA-WLMMSE receivers is investigated. For convenience, we assume that all UTs and BS antenna branches have the same phase and amplitude mismatches, i.e.,