Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels under Limited Feedback

Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels under Limited Feedback


In this paper, we study the sum rate performance of zero-forcing (ZF) and regularized ZF (RZF) precoding in large MISO broadcast systems under the assumptions of imperfect channel state information at the transmitter and per-user channel transmit correlation. Our analysis assumes that the number of transmit antennas and the number of single-antenna users are large while their ratio remains bounded. We derive deterministic approximations of the empirical signal-to-interference plus noise ratio (SINR) at the receivers, which are tight as . In the course of this derivation, the per-user channel correlation model requires the development of a novel deterministic equivalent of the empirical Stieltjes transform of large dimensional random matrices with generalized variance profile. The deterministic SINR approximations enable us to solve various practical optimization problems. Under sum rate maximization, we derive (i) for RZF the optimal regularization parameter, (ii) for ZF the optimal number of users, (iii) for ZF and RZF the optimal power allocation scheme and (iv) the optimal amount of feedback in large FDD/TDD multi-user systems. Numerical simulations suggest that the deterministic approximations are accurate even for small .


pioneering work in [1] and [2] revealed that the capacity of a point-to-point (single-user (SU)) multiple-input multiple-output (MIMO) channel can potentially increase linearly with the number of antennas. However, practical implementations quickly demonstrated that in most propagation environments the promised capacity gain of SU-MIMO is unachievable due to antenna correlation and line-of-sight components [3]. In a multi-user (MU) scenario, the inherent problems of SU-MIMO transmission can largely be overcome by exploiting multi-user diversity, i.e., sharing the spatial dimension not only between the antennas of a single receiver, but among multiple (non-cooperative) users. The underlying channel for MU-MIMO transmission is referred to as the MIMO broadcast channel (BC) or MU downlink channel. Although much more robust to channel correlation, the MIMO-BC suffers from inter-user interference at the receivers which can only be efficiently mitigated by appropriate (i.e., channel-aware) pre-processing at the transmitter.

It has been proved that dirty-paper coding (DPC) is a capacity achieving precoding strategy for the Gaussian MIMO-BC [4]. However, the DPC precoder is non-linear and to this day too complex to be implemented efficiently in practical systems. It has been shown in [4], that suboptimal linear precoders can achieve a large portion of the BC rate region while featuring low computational complexity. Thus, a lot of research has recently focused on linear precoding strategies.

In general, the rate maximizing linear precoder has no explicit form. Several iterative algorithms have been proposed in [12], but no global convergence has been proved. Still, these iterative algorithms have a high computational complexity which motivates the use of further suboptimal linear transmit filters (i.e., precoders), by imposing more structure into the filter design. A straightforward technique is to precode by the inverse of the channel. This scheme is referred to as channel inversion or zero-forcing (ZF) [4].

Although [12] assume perfect channel state information at the transmitter (CSIT) to determine theoretically optimal performance, this assumption is untenable in practice. It is indeed a particularly strong assumption, since the performance of all precoding strategies is crucially depending on the CSIT quality. In practical systems, the transmitter has to acquire the channel state information (CSI) of the downlink channel by feedback signaling from the uplink. Since in practice the channel coherence time is finite, the information of the instantaneous channel state is inherently incomplete. For this reason, a lot of research has been carried out to understand the impact of imperfect CSIT on the system behavior, see [14] for a recent survey.

In this contribution, we focus on the multiple-input single-output (MISO) BC, where a central transmitter equipped with antennas communicates with single-antenna non-cooperative receivers. We assume , i.e., we do not account for user scheduling, and consider ZF and regularized ZF (RZF) precoding under imperfect CSIT (modeled as a weighted sum of the true channel plus noise) as well as per-user channel correlation, i.e., the vector channel of user () satisfies and . To obtain insights into the system behavior, we approximate the signal-to-interference plus noise ratio (SINR) by a deterministic quantity, where the novelty of this study lies in the large system approach. More precisely, we approximate the SINR of user by a deterministic equivalent such that almost surely, as the system dimensions and go jointly to infinity with bounded ratio . Hence, becomes more accurate for increasing . To derive , we apply tools from the well-established field of large dimensional random matrix theory (RMT) [15]. Previous work considered SINR approximations based on bounds on the average (with respect to the random channels ) SINR. The deterministic equivalent is not a bound but is a tight approximation, for asymptotically large . Furthermore, the RMT tools allow us to consider advanced channel models like the per-user correlation model, which are usually extremely difficult to study exactly for finite dimensions. Interestingly, simulations suggest that is very accurate even for small system dimension, e.g., . Currently, the 3GPP LTE-Advanced standard [17] already defines up to transmit antennas further motivating the application of large system approximations to characterize the performance of wireless communication systems. Subsequently, we apply these SINR approximations to various practical optimization problems.

1.1Related Literature

To the best of the authors’ knowledge, Hochwald et al. [18] were the first to carry out a large system analysis with and finite ratio for linear precoding under the notion of “channel hardening”. In particular, they considered ZF precoding, called channel inversion (CI), for under perfect CSIT, and showed that the SINR for independent and identical distributed (i.i.d.) Gaussian channels converges to , where is the signal-to-noise ratio (SNR), independent of the applied power normalization strategy. They go on to derive the sum rate maximizing system loading for a fixed . Their results are a special case of our analysis in Section 3.2 and Section 5.1. The authors in [18] conclude by showing that for , ZF achieves a large fraction of the linear (with respect to ) sum rate growth. The work in [9] extends the analysis in [18] to the case and shows that the sum rate of ZF is constant in as , i.e., the linear sum rate growth is lost. The authors in [9] counter this problem by introducing a regularization parameter in the inverse of the channel matrix. Under the assumption of large , perfect CSIT and for any rotationally-invariant channel distribution, [9] derives the regularization parameter that maximizes the SINR. Note here that [9] does not apply the classic tools from large dimensional RMT to derive their results but rather find the solution by applying various expectations and approximations. In the present contribution, the RZF precoder of [9] is referred to as channel distortion-unaware RZF (RZF-CDU) precoder, since its design assumes perfect CSIT, although in practice, the available CSIT is erroneous or distorted. It has been observed in [9] that the RZF-CDU precoder is very similar to the transmit filter derived under the minimum mean square error (MMSE) criterion [19] and both become identical in the large limit. Likewise, we will observe some similarities between RZF and MMSE filters when considering imperfect CSIT. The RZF precoder in [9] has been extended in [20] to account for channel quantization feedback under random vector quantization (RVQ). The authors in [20] do not apply tools from large RMT but use the same techniques as in [9] and obtain different results for the optimal regularization parameter and SINR compared to our results in Section 6.

The first work applying tools from large RMT to derive the asymptotic SINR under ZF and RZF precoding for correlated channels was [21]. However, in [21] the regularization parameter of the considered RZF precoder was set to fulfill the total average power constraint. Similar work [22] was published later, where the authors considered the RZF precoder in [9] and derived the asymptotic SINR for uncorrelated Gaussian channels. Moreover, they derived the asymptotically optimal regularization parameter , already derived in [9], which is a special case of the result derived in Section 4. Another work [23], reproducing our results, noticed that the optimal regularization parameter in [9] is independent of transmit correlation when the channel correlation is identical for all users.

In the large system limit and for channels with i.i.d. entries, the cross correlations between the user channels, and therefore the users’ SINRs, are identical. It has been shown in [24] that for this symmetric case and equal noise variances, the SINR maximizing precoder is of closed form and coincides with the RZF precoder. Recently, the authors in [25] claimed that indeed the RZF precoder structure emerges as the optimal precoding solution for . This asymptotic optimality further motivates a detailed analysis of the RZF precoder for large system dimensions.

1.2Contributions of the Present Work

In this paper, we provide a concise framework that directly extends and generalizes the results in [18] by accounting for per-user correlation and imperfect CSIT. Furthermore, we apply our SINR approximations to several limited-feedback scenarios that have been previously analyzed by applying bounds on the ergodic rate of finite dimensional systems. Our main contributions are summarized as follows:

  • Motivated by the channel model, we derive a deterministic equivalent of the empirical Stieltjes transform of matrices with generalized variance profile, thereby extending the results in [27].

  • We propose deterministic equivalents for the SINR of ZF () and RZF () precoding under imperfect CSIT and channel with per-user correlation, i.e., deterministic approximations of the SINR, which are independent of the individual channel realizations, and (almost surely) exact as .

  • Under imperfect CSIT and common correlation (), we derive the sum rate maximizing RZF precoder called channel-distortion aware RZF (RZF-CDA) precoder.

  • For ZF and RZF, under common correlation and different CSIT qualities, we derive the optimal power allocation scheme which is the solution of a water-filling algorithm.

For uncorrelated channels, we obtain the following results:

The remainder of the paper is organized as follows. Section 2 presents the transmission model and channel model. In Section 3, we propose deterministic equivalents for the SINR of RZF and ZF precoding. In Section 4, we derive the sum rate maximizing regularization under RZF precoding. Section 5 studies the sum rate maximizing number of users for ZF precoding and the optimal power allocation when the CSIT quality of the users is unequal. Section 6 analyses the optimal amount of feedback in a large FDD system. In Section 7, we study a large TDD system and derive the optimal amount of uplink channel training. Finally, in Section 8, we summarize our results and conclude the paper.

Most technical poofs are presented in the appendix. In these proofs, we apply several lemmas collected in Appendix Section 14.

Notation: In the following, boldface lower-case and upper-case characters denote vectors and matrices, respectively. The operators , and denote conjugate transpose, trace and expectation, respectively. The identity matrix is denoted , is the natural logarithm and is the imaginary part of . and are the spectral radius and the minimum eigenvalue of the Hermitian matrix , respectively. The imaginary unit is denoted . The sets and are defined as and . A random vector is complex Gaussian distributed with mean vector and covariance matrix .

2System Model

This section describes the transmission model as well as the underlying channel model.

2.1Transmission Model

Consider a MISO broadcast channel composed of a central transmitter equipped with antennas and of single-antenna non-cooperative receivers. We assume , thus user scheduling is not taken into account. Furthermore, we suppose narrow-band transmission. The signal received by user at any time instant reads

where is the random channel from the transmitter to user , is the transmit vector and the noise terms are independent. We assume that the channel evolves according to a block-fading model, i.e., the channel is constant at every time instant but varies independently from one time instant to another.

The transmit vector is a linear combination of the independent user symbols and can be written as

where and are the precoding vector and the signal power of user , respectively. Subsequently, we assume that user has perfect knowledge of and the effective channel . In particular, an estimate of can be obtained through dedicated downlink training by precoding the pilots of user by . The precoding vectors are normalized to satisfy the average total power constraint

where , and is the total available transmit power.

Denote the SNR. Under the assumption of Gaussian signaling, i.e., and single-user decoding with perfect channel state information at the receivers, the SINR of user is defined as [29]

The rate of user is given by

and the ergodic sum rate is defined as

where the expectation is taken over the random channels .

2.2Channel Model

Each user channel is modeled as

where is the channel correlation matrix of user and has i.i.d. complex entries of zero mean and variance . The channel transmit correlation matrices are assumed to be slowly varying compared to the channel coherence time and thus are supposed to be perfectly known to the transmitter, whereas receiver has only knowledge about . Moreover, only an imperfect estimate of the true channel is available at the transmitter which is modeled as [30]

where , has i.i.d. entries of zero mean and variance independent of and . The parameter reflects the accuracy or quality of the channel estimate , i.e., corresponds to perfect CSIT, whereas for the CSIT is completely uncorrelated to the true channel. The variation in the accuracy of the available CSIT between the different user channels arises naturally. Firstly, there might be low mobility users and high mobility users with large or small channel coherence intervals, respectively. Therefore, the CSIT of the high mobility users will be outdated quickly and hence be very inaccurate. On the other hand, the CSIT of the low mobility users remains accurate since their channel does not change significantly from the time of the channel estimation until the time of precoding and coherent data transmission. Secondly, different CSIT qualities arise when the feedback rate varies among the users. For instance, if the CSIT is obtained from uplink training, the training length of each user could be different, leading to different channel estimation errors at the transmitter. Similarly, if the users feed back a quantized channel, they could use channel quantization codebooks of different sizes depending on their channel quality and the available uplink resources. However, for simplicity, we assume identical CSIT qualities for the optimization problems considered in Section 6 and Section 7.

Define the compound estimated channel matrix . Therefore, the matrix can be written as

The per-user channel correlation model (also called generalized variance profile) is very general and encompasses various propagation environments. For instance, all channel coefficients of the vector channel may have different variances resulting from different attenuation of the signal while traveling to the receivers. This so called variance profile of the vector channel is obtained by setting , see [27]. Another possible scenario consists of an environment where all user channels have identical transmit correlation , but where the users are heterogeneously scattered around the transmitter and hence experience different channel gains . Such a setup can be modeled with . From a mathematical point of view, a homogeneous system with common user channel correlation is very attractive. In this case, the user channels are statistically equivalent and the deterministic SINR approximations can be computed by solving a single implicit equation instead of multiple systems of coupled implicit equations. A further simplification occurs when the channels are uncorrelated , in which case the approximated SINRs are given explicitly.

The model in has never been considered in large dimensional RMT and therefore no results are available. The most general model studied, assumes a variance profile, first treated in [27] and extended in [28], which is a special case of the model in . Therefore, to be able to derive deterministic equivalents of the SINR, we need to extend the results in [27] to account for the per-user correlation model in , which is done in the next section.

3A Deterministic Equivalent of the SINR

This section introduces deterministic approximations of the SINR under RZF and ZF precoding for various assumptions on the transmit correlation matrices . These results will be used in Sections Section 4-Section 7 to solve practical optimization problems.

The following theorem extends the results in [35] by assuming a generalized variance profile. This theorem is required to cope with the channel model in and forms the mathematical basis of the subsequent large system analysis of the MISO BC under RZF and ZF precoding.

Note that forms a system of coupled equations, from which is given explicitly.

To derive a deterministic equivalent of the SINR under RZF and ZF precoding, we require the following assumptions on the correlation matrices and the power allocation matrix .

3.1Regularized Zero-forcing Precoding

Consider the RZF precoding matrix

where is the channel estimate available at the transmitter, is a normalization scalar to fulfill the power constraint and is the regularization parameter. Here, is scaled by to ensure that itself converges to a constant, as .

From the total power constraint , we obtain as

where we defined . Denoting , the SINR of user in under RZF precoding takes the form

where and .

To derive a deterministic equivalent of the SINR defined in such that , almost surely, we require the following assumption.

A deterministic equivalent of is provided in the following theorem.

Note that under Assumption ?, the term in can be omitted since the convergence in still holds true. We will make use of this simplification when studying different applications of the SINR approximations.

In particular, we will consider two different RZF precoders. The first RZF precoder is defined by and is referred to as RZF channel distortion unaware (RZF-CDU) precoder. Under imperfect CSIT the RZF-CDU precoder is mismatched to the true channel. The second RZF precoder is called RZF channel distortion aware (RZF-CDA) precoder and does account for imperfect CSIT. The optimal regularization parameter for the RZF-CDA precoder is derived in Section 4.

Moreover, there are two limiting cases of the RZF precoder corresponding to and . For the RZF precoder converges to the matched filter (MF) precoder . A deterministic equivalent for the MF precoder can be derived by taking the limit . However, since the performance of the MF precoder is rather poor and does not involve Stieltjes transforms anymore, we will not discuss this precoding scheme in the present work. The reader is referred to [38] or [39] for a detailed large system analysis of the MF precoder. In the case of , the RZF precoder converges to the ZF precoder, which is discussed in the next section.

3.2Zero-forcing Precoding

For , the RZF precoding matrix in reduces to the ZF precoding matrix which reads

where is a scaling factor to fulfill the power constraint and is given by

where . Defining , the SINR of user in under ZF precoding reads

To obtain a deterministic equivalent of the SINR in , we need to ensure that the minimum eigenvalue of is bounded away from zero for all large , almost surely. Therefore, the following assumption is required.

Furthermore, we require the following assumption for the channel model with per-user correlation.

3.3Rate Approximations

We are interested in the individual rates of the users as well as the average system sum rate . Since the logarithm is a continuous function, by applying the continuous mapping theorem [40], it follows from the almost sure convergence , that

almost surely, where . An approximation of the ergodic sum rate is obtained by replacing the instantaneous (i.e., without averaging over the channel distribution) SINR with its large system approximation , i.e.,

It follows that

holds true almost surely.

Another quantity of interest is the rate gap between the achievable rate under perfect and imperfect CSIT. We define the rate gap of user as

where is the rate of user under perfect CSIT, i.e., for . Then, from it follows that a deterministic equivalent of the rate gap of user such that

almost surely, is given by

where is a deterministic equivalent of the rate of user under perfect CSIT.

Since we will require the per-user rate gaps for uncorrelated channels () in the limited feedback analysis in Sections Section 6 and Section 7, we introduce hereafter for RZF-CDU and ZF precoding.

3.4Numerical Results

We validate Theorem ? and Theorem ? by comparing the ergodic sum rate , obtained by Monte-Carlo (MC) simulations of i.i.d. Rayleigh block-fading channels, to the large system approximation , for finite system dimensions and equal power allocation .

The correlation of the th user channel is modeled as in [41] by assuming a diffuse two-dimensional field of isotropic scatterers around the receivers. The waves impinge the receiver uniformly at an azimuth angle ranging from to . Denoting the distance between transmit antenna and , the correlation is modeled as

where denotes the signal wavelength. The users are assumed to be distributed uniformly around the transmitter at an angle and as a simple example, we choose and . Note that for small (in our example for small values of ), the corresponding signal of user is highly correlated since the signal arrives from a very narrow angle. Thus, the correlation model yields rank-deficient correlating matrices for some users. The transmitter is equipped with a uniform linear array (ULA). To ensure that is bounded as grows large, we assume that the distance between adjacent antennas is independent of , i.e., the length of the ULA increases with .

Figure 1: RZF, (R_{\rm sum}-R_{\rm sum}^\circ)/R_{\rm sum} vs. M for a fixed SNR of \rho=10 dB with M\!=\!K, \alpha=1/\rho.
Figure 1: RZF, vs. for a fixed SNR of dB with , .
Figure 2: RZF, sum rate vs. SNR with M\!=\!K\!=\!30 and \alpha=1/\rho, simulation results are indicated by circle marks with error bars indicating the standard deviation.
Figure 2: RZF, sum rate vs. SNR with and , simulation results are indicated by circle marks with error bars indicating the standard deviation.
Figure 3: ZF, sum rate vs. SNR with M\!=\!30, K\!=\!15, simulation results are indicated by circle marks with error bars indicating the standard deviation.
Figure 3: ZF, sum rate vs. SNR with , , simulation results are indicated by circle marks with error bars indicating the standard deviation.

The simulation results presented in Figure 1 depict the absolute error of the sum rate approximation compared to the ergodic sum rate , averaged over independent channel realizations. The notation “” indicates that is modeled according to with . From Figure 1, we observe that the approximated sum rate becomes more accurate with increasing .

Figures Figure 2 and Figure 3 compare the ergodic sum rate to the deterministic approximation under RZF and ZF precoding, respectively. The error bars indicate the standard deviation of the MC results. It can be observed that the approximation lies roughly within one standard deviation of the MC simulations. From Figure 2, under imperfect CSIT (), the sum rate is decreasing for high SNR, because the regularization parameter does not account for and thus the matrix in the RZF precoder becomes ill-conditioned. Figure 3 shows that, for , the sum rate is not decreasing at high SNR, because the CSIT is much better conditioned. The optimal regularization is discussed in Section 5. Further observe that in Figure 2 the deterministic approximation becomes less accurate for high SNR. The reason is that in the derivation of the approximated SINR, we apply Theorem ? in and thus the bounds in Proposition ? (Appendix Section 9.1) are proportional to the SNR. Therefore, to increase the accuracy of the approximated SINR, larger dimensions are required in the high SNR regime.

We conclude that the approximations in Theorems ? and ? are accurate even for small dimensions and can be applied to various optimization problems discussed in the sequel.

4Sum Rate Maximizing Regularization

The optimal regularization parameter maximizing is defined as

In general, the optimization problem is not convex in and the solution has to be computed via a one-dimensional line search.

In the following, we confine ourselves to the case of common correlation , since for per-user correlation a common regularization parameter is not optimal anymore [12]. Under common transmit correlation, we subsequently assume that the distortions of the CSIT are identical for all users, since the users’ channels are statistically equivalent. Under these conditions maximizes and the optimization problem has the following solution.

Note that the solution in Proposition ? assumes a fixed distortion . Later in Section 6 the distortion becomes a function of the quantization codebook size and in Section 7 it depends on the uplink SNR as well as on the amount of channel training.

Under perfect CSIT (), Proposition ? simplifies to the well-known solution , independent of , which has previously been derived in [9]. As mentioned in [9], for large the RZF-CDA precoder is identical to the MMSE precoder in [43]. The authors in [26] showed that, under perfect CSIT, is independent of the correlation . However, for imperfect CSIT (), the optimal regularization parameter depends on the transmit correlation through and . For uncorrelated channels (), we have and and therefore the explicit solution

Note that in this case, it can be shown that in is the unique positive solution to .

For imperfect CSIT (), the RZF-CDA precoder and the MMSE precoder with regularization parameter [43] are not identical anymore, even in the large limit. Unlike the case of perfect CSIT, now depends on the correlation matrix through and . The impact of and on the sum rate of RZF-CDA precoding is evaluated through numerical simulations in Figure 5. Further note that since and are bounded from above under the conditions explained in Remark ? below, at asymptotically high SNR the regularization parameter in converges to , where is a positive solution of

For uncorrelated channels, the limit in takes the form

Thus, for asymptotically high SNR, RZF-CDA precoding is not the same as ZF precoding, since the regularization parameter is non-zero due to the residual interference caused by the imperfect CSIT. Similar observations have been made in [43] for the MMSE precoder.

For various special cases, substituting into the deterministic equivalent of the SINR in yields the following simplified expressions.

For uncorrelated channels , the solution to is explicit and summarized in the following corollary.

A deterministic equivalent of the rate gap under RZF-CDA precoding is provided in the following corollary.

The impact of the regularization parameter on the ergodic sum rate is depicted in Figures Figure 4 and Figure 5.

Figure 4: RZF, ergodic sum rate vs. SNR with M\!=\!K \!=\!5, {\bm{\Theta}}_k\!=\!\mathbf{I}_M~\forall k, \mathbf{P}=\frac1K\mathbf{I}_K and \tau^2\!=\!0.1.
Figure 4: RZF, ergodic sum rate vs. SNR with , , and .

In Figure 4, we compare the ergodic sum rate performance for different regularization parameters with CSIT distortion . The upper bound is obtained by optimizing for every channel realization, whereas maximizes the ergodic sum rate. It can be observed that both and perform close to the optimal . Furthermore, if the channel quality is unknown at the transmitter (and hence assumed to be equal to zero), the performance is decreasing as soon as dominates (i.e. the inter-user interference limits the performance) the noise power and approaches the sum rate of ZF precoding for high SNR. We conclude that (i) adapting the regularization parameter yields a significant performance increase and (ii) that the proposed RZF-CDA precoder with performs close to optimal even for small system dimensions.

Figure 5: RZF, ergodic sum rate vs. SNR with M\!=\!K \!=\!5, \mathbf{P}=\frac1K\mathbf{I}_K and \tau^2\!=\!0.05.
Figure 5: RZF, ergodic sum rate vs. SNR with , and .

In Figure 5, we simulate the impact of transmit correlation in the computation of on the sum rate. For this purpose, we use the standard exponential correlation model, i.e.,

We compare two different RZF precoders: A first precoder coined RZF common correlation aware (RZF-CCA) that takes the channel correlation into account and computes according to , and a second precoder, called RZF common correlation unaware (RZF-CCU) that does not take into account and computes as in . We observe that for high correlation, i.e., , the RZF-CCA precoder significantly outperforms the RZF-CCU precoder at medium to high SNR, whereas both precoders perform equally well at low SNR. Therefore, we conclude that it is beneficial to account for transmit correlation, especially in highly correlated channels. Further simulations (not provided here) suggest that the sum rate gain of RZF-CCA over RZF-CCU precoding is less pronounced for lower CSIT qualities (i.e., increasing ), because in this case the impact of the CSIT quality is more significant than the impact of on the sum rate.

5Optimal Number of Users and Power Allocation

In this section, we address two problems: (i) the determination of the sum rate maximizing number of users per transmit antenna for a fixed and (ii) the optimization of the power distribution among a given set of users with unequal CSIT qualities.

Consider problem (i). Intuitively, an optimal number of users exists because serving more users creates more interference which in turn reduces the rates of the users. At some point the accumulated rate loss, due to the additional interference caused by scheduling another user, will outweigh the sum rate gain and hence the system sum rate will decrease. In particular, we consider a fair scenario where the SINR approximation of all users are equal. Here, the (approximated) optimal solution can be expressed under a closed form for ZF precoding.

In problem (ii), we optimize the power allocation matrix for a given . More precisely, we focus on common correlation with different CSIT qualities , since in this case the (approximated) optimal power distribution is the solution of a classical water-filling algorithm.

5.1Sum Rate Maximizing Number of Users

Consider the problem of finding the system loading maximizing the approximated sum rate per transmit antenna for a fixed , i.e.,

where denotes either with or with . In general has to be solved by a one-dimensional line search. However, in case of ZF precoding and uncorrelated antennas, the optimization problem has a closed-form solution given in the following proposition.

For , we have and . In this case is a well-defined function. If , we obtain the results in [18], although in [18] they are not given in closed form. Note that for , we have , i.e., the optimal system loading tends to one. Further note that only integer values of are meaningful in practice.

5.2Power Optimization under Common Correlation

From Corollaries ? and ?, the approximated sum rate for both RZF and ZF precoding takes the form

with , where the only dependence on user stems from . The user powers that maximize , subject to , , are thus given by the classical water-filling solution [44]

where and is the water level chosen to satisfy . For , the optimal user powers are all equal, i.e., and . In this case though, it could still be beneficial to adapt the number of users as discussed in Section 5.1.

5.3Numerical Results

Figure 6 compares the optimal number of users in to obtained by choosing the such that the ergodic sum rate is maximized, whereas Figure 7 depicts the impact of a suboptimal number of users on the ergodic sum rate of the system.

From Figure 6, it can be observed that (i) the approximated results do fit well with the simulation results even for small dimensions, (ii) increase with the SNR and (iii), for , saturate for high SNR at a value lower than . Therefore, under imperfect CSIT, it is not optimal anymore to serve the maximum number of users for asymptotically high SNR. Instead, depending on , a lower number of users should be served even at high SNR which implies a reduced multiplexing gain of the system. The impact of different numbers of users on the sum rate is depicted in Figure 7.

Figure 6: ZF, sum rate maximizing number of users vs. SNR with {\bm{\Theta}}_k=\mathbf{I}_M~\forall k, \tau^2=0.1 and \mathbf{P}=\frac1K\mathbf{I}_K.
Figure 6: ZF, sum rate maximizing number of users vs. SNR with , and .
Figure 7: ZF, R_{\rm sum} vs. SNR with M\!=\!16, {\bm{\Theta}}_k=\mathbf{I}_M~\forall k, \mathbf{P}=\frac1K\mathbf{I}_K and \tau^2\!=\!0.1.
Figure 7: ZF, vs. SNR with , , and .
Figure 8: RZF-CDU, R_{\rm sum} vs. \rho with \alpha\!=\!1/\rho, {\bm{\Theta}}_k\!=\!\mathbf{I}_M~\forall k, P\!=\!1 and \tau_k^2\!\in\!\mathcal{T}_1,\cup_{k=1}^3\tau_k^2=\mathcal{T}_1 (M\!=\!5) and \tau_k^2\!\in\!\mathcal{T}_2,\cup_{k=1}^3\tau_k^2=\mathcal{T}_2 (M\!=\!3).
Figure 8: RZF-CDU, vs. with , , and () and ().

From Figure 7 we observe that (i) the approximate solution achieves most of the sum rate and (ii) adapting the number of users with the SNR is beneficial compared to a fixed . Moreover, from Figure 6, we identify as an optimal choice (for ) for medium SNR and, as expected, the performance is optimal in the medium SNR regime and suboptimal at low and high SNR. From Figure 6 it is clear that is highly suboptimal in the medium and high SNR range and we observe a significant loss in sum rate. Consequently, the number of users must be adapted to the channel conditions and the approximate result is a good choice to determine the optimal number of users.

In Figure 8, under RZF-CDU precoding, we compare the ergodic sum rate performance with power allocation from to equal power allocation . We consider a system with , where the CSIT qualities vary significantly among the users, i.e., with , . We observe a significant gain over the whole SNR range when optimal power allocation is applied. In contrast, if the CSIT distortion of the users’ channels with does not differ considerably (, with ), we only observe a small gain at high SNR. For increasing SNR, the SINRs become increasingly distinct depending on the . Therefore, it might be optimal to turn off the users with lowest CSIT accuracy as the SNR increases, which explains why the sum rate gain is larger at high SNR than at low SNR. However, recall that the water-filling solution is optimal under Assumption ? () and large . We thus conclude that the optimal power allocation proposed in achieves significant performance gains, especially at high SNR, when the quality of the available CSIT varies considerably among the users’ channels.

6Optimal Feedback in Large FDD Multi-user Systems

Consider a frequency-division duplex (FDD) system, where the users quantize their perfectly estimated channel vectors and send the codebook quantization index back to the transmitter over an independent feedback channel of limited rate. The feedback channels are assumed to be error-free and of zero delay. The quantization codebooks are generated prior to transmission and are known to both transmitter and respective receiver. Due to the finite rate feedback link, imposing a finite codebook size, the transmitter has only access to an imperfect estimate of the true downlink channel. To obtain tractable expressions, we restrict the subsequent analysis to i.i.d. Gaussian channels .

In the sequel, we follow the limited feedback analysis in [45], where each user’s channel direction is quantized using bits which are subsequently fed back to the transmitter. Under Rayleigh fading, the channel can be decomposed as , where we suppose that the channel magnitude is perfectly known to the transmitter since it can be efficiently quantized with only a few bits [45]. Without loss of generality,1 we assume random vector quantization (RVQ), where each user independently generates a random codebook containing vectors that are isotropically distributed on the -dimensional unit sphere. Subsequently, user quantizes its channel direction to the closest according to

Under RVQ, the quantized channel direction is isotropically distributed on the -dimensional unit sphere due to the statistical properties of both, the random codebook and the channels . Thus, for fine quantization with small errors, the entries of both and can be modeled with good approximation as i.i.d. Gaussian of zero mean and unit variance. The quantization error vector can be approximated as [46] and we can write

where is the quantization error variance. The scaling in is required to ensure that the elements of have unit variance. Therefore, the effect of imperfect CSIT under RVQ in is captured by the channel model . For RVQ, the quantization error can be upper bounded as [45]

The bound in is tight for large [45]. Moreover, since the quantization codebooks of the users are supposed to be of equal size, the resulting CSIT distortions can be assumed identical, i.e., . Under this assumption and equal power allocation, for large , the SINR is identical for all users and, hence, optimizing is equivalent to optimizing the per-user rate bits/s/Hz and the sum rate .

In the following, in particular under RVQ, we will derive the necessary scaling of the distortion to ensure that

almost surely, where is defined in and . That is, a constant rate gap of is maintained exactly as . A constant rate gap ensures that the full multiplexing gain of is achieved. Thus, the proposed scaling also guarantees a larger but constant rate gap to the optimal DPC solution with perfect CSIT. The choice of a rate offset is motivated by mere mathematical convenience to avoid terms of the form and to be compliant with [45].

With this strategy we closely follow [45]. In [45], the author derived an upper bound of the ergodic per-user gap for ZF precoding with and unit norm precoding vectors under RVQ, which is given by

We cannot directly compare the deterministic equivalents to the upper bound in for two reasons, (i) under ZF precoding and , a deterministic equivalent for the per-user rate gap does not exist and (ii) [45] considers unit norm precoding vectors, whereas in this paper we only impose a total power constraint . Concerning (i), at high SNR, we can use the deterministic equivalent for RZF-CDU precoding given in Corollary ? as a good approximation for ZF precoding, since for high SNR the rates of RZF-CDU and ZF precoding converge. Regarding (ii), deriving a deterministic equivalent of the SINR under linear precoding with a unit norm power constraint on the precoding vectors is difficult, since it introduces an additional non-trivial dependence on the channel. However, it is useful to compare the accuracy of the upper bound in and the deterministic equivalent in Corollary ? at high SNR.

Figure 9, depicts the per-user rate gap as a function of the feedback bits per user under ZF precoding at a SNR of 25 dB. We simulated the ergodic per-user rate gap and of ZF precoding with unit norm precoding vectors and total power constraint, respectively. We compare the numerical results to the upper bound and to the deterministic equivalent for and . For both system dimensions and are close, suggesting that our results derived under the total power constraint may be good approximations for the case of unit norm precoding vectors as well. As mentioned in [45], the accuracy of the upper bound increases with increasing but the deterministic equivalent appears to be more accurate for both and . In fact, for , approximates the per-user rate gap significantly more accurately than the upper bound for the given SNR. We conclude that the proposed deterministic equivalent is sufficiently accurate and can be used to derive scaling laws for the optimal feedback rate.

Figure 9: ZF, per-user rate gap vs. number of bits per user with \rho=25 dB, {\bm{\Theta}}_k\!=\!\mathbf{I}_M~\forall k.
Figure 9: ZF, per-user rate gap vs. number of bits per user with dB, .

In the following, we compare the scaling of under RZF-CDA, RZF-CDU and ZF () precoding to the upper bound given for ZF () precoding in [45]. For the sake of comparison, we restate [45].

where . It is also mentioned that the result in [45] holds true for RZF-CDU precoding for high SNR, since ZF and RZF-CDU precoding converge for asymptotically high SNR. Furthermore, it is claimed, corroborated by simulation results, that [45] is true under RZF-CDU precoding for all SNR.

In order to correctly interpret the subsequent results, it is important to understand the differences between our approach and the approach in [45]. The scaling given in [45] is a strict upper bound on the ergodic per-user rate gap for all SNR and all under a unit norm constraint on the precoding vectors. In contrast, our approach yields a necessary scaling of that maintains a given instantaneous target rate gap exactly as under a total power constraint. Therefore, our results are not upper bounds for small , i.e., we cannot guarantee that for small dimensions. But since for asymptotically large , the rate gap is maintained exactly and we apply an upper bound on the CSIT distortion under RVQ , it follows that our results become indeed upper bounds for large . Simulations reveal that under the derived scaling of , the per-user rate gap is very close to even for small dimension, e.g., . Concerning the ergodic and instantaneous per-user rate gap, the reader is reminded that our results hold also for ergodic per-user rates as a consequence of the dominated convergence theorem, see Remark ?.

Consequently, a comparison of the results in [45] to our solutions is meaningful, especially for larger values of where our results become upper bounds.

In the following section, we apply the deterministic equivalents of the per-user rate gap under RZF-CDA, RZF-CDU and ZF precoding provided in Corollaries ?, ? and ?, respectively, to derive scaling laws for the amount of feedback necessary to achieve full multiplexing gain.

6.1Channel Distortion Aware Regularized Zero-forcing Precoding

Although the proposed scaling of in converges to zero for asymptotically high SNR, we can approximate the term in the high SNR regime.

To compare Proposition ? to [45], we use the upper bound on the quantization distortion , i.e., , where is the number of feedback bits per user under RZF-CDA precoding. Thus, can be rewritten as

6.2Channel Distortion Unaware Regularized Zero-forcing Precoding

Although the RZF-CDU precoder is suboptimal under imperfect CSIT, the results are useful to compare to the work in [45].

An approximation of the term at high SNR is given in the following proposition.

Applying the upper bound on the CSIT distortion under RVQ with bits per user, we obtain

6.3Zero-forcing Precoding

The following results are only valid for and thus, they cannot be compared to [45] which are derived under the assumption . However, for high SNR the results for the RZF-CDU precoder are a good approximation for the ZF precoder as well, even for .

Under RVQ with feedback bits per user, we have

6.4Discussion and Numerical Results

At this point, we can draw the following conclusions. The optimal scaling of the CSIT distortion is lower for compared to . For , the optimal scaling of the feedback bits , and for ZF in [45] are different, even at high SNR. In fact, for large , under RZF-CDU precoding and ZF precoding, the upper bound in [45] appears to be too pessimistic in the scaling of the feedback bits. From and , a more accurate choice may be

i.e., bits less than proposed in [45]. However, recall that becomes an upper bound for large and a rate gap of at least bits/s/Hz cannot be guaranteed for small values of . Moreover, for high SNR, and large , to maintain a rate offset of , the RZF-CDA precoder requires bits less than the RZF-CDU and ZF precoder and bits less than the scaling proposed in [45].

In contrast, for and high SNR, we have . Intuitively, the reason is that, for , the channel matrix is well conditioned and the RZF and ZF precoders perform similarly. Therefore, both schemes are equally sensitive to imperfect CSIT and thus the scaling of is the same for high SNR.

Note that our model comprises a generic distortion of the CSIT. That is, the distortion can be a combination of different additional factors, e.g., channel estimation at the receivers, channel mismatch due to feedback delay or feedback errors (see [47]) as long as they can be modeled as additive noise . Moreover, we consider i.i.d. block-fading channels, which can be seen as a worst case scenario in terms of feedback overhead. It is possible to exploit channel correlation in time, frequency and space to refine the CSIT or to reduce the amount of feedback.

Figure 10: RZF, ergodic sum rate vs. SNR under RZF precoding and RVQ with B feedback bits per user, where B is chosen to maintain a sum rate offset of K\log_2b\!=\!10, {\bm{\Theta}}_k=\mathbf{I}_M~\forall k and M=K=10.
Figure 10: RZF, ergodic sum rate vs. SNR under RZF precoding and RVQ with feedback bits per user, where is chosen to maintain a sum rate offset of , and .
Figure 11: RZF, B feedback bits per user vs. SNR, with B to maintain a sum rate offset of K\log_2b\!=\!10 and {\bm{\Theta}}_k\!=\!\mathbf{I}_M~\forall k, M\!=\!K \!=\!10.
Figure 11: RZF, feedback bits per user vs. SNR, with to maintain a sum rate offset of and , .

Figures Figure 10 and Figure 11 depict the ergodic sum rate of RZF precoding under RVQ and the corresponding number of feedback bits per user , respectively. To avoid an infinitely high regularization parameter , the minimum number of feedback bits is set to one.

In Figure 10, we plot the ergodic sum rate for RZF precoding under perfect CSIT with total power constraint (red solid lines) and unit norm constraint on the precoding vectors (red dashed line). We observe, that the sum rate under unit norm constraint is slightly larger at high SNR, suggesting that our scaling results for RZF precoding derived under a total power constraint become inaccurate under the unit norm constraint at high SNR. Hence, one has to be cautious when comparing the scaling in [45] directly to the scaling derived with the large system approximations at high SNR. From Figure 10, we further observe that (i) the desired sum rate offset of bits/s/Hz is approximately maintained over the given SNR range when is chosen according to and the high SNR approximation in under RZF-CDA and RZF-CDU precoding, respectively, (ii) given an equal number of feedback bits , the RZF-CDA precoder achieves a significantly higher sum rate compared to RZF-CDU for medium and high SNR, e.g., about 2.5 bits/s/Hz at dB and (iii) to maintain a sum rate offset of bits/s/Hz, the proposed feedback scaling of for unit norm precoding vectors [45] is very pessimistic, since the sum rate offset to RZF with total power constraint and unit norm constraint is about bits/s/Hz and bits/s/Hz at dB, respectively.

We conclude that the proposed RZF-CDA precoder significantly increases the sum rate for a given feedback rate or equivalently significantly reduces the amount of feedback given a target rate. Moreover, the scaling of the number of feedback bits under RZF-CDU precoding proposed in [45] appears to be less accurate under a total power constraint than our large system approximation in .

7Optimal Training in Large TDD Multi-user Systems

Consider a time-division duplex (TDD) system where uplink (UL) and downlink (DL) share the same channel at different times. Therefore, the transmitter estimates the channel from known pilot signaling of the receivers. The channel coherence interval , i.e., the amount of channel uses for which the channel is approximately constant, is divided into channel uses for UL training and channel uses for coherent transmission in the DL. Note that in order to coherently decode the information symbols, the users need to know their effective (precoded) channels. This is usually accomplished by a dedicated training phase (using precoded pilots) in the DL prior to the data transmission. As shown in [48], a minimal amount of training (at most one pilot symbol) is sufficient when data and pilots are processed jointly. Therefore, we assume that the users have perfect knowledge of their effective channels and we neglect the overhead associated with the DL training.

In the considered TDD system, the imperfections in the CSIT are caused by (i) channel estimation errors in the UL, (ii) imperfect channel reciprocity due to different hardware in the transmitter and receiver and (iii) the channel coherence interval . In what follows, we assume that the channel is perfectly reciprocal and we study the joint impact of (i) and (iii) for uncorrelated channels ().

7.1Uplink Training Phase

In our setup, the distortion of the CSIT is solely caused by an imperfect channel estimation at the transmitter and is identical for all entries of . To acquire CSIT, each user transmits the same amount of orthogonal pilot symbols over the UL channel to the transmitter. Subsequently, the transmitter estimates all channels simultaneously. At the transmitter, the signal received from user is given by

where we assumed perfect reciprocity of UL and DL channels and is the average available transmit power at the receivers. That is, the UL and DL channel coefficients are equal and the UL noise is assumed identical for all users and statistically equivalent to its DL analog. Subsequently, the transmitter performs an MMSE estimation of each channel coefficient (, ). Due to the orthogonality property of the MMSE estimation [49], the estimates