# Downlink Assisted Uplink Zero Forcing for TDD Multiuser MIMO Systems

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## Abstract

This paper proposes practical coordinated linear transmit-receive processing schemes for the uplink (UL) of multiuser multiple-input multiple-output (MIMO) systems in the time division duplex (TDD) mode. The base station (BS) computes the transmission parameters in a centralized manner and employs downlink (DL) pilot signals to convey the information of the beam selection and beamformers to be used by the terminals. When coexisting with the DL transmit-receive zero forcing, the precoded DL demodulation pilots can be reused for UL beam allocation so that no additional pilot overhead is required. Furthermore, the locally available channel state information (CSI) of the effective MIMO channel is sufficient for the terminals to perform transmit power and rate allocation independently. In order to reduce the UL pilot overhead as well, we propose reusing the precoded UL demodulation pilots in turn for partial CSI sounding. The achievable sum rate of the system is evaluated in time-varying fading channels and with channel estimation. According to the results, the proposed UL transmission strategy provides increased rates compared to single-user MIMO transmission combined with user selection as well as to UL antenna selection transmission, without being sensitive to CSI uncertainty.

## Keywords

Channel State Information Pilot Symbol Antenna Selection Zero Force Linear Minimum Mean Square Error## 1. Introduction

In order to attain all the capacity gains available in multiple-input multiple-output (MIMO) communication systems, channel state information in the transmitter (CSIT) should be utilized. CSIT is available in time division duplex (TDD) systems, provided that the channel does not change significantly between the receive and transmit periods. Due to the channel reciprocity, the receiving node can estimate the state of the channel during one frame, and use that knowledge for the purposes of MIMO transmission in the next one. CSI can be estimated from pilot symbols that are known to the receiver. The pilots are also necessary for performing coherent demodulation in the receiver side. In order to keep the pilot overhead as low as possible, it is desirable that the same pilot symbols are a useful reference for both reception and transmission.

In a cellular multiuser MIMO system, the downlink (DL) comprises a broadcast channel (BC), whereas the uplink (UL) is a multiple access channel (MAC). The channel reciprocity leads into duality properties between the BC and MAC [1, 2]. When designing the user multiplexing strategy for a MIMO system, both directions need to be taken into account together. A distinctive difference between the base station (BS) and the user terminals is that the BS can have the CSI of the channels to all the terminals, while the terminals only have access to the CSI of their individual radio channels. Thus, the BS is capable to centralized processing to attain space division multiple access (SDMA). On the other hand, the terminals can attempt SDMA like transmission only based on the information contained in the signal received in the DL.

TDD is one of the modes included in the cellular 3GPP Long-Term Evolution (LTE) standard, and it is best applicable to urban, local area or office deployments, where the transmit powers, mobile speeds, and the channel propagation delays are relatively low. The TDD mode can well facilitate advanced multiuser MIMO DL transmission methods, if the terminals provide CSI to the BS by transmitting channel sounding pilots in the UL [3]. The motivation of this paper is to study the DL transmission, and to propose a practical matching UL beamforming method for improving the capacity of the cellular system. The underlying assumption is that both the DL and the UL employ orthogonal frequency division multiplexing (OFDM), where the frequency-time resource blocks experience essentially flat fading.

Zero forcing DL transmission by a multiantenna BS provides SDMA in which intracell multiuser interference is nulled. For single-antenna terminals, zero forcing (ZF) is achieved simply by channel inversion in the transmitter [4]. Coordinated transmit-receive processing with block diagonalization (BD) is a zero forcing SDMA scheme that supports also multiantenna user terminals [5]. It decouples the MIMO channels of different users so that precoding based on singular value decomposition (SVD) can be carried out individually for each user. Our preferred transmit-receive solution is obtained when the terminals employ conventional maximal ratio receivers (MRCs) as suggested in [6]. In that case, the ZF solution can be found via an iterative algorithm that was proposed in [7], and further studied in [8]. While corresponding general closed form solutions have not been presented, in [9] it was derived for a two-user case and in [10] the solutions for a three-user setup were studied.

It is beneficial to combine multiuser beamforming with greedy beam selection [11]. In the context of multiuser MIMO DL with coordinated transmit-receive processing, greedy beam selection was studied in [12, 13].

In a time-varying fading radio channel the CSI obtained during the TDD receive frame is already partially outdated when the transmit frame starts. Therefore, the CSI contains a lag error that has a decremental impact on the system performance. The effect of delayed CSI in case of single-user MIMO communications was studied in [14], and in case of DL multiuser MIMO systems in [15]. In addition to the lag error, the effect of noisy CSI estimation on multiuser multiple antenna systems was analyzed in [16].

Based on the principles of DL multiuser transmit-receive zero forcing and beam selection, in this paper, we propose a corresponding communication strategy for the UL. In [17], we presented a similar approach based DL BD by transmit processing only. While in that simple form of BD, the number of antennas in the BS must always be equal to or larger than the aggregate number of antennas in the user terminals [5], the strategy described here can support more general antenna setups and resource allocation methods. We also evaluate by simulations the impact of imperfect CSI estimation as well as lag error on the achievable rates in the system.

While the algorithms for multiuser processing and beam selection are known from literature, the main contribution of our work consists of two novel signaling concepts. The first concept is to convey the UL beamforming parameters to the terminals by means of DL pilot signals. The second concept is to append the UL demodulation pilot signal with additional pilot beams so that the combined signal serves as a full CSI sounding pilot. While the both new techniques can be applied in TDD systems separately, we introduce them as features supporting a combined uplink-downlink strategy with reduced pilot overhead. As a result, the precoded pilot symbols are sufficient in both UL and DL to satisfy the needs of both transmission and reception.

The paper is organized as follows. In Section 2, the generic uplink-downlink multiuser MIMO system model is described. Section 3 summarizes the ideas of coordinated transmit-receive processing and beam selection. Section 4 presents the details of the proposed uplink-downlink beamforming scheme, and in Section 5, numerical capacity analysis results are given. Finally, Section 6 concludes the paper.

## 2. System Model

We consider a MIMO system with one base station having Open image in new window antenna elements, and Open image in new window user terminals with Open image in new window antenna elements each. Furthermore, we assume the users are symbol synchronous, and that each user Open image in new window is allocated with Open image in new window data streams in both UL and DL, where Open image in new window . We denote the set of active, that is, scheduled users as Open image in new window Open image in new window .

where Open image in new window is the UL transmit precoder matrix with unit norm column vectors, and Open image in new window is the diagonal transmit amplitude matrix. Here, Open image in new window denotes matrix transpose, and for complex conjugation and conjugate transposition, notations Open image in new window and Open image in new window are used, respectively. The signal model is free from intersymbol interference; this can be realized, for example, by OFDM.

where the matrices Open image in new window , Open image in new window , and Open image in new window contain, respectively, the left and right singular vectors and singular values in nonascending order, corresponding to the nonzero eigenmodes. Note that we excluded the null space from the decomposition. In physical channels, the number of nonzero singular values is typically Open image in new window .

which is also an upper bound for the achievable data rate.

## 3. Coordinated Transmit-Receive Processing

where Open image in new window , in order to apply SVD precoding so that the DL precoding matrix for user Open image in new window is Open image in new window and the corresponding receiver Open image in new window .

where Open image in new window is the transmit power allocated to the eigenmode Open image in new window of user Open image in new window .

In the coordinated transmit-receive processing, the BS computes all the transmitters and corresponding receivers in a centralized manner, based on the CSI of the selected users. In this section, the processing is described with the assumption that the channel matrices Open image in new window are known. In Section 4 we explain how the UL pilot responses of our proposed strategy can be applied as a reference instead.

### 3.1. Closed-Form ZF Solution

The solution for (7) is not unique, as the receive processors Open image in new window can be selected in multiple ways. One simple choice is to choose the column vectors associated to the strongest singular values from matrix Open image in new window in (3) as suggested in [5]. Let Open image in new window contain the Open image in new window selected left singular vectors and Open image in new window the corresponding right singular vectors. The zero forcing criterion becomes Open image in new window , which can be shown to be equivalent to Open image in new window .

The decomposition (8) lends itself for the purposes of UL transmission as well, as the effective UL MIMO channel is a transposed version of the DL so that Open image in new window . Thus our proposed UL signal processing chain is ideally a reversed version of the DL so that the receivers become transmitters and vice versa, as shown in Figure 1(b). Consequently, the zero forcing criterion in the UL is equivalent to (7), that is, Open image in new window . Since in both directions the eigenmodes of the effective MIMO channels are the same, and as the interference is nulled both ways, for each user the UL and DL are essentially equal. The achievable rates differ only if different transmit powers are applied or if the background noise levels seen by the BS and the terminal are different.

### 3.2. Iterative ZF Solution

The iterative solution for (7) has two desirable properties. Firstly, the performance in terms of achievable rates compared to the closed form solution is improved. Secondly, the optimal receivers in user terminals are filters matched to the received stream responses so that ideally, the terminal side needs not actively estimate and suppress interference.

In the iterative algorithm the processors Open image in new window are initialized by matrix Open image in new window , and then the transmitter Open image in new window and receiver Open image in new window processors for each user are optimized successively until orthogonality between the users is achieved [7, 8]. After convergence, the received DL stream responses dedicated to user Open image in new window are Open image in new window , which implies that the final zero forcing receiver matrix is a set of matched filters.

In our simulations, in the case of Open image in new window , Open image in new window and Open image in new window , the iterative algorithm converged on the average in less than five iterations. Our stopping condition of the algorithm required that the sum of the absolute values of all cross terms Open image in new window must be less than Open image in new window .

### 3.3. Greedy Beam Selection

Greedy beam selection is a process of allocating beams to the users based on their individual channel conditions and spatial compatibility [11]. In the context of the multiuser MIMO system and zero forcing, beam selection has been studied in [12, 13]. The algorithm consecutively selects at most Open image in new window eigenbeams from the total set of Open image in new window to be allocated. Number Open image in new window indicates the number of degrees of freedom available in the system.

where matrix Open image in new window contains as columns all the right singular vectors Open image in new window corresponding to the previously selected eigenbeams. Note that the Open image in new window eigenbeams selected for user Open image in new window are not necessarily the strongest, since weaker beams may be preferred due to their better spatial compatibility properties.

The selection process stops if the calculated capacity of the system is reduced compared to the previously selected beam set. Thus, there may be fewer active streams in the system than there are degrees of freedom. In this paper, the stopping condition is always calculated based on the closed-form zero forcing solution in order to avoid multiple zero forcing iteration rounds.

The role of the beam selection is to make the problem of zero forcing relatively easy, by ensuring that the selected eigenbeams are nearly orthogonal so that the zero forcing loss remains acceptable. The stopping condition of the selection has a similar effect, as the algorithm rather stops than chooses more linearly dependent eigenbeams.

A straightforward simplification to the multiple access protocol can be introduced by restricting the maximum number of beams per user to be one, that is, Open image in new window . Especially when the number of users is high, the effect of the restriction on the system throughput is minor. However, by allowing multiple data streams per user, higher user peak data rates can be provided.

## 4. Uplink-Downlink Beamforming Strategy

The main contribution of this paper consists of two novel concepts. The first concept is to convey the uplink (UL) beamforming parameters to the terminals by means of downlink (DL) pilot signals. The second one is to append the UL demodulation pilot signal with additional pilot beams so that the combined signal serves as a CSI sounding pilot. While the both new techniques can be applied in TDD systems separately, we introduce them as features supporting a combined uplink-downlink strategy with reduced pilot overhead.

Most of the intelligence as well as the computational complexity of the proposed strategy lie in the base station (BS) that carries out the multiuser processing, including beam selection and precoding. On the other hand, the terminals essentially perform single-user MIMO processing in conjunction with interference suppression.

### 4.1. Signaling for Uplink Beamforming

The resource allocation and pilot signaling in TDD mode are in general open research problems and standardization issues. Due to the TDD channel reciprocity, the need for CSI quantization can be avoided unlike in the FDD mode. Thus, in principle, TDD can support more advanced spatial signal processing methods than FDD. However, reasonable pilot signal overhead is still required, and due to estimation errors CSI is not perfect. In order to facilitate fast advanced centralized processing in the BS, antenna-specific UL CSI sounding pilots are needed [3]. These pilots enable any form of multiuser MIMO precoding in the DL.

UL MU beamforming approaches.

Method | UL signaling | DL signaling | Power and rate control |
---|---|---|---|

Unquantized precoding | CSI sounding pilot | Beam allocation pilots | May be locally decided by terminal |

Quantized precoding | CSI sounding pilot | Precoder indexes and rate parameters | Signalled by BS |

One more obvious method to facilitate UL precoding is to employ a DL common pilot so that each terminal can form beams based on the knowledge of its individual MIMO channel. However, this mode does not easily allow centralized multiuser control, and the resulting UL beams may end up undecodable if they are not spatially compatible.

### 4.2. Combined Uplink-Downlink Signaling

When applying multiuser MIMO precoding in the DL, the DL demodulation pilots may be reused as beam allocation pilots as shown in Figure 3(b). In this approach, the same spatial beams are active in both directions, and the need for specific DL signaling of the desired UL precoders is removed. On the other hand, the UL demodulation pilots can be reused for partial CSI sounding. By adding parallel pilot beams, full CSI sounding can be achieved, as described in the following subsection. As a result, the amount of required specific CSI sounding pilot overhead is reduced.

For example, in our simulation setup with Open image in new window , Open image in new window , and Open image in new window , coupling of the UL and DL beamforming halves the required DL pilot overhead. At the same time, the UL pilot overhead is reduced approximately by one third.

Obviously, the combined strategy sets constraints to the overall resource allocation of the system, as the same frequency resource blocks are assumed to be allocated to the same users in both UL and DL. Therefore, the concept is at its most efficient when the offered data traffic loads in both directions are approximately equal. In the system level, the possible asymmetry of the traffic can be treated in time domain, for example, by allocating more time frames to the DL than UL. Furthermore, the concept of reusing the demodulation pilot signals for CSI sounding and beam allocation can be utilized whenever the receive frame is close enough to the corresponding transmit frame. In other times, separate sounding pilots need to be employed.

### 4.3. Pilot Responses

Pilot symbols transmitted with beamforming via the same precoders as data are necessary in order to facilitate coherent demodulation. However, unlike data, we propose that the pilots have equal power allocation per stream. This way the channel gains can be correctly observed from the received signal without getting mixed with the amplitude adjustment caused by power allocation, and the pilot responses can be utilized for the purpose of transmit precoding as well.

where Open image in new window is the data precoder matrix, and Open image in new window contains the precoders for the additional pilot streams. On the other hand, in the DL it suffices to transmit just as many pilot streams as there are data streams.

for DL and UL, respectively. In the DL, Open image in new window denotes the response seen by user Open image in new window of the signal transmitted to user Open image in new window .

The number of required pilot streams in UL is Open image in new window and increases with the number of simultaneous users, whereas for DL Open image in new window pilot streams always suffice. Thus, the UL limits the practical number of users to be included in the same spatial processing group.

### 4.4. Base Station Processing

Section 3 described how the coordinated transmit-receive processing and beam selection are carried out by the BS, based on the knowledge of the MIMO channels Open image in new window . However, the same computations can be realized by replacing the channel matrices with the UL pilot responses Open image in new window as well, since the right singular vectors (3), forming the transmit signal space, and the corresponding singular values are invariant to the multiplication by the unitary pilot precoder matrix. As a result, the BS obtains the same set of transmit precoders and powers as when applying the channel matrices directly. On the other hand, the set of receiver processors the algorithm assumes will be different.

Furthermore, let Open image in new window be the receiver processor the user terminal Open image in new window applies in order to reject multiuser interference. This processor must satisfy Open image in new window . By comparing to (13) we can see that Open image in new window is the valid orthonormal zero forcing processor at the terminal.

respectively. Here, the user-specific receivers are stacked in the large result matrix as Open image in new window . Note that for our proposed UL precoding, the ZF receiver is ideally equivalent to the corresponding DL precoder Open image in new window . In practice, however, due to estimation errors, channel time-variations and other nonidealities, the receiver must always rely on the received stream responses.

### 4.5. Terminal Processing

where Open image in new window . For the iterative zero forcing transmit-receive processing, in an ideal case, both the ZF/LN and the LMMSE receiver are equivalent to the matched filter (MF) Open image in new window .

while applying the individual power constraint Open image in new window .

However, if the DL precoding was not ideal, or the terminal receiver is formulated based on estimated channel, the receive beamformers of user Open image in new window do not necessarily remain orthogonal to each other. A conceptually straightforward way to orthonormalize the receive beamformers, and to simultaneously obtain the additional Open image in new window UL pilot precoders, is to perform full SVD as Open image in new window , and to set Open image in new window , where the first Open image in new window columns correspond to the data streams. This method was used in the simulations of this paper.

It is worth noting that even when the terminal employs the LMMSE receiver, in the closed-form transmission mode, the transmit precoders are still calculated based on the ZF/LN receivers. In the iterative zero forcing mode, when operating with estimated CSI, it turned out that the MF receiver is the best reference for UL precoding, even though as a receiver ZF/LN performs better.

### 4.6. CSI Uncertainty

The treatment in the previous sections considered error-free CSI. In practice the beam selection, transmit precoding, and receiving have to be carried out based on noisy channel responses experienced during the latest received frame prior to transmission. In a time-varying channel this results in a lag error in transmit CSI. As a result, the orthogonality between users and streams in DL is partially lost. Also in the UL, the channel reciprocity is reduced. In the receiver side, the pilot reference is timely and correct so that both the desired signal and interference responses can be estimated and utilized without lag error.

We assume that the pilot symbol sequences associated with different streams and users are all mutually orthogonal, which accommodates interference free channel or pilot response estimation. For zero forcing transmit and receive processing, the estimation of the pilot responses Open image in new window and Open image in new window is adequate. On the other hand, in order to construct LMMSE receivers, the spatial signal covariance or the transmit amplitudes Open image in new window and Open image in new window need to be known or estimated. For our simulations, the estimation of signal covariance is carried out as described in [17].

where Open image in new window is the frame index, and Open image in new window and Open image in new window denote the precoding algorithms running in the BS and in the terminals, respectively. Let Open image in new window be the channel lag error so that Open image in new window . By denoting estimation noise Open image in new window , the estimates in BS become

which indicates that the error sources seen in both UL and DL accumulate to affect the UL transmission.

## 5. Numerical Results

Different multiuser MIMO scenarios were simulated in frequency flat fading with Jakes' Doppler spectrum and uncorrelated channels between antennas. We denote the Doppler spread Open image in new window where Open image in new window is the maximum Doppler shift. The equal length UL and DL TDD frames of duration Open image in new window follow each other consecutively as illustrated in Figure 3(b). Each simulation comprises 20 000 randomly generated, independent channel process bursts of several frames. The channel coefficients remain constant over each frame. System signal-to-noise-ratio SNR was set to 10 dB, and it is defined as Open image in new window All the methods compared employ the same sum transmit power.

where Open image in new window is the total transmitted power in the cell.

One of the simulated benchmark methods is the UL antenna selection transmission, where the BS chooses a subset of terminal antennas that simultaneously transmit one independent unprecoded data stream each. Here, the greedy selection algorithm (10) is applied so that the channel singular vectors are replaced by channel vectors, that is, by rows from matrices Open image in new window . Thus, centralized multiuser control is exercised in order to ensure the spatial compatibility of the concurrent transmissions. Equal transmit power per antenna is allocated, and multiple data streams per user are allowed. While antenna selection is simpler compared to the UL beamforming, it offers no reduction to the required pilot overhead, since the UL CSI sounding pilots are still needed for reference.

Another comparison scheme is the single-user MIMO transmission, "best-user SVD'', where the user with the strongest MIMO channel is always chosen for single-user MIMO transmission by SVD precoding. In that frame, the transmit power of the cell is allocated to one user.

## 6. Conclusion

We have presented practical linear coordinated transmit-receive zero forcing schemes for the uplink of cellular multiuser MIMO systems in the TDD mode. Beam selection is an integral part of the strategy, as it helps to avoid excessive zero forcing loss while achieving gain from multiuser diversity. The BS computes the transmission parameters in a centralized manner and employs DL pilot signals to convey the information of the beam selection and beamformers to be used by the terminals. When coexisting with the DL transmit-receive zero forcing, the precoded DL demodulation pilots can be reused for UL beam allocation so that no additional pilot overhead is required. In order to reduce the UL pilot overhead as well, we proposed reusing the precoded UL demodulation pilots in turn for partial CSI sounding. As a result, only the precoded pilot symbols are needed in both UL and DL to satisfy the needs of both transmission and reception. The system is readily scalable, since any combination of base station and terminal antenna array setups can be supported.

In zero forcing, the multiuser MIMO channel is decoupled into noninterfering parallel channels by linear processing. Thus, the strategy lends itself to straightforward power and rate allocation as well as coding and modulation. Furthermore, the system works well with suboptimal linear receivers that can be easily constructed based on simple CSI estimation tasks. The use of more complex nonlinear successive interference cancellers or turbo receivers is not necessary, which further increases the robustness of the system, as the possible error propagation between the users' signals is avoided.

We evaluated the performance of the strategy in time-varying fading channels and with CSI estimation. The largest gains from multiuser MIMO communication are obtained when the fading is slow, and when the quality of CSIT at the BS is good. It is worth noting that UL beamforming is not sensitive to the quality of CSIT at the terminals, and even the simple antenna selection transmission performs adequately in multiuser environments. Obviously, the benefit of beamforming grows with the number of terminal antenna elements.

From the results we conclude that multistream precoding also in the UL is in practice feasible, robust and beneficial from the system capacity point of view. Due to its practical nature, the proposed concept is a promising candidate for the evolution steps of future cellular systems such as 3GPP LTE.

The uplink-downlink beamforming concept is at its most efficient when the offered data traffic loads in both directions are approximately equal. The possible asymmetry of the traffic can be treated in time domain, for example, by allocating longer time frames to the DL than UL. In the extreme case, UL beamforming can be decoupled from the DL data transmission completely. In this case, the BS would merely arrange the UL multiuser transmission by communicating the beam selection to the terminals via DL pilots.

## Notes

### Acknowledgments

This work has been supported by the Finnish Funding Agency for Technology and Innovation (Tekes), Nokia, Nokia Siemens Networks, Elektrobit and Tauno Tönning Foundation. This work has been performed in part in the framework of the CELTIC Project CP5-026 WINNER+. The authors would like to acknowledge the contributions of their colleagues.

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