- Research Article
- Open Access
Efficient Transmission Schemes for Multiuser MIMO Downlink with Linear Receivers and Partial Channel State Information
© M. Eslami and W. A. Krzymień. 2010
- Received: 18 August 2009
- Accepted: 10 May 2010
- Published: 7 June 2010
Downlink of a multiuser MIMO system is considered, in which the base station (BS) and the user terminals are both equipped with multiple antennas. Efficient transmission schemes based on zero-forcing (ZF) linear receiver processing, eigenmode transmission, and partial channel state information (CSI) at the BS transmitter are proposed. The proposed schemes utilize a handshaking procedure between the BS and the users to select (schedule) a subset of users and determine the precoding matrix at the BS. The advantage of the proposed limited feedback schemes lies in enabling relatively low-complexity user scheduling algorithms and high sum-rate throughput, even for a small pool of users. For large user pools and when the number of antennas at each user terminal is at least equal to the number of antennas at the BS, we show that the proposed scheme is asymptotically optimum.
- Channel State Information
- Channel Quality Indicator
- User Terminal
- Precoding Matrix
- Time Division Multiplex
Increasing demand for broadband wireless services calls for much higher throughputs in future wireless communication systems. It has been shown that with the use of multiple antennas at the transmitter (Tx) and the receiver (Rx), the capacity of a point-to-point communication link increases linearly with min where is the number of Tx antennas and is the number of Rx antennas [1, 2]. Recently, there has been a great interest in multiuser multiple-input multiple-output (MU-MIMO) systems and transmission strategies that would enable similar capacity gains in multiuser environment [3–5]. In a multiuser downlink with the base station (BS) equipped with multiple antennas, multiple users can be served simultaneously. In fact, it has been shown that to obtain the MU-MIMO downlink sum capacity, transmitting to several users simultaneously must be considered . Since the number of users in the system is usually greater than the maximum number that can be served simultaneously through spatial multiplexing, user selection is required. User selection (or scheduling) favours users, which experience better propagation condition while being sufficiently separated in space. Such user scheduling leads to multiuser diversity gain [7, 8], which increases with increasing number of users awaiting transmission.
It has been shown that the capacity of the MU-MIMO downlink can be achieved by dirty paper coding (DPC) , which is a transmitter multiuser encoding strategy based on interference presubtraction. DPC requires nonlinear search for optimal precoding matrices as well as noncausal channel coding for these users, which is practically impossible in real-time systems. Therefore, suboptimum transmission strategies such as different forms of beamforming have been considered in the literature. In MU-MIMO beamforming, linear or nonlinear transmitter precoding algorithms together with user scheduling are designed to maximize the system's sum rate or some other related objective function (e.g., sum rate under fairness constraint). Unfortunately, most beamforming algorithms considered assume availability of perfect channel state information at the transmitter, which presents a big challenge to their practical implementation (references [9, 10] and references therein contain an overview of the subject).
Transmission schemes based on availability of quantized channel state information at the BS: the quantized CSI is used to utilize a variant of beamforming at the BS. See  and references therein for further details.
User scheduling and precoder selection from a codebook of vectors/matrices known a priori to both the BS and the users based on partial CSI: the scheme proposed in  called transmit beam matching (TBM) is one example, which extends the per-user unitary rate control (PU RC) [12, 24] approach to multiple antenna users. PU RC is Samsung Electronic's proposal to the 3rd Generation Partnership Project (3GPP). The proposed approach is characterized by the relatively low complexity structure of PU RC, and it uses channel matrix pseudoinverse operation in order to minimize interstream interference at each user's terminal. However, when users have fewer antennas than the base station, the pseudoinverse operation can not completely eliminate interstream interference, which leads to some performance degradation. A similar approach called random precoding has been introduced in .
Eigenmode transmission with limited feedback: One example is , which employs singular value decomposition (SVD) of user channel matrices and data transmission on the eigenmode with the largest gain. Another example is , in which the authors propose a combination of zero-forcing beamforming (ZF-BF) with eigenmode transmission.
All schemes mentioned above use precoding at the BS. In addition to precoding at the BS, multiple antenna users can use their antennas to process their received signal vector using relatively low-complexity linear schemes such as zero-forcing (ZF) and minimum mean squared error (MMSE) processing and send back some sort of channel quality indicator (CQI), for example, SINR or rate, to the BS. One example is , in which a MIMO downlink scheme with opportunistic feedback is proposed. In this scheme users use ZF linear processors and send back the quality indicator for each spatial channel to the base station according to an opportunistic feedback protocol. The main contribution of  lies in its feedback protocol and not the transmission scheme itself.
In this paper, we present a transmission scheme for MU-MIMO downlink using eigenmode transmission, and ZF linear processing, which only requires partial CSI and falls under the third category mentioned above. We assume that all users have the same number of Rx antennas. With this assumption and the number of Rx antennas of each user terminal being less or greater than the number of transmit antennas, two transmission strategies are proposed. For systems where the number of Rx antennas is greater than or equal to the number of Tx antennas, one user is selected to receive data through eigenmode transmission and its right eigenvector matrix is used for precoding, while other selected users use ZF linear processing. When the number of Rx antennas of each user terminal is less than the number of Tx antennas at the base station, partial CSI at the base station is used to design a precoding matrix such that the number of interfering streams at the selected user terminals (after Rx preprocessing) is reduced to the number of Rx antennas, and ZF receiver processing can be efficiently applied. Analytical expressions and approximations are derived for the sum rate of the proposed scheme and also for time division multiplexing (TDM) with eigenmode transmission.
For the case of ( denotes the number of Rx antennas at each user terminal; denotes the number of Tx antennas at the BS), our work is distinct from  in the following aspects. ( ) In our proposed scheme the users do not need to send back their channel singular vectors as required in the scheme of ; only one user is asked to send back its right singular vector matrix. ( ) The scheme presented here results in zero interuser and interstream interferences, whereas the scheme of  does not. ( ) In our scheme user selection criterion is straightforward and there is no need for a greedy search algorithm to select users as required by the scheme introduced in . Compared to , what distinguishes our work is the use of ZF receiver processing and the lower complexity of our user scheduling and eigenmode assignment to selected users compared to the high complexity of exhaustive search to find the threshold value (denoted by in ). Parts of this work have been presented in [26, 27]. Nevertheless, this paper generalizes our proposed scheme to any number (greater than one) of Tx (at the BS) and Rx (at each user terminal) antennas and provides further analysis on the proposed scheme's sum rate.
The paper is organized as follows. In Section 2, the system model for multiuser MIMO downlink is described. Two well-known transmission schemes based on limited feedback are briefly outlined in Sections 3 and 4. Section 5 describes the proposed transmission techniques along with asymptotic analysis for the case of . Numerical results are provided in Section 6, and Section 7 concludes the paper.
Throughout this paper, upper case and lower case bold characters denote matrices and vectors, respectively. denotes the conjugate transpose of the matrix argument. is the expectation operation. denotes the trace of the matrix argument.
where is the th eigenvalue of while is the th singular value of . denotes the power given to the th data stream and . The optimum power distribution over the spatial channels is obtained through water-filling . For the case of equal power allocation we have . This transmission scheme has been considered within the context of time-division multiplexing (TDM) where the users send back their achievable rate, , to the base station and the base station selects the user with the largest achievable transmission rate in each time slot. Compared to multiuser MIMO schemes in which multiple users are served simultaneously, this scheme is very suboptimal as it does not take full advantage of multiuser diversity, which implies that some of the eigenmodes of the selected user's channel matrix might be very weak.
In case of , with spatial multiplexing at the base station when an independent data stream is transmitted from each Tx antenna and ZF receiver processing is used at each user terminal, the scheduled users can detect their data without interstream interference.
where is the sum rate of the DPC scheme, for a small pool of users it achieves a relatively poor sum rate.
In the next subsections our proposed transmission scheme is presented for two scenarios. In the first scenario, each user terminal has the number of antennas at least equal to that of the base station ( ), and in the second scenario the base station has more antennas than each user terminal ( ).
5.1. Case : Precoding with Right Singular Vector Matrix
- (1)All the users perform SVD of their own channel and report back a single rate value evaluated according to
where . The parameter is evaluated beforehand based on the system parameters and will be discussed in the next subsection. s are the ordered eigenvalues of the matrix which is a complex Wishart matrix . is the largest eigenvalue.
The base station scheduler selects the user with the largest (user ) and asks that user to send its matrix to the BS. The matrix is obtained through the SVD of the selected users' channel matrix. The matrix is then used as the precoding matrix, . User will receive its data through the first data streams (encompassing data symbols ), using as its receiver processing matrix (eigenmode transmission).
User ( ) will estimate its equivalent channel, which at this stage is . Then all users (except user ) will apply ZF linear processing using the estimated equivalent channel and send back the postprocessing SNR of data streams to to the base station.
For each of the remaining data streams, the base station selects the user with the highest postprocessing SNR.
5.1.1. Finding the Optimum Number of Eigenmodes ( )
where is the product of Gamma functions.
For , a closed form analytical expression for the average throughput contribution from eigenmode transmission, , is very complicated to evaluate. However, a close approximation for can be obtained using the following proposition.
Proposition 1 :.
and is the achievable rate on the th eigenmode.
See the appendix.
In summary, to find the optimum , one has to find the smallest eigenvalue, , for which . Then the optimum value for is . To obtain , the marginal pdf, CDF, and joint pdfs of are required, which can be obtained using (14). is then approximated using (15). Based on (12) and (15), the optimum value depends on , , and . For a system with specific number of Tx and Rx antennas, can be evaluated for different values of and beforehand and stored in a lookup table to be used later.
5.1.2. Scaling Law of Sum Rate of the Proposed Scheme
In this subsection, the asymptotic behaviour of the average sum rate of the proposed scheme described in 5.1 is investigated for systems with a large number of users. First we start with the following lemma,
where is the natural logarithm.
and that completes the proof.
As the sum capacity (achievable with DPC) for data streams asymptotically increases with , , in general is not asymptotically optimum. However, for the case of we present the following theorem.
and since DPC has the optimum scaling sum rate, the ratio in the above equation can not be greater than one.
The above lemma and theorem make one expect that as the number of users increases, the optimum value will decrease to one, which is confirmed by simulations in Section 6.
5.2. Case : Null Space Precoding with Singular Vector Selection
In this section, the general algorithm proposed for this case is presented, before a novel scheme for the specific case of Tx and Rx antennas is discussed.
where is a size vector and is obtained by eliminating terms from . Then user uses as its receiver processing matrix to detect out of the total transmitted data streams.
For the th receiver to be able to detect its data using ZF receiver processing, the number of interfering data streams (after Rx pre-processing) must not be greater than . In other words, the matrix must have zero columns. This further implies that the precoding matrix needs to contain basis vectors of the null space (space spanned by the rightmost vectors of ) of each selected user's channel matrix. Therefore, users can be served simultaneously ( denotes floor of its argument). Therefore, to be able to take greater advantage of multiuser diversity, should be as close as possible to with the best case being . When this scheme becomes identical to TDM.
Since the postprocessing SNR of each data stream in this case depends on the precoding matrix and each selected user's and matrices, finding users with channel conditions that maximize the sum rate based on partial CSI turns out to be not straightforward. Nevertheless, a heuristic approach would be to adopt a two-stage user selection, where in the first stage a set of users is selected based on a channel quality indicator (CQI), for example, the largest singular value. In the next stage, the selected users send back their full CSI to the BS, and the BS broadcasts their CSI to all users. Then, knowing the CSI of the selected users, each user (outside of the set of selected users) substitutes itself sequentially for each of the selected users and evaluates the resulting sum rate for each substitution. If a user finds that by substituting itself for one of the selected users, the sum rate increases, it will inform the BS of it. The BS will update the user set according to the suggestion of the user which has reported the maximum increase in the sum rate. Our results show that the sum rate obtained by adopting this scheme and user selection based on the largest eigenvalue achieve a higher sum rate compared to TDM, while the gap between the sum rate of this scheme and the optimum DPC increases as the number of antennas increases. In the following subsection we present an efficient transmission scheme for the special case of and .
The Case of and
Each user performs the SVD of its channel matrix and sends back to the base station.
- (2)The base station selects the user with the largest , user , and asks that user for matrix. To detect its data, user uses as its receiver processing matrix,where , , , and . As seen in (31), the interference caused by the first data stream to the second and third data streams after Rx processing at user has been canceled. Therefore, a ZF linear receiver can be used and for the second data stream we have 
where . Thus, the achievable rate for this user will be .
The base station broadcasts and to all users.
- (4)For now, let us assume that user is the second selected user. Then the precoding matrix will be
where , , and . It is evident that the interfering effect of on the other data streams is canceled, and the first data stream can be detected using a matched filter, which results in as postprocessing SNR for the first data stream ( ).
To detect the third data stream, the effect of the first detected data stream is subtracted out, that is, ( denotes the first detected data symbol). Canceling the effect of the the first data stream is possible due to the knowledge of at user which enables it to evaluate . The SNR for the third data stream, , after interference cancelation and matched filtering, is obtained as (ignoring error propagation).
Considering (32) and the third step of the algorithm, user has all the required information to evaluate the rate of user as well as its own rate. Therefore, it will send back a sum rate value, , that is achieved by scheduling data transmission to itself and user .
The base station selects the second user, user , which has the largest and asks that user to send back and vectors.
At this stage data transmission to the selected users begins. User will receive its data from the first and third Tx antennas, and user will receive its data from the second Tx antenna.
In this section, the expected throughputs of the proposed schemes are compared to limited feedback MIMO-downlink techniques using transmit beam matching (TBM) , which is a modified version of PU RC for multiple antenna users, zero-forcing beamforming (ZF-BF) using channel vector quantization (CVQ) [18, 37, 38], spatial multiplexing with zero-forcing receiver processing, and TDM with eigenmode transmission for different numbers of antennas, users, and SNR values. The throughput of the DPC scheme is also given as an upper bound on the sum rate. The sum rate curves for DPC have been obtained using the iterative water-filling algorithm introduced in . In the following, we consider two case examples, in which , and one example for the case .
The Case of and
Selecting user which has the largest (8) and only serving that user in each time slot (TDM with eigenmode transmission).
where and are obtained using (16).
ZF receiver Rx processing with partial CSI.
The case of and
The proposed scheme with .
with the pdf and CDF of given as follows
The proposed scheme with
where and are obtained using (16), and marginal and joint eigenvalue distributions are given by
Selecting user which achieves the largest rate and only serving that user in each time slot (TDM with eigenvalue distribution). The average sum rate in this case is approximated by (15) with and using  where more simplified expressions (for case ) have been given for and .
ZF receiver processing scheme using partial side information.
Optimum values and the percentage increase of the proposed scheme's sum rate over ZF and TDM schemes for different numbers of antennas.
Percentage increase over ZF ( )
Percentage increase over TDM ( )
The Case of and
6.1. Comparison of Feedback Requirement for Different Schemes
In limited feedback schemes, there is usually a tradeoff between the sum rate and feedback load. An example of this tradeoff is seen in the PU RC scheme where there are two feedback modes. In one mode which achieves higher average sum rate, the SINRs of all codewords are sent back to the base station, and in the other mode only the largest SINR and the index of its corresponding codeword are sent back to the base station. In ZF-BF with CVQ each user sends back the index of a selected quantization vector along with its corresponding SINR lower bound [18, 37]. In the transmission scheme based on spatial multiplexing at the base station with linear receiver processing at each user terminal, each user sends back SNR values to the base station. In TDM with eigenmode transmission, each user sends back only one real value (a rate value), before the user with the highest reported rate is asked to send back its right singular matrix, which for a system with has real terms.
In our proposed scheme and for the case of , users send back information in three stages. At the first stage all users send back a single rate value, in the second stage one user sends back an matrix of complex values, and in the third stage all users except one send back SNR values. This amount of feedback is larger than the amount required in TDM with eigenmode transmission, yet it is comparable to PU RC and spatial multiplexing at the base station with ZF receiver processing schemes at user terminals described in Section 4.
For the proposed scheme in case of and , each user needs to feedback only one real value to the base station in the first stage. In the second stage, one user needs to send back a matrix, and in the third stage all users except one need to send back one rate value. Finally, the second selected user sends back two vectors to the base station. This amount of feedback is larger than the amount required in TDM with eigenmode transmission. Yet, it is less than ZF-BF with CVQ , since except for the two users, all other users send back only two real values in two stages.
We have proposed limited-feedback MIMO downlink transmission schemes for a system in which the base station and each user terminal are equipped with and antennas, respectively. For the case of , one user receives data through eigenmode transmission on its strongest eigenmodes ( is a predetermined value, which maximizes the average sum rate) while each of the remaining data streams is assigned to a user with the highest ZF receiver postprocessing SNR. We have shown that in this case the average sum rate of the proposed scheme scales with ( is the number of users in the system), which is asymptotically optimal. In case of , the precoding matrix consists of right singular vectors of at least two and at most users such that the number of interfering streams at each selected user terminal is reduced to the number of its receive antennas, and hence, the interstream interference can be effectively removed using ZF receiver processing. The results show that the proposed schemes lead to a higher average sum rate compared to a number of well-known limited feedback schemes, especially for a small pool of users.
Funding for this work has been provided by TRLabs, Huawei Technologies, the Rohit Sharma Professorship, and the Natural Sciences and Engineering Research Council (NSERC) of Canada.
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