- Research Article
- Open Access

# A Comparison of Scheduling Strategies for MIMO Broadcast Channel with Limited Feedback on OFDM Systems

- Ermanna Conte
^{1}Email author, - Stefano Tomasin
^{1}and - Nevio Benvenuto
^{1}

**2010**:968703

https://doi.org/10.1155/2010/968703

© Ermanna Conte et al. 2010

**Received:**14 October 2009**Accepted:**22 March 2010**Published:**4 May 2010

## Abstract

We consider a multiuser downlink transmission from a base station with multiple antennas (MIMO) to mobile terminals (users) with a single antenna, using orthogonal frequency division multiplexing (OFDM). Channel conditions are reported by a feedback from users with limited rate, and the base station schedules transmissions and beamforms signals to users. We show that an important set of schedulers using a general utility function can be reduced to a scheduler maximizing the weighted sum rate of the system. For this case we then focus on scheduling methods with many users and OFDM subcarriers. Various scheduling strategies are compared in terms of achieved throughput and computational complexity and a good tradeoff is identified in greedy and semiorthogonal user selection algorithms. In the greedy selection algorithm, users are selected one by one as long as the throughput increases, while in the semiorthogonal approach users are selected based on the channel correlation. An extension of these approaches from a flat-fading channel to OFDM is considered and simplifications that may be useful for a large number of subcarriers are presented. Results are reported for a typical cellular transmission of the long-term evolution (LTE) of 3GPP.

## Keywords

- Orthogonal Frequency Division Multiplex
- Channel State Information
- Orthogonal Frequency Division Multiplex System
- Orthogonal Frequency Division Multiplex Symbol
- Resource Block

## 1. Introduction

Next generation wireless cellular systems are expected to support high-quality multimedia services; this motivates the interest in multiantenna (MIMO) systems, where both spatial diversity and multiplexing can be used to increase the achievable throughput. In fact, it has been shown that the downlink capacity of a MIMO system with perfect channel state information (CSI) scales as a linear function of the number of transmit antennas [1]. Although nonlinear dirty paper coding scheme achieves the system capacity, it has a high computational cost [2], and simpler solutions have been investigated. Linear beamforming has been shown [3] to achieve a large part of dirty paper coding capacity; in particular, zero forcing beamforming matched to an opportunistic scheduling is widely used [3].

However, benefits of MIMO are obtained only by a proper scheduling of transmissions, which opportunistically exploits channel conditions in order to increase throughput, while ensuring quality of service (QoS). Several scheduling techniques have been proposed for MIMO single carrier systems on flat fading channels based on various approaches, including clique search [4], maximization of the Frobenius norm of the composite channel matrix [5, 6], user channel orthogonality [7–9], single bit feedback [10], waterfilling [11], tree search [12], evolutionary algorithms [13], and greedy scheduling [14] extended to the case of limited feedback in [15]. In some cases, joint optimization of scheduling and power allocation is performed [4–6, 10, 11, 13], while in other cases only scheduling is considered [7–9, 14]. Moreover, QoS-oriented multiuser scheduling and beamforming have been investigated in [16], in order to conciliate the request of high throughput with low packet delays. An overview of research on cross-layer scheduling for multiuser MIMO single-carrier systems is given in [17]. A similar problem to multiuser MIMO scheduling can be found in other transmission systems, such as multicarrier code- or frequency-division multiple access [18].

In frequency selective channels, single carrier modulation is often replaced by orthogonal frequency division multiplexing (OFDM) due to its efficiency in overcoming multipath fading. In fact, the combination of MIMO and OFDM seems to be the technology of future wireless cellular systems, as it has been proposed for downlink in the long term evolution (LTE) release of 3GPP standard [19, 20]. When MIMO OFDM is considered, scheduling becomes more complex, as the number of resources to be allocated, that is, the number of subcarriers, increases and only suboptimal approaches are viable [12]. Complexity is further increased in a frequency division duplexing system, where CSI is provided to the base station by each user (mobile terminal) through a feedback channel. In fact, due to the limited feedback rate, only a partial CSI is available at the base station and additional processing is required to compensate the channel uncertainty. Some of the scheduling techniques considered for single carrier transmissions can be extended to OFDM. For example, in [21] a scheduling algorithm has been proposed for MIMO OFDM systems which extends method [14] for single-carrier systems: the set of scheduled users on each subcarrier is built in a greedy fashion, by adding one user at a time with the aim of maximizing a weighted sum rate (WSR). In [22] this approach has been further simplified to avoid the need of computing a new beamforming matrix upon the insertion of a new candidate in the set of scheduled users. A further simplification of the scheduling is achieved by computing an estimate on their signal to interference ratio which is then used to exclude users that would not contribute to the WSR, by introducing a threshold to their signal to noise plus interference ratio.

In this paper, we first show that any scheduler maximizing a wide class of utility functions can be reduced to a scheduler maximizing the weighted sum rate, where the weights are suitably chosen according to the utility function. Then we revise scheduling techniques proposed in the literature that maximize the weighted sum rate for a multiuser MIMO OFDM system with limited feedback and compare them in a LTE 3GPP scenario in terms of (i) computational complexity, (ii) memory requirements, and (iii) achievable throughput.

The rest of the paper is organized as follows. In Section 2 we describe the downlink MIMO OFDM system model. In Section 3 a general scheduling method is derived and algorithms [14, 21] are revised. Sections 4 and 5 present, respectively, the user selection and user preselection strategies of [22]. In Section 6 the complexity of the various strategies is investigated. In Section 7 simulation results are illustrated and Section 8 outlines main conclusions.

Notation 1.

Bold upper and lower letters denote matrices and vectors, respectively; denotes Hermitian operation (transpose complex conjugate), while denotes transpose; is the vector norm, and stands for expectation.

## 2. System Model

*stream*the (user, resource block) pair . Let also be the set of streams scheduled at slot , that is,

where the expectation is taken only with respect to , and is the th entry of .

### 2.1. Feedback Information

*reconstructed*channel vectors . Using the partial CSI, base station evaluates an estimate of the SNIR of stream as will be seen in Section 4. Zero-forcing beamforming with equal power distribution among streams is implemented for each resource block, hence the beamforming matrix is

with suitable weights that take into account fairness and QoS constraints.

### 2.2. Exhaustive Search Scheduling

At each slot, we aim at scheduling the set of streams that maximizes WSR.

This problem can be solved by considering all
possible sets and evaluating the WSR achieved by each candidate set. Unfortunately, this *exhaustive search* (ES) scheduling has a high computational cost, which becomes infeasible for an increasing number of users and subcarriers. Simpler and suboptimal scheduling methods are investigated in Section 4.

## 3. Maximum Utility Scheduler

In order to balance the opportunistic use of channel resources with fairness among users, we consider a multiuser scheduler. We first consider in this section general criteria for the choice of weights of the WSR and we derive the optimum maximum utility scheduler weights for a general utility function. Then we specialize the result for the maximum sum rate scheduler and the proportional fair scheduler.

### 3.1. General Multiuser Scheduling

where
is a fairness parameter to be chosen according to the desired scheduling policy. For example, for
we obtain the *proportional fair* scheduler (PFS). For
we obtain the utility function of the *maximum sum rate* scheduler. When
, (13) becomes the utility function of the *max-min* scheduler.

where is the set of all possible streams. Note that for , (16) boils down to the maximum utility scheduler of [25].

### 3.2. Maximum Sum Rate Scheduling

### 3.3. Proportional Fair Scheduling

The multiuser multicarrier *proportional fair* scheduling (MMPFS) algorithm [26] is an extension to the OFDM multiuser scenario of the PFS algorithm.

and MPFS (18) coincides with the maximization of the WSR (15) with weights (16), and .

## 4. Greedy Scheduling Strategies

Now that we have established that maximizing the weighted sum rate is equivalent to the maximization of a wide set of utility functions, we focus on methods that allow to achieve this goal. In the following we investigate suboptimal solutions to problem (15) for a small number of users , when the probability of having a fully loaded system is small. In fact, in this scenario power distribution has an important role in selecting the optimal user set. In Section 4.3 we will consider the case of a high number of users , and in this case a simplification of scheduling is possible. For ease of notation we drop both slot ( ) and OFDM symbol ( ) index in the remaining of the paper.

### 4.1. Multicarrier Greedy (MG)

In [14], a greedy scheduling algorithm in a single-carrier flat-fading system has been proposed, where users are selected one by one as long as the throughput increases and it has been then extended to an OFDM system in [21] and denoted here multicarrier greedy (MG).

where is given by (6) while is the th column of the beamforming matrix for users scheduled at step . Note that total power has been divided by in order to obtain the per stream power .

### 4.2. Projection-Based Greedy (PBG)

- (1)
the power is redistributed among all streams;

- (2)
beamforming of streams already scheduled on the same resource block is modified.

By using (26) and (28), there is no need to determine a new beamformer in correspondence of each candidate stream; instead, only the basis needs to be updated at each step, and this requires only few vector multiplications. Note that the computation of is based on the projection of the candidate vector on the basis, as from the acronym PBG. Once all streams have been scheduled, a beamformer is computed to perform transmission.

### 4.3. Greedy Scheduling Strategies in the High Scenario

Scheduling can then be simplified by operating independently on each resource block.

### 4.4. Multicarrier Semiorthogonal User Selection Algorithm (MSUS)

where is a design parameter that sets the maximum correlation allowed between the quantized channel vectors of the selected users. We note that in MSUS we apply single carrier SUS in parallel, one for each resource block. Also in this case the number of steps is random as the algorithm ends when set is empty. Once users have been scheduled, the total power is equally distributed among the scheduled streams according to (4).

## 5. Preselection Methods

From (23) we obtain that this condition is satisfied only if the SNIR is high enough to compensate for losses incurred by the insertion of a new scheduled stream, that is, the power redistribution and the beamforming modification, as described by conditions (1) and (2) of Section 4.2. This observation suggests a further simplification of the PBG algorithm, by a-priori excluding the streams whose SNIR is below a certain threshold. Indeed, as for each candidate stream the SNIR (26) must be evaluated, by excluding streams that could never be inserted, the scheduling procedure can be fastened [22].

Note that the idea of preselecting users has been first introduced in [28], by letting users feeding back their CSI and rate request only if the quality of their channel is above a threshold. On the other hand, we use preselection as a technique to simplify scheduling rather than reducing the feedback rate. Moreover, in our case the preselection is not based only on the channel quality but also on the correlation with other users' channels.

### 5.1. Preselection PBG (PPBG)

Then by considering only streams satisfying (35), we decrease the number of comparisons and SINR updates at each step of PBG. In the high scenario the preselection technique is not feasible; in fact, as illustrated in Appendix , for , and therefore (35) is verified by all streams.

We further note that is an increasing function of ; hence, streams whose CQI is below the threshold at step can be neglected also in the next steps.

### 5.2. Simplified Preselection PBG (S-PPBG)

Within PBG methods, we note that this approach becomes optimal when the scheduling objective coincides with the maximization of the SR. However, for the maximization of the WSR, S-PPBG is in general suboptimal.

## 6. Complexity Analysis

We analyze the worst case complexity of the various approaches, in terms of both computational complexity and memory requirement.

### 6.1. Computational Complexity

We first observe that all considered algorithms select one stream per step, until at most streams are allocated on each resource block, thus in general . At step , streams are considered for insertion in . Furthermore, at each step, the per stream power is adapted, due to the insertion of a candidate stream in .

#### 6.1.1. MG Complexity

where denotes the resource block of the stream selected at step . The first term in (39) accounts for the selection of the stream with maximum CQI. The remaining terms account for step through , with (a) update of SNIR estimate of the already scheduled streams, (b) computation of a new beamformer for each of the candidate streams on subcarrier , (c) evaluation of , (d) update of the SNIR estimates, and (e) evaluation of the WSR. Lastly, the algorithm determines the stream which maximizes the WSR at step and checks condition (21).

#### 6.1.2. PBG Complexity

In fact, the PBG algorithm for each candidate stream on resource block (a) performs the projection of channel vector on the orthogonal basis and (b) updates the SNIR estimate. At each step, the basis is also updated according to the channel vector of last scheduled stream. At the end, the beamforming matrix is computed according to the set of scheduled streams.

since scheduling can be performed in parallel on all resource blocks.

#### 6.1.3. PPBG Complexity

#### 6.1.4. S-PPBG Complexity

### 6.2. Asymptotic Complexity Analysis

### 6.3. Memory Occupation

as PBG stores (a) the value , (b) total rate provided by each candidate stream ( CLS), and (c) orthogonal basis ( CLS).

with respect to PBG it needs to store also ( CLS as worst case).

as MSUS stores (a) correlations of candidate streams and last inserted stream ( CLS), (b) the value of ( CLS), and (c) the set of total rates of each candidate ( CLS as worst case).

## 7. Simulation Results

We compare the scheduling algorithms in terms of average sum rate (SR) and complexity requirements. All users are uniformly distributed in a cell of radius 500 m, as in [29]; we consider an average of 15 dB per resource block at the cell border and path loss is included in the channel model. We assume also a realistic MIMO channel with time, frequency, and spatial correlation among the elements of . The channel is modeled as slowly time-variant, frequency selective Rayleigh fading as from the spatial channel model (SCM) [30]. According to the LTE release, we set transmission bandwidth to 2.5 MHz, divided into resource blocks and centered at the carrier frequency of 2 GHz. The base station is equipped with antennas spaced by 10 wavelength. Scheduling and beamforming are performed once a slot, and each slot is composed of 7 adjacent OFDM symbols. CSI feedback is performed with a variable number of bits using an optimized codebook, as detailed in [31].

### 7.1. Performance Comparison

We note also in Figure 3 that preselection applied to PBG provides slightly better performance, despite the fact that it considers a lower number of candidate sets. In fact, preselection aims at excluding from scheduling streams that would not increase the WSR and prevents the scheduler from inserting them for fairness reasons.

### 7.2. Complexity Comparison

In the high scenario, simulations confirm the analysis; in fact, for we have , , , and . We underline that in the high regime S-PPBG complexity is higher than that of PBG because of the required power distribution; indeed simplification of preselection does not compensate the need of redistributing the total power. On the other hand, we note that the high complexity required by MG is mainly due to the evaluations of the beamformer at each step.

Memory requirements, investigated in Section 6, do not prefigure large differences between different methods; for required memory locations are 35890 for MG, 29682 for PBG, 29730 for S-PPBG, and 33841 for MSUS. Hence, the simplified techniques achieve a reduction of memory requirement with respect to existing algorithms.

## 8. Conclusions

This paper has provided an overview of scheduling problems for multiuser downlink MIMO OFDM systems. We first have shown that scheduling according to a wide class of utility functions can be reduced to a scheduling problem aiming at maximizing the weighted sum rate of the system, under a proper choice of the weighting function. Then we have compared scheduling algorithm having as objective the maximization of the weighted sum rate, including greedy algorithms, based on throughput maximization and algorithms based on the semiorthogonality among MIMO channels. Extensions to a OFDM scenario of algorithms originally devised for flat-fading single-carrier systems have been investigated. The comparison has been carried out both in terms of computational complexity and in terms of achievable throughput.

Several insights on the performance of the state of the art scheduling algorithms can be highlighted from the numerical results. Firstly, the MG approach achieves an average sum rate which is very close to the maximum value achieved by ES, over a wide range of cell loads. When compared against MSUS, the proposed MG technique has a gain of about 50% in terms of average sum rate in most network conditions. Moreover, MG requires a significantly lower complexity than that of ES and only 30% additional CMUXs than MSUS. Hence, we believe that MG provides a good trade-off between performance and complexity.

Lastly, limitations in the feedback rate have a severe impact on the performance of all scheduling approaches. Indeed we have seen that all schedulers yield an average sum rate that increases linearly with the number of bits used to feedback the CSI with an increase of about 1 bit/s/Hz for each additional feedback bit.

## Appendices

### A. Proof of (15)

where indicates that user is scheduled, that is, if and otherwise.

Therefore, by inserting (A.5) into (A.4) we obtain (15).

### B. Proof of (36)

## Declarations

### Acknowledgment

The authors thank the editor and the reviewers for their comments on the manuscript.

## Authors’ Affiliations

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