Dynamic Algorithms in Multi-User OFDM Wireless Cells? James Gross Telecommunication Networks Group Technische Universit¨at Berlin Einsteinufer 25, 10587 Berlin, Germany
[email protected]
Abstract. This paper presents several results on dynamic OFDMA systems. It addresses especially the algorithmic complexity involved with several resource allocation approaches, sub-optimal heuristics for the use in practical systems, the related signaling overhead and modifications to the IEEE 802.11 protocol stack. It is argued that for the generation of dynamic OFDMA resource allocations very good sub-optimal methods exist while the loss due to signaling can be kept low for a large range of system parameters. Finally, an outline of the integration of dynamic OFDMA schemes into OFDM-based IEEE 802.11 systems is presented, providing significant performance benefits for such wireless local area networks.
1 Introduction During the last ten years orthogonal frequency division multiplexing (OFDM) has become a very popular transmission scheme for frequency-selective broadband communication channels. Example systems that feature OFDM are digital video and audio broadcasting (DVB and DAB), wireless local area networks like IEEE 802.11a/g/n, wireless metropolitan area networks like IEEE 802.16, the upcoming extension of highspeed down-link packet access (HSDPA) in 3G cellular networks but also wired access systems like the digital subscriber line (DSL). OFDM offers the advantage of mitigating intersymbol interference (ISI). By means of advanced signal processing the broadband channel is split into N narrowband subcarriers (where N takes values between 52 and 2048). Each sub-carrier exhibits a frequency-flat behavior. Thus, instead of transmitting many digital symbols sequentially (as in a broadband single carrier system), N symbols are transmitted in parallel. Therefore, an OFDM system with an equivalent gross symbol rate can afford an N -times increased symbol time per sub-carrier, which reduces the impact of ISI significantly. However, there is still frequency selectivity in the system as the channel gain varies over a larger set of sub-carriers. In addition, in a multi-user scenario, for example the down-link of a cell, the gain for each sub-carrier varies also regarding different terminals (referred to as multi-user diversity). This offered diversity can be exploited by ?
This work has been supported partially by the German research funding agency ’Deutsche Forschungsgemeinschaft (DFG)’ under the graduate program ’Graduiertenkolleg 621 (MAGSI/Berlin)’.
dynamic resource allocation schemes. Given the knowledge of sub-carrier gains at the transmitter, modulation/coding combinations as well as the transmit power can be allocated dynamically per sub-carrier (commonly referred to as bit- and/or power loading). Moreover, disjoint sub-carrier sets can be assigned to different terminals (known as dynamic sub-carrier assignments). These dynamic allocation schemes for multi-user scenarios, often referred to as dynamic OFDMA schemes, can improve various transmission metrics such as the total transmit power, error rates or system throughput. Despite this fact, many issues remain open regarding the application of such dynamic OFDM(A) schemes in practical systems. In this paper three such aspects are addressed, summarizing the major contributions of [1]. Initially, the question arises how dynamic resource allocations should be performed in order to serve the data flows of several terminals best (i.e. maximizing the number of flows that can be served while maintaining fairness). Given this objective, an important issue for practical systems is how to generate such allocations in real time. As the sub-carrier gains are only stable for several milliseconds, potential allocation algorithms have to terminate quite fast. This problem is addressed in Section 3. Once resource allocations have been determined at some central point in the network, a further issue is how to convey necessary control information to the terminals (as they have to be informed of their next allocations). This so called signaling problem is discussed in Section 4. Finally, the integration of dynamic OFDMA schemes into OFDM based wireless local area networks (i.e. IEEE 802.11 a/g) is studied. Here, apart from the issue of computational complexity and the signaling overhead also backward compatibility and other protocol aspects play an important role (Section 5). Finally, some conclusions are drawn and future work is presented (Section 6).
2 System Model Consider the down-link of a single wireless cell. The access point serves J terminals which all receive data flows (consisting of packets queued at the access point). A slotted system is assumed in which time is divided into units (frames) of duration T f . A total bandwidth of B [Hz] at the center frequency fc [Hz] is available for data transmission (maximum total transmit power of Pmax ). The given bandwidth is split into N OFDM sub-carriers, each one featuring a fixed symbol duration N B = Ts . Each sub-carrier gain varies due to path loss, shadowing and fading, hence the perceived signal quality (SNR) per sub-carrier varies from frame to frame (but is assumed to be constant during one frame). Depending on the SNR, M different amounts of bits might be represented per symbol for each sub-carrier (i.e. M different modulation/coding combinations are available). The choice of an adequate modulation/coding combination is determined by a constraint on the error probability. Each frame is split into a down-link and an up-link phase. OFDMA is applied during the down-link phase. The duration of one down-link phase is denoted by T d . A total of S = Td /Ts symbols per sub-carrier can be transmitted during that down-link period. Prior to each down-link phase, the access point generates new assignments of subcarriers to terminals based on the knowledge of the sub-carrier states (i.e. the SNR).
Perfect channel knowledge at the access point is assumed (by estimating the sub-carrier states during the previous up-link phase and assuming a reciprocal channel gain).
3 Dynamic Algorithms for Resource Allocation Assume initially that the transmit power per sub-carrier is statically distributed. Then the gain per sub-carrier and terminal directly yields the SNR. Given the SNR, a certain modulation/coding combination is obtained, as the transmission is subject to a certain error constraint and simply the “best” modulation/coding combination (i.e the one with (t) the highest throughput) is chosen which fulfills the error constraint. Denote by b j,n the amount of bits that can be transmitted to terminal j on sub-carrier n during down-link phase t . How should sub-carriers be assigned to terminals? From a system point of view a basic constraint is that each sub-carrier should only be (t) assigned once. Denote by the binary variable xj,n the decision if a sub-carrier/terminal pair is fixed as assignment during down-link phase t . Given this framework, a simple assignment strategy is to allocate sub-carriers per down-link phase such that the sumrate of the cell is maximized [2]: X (t) X (t) (t) xj,n ≤ 1 ∀n . bj,n · xj,n s. t. max (SUMRATE) j
j,n
However, in a cell typically several terminals are located farer away from the access point than other stations. Hence, a subset of all terminals always has a much better SNR per sub-carrier than the remaining stations. As a consequence, terminals with a significantly lower sub-carrier gain obtain only occasionally a sub-carrier, which leads to starvation of their flows. In order to avoid this situation, a more sophisticated approach is required. Ideally, the assignment scheme should maximize the throughput of each terminal equally. This ”max-min”formulation of sub-carrier assignments [3] is given by:
s. t.
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Above αj gives the opportunity to scale the minimum throughput per terminal j and down-link phase t . Hence, given the queue state of each terminal at the access point, the assigned rates for the next down-link phase can be scaled according to the queue sizes. Compared to a static approach (for example, a TDM approach where each terminal receives all sub-carriers for some down-link phase in a round-robin manner) it can be shown that a dynamic approach according to (MAXMIN) provides a much higher throughput per terminal [4]. However, the question remains open how the assignments (t) can be generated in practical systems given the values bj,n . This question can be answered by considering the computational complexity. In fact it turns out that (MAXMIN) is NP-hard. PARTITION reduces in polynomial time to the decision problem of (MAXMIN) [1]. In addition, a broader set of dynamic OFDMA optimization problems all turn out to be NP-hard (obviously (MAXMIN) with dynamic
power allocation, but also optimization problems where the transmit power is reduced for a given set of rates per terminal, referred to as margin-adaptive allocation problems [6]). Finally, practical instances of (MAXMIN) indeed turn out to be diffcult to solve by software for linear integer programming problems (an example instance has been added to a database of difficult integer programming problems, cf. opt1217 of [5]). Thus, sub-optimal schemes are required for the application in real systems. One particular good sub-optimal scheme is relaxation. By initially relaxing the in(t) teger constraints on the assignments xj,n , (MAXMIN) becomes a pure linear programming (LP) problem. For LPs polynomial time algorithms are publicly available which perform quite promising even for bigger instances of (MAXMIN) (in the range of tens of milliseconds on standard computers without applying customization). Once the relaxed solution is obtained, a necessary step is to find a good feasible solution consisting only of integer assignments. For the case of static power distribution among the subcarriers, a simple rounding approach performs well (cf. Figure 1). In case that the power is distributed dynamically, a more advanced approach is required (cf. Figure 1). In summary, relaxation has been found to provide the best performance of all so far published schemes. 3.4 Average minimum throughput [MBit/s]
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Fig. 1. Minimum throughput per cell for an IEEE 802.11a like wireless OFDM system with J = 8 terminals (for more information on the simulation scenario refer to [4]). Comparison of the IP optimum with the relaxation approach and several other sub-optimal schemes proposed in the literature. Left graph: static power distribution – Right graph: dynamic power distribution.
4 Signaling Overhead Signaling is a problem which applies to many dynamic approaches in OFDM systems. It relates to the need to inform terminals about the next resource allocation such that the payload transmission can be successfully received. The access point basically has to
inform each terminal which sub-carriers have been assigned to it during the next downlink phase. In addition, the terminals also have to be informed of the respective modulation/coding combination chosen per sub-carrier. A fundamental question regarding signaling is if the resulting overhead outweighs the benefits from dynamic OFDMA. Initially, the following straightforward model is used to study the overhead. At the beginning of each down-link phase a signaling phase is inserted. The signaling information is broadcasted on all sub-carriers during this signaling phase using a predefined, fixed modulation type with bit rate bsig . For each assignment the triple hsub-carrier, terminal, modulationi has to be transmitted. However, by transmitting all assignments per down-link phase, the sub-carrier reference can be omitted as the position of the tuple hterminal, modulationi indicates the sub-carrier it refers to. Therefore, a fixed number of N · (dlog2 (J)e + dlog2 (M )e)/(N · bsig ) signaling bits is required per down-link phase leading to ς = d(dlog2 (J)e + dlog2 (M )e)/bsig e OFDM symbols of overhead. For payload communication S − ς symbols remain per down-link phase. This scheme is referred to as fixed-size signaling field model (FSSF). From these basic considerations it is clear that the signaling overhead depends on various system parameters such as the number of sub-carriers N , the number of terminals J , the number of modulation/coding combinations M , but also on the length of a down-link phase Td (as this determines the number of OFDM symbols S ). Moreover, it turns out that there exists a trade-off between the control overhead and the achieved performance, as for example a larger number of sub-carriers leads to a higher average throughput per terminal (due to the increase of the symbol times but a fixed guard period setting) while also increasing the signaling loss. An example result of this is shown in the left graph of Figure 2. Clearly, the net throughput of the dynamic OFDMA scheme still outperforms the static approach. However, for an increasing number of sub-carriers the net average throughput per terminal first increases up to an optimum and sharply decreases thereafter. For a large number of sub-carriers the net throughput is even worse than for the static approach. Also note that considering a dynamic OFDMA system without signaling cost leads to very different system results. This is true for a broad set of system and environment parameters. The most important parameters for the overhead are the system bandwidth, the duration of a down-link phase and the number of sub-carriers. This observed performance behavior motivates the investigation of the optimal net performance. In order to judge the efficiency of the FSSF model, a lower bound on the signaling overhead can been derived. The outcome of the assignment algorithm at the access point can be interpreted as an stochastic source of information (generating symbols from a discrete set). By exploiting the first- and second-order statistical properties of this information source, it is possible to derive the entropy (defined as the minimal binary rate required to represent the source without losses). This yields an upper bound on the net throughput. In the left graph of Figure 2 the upper bound on the net throughput from the entropy rate is shown as well, indicating that there is significant room for improvement. This performance improvement stems from correlation. Sub-carrier states are correlated in time and frequency. Hence, the stronger the correlation in time and frequency, the more likely is a correlated assignment of sub-carriers. The entropy indicates that for the considered scenario already a significant amount of correlation is
present in the assignments. In order to exploit this correlation in practice, several options arise. First of all, the assignments could be better encoded by a more sophisticated representation. For example, only “new” assignments could be signaled, where “new” refers to the change of a sub-carrier assignment from down-link phase t to the next one t + 1. A somewhat similar approach can be found for exploiting the correlation in frequency as well as exploiting the correlation in both dimensions. However, the downside of these schemes is that per single sub-carrier assignment more bits are required to represent them. Thus, if the correlation is low, the resulting overhead is high. However, given these representations building on top of the correlation of assignments, a further option is to stimulate correlated assignments within the assignment algorithm itself. This can be driven to optimize the net throughput for a given representation. Assume that the bit cost per signaled assignment is given by C sig (the more the representation exploits correlation, the higher is this cost). Then, for a given sub-carrier assignment (and possibly a given previous assignment), the total number of signaling symbols ς required to transmit the signaling information can be obtained. The net throughput depends on the amount of bits that can be transmitted per OFDM symbol during the payload phase but also on the remaining duration of the payload communication S − ς. Ultimately, this leads to a quadratic, integer optimization problem for which an iterative algorithm has been developed [7]. This approach can achieve a significant performance increase for dynamic OFDMA systems as demonstrated in the right graph of Figure 2 (exploiting only the correlation in time and the correlation in time and frequency). Note that even with these optimization approaches a system can be subject to such rapid channel changes that the resulting signaling cost is too high. Then a static approach should be preferred.
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Fig. 2. The impact of signaling cost for different models (more information on the simulation scenario is given in [7]). Left graph: Basic comparison of the signaling cost modeled by the FSSF versus the cost derived from entropy analysis versus a static and dynamic OFDMA system without any signaling cost. Right graph: Two optimization approaches reducing the signaling overhead of the FSSF by exploiting the correlation in time and in time/frequency.
5 Integration into IEEE 802.11 a/g Having discussed various aspects regarding dynamic resource allocation algorithms in OFDM systems, the question arises how such schemes could be incorporated in upcoming or even existing OFDM based wireless networks. This will be discussed considering the IEEE 802.11 wireless local area network standard, focusing on the OFDMbased implementations (802.11a and 802.11g). For the infrastructure mode a dynamic OFDMA scheme fits best to the down-link transmissions from the access point to several stations. In the following the focus is only on this transmission direction. Clearly, the OFDM-based IEEE 802.11 link layer and physical layer protocols have to be changed such that the access point can acquire the channel knowledge, afterwards calculate the sub-carrier assignments and finally transmit the payload (prepending a signaling field). In order to perform these tasks the transmission order is modified, as indicated in Figure 3. Initially, the access point transmits a Beacon frame as this allows it to access the medium with a higher priority. Immediately after the transmission of a Beacon a modified RTS frame is transmitted, which polls all stations to be included in the following OFDMA burst transmission. The stations reply with a legacy CTS frame which is used at the access point to estimate the sub-carrier states. Then the assignments are calculated and an OFDMA burst frame is generated which holds the signaling information. After the payload transmission the stations have to acknowledge the reception of their frames. Finally, the access point ends the transmission cycle by a RTS to itself. Some issues come up when evaluating this modified layout. First of all, at the beginning of the transmission sequence the access point does not know the exact duration the medium will be busy, as the sub-carrier states are not known yet. Hence, the virtual allocation vector, referred to as NAV (network allocation vector), can not be set to the precise end of the transmission cycle. This potentially harms legacy devices. The problem can be resolved by initially (during the transmission of the Beacon frame at the beginning) setting the NAV to a very large value, blocking the channel for a long time. Once the precise length of the busy period is known (when the OFDMA burst frame is generated) the NAV is reset by all acknowledgement frames and by the final RTS frame. Hence, even the NAV of legacy devices can be controlled by this scheme. A further issue relates to the calculation time at the access point. Obviously, after acquiring the channel knowledge, the access point will have to calculate the assignments. However, the medium access scheme enables stations to access the wireless medium if it is not busy for a time span of DIFS (which is in the range of 30 µs for IEEE 802.11a/g). Hence, the calculation of the assignments and the generation of the OFDMA burst must not require more than this time span - otherwise some stations might interfere with the OFDMA burst transmission. This can be resolved by either transmitting a busy tone (which degrades the performance of the system, though) or by pipelining the assignment calculation. In this case the access point starts to compute the assignments for a subset of stations after acquiring their channel knowledge. Regarding the performance, several issues have to be taken into account. Compared to a sequential transmission of the packets in the legacy mode, there is a higher overhead by the new scheme due to the initial Beacon frame, the signaling part and the trailing RTS to itself. In addition, for short packets in the legacy case no RTS/CTS
frame exchange takes place, favoring the legacy system even more. However, only a single medium access takes place in this new proposal which gives it a significant performance improvement. In addition, during the payload transmission data is transmitted with a much better efficiency. Finally, the new scheme controls the packet error rate as the modulation types are adapted per sub-carrier, leading to much less retransmissions especially compared to the legacy mode without RTS/CTS frame exchange. Initial results on this new mode show that it is very promising [1]. DIFS
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Fig. 3. New transmission sequence for a down-link data transmission in OFDM-based IEEE 802.11 systems featuring dynamic OFDMA.
6 Conclusions This paper has summarized major results from [1] regarding dynamic OFDMA systems. Various aspects regarding the algorithmic complexity, the signaling overhead and the incorporation into existing protocol stacks have been addressed. It turns out that dynamic OFDMA systems provide a superior performance while the increase in complexity is much lower than at first hand assumed. Thus, dynamic OFDMA is a promising technique for future wireless systems.
References [1] Gross, J.: Dynamic Algorithms in Multi-User OFDM Wireless Cells. PhD Thesis, Technische Universit¨at Berlin (2006) [2] Jang, J., Lee, K.: Transmit Power Adaption for Multiuser OFDM Systems. IEEE J. Select. Areas Commun. No. 2 (2003) 171–178 [3] Ergen, M., Coleri, S., Varaiya, P.: QoS Aware Adaptive Resource Allocation Techniques for Fair Scheduling in OFDMA Based Broadband Wireless Access Systems. IEEE Trans. Broadcast. No. 4 (2003) 362–370 [4] Bohge, M., Gross, J., Wolisz, A.: The Potential of Dynamic Power and Sub-Carrier Assignments in Multi-User OFDM-FDMA Cells. Proc. of IEEE Globecom, November 2005 [5] Koch, T.: MIPLIB. Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin (ZIB), http://miplib.zib.de/miplib2003.php, November 2006. [6] Wong, C., Cheng, R., Letaief, K., Murch, R.: Multiuser OFDM with Adaptive Subcarrier, Bit and Power Allocation. IEEE J. Select. Areas Commun. No. 10 (1999) 1747-1758 [7] Gross, J., Geerdes. H., Karl, H., Wolisz, A.: Performance Analysis of Dynamic OFDMA Systems with Inband Signaling. IEEE J. Select. Areas Commun. No. 3 (2006) 427-436
