Fast Prioritized Bit-loading and Subcarriers Allocation for Multicarrier Systems

Transcription

1 Fast Prioritized Bit-loading and Allocation for Multicarrier Systems Khaled Hassan and Werner Henkel School of Engineering and Science, Jacobs University, Transmission Systems Group (TrSys) 8759 Bremen, Germany, {k.hassan & Abstract Unequal Error Protection (UEP) is the key to future multi-layer and scalable data and video transmission. This paper presents a novel bit-loading and channel adaptation technique to realize UEP properties in the physical transport using greedy and sub-optimal bit-loading algorithms. In this case, we exploited the power minimization approach developed by the greedy algorithm to allow for an arbitrary number of classes, arbitrary SNR margins between the classes, and arbitrary numbers of bits per class. We achieved similar results using a fast sub-optimum power minimization bit-loading algorithm based on Campello s algorithm. Our results show the suitability of this technique to realize UEP for a single user case using orthogonal frequency division multiplexing (OFDM). Additionally, we extended the results to demonstrate the performance of allocating resources to two users with different quality of service (QoS), each of which requires data of different priorities. Index Terms UEP, adaptive modulation, power minimization, margin adaptation, QoS I. INTRODUCTION Historically, in single-link single-priority communication, a number of algorithms have been introduced to adapt the multicarrier system transmission to varying channel conditions. The classical Hughes-Hartogs algorithm is well known in literature as an optimum bit-loading algorithm. It is based on a greedy optimization method that aims at finding a global minimum power by adding a single bit to the subcarrier that requires the minimum incremental power at each iteration ], 5]. Therefore, it is optimal for minimizing the energy or maximizing the system margin by reallocating the leftover power. However, due to complex search operations in Hughes- Hartogs, less complex (and also sub-optimal) methods have been proposed later. Campello bit-loading 6], which is a linear representation of the Levin bit-loading algorithm 7], is a simple alternative to Hughes-Hartogs. It achieves almost the same optimum power allocation requiring only a fraction of the complexity. The simplification here lies in the quantization of the channel-gain-to-noise ratio g k and grouping similar levels of g k into L groups, where L is much smaller than the number of subcarriers N. Thereafter, the number of bits are computed based on the quantized channel capacity formula as in 8]. In modern communication systems, unequal error protection (UEP) transmissions are more requested to deliver scalable qualities of service (QoSs), e.g., scalable multimedia videos and prioritized communication for single and multiple users. UEP adaptation schemes allow different shares of given data to acquire different link qualities, e.g., the algorithm in 1] and ] which follows the Chow bit-loading algorithm 8]. This scheme allocates different parts of the data stream to different subcarriers (with different bit-rate and error probabilities) according to the required QoS. However, for greedy algorithms and the algorithms that follow the power minimization criterion, the partitioning scheme proposed in 1] becomes a very complex and impractical approach. Therefore, we developed a partitioning scheme that efficiently approximates the number of subcarriers required for each priority class exploiting the relation between the required QoS, the target bitrate, and the given number of subcarriers. First, we modified the heaviest -and the optimum- greedy bit-loading algorithm developed by Hughes-Hartogs ]. Therein, the required target rates are only allowed to allocate the pre-calculated subcarrier partitions. Second, we modified the less complex (and suboptimal) algorithm by Campello in 6]. It achieves almost the same optimum power allocation, however, with only a fraction of its complexity. The modified UEP Campello bitloading can be thought of as a practical solution for limited (quantized) channel feedback systems. In this paper, we reduce the complexity even further to make it more suitable for wireless applications. Finally, this paper is organized as follows. Section discusses our system model. Section introduces our QoS adaptive modulation using a Hughes-Hartogs-like channel adaptation method. Section introduces our fast prioritized adaptive modulation using a suboptimal Campello-like channel adaptation algorithm. Section 5 discusses the most important results. Finally, we conclude our findings in the last section. II. SYSTEM MODEL A. Channel Model The channel impulse response follows a Rayleigh fading model with an exponential power delay profile as in ], where the channel entries are modeled as independent Rayleigh fading blocks composed of L m different paths with amplitude β l, delay τ l, and random uniform phase shift θ l, π). The time-variant channel impulse response for each channel matrix element can be defined following 9] as h(τ) = L m 1 l= β l p l e jθ l δ(τ τ l ), (1) /1/$6. 1 IEEE

2 where t is the observation time, β l is an i.i.d. zero mean random complex Gaussian variable, and p l is an exponentially decaying factor. B. Multicarrier System Model with Prioritized Information In this paper, we assume a multicarrier system that is composed of N subcarriers. The channel state information (CSI) of each subcarrier is assumed to be available accurately at the transmitter in the form of signal-to-noise ratios (SNRs). To realize arbitrary UEP classes, we have to select different numbers of bits with different spacing in the SNR or equivalently, with different bit-error ratios (BERs). This generates arbitrary margin separation between classes 1], ], where these separations between can easily be adjusted using the modified Shannon capacity formula in 1]. Similar to 1], we assume M different classes with a known margin separation equal to Δγ j, where j 1,..., M. III. UEP USING OFDM ADAPTIVE MODEL A. Modified Hughes-Hartogs Algorithm for UEP Bit-loading Hughes-Hartogs (greedy) bit-loading algorithm cannot follow the partitioning scheme in 1], ]. The reason is that the greedy algorithms allocate bits to the subcarriers with the minimum incremental power first. Therefore, we approximate the number of subcarriers required for each class by finding a hypothetical threshold between this class and the successive classes as will be discussed later. Our approach follows the margin-adaptive criterion, which is defined as: N min E σ = E k () E k subject to : N ( log 1+ E ) kg k Γ = B T, () where E k is the power allocated to the k th subcarrier, E σ is the accumulated power, G k is the channel coefficient square ( h k ) to noise (σ n) ratio, N is the total number of usable subcarriers, and the gap approximation is given by Γ = )], ]. If the achieved rate is fixed to be BT ( erfc 1 Pej > E σ, where E tot is the given target power, then and E tot the performance can be further enhanced by scaling up the effective power allocation E k by the ratio E tot /E σ.thisis called margin maximization. The maximum system margin is defined as γ max = E tot. () E σ In ], the bits are allocated in a hierarchical fashion due to the criterion of allocating the subcarriers with the lower incremental power first and in order to avoid the lengthy iterations needed for finding hypothetical thresholds between classes. However, hierarchical modulation already reduces the overall performance ]. Therefore, one needs to sub-divide the given set of subcarriers amongst the given priority classes in order to avoid hierarchical modulation without introducing more complexity. 1) Partitioning Scheme: Based on the required and the target bit rates T j for each class, we compute equivalent bit rates B Tj in Appendix A, assuming that each class can freely allocate the entire number of subcarriers. We compute the required subcarriers for each class by accumulating bits until T j is fulfilled. This subcarrier is set as a hypothetical threshold for this class. For the next class, we start accumulating bits from the next subcarrier position, and stop at the position where the accumulated number of bits is T j+1. The following are the steps to subdivide the given subcarriers among the required classes without extra iterations: (a) Assume that each priority class is allowed to consume all the N subcarriers (sorted in descending order) to allocate B Tj bits simultaneously. (b) As in Fig. 1.a (part 1), the first, the second, and the third priority classes will allocate B T N, B T, and B T + N bits, respectively. See Appendix A for more details. (c) Starting from the first class and the highest SNR subcarrier, bits are accumulated using a cumulative summation (cum-sum) process. (d) The cum-sum will stop accumulating bits when the output is greater than or equal T. Hence, the current subcarrier will be the hypothetical threshold subcarrier θ 1. (e) For the next class, the cum-sum will accumulate the bits of the second class starting from the next subcarrier (θ 1 +1), and continue accumulating until the output T 1 as in Fig. 1.a. (Part ). (f) The same steps are repeated for the remaining classes, see Fig 1.a (Part ), to find θ j. ) UEP Bit-loading Steps Based on Hughes-Hartogs Algorithm: The complete Hughes-Hartogs-like bit-loading algorithm steps are as follows: (a) Initialize the zero matrices B (bits) and E (power) with N rows and M columns. (b) Compute the incremental power steps ΔE k,j for M given classes in case of incrementing each subcarrier by only one bit as in ] ΔE k,j = erfc 1 ( Pe G k )] bit increment {}}{ Bk,j+1 B k,j, (5) Δ γ j where P e is the probability of error of the first class. (c) Find k with the minimum ΔE k,j among all subcarriers in each class; increment B(k, j) by one bit such that it does not exceed the maximum of bits/subcarrier b max. (d) Increment the power of the k th subcarrier by ΔE k,j, i.e., E k,j = E k,j +ΔE k,j. (e) If the achieved rates of the j th class is less than B Tj, return to b), else continue. (f) Find θ j such that M j b k = T j, see Fig. 1.a for more details. (g) Calculate the power for the k th subcarrier and the j th class

3 no. of bits no. of bits no. of bits Fig. 1. B Tj BT BT1 BT T Class T θ1 θ1 T1 Part 1 Part Part Class Class 1 Class θ Partitioning using our approximation, cum-sum, and the calculated and update E using E k,j = erfc 1 ( Pe G k T )] ( B k ) 1. (6) Δ γj (h) Rescale the allocated power such that E k,j = E k,j E k,j /E tot. (7) B. Modified Campello Algorithm for UEP Bit-loading In ], we developed a sub-optimal bit loading algorithm based on the one described by Campello in 6] in order to realize an adaptive hierarchical modulation. This has been done to avoid the iterative QoS based partitioning in 1], ]. However, since we are now able to perform a simple QoS partitioning as discussed before, it is now easier to modify the same sub-optimal bit-loading algorithm to realize UEP using a non-hierarchical modulation. This will be more simple and efficient than the previous greedy bit-loading algorithm. The simplification here lies in the quantization of the channel-gainto-noise ratio g k. Where similar levels of g k can be gathered into L groups, where L is much smaller than the number of subcarriers N. We simplified it further by introducing a linear solver that computes the approximate bit load for these L levels, instead of the consecutive logical iterations in the original algorithm proposed in 6]. As in 6] the discrete bit-loading b Z bmax is said to be energy-efficient, assuming two different subcarriers n & m, if max n(b n ) < min m(b m +1) n, m =1,..., N. (8) n m This means that the maximum incremental power achieved (to place b n bits) is less than the minimum power required to add an extra bit on any other subcarrier (min m ΔE m (b m +1)). As in ], the approximated discrete bit-allocation b k,j that satisfies Eq. (8) is given by 6] ] b k,j = log g jk + i bmax B opt, (9) N where i Bopt Z is selected such that k b k,j B Tj and g j k = G k/γ j is the gain to noise ratio with the gap Γ j of each class j. In6],thevaluei Bopt guarantees only a local optimum value. Therefore, in ], we relaxed this expression to include i Bopt, i Bopt +1, and i Bopt 1. Certainly, the one that approaches the target rate is selected. This reduces the total number of iterations required to brute-force the overall bit-loads in order to achieve a strict target rate. Γ j,asin5], is defined as Γ j = ( )] erfc 1 Pej. (1) The floor operator ensures, first, a lower bound for the bitloading and, second, that subcarriers with different g k,j are allocated to the same number of bits b i, where i is the index of the similar bit-loading. Assuming that the number of similar bit-loading is R; letv be a vector of length R, where each index contains a pointer to a vector u j i which contains the subcarriers with similar bit-loading such that for class j, each index in V j is V j i = {u j i {1,..., N} : log gu j i = b i }. (11) Since log g j k can result in negative values, one has to scale it by the minimum value log g j min such that Ĝ j i = log g j k log g j min, are all positive values. The quantization error due to the floor operator is Δ j k = {log g j k log g j min } Ĝj i. The total bit-rate (computed using (9)) is N 1 k= L 1 b k,j = i= Ĝj i V i + i j B opt V i ] bmax. (1) After relaxing (1) as in ], the approximated value for i Bopt is given as i j B = BTj L 1 i= Ĝj i V ] j 1 i + Z. (1) L j +1 Therefore, i Bopt is selected from the i j B three indices in order to minimizes the following N j min B Tj b k,j, (1) i B where b k,j is computed as Eq. (9). The pseudo-code of the 1 st method: (a) Initialize B and E as in the first alg.. (b) Compute Ĝj i i,r 1] for all j classes, then compute i Bopt using (1) and B Tj in the previous alg.; compute the bit-loading b k,j using (9) and fill B k,j. (c) If B Tj < (>) k b k,j, decrement (increment) b k at the subcarriers with the smallest (highest) Δ k, and change Δ k by +1 ( 1). (d) The energy is calculated using (6) and re-scaled according to the previous alg..

4 One has to consider these two issues while brute-forcing using the quantization error only: 1) the quantization error on each subcarrier can only be used once, i.e., since only the first time minimizes the energy for sorted subcarriers 6]. ) the quantization error on weak subcarriers may lead to allocate non-feasible subcarriers or over-load others. This leads to a huge power consumption. Therefore, we implemented the same steps as before except for step c), which is modified to fulfil Eq. (8) as follows The pseudo-code of the nd method: (c) If B Tj < (>) k b k,j, decrement (increment) the m th subcarrier with that has the maximum (minimum) incremental power, i.e., max ΔE m (b m 1) (min ΔE m (b m +1)). Thereafter, update ΔE m such that ΔE m = ΔE m max ΔE m (b m 1) or ΔE m =ΔE m + min ΔE m (b m +1), respectively. IV. RESULTS AND DISCUSSION To evaluate our algorithm, we consider three priority classes with three different priorities. As can be seen from Fig., assuming 8 bits/class, the db margin separation in case of perfect channel state information is preserved with almost no difference in performance between the modified Hughes- Hartogs and the modified Campello algorithms Class Class 1 Hughes Hartogs like Campello like Class SNR db] Fig.. Adaptive UEP OFDM with Δ γ = db, 8 subcarriers, and 8 bits/class. For channel uncertainties, we adopted the uncertainty model in ], where a Rayleigh channel with complex components, each real and imaginary is assumed to be a Gaussian variable with zero mean and variance equal to 1 is assumed, i.e., CN(, 1). The errors are assumed to be Gaussian as well with zero mean and 5% deviation from the channel variance. In Fig., the performance of our modified Campello alg. with channel uncertainty and the performance in case of rate reduction are depicted. One can see that that reducing the overall rate by 5% can enhance the performance. The difference between the proposed methods can be seen in Fig Reduced rate Fixed date rate 6 1 SNR db] Fig.. Adaptive UEP with 5% channel uncertainty using Campello-like alg., nd method. # bits # bits st Class 1 st Class nd Class Campello s nd nd Class rd Class Campello s 1 st rd Class Hughes Hartogs Fig.. UEP bit-loading using Campello-like alg. using the 1 st and nd method and Hughes-Hartogs-like alg.. In this figure, the upper plot represents Campello-like UEP bit-loading using the nd method (the solid-dark lines). For the third class, we intentionally plot the bit-loading of the 1 st method (light-grey lines) together with the nd method to show that the quantization error alone is not sufficient to determine the brute-force final bit distribution at the low-snr values. We further verify this issue by plotting the power allocation of these three schemes in Fig. 5. In this figure, it is clear that the 1 st method of Campello-like UEP bit-loading consumes more power at the weaker subcarriers. However, Hughes-Hartogslike and Campello-like, using the nd method, are distributing the power more efficiently. In these two schemes, a jump by a factor of db is seen at any bit-loading transition. A. Multiuser Resource Allocation Example We implemented a two user case sharing 8 subcarriers; each sub carrier can be dedicated to a single user at a time. Each user has two classes of service. The margin separations between each two classes and between the two users is db. The algorithm in III-B is used to allocate these subcarriers

5 value db] Fig. 5. Campello s 1 st Campello s nd Hughes Hartogs Power allocation using Hughes-Hartogs and Campello like alg... assuming an equivalent data classes; two classes per user. Since the number of classes are even, one has to interpolate extra classes to compute the equivalent number of bits of these four classes. According to Appendix A, the first and the second classes of the first user and the second user are B T /N, B T 1/N, B T +1/N, and B T +/N, respectively. The highest priority user sorts his channel subcarriers and allocates his two classes to the highest ones. The second user sorts the left over subcarrier indices (of his channel) and allocates his classes to them as in III-B Class Class 1 User 1 db db > db User SNR db] Fig. 6. Adaptive OFDMA with two different QoS user, each of which has two classes of protection levels with 156 bits each, i.e., 61 bits in total. Figure 6 shows the performance of this multiuser example, where the db is preserved between the data classes of each user. However, the margin separation between the two users varies from low to high SNR. At low SNR, the margin separation is slightly less than db. This is due to multiuser diversity, where switching opportunistically to another channel brings in good sub-channels. However, at higher SNR, the users order is kept safe, while the db becomes slightly wider. V. CONCLUSIONS We implemented a UEP bit-loading that realizes a fixed noise margin using Hughes-Hartogs and Campello algorithms. Both algorithms realize UEP bit-loading in a non-hierarchical fashion using a simple QoS based partitioning. The UEP Campello-like bit-loading can be thought of as a practical solution for wireless systems with limited (quantized) channel feedback, due to the simplified calculations and the reduced number of iterations, especially, after combining it with the non-iterative subcarrier grouping method. Furthermore, a twousers example with different QoS shows that this technique is capable of realizing modern prioritized wireless multiuser systems. VI. ACKNOWLEDGMENT This work is funded by the German National Science Foundation (Deutsche Forschungsgemeinschaft, DFG). REFERENCES 1] Henkel, W. and Hassan, K., OFDM (DMT) Bit and Power Loading for Unequal Error Protection, 11 th International OFDM-workshop 6, Hamburg, Germany, Aug. 6, pp. 6-. ] Hassan, K. and Henkel, W., Unequal Error Protection with Eigen Beamforming for Partial Channel Information MIMO-OFDM, Sarnoff Symposium, 7, Princton, NJ, USA, Apr. - May, 7, pp ] Hassan, K. and Henkel, W., UEP with Adaptive Multilevel Embedded Modulation for MIMO-OFDM Systems, 1 th International OFDM- Workshop 8, Hamburg, Germany, August 7-8, 8. ] Hughes-Hartogs, D., Ensemble Modem Structure for Imperfect Transmission Media, U.S. Patents Nos.,679,7 (July 1987),,71,816 (March 1988), and,8,76 (May 1989). 5] Pfletschinger, S., Multicarrier Modulation for Broadband Return Channels in Cable TV Networks, Ph.D. thesis, Institute of Communications, University of Stuttgart, Stuttgart, Germany, Feb. 7,. 6] Campello, J., A Practical Bit Loading for DMT, proc. ICC 1999, Vancouver, June 1999, vol., pp ] Levin, H.E., A complete and optimal data allocation method for practical discrete multitone systems, GLOBECOM 1, San Antonio, TX, USA, Aug. Nov. 1, Vol. 1, pp ] Chow, P.S., Cioffi, J.M., Bingham, J.A.C., A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels, IEEE Transactions on Communications, Vol., No., Feb- Mar-Apr 1995, pp ] Hoeher, P., A statistical discrete-time model for the WSSUS multipath channel, proc. IEEE Transactions on Vehicular Technology, Nov. 199, Vol. 1, pp APPENDIX A Assuming an odd number of classes (e.g., ) with individual target rates T j (B T = j= T j) and db margin separations, i.e., linear Δ γ = such that γ j = γ j+1. Following the approximation in ], θ 1 1 θ 1 η k B T = log 1 + η k log 1/ + N η k log 1/ k=θ 1 k=θ = B T +N θ θ 1 1 B T + N, (15) where η k = h k γ σ such that h n k is the channel coefficient squared. Following the same approximation in Eqn. (15) and starting with γ 1 (middle class), we find that, B T B T1. Similarly, when we start with γ we obtain B T B T N. For even number of classes, an extra interpolation, one hypothetical class between each two, is need to convert it into odd. This make it similar to the previous computation. Example: For levels with db separation, one has to interpolate extra levels, i.e., a level between each two. The separation between the resultant level is only 1.5 db, i.e., linear Δ γ =. Similar to the previous calculations, the equivalent rates of the four original classes are B T B T /N, B T1 B T 1/N, B T B T +1/N, and B T B T +/N