Simple Bit Allocation Algorithms with BER-Constraint for OFDM ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Simple Bit Allocation Algorithms with BER-constraint for OFDM-based Systems Hyeonmok Ko, Seungyoul Oh, Bongsu Kim † , Cheeha Kim Department of Computer Science and Engineering, Pohang University of Science and Technology (POSTECH) Pohang, Gyungbuk, Korea E-mails: {felix, ohsy, chkim}@postech.ac.kr † Corporate Research & Development Division, Hyundai Motor Company R&D Center, Hwaseong, Gyeonggi, Korea E-mail: [email protected]

Abstract—Orthogonal Frequency Division Multiplexing (OFDM) has been thoroughly investigated as an enabling technology for future broadband multimedia communication. In this paper, we suggest adaptive bit allocation algorithms that operate in a frequency selective fading channel environment by exploiting channel state information obtained through a feedback channel. The proposed algorithms try to maximize the overall throughput of the system with significantly reduced complexity while guaranteeing that mean bit error rate (BER) of all sub-channels remains below the pre-defined BER threshold. To do that, the first proposed algorithm divides all sub-channels into several groups according to their signal-to-noise ratio (SNR). The grouping criteria are adaptive to the current channel state and BER constraint. The second algorithm tries to find the appropriate constellation size for each sub-channel. The proposed algorithms were compared with existing algorithms through simulation. The simulation results show that the proposed algorithms are close to optimum solution with significantly lower complexity. Index Terms—Adaptive bit loading, flat fading channel, multicarrier modulation, wireless communications.

I. I NTRODUCTION OFDM has been adopted in many wireless standards, and has been intensively studied as an enabling technology for future broadband multimedia communication. OFDM divides the available channel bandwidth into multiple sub-channels that are mutually orthogonal. One of the main advantages of OFDM is that inter-symbol interference can be practically eliminated by introducing a guard interval between consecutive OFDM symbols. Because these sub-channels do not interfere with each other, channels can be utilized more efficiently. Also, a frequency-selective fading channel becomes a near flat fading channel. Many contemporary OFDM-based systems employ the conventional multicarrier modulation that uses the same signal constellation size for all sub-channels. However, the performance of these systems is dominated by the sub-channel(s) with the highest error probability. Hence, in order to improve the system-wide error probability, power or bit needs to be adaptively allocated to each sub-channels based on their channels states. According to the system performance improvement factors

and system constraints, the bit loading algorithms can be categorized as follows. 1) Minimizing the average BER - an algorithm that minimizes the average BER while satisfying target throughput or power requirements or both of them. [6] 2) Minimizing the transmission power - an algorithm that minimizes the transmission power while satisfying throughput or BER requirements or both of them. [4] [7] 3) Maximizing the throughput - an algorithm that maximizes the system throughput while satisfying power or BER requirements or both of them. [3] [5] In addition, the bit allocation algorithms can also be categorized as follows according to their operation: incremental algorithm (greedy algorithm) [3] [5], channel capacity approximation-based allocation [4] [7], and bit error probability expression-based allocation [6]. Incremental algorithms show optimal performance at the cost of high complexity. In contrast, the other two algorithms compromise performance for lower complexity. The main factor of adaptive bit allocation in dynamic channel condition is to dynamically assign a modulation scheme to each sub-channel according to Channel State Information (CSI). Therefore, the computation complexity of the bit allocation algorithm determining an appropriate modulation scheme for each sub-channel is important. So far, many previous algorithms have tried to reduce the computation complexity. Our goal is to maximize the system throughput with significantly reduced computational complexity while guaranteeing that the mean BER of all sub-channels remains below the pre-defined BER threshold. To do that, we group together sub-channels based on their SNRs. The main contribution of this work is to devise an algorithm that performs near optimally with significantly low complexity. The rest of this paper is organized as follows. In Section 2, we present adaptive bit allocation algorithm related works. In Section3, a system model is introduced. II. R ELATED W ORKS Wyglinkski et.al [5], proposed an incremental algorithm to maximize the throughput satisfying the requirement of the pre-

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

defined BER threshold in OFDM-based systems. The proposed incremental algorithm assigns the largest available signal constellation size to all sub-channels, and then computing the mean BER of all sub-channels. If the mean BER is above the pre-defined BER threshold, find the sub-channel(s) which have the worst performance and reduce the signal constellation size of the sub-channel(s). These steps are repeated until the mean BER of all sub-channels is below the pre-defined BER threshold. In channel capacity approximation-based allocation algorithm, a closed-form expression is used for bit or power allocation. For example, to find the number of bits for subchannel i according to SNR the following expression given by [1] can be used:   SN Ri (1) bi = log2 1 + Γ · γm where the number of allocated bit(s) in sub-channel i is denoted by bi ; SN Ri denotes a SNR value of sub-channel i (in dB); Γ denotes the SNR Gap (in dB); The system performance margin is denoted by γm (in dB). SNR Gap given by [1] presents how far the system is from the achieving capacity.  2  SER 1 Q−1 (2) Γ= 3 4 where Q−1 denotes the inverse q-function; SER denotes the symbol error rate. The algorithm proposed in [4] uses the modified expression for the noise-to-signal ratio (NSR) to obtain a constant uniform power allocation across all subchannels. The proposed algorithm computes all sub-channels ’SNR for unit input energy, and then the sub-channels with SNR value below a certain threshold are turned off. And then the total energy budget is assigned to all turned-on channels using optimal water-filling distribution. The bit loading algorithm proposed by J. Campello [7] uses channel capacity approximation for computing the bit allocation in all sub-channels guaranteeing the target bit rate. To minimize the average BER, the algorithm proposed by Fischer et.al [6] uses a closed-form of the bit error probability. In order to obtain the probability of bit error according to the signal constellation size, the closed-form expression given by [1] is used. For example, the bit error probability of BPSK can be approximated as follows:  (3) BERi ∼ = Q( 2SN Ri ) The bit error probability of Rectangular QAM, such as 4QAM, 16-QAM, ... , is described as follows: BERi,Mi = 1 −

2     2 1 3log2 Mi · SN Ri 1− 1− √ Q (4) log2 Mi Mi − 1 Mi where the signal constellation size of sub-channel i is denoted by Mi . In [6], BER is assumed to be constant for all

sub-channels. And then the algorithm allocates bit or power to each sub-channel to minimize the average BER. III. S YSTEM M ODEL OFDM-based systems divide the available bandwidth into N orthogonal sub-channels. With a sufficiently long cyclic prefix playing the role of a guard interval, these N sub-channels can be treated as independent parallel locally-flat channels corrupted by additive White Gaussian noise (AWGN). We assume that the bandwidth of all sub-channels is narrower than the coherence bandwidth. Therefore, we treat all sub-channels as flat fading channels. In addition, our system has point-to-point wireless link which consists of transmitter and receiver pairs. (called the single user OFDM system) For dynamic modification of bit allocation, we assume that the transmitter can obtain the channel information using a training signal which is known to the receiver via a feedback channel. H. Minn et.al [8] addressed optimal training signals in AWGN. The process for obtaining CSI is described as follows. The transmitter sends a pre-defined training signal on the specific sub-channel, and then the receiver performs channel estimation. The received signal is compared with the predefined training signal which is already known to the receiver. Then, the estimated CSI is fed back by the receiver. IV. P ROPOSED A LGORITHM We present two different algorithms in this section. The first algorithm tries to find the appropriate constellation size of each sub-channel while guaranteeing that each sub-channel’s BER remains below the pre-defined BER threshold. The second algorithm divides all sub-channels into L groups according to their SNR values. A. Problem Formulation The proposed algorithms try to solve the following equation. M aximize

N

bi

i=1

subject to

P¯ =

N

BERi · bi ≤ PT

N i=1 bi

i=1

(5)

where mean BER of all sub-channels is denoted by P¯ ; predefinded BER threshold PT ; BER of sub-channel i BERi . B. Simple Bit Allocation Algorithm In this algorithm, each sub-channel is assigned with the most appropriate the signal constellation size that satisfies the BER constraint. To obtain such a constellation size, we can use the closed-form expression of bit error probability. The detailed operation of this algorithm is described as follows. 1) Initialization Step: assign the largest constellation size to all sub-channels. 2) Calculate P¯ using (5). 3) Compare P¯ with PT . If P¯ is less than or equal to PT , the current configuration is kept and the algorithm ends, otherwise go to step 4.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

4) For k = 1 to N (N: total number of sub-channels) a) Calculate the BER of kth sub-channel with current SNR value using (3) or (4), by applying the largest constellation size b) If the resulting BER is above the PT , reduce the constellation size until the resulting BER is less than or equal to PT . If the resulting BER employing the smallest constellation size is above the PT , turn off the current sub-channel. C. Fast Bit Allocation Algorithm We present the group-based adaptive bit allocation algorithm that tries to maximize the overall throughput of the system while guaranteeing that the mean BER remains below the pre-defined BER threshold. In this algorithm, sub-channels are divided into several groups according to their SNR values. Sub-channels in each group are employed with the same modulation scheme. Grouping of the sub-channels is intended to reduce the computational complexity.     SN Rj Ri is greater than 1 + (1) shows that if 1 + SN Γ·γm Γ·γm by 2β , sub-channel i can carry β more bit(s) than sub-channel j. Using this, we can derive Dj (β) (in dB) which represents the minimum required SNR difference to carry β more bit(s) than sub-channel j.   β (2 − 1) · Γ · γm β (6) +2 Dj (β) = SN Rj When the SNR values of all sub-channels are within the range of SN Rmin to SN Rmax , we can divide all subchannels using SN Rmin as a criterion. In fact, the subchannel(s) with SN Rmin may not carry even 1 bit. To increase the overall system performance, this bad-channel should be turned off. Given the SNR gap and the system performance margin, we ´ min ) to can calculate the minimum required SNR value (SN R carry at least 1 bit using following equation derived from (1): ´ min = Γ · γm SN R

(7)

´ min , sub-channels whose If SN Rmin is less than the SN R ´ SNR values is less than SN Rmin are turned off during the initialization step, and then SN Rmin is replaced with ´ min . SN R Prior to grouping of sub-channels, we have to determine the maximum number of groups with a set of available signal constellation size and a SNR difference between SN Rmax and SN Rmin The number of groups can be obtained by using following described steps: ˆ that satisfies (8). L ˆ represents that 1) Obtain the L how many groups we can make when α is determined. e.g., Given allowable signal constellation set = {2, 4, 16, 64}, if α = 4, i(α) = 3 and then, we can make at most two groups using (8) because of

the number of rested available modulation scheme(s) (n − i(α) = 1) . ˜ that satisfies (9) among possible 2) Obtain the minimum L ˜ values. L represents that how many groups we can make for a given GAP. If (9) cannot be satisfied with ´ ˜ is n − i(α). maximum L(= n − i(α)), L 3) Obtain the number of groups using (10) ˆ = (n − i(α)) + 1 such that L βi(α) = α f or 1 ≤ i(α) ≤ n

(8)

˜ = minimum L ´ such that L ´ ≤ n − i(α) (9) GAPmax < Di(min) (ΔL´ ) f or 1 ≤ L ˆ L] ˜ L = minimum[L,

(10)

where i(min) denotes the index of sub-channel(s) whose SNR value is equal to SN Rmin ; α denotes the maximum number of bit(s) which can be carried by sub-channel i(min); β denotes a set of available bit(s); i(α) denotes the index of β’s element whose value is equal to α; n = |β|; GAPz denotes SN Rz − SN Rmin ; Δk = βi(α)+k − βi(α) . e.g., if available constellation size = {2, 4, 16, 64} and α = 1, then β = {1, 2, 4, 6}, n = 4, i(α) = 1, Δ1 = 1, Δ2 = 3 and Δ3 = 5. Similarly to obtaining the number of groups, we can divide all sub-channels into L group(s) and assign the most appropriate constellation size to each group by using the following equations: Groupk = {z|Di(min) (Δk−1 ) ≤ GAPz < Di(min) (Δk )} f or 1 < k < L, z ∈ all sub − channel (11) T he constellation size of Groupk = 2βi(α)+k−1

(12)

The detailed algorithm is described as follows. 1) Calculate BER of the sub-channel i(min) using (3) or (4), by applying the largest available signal constellation size. 2) If the resulting BER is above PT go to step 3, otherwise the largest constellation size is assigned to all subchannels and the algorithm ends. 3) Reduce the constellation size until satisfying that the resulting BER is less than or equal to PT , and then assigns α = log2 (current constellation size). 4) Grouping Step: Using (11) and (12), divides all subchannels into L group(s) and assigns the appropriate signal constellation size to each group. 5) Intermediate Bit Allocation Step: Calculate P¯ of all sub-channels using (5). If the resulting P¯ is above PT go to step 7, if not go to step 6. 6) Maximization Step: Search for the sub-channel j with the best resulting BER and increase the signal constellation size. If the available largest constellation size had

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

V. S IMULATION

350

Throughput(bit(s) per sym bol)

been assigned to the sub-channel already, find a next candidate. Repeat this step while P¯ ≤ PT 7) Fine-tuning Step: Search for the sub-channel j with the worst resulting BER and reduce the signal constellation size. If the smallest constellation size had been assigned to the sub-channel already, null the sub-channel (i.e., turn off the sub-channel). Repeat this step while P¯ > PT

Fast

250

Exhaustive

200

Wyglinkski

150

100 50 0

A. Simulation Set Up

The simulation results of 52 sub-channels with BER threshold 10−6 and 10−7 , as presented in Fig. 1 and Fig. 2 respectively, show that all algorithms except the simple algorithm reach the optimal overall throughput obtained by an exhaustive search. The incremental algorithm proposed by Wyglinski et.al [5] reaches the same throughput. We considered grouping all of the sub-channels to reduce the computational complexity. We reduced the computational times by assigning the most appropriate signal constellation size to each sub-channel according to its SNR value. The results of the computational times, as shown in Fig. 3 and Fig. 4 respectively, show that our algorithms executes faster than Wyglinski’s algorithm. Our proposed algorithms have nearconstant computational times over all SNR values. However, the algorithm of Wyglinski has higher computational times than our proposed algorithms when SNR values is smaller than 25 dB. This is why our proposed algorithm does not need to compute P¯ in every bit allocation step no matter whether P¯ violates the constraint or not. Up to now, we have considered the same SNR values ranging from 0 to 30dB across all sub-channels. To simulate

2.5

350

Throughput(bit(s) per sym bol)

5

7.5

10 12.5 15 17.5 20 22.5 25 27.5 30 dB,Eb/No

The overall throughput for 52 sub-channels with PT = 10−6

Fig. 1.

Simple

300

Fast

250

Exhaustive

200

Wyglinkski

150 100 50 0 0

2.5

5

7.5

10 12.5 15 17.5 20 22.5 25 27.5 30 dB,Eb/No

The overall throughput for 52 sub-channels with PT = 10−7

Fig. 2. 0.18

Wyglinsky

0.16

Com putation Tim e (sec)

B. Simulation Result

0

Simple

0.14

Fast

0.12 0.1 0.08 0.06

0.04 0.02 0 0

2.5

5

7.5

10 12.5 15 17.5 20 22.5 25 27.5 30 dB,Eb/No

The computation times for 52 sub-channels with PT = 10−6

Fig. 3. 0.2

Wyglinsky

0.18

Com putation Tim e (sec)

The performance of the proposed algorithm is evaluated in comparison with other algorithms. The OFDM-based systems that we employed are based on The IEEE Standard 802.11a system [2]. However, unlike the standard, our algorithm can employ a different constellation size. We assume that the available signal constellation sizes are {2, 4, 16, 64} in this simulation. According to the signal constellation size, the available modulation schemes are {BPSK, 4-QAM, 16-QAM, 64-QAM} and βk are {1, 2, 4, 6}. The channel model we used in this simulation is the frequency selective Rayleigh fading channel similar to the well-known Jakes channel model. For practical channel realization, each sub-channel operates under different SNR values ranging from 0 to 30 dB. Simulations are repeated 10000 times under different channel condition and the results are averaged. We evaluate the performance of our algorithm in comparison with other previous algorithms using averaged results. 10−6 and 10−7 are used as the pre-defined BER threshold. We assume that all sub-channels use an uncoded modulation schemes. 8.8 dB for 10−6 and 9.5 dB for 10−7 are used as the SNR gap.

Simple

300

0.16

Simple

0.14

Fast

0.12 0.1 0.08

0.06 0.04 0.02

0 0

Fig. 4.

2.5

5

7.5

10 12.5 15 17.5 20 22.5 25 27.5 30 dB,Eb/No

The computation times for 52 sub-channels with PT = 10−7

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

TABLE I AVERAGED R ESULT O F 10000 R EPEATED S IMULATION BER threshold (PT ) 10−6 10−7

Algorithms

Mean BER (P¯ )

Simple Fast Wyglinski Simple Fast Wyglinski

8.84E-9 7.97E-6 7.97E-6 1.60E-10 6.86E-8 6.86E-8

Mean Averaged Computation Throughput Time (sec) 131.68 0.03011 141.72 0.03038 141.72 0.09984 117.53 0.03204 124.98 0.03236 124.98 0.10748

an extreme channel condition, we randomly choose the SNR value of each channel within the range of -10 to 30dB. Table.I shows the averaged results of 10000 repeated simulations. All algorithms except the simple algorithm reach the same overall throughput, and the mean BER of all sub-channels do not violate PT . The simple and the fast algorithms reduce the computational times by about 70% and about 72% respectively, compared to the algorithm proposed by Wyglinsky et.al [5]. VI. C ONCLUSION In this paper, we deal with the problem of bit allocation for point-to-point wireless links which consist of single transmitter and single receiver pairs. Our main idea is to group together sub-channels according to their SNR values. The proposed algorithm tries to maximize the overall throughput of the system while guaranteeing that mean BER of all subchannels remains below the pre-defined BER threshold. The grouping is adaptive to the current channel state. We showed that the proposed scheme has outstanding performance when compared to other schemes in terms of computational times. In addition, the proposed algorithm is close to the optimum overall throughput with significantly lower complexity. The aim of our future work is to design another bit loading scheme for OFDM in consideration of power-metric. ACKNOWLEDGMENT This works was supported by Hyundai Motor Company. R EFERENCES [1] J. G. Proakis, Communication Systems Engineering, 2nd ed. PrenticeHall, 2002. [2] ’IEEE Std. 802.11a: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: high-spped physical layer in the 5 GHz band’, 1999. [3] B. Fox, ”Discrete optimization via marginal analysis,” Manage. Sci., vol. 13, no. 3, pp. 210-216, Nov. 1966. [4] A. Leke and J. M. Cioffi, ”A maximum rate loading algorithm for discrete multitone modulation systems,” in Proc. IEEE Global Telecommunications Conf., Phoenix, AZ, 1997, vol. 3, pp. 1514-1518. [5] A.M. Wyglinski, F. Labeau, and P. Kabal, ”Bit loading with BERconstraint for multicarrier systems,” IEEE Trans. Wirel. Commun., 2005, vol 4, no. 4, pp. 1383 - 1387. [6] R. Fischer, ”A new loading algorithm for discrete multitone modulation,” in Proc. IEEE Global Telecommunications Conf., pp. 724-728m March. 1996, London, IK. [7] J. Campello ”Practical bit loading for DMT,” in Proc. IEEE Int. Conf. Communications, Vancouber, Canada, 1999, col.2, pp.801-905

[8] H.Minn and S.Xing, ”An optimal training signal structure for frequency offset esimation,” IEEE Trans. Commun., vol. 53, no. 3, pp.343-355, Feb. 2005