Flexible Adaptive-Modulation-and-Coding Tables for a Wireless Network

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Flexible Adaptive-Modulation-and-Coding Tables for a Wireless Network ∗

Edward W. Jang∗ , Chan-Soo Hwang† , and John M. Cioffi∗

STAR Laboratory, Stanford University, Stanford, CA 94305, U.S.A. Email: {ej1130, cioffi}@stanford.edu † Communication & Network Lab., Samsung Advanced Institute of Technology, Yongin, Korea Email: [email protected]

Abstract— This paper proposes to use flexible adaptivemodulation-and-coding (AMC) tables in a wireless network. To support the flexibility of AMC tables, a low-complexity AMC table optimization algorithm is also proposed. The proposed algorithm iteratively optimizes the switching levels of an AMC table according to channel environments, and has fast convergence speed. Computer simulation results show that the proposed algorithm optimizes AMC tables according to different cell characteristics, and that using flexible AMC tables for a wireless network achieves near-optimal average spectral efficiency even with a small feedback load.

I. I NTRODUCTION Many recent cellular systems, such as IEEE 802.16e standard [1], employ an adaptive-modulation-and-coding (AMC) scheme to adjust the transmission rate according to the channel state information (CSI), thereby achieving high spectral efficiency while guaranteeing sufficient reliability [2]. Specifically, an AMC scheme assigns the CSI intervals, bounded by switching levels, to combinations of a modulation size and a code rate based on an AMC table. When a user’s CSI falls in an interval, the user feeds the interval’s index back to the base station (BS), and the BS transmits a packet to the user with the corresponding transmission rate. With more switching levels in the AMC table, the spectral efficiency increases because round-off errors decrease; however, the feedback load also increases with the number of switching levels. This feedback overhead is especially problematic for an orthogonal-frequency-division multiple-access (OFDMA) system, where each user needs to feed AMC table indexes for multiple subchannels. Therefore, a judicious choice of the AMC table’s switching levels is crucial to increase the spectral efficiency while minimizing the feedback load. The optimal set of switching levels of an AMC table crucially depends on many channel characteristics, such as a users’ geographical distribution, a coverage radius, path loss, shadowing, and transmit power. Since a wireless network often comprises channels with different characteristics, e.g., a cellular system has urban microcells as well as suburban macrocells, employing flexible AMC tables optimized for different channel characteristics significantly improves the overall spectral efficiency of a wireless network compared to the case of employing a fixed AMC table across channels with different characteristics. However, using flexible AMC tables requires

an efficient algorithm that optimizes the switching levels for arbitrary channel state distributions. Currently applicable algorithms require a large computational complexity to optimize an AMC-table, i.e., to find the switching levels that maximize the average spectral efficiency. This optimization problem was first considered by Holm et al., who found a recursive solution for a Rayleigh fading channel in [3]. However, the AMC-table optimization for a general channel state distribution has remained an open problem. Other algorithms can be applied by relating this AMC-table optimization problem to the random variable quantization problem. However, the widely-known Lloyd algorithm [4] or the algorithms using sufficient conditions for the optimality [5] are not straightforwardly applicable because the distortion measure for this optimization problem is asymmetric. The dynamic-programming-based quantization algorithm [6], which includes asymmetric distortion measure cases, is applicable for this optimization problem; however, its complexity is prohibitively high due to the lack of a boundary condition for switching levels. To solve this problem, this paper proposes an iterative flexible-AMC-table optimization (IFAO) algorithm that iteratively finds one switching level of an AMC table per step by using the necessary conditions for the optimality. This iterative approach significantly reduces the computational complexity compared to other algorithms that simultaneously optimize all switching levels [3], [6]. This is because using only the necessary conditions reduces the computational complexity that would have incurred if both the necessary and sufficient conditions were used to solve the problem. Simulation results show that the proposed IFAO algorithm optimizes an AMC table that achieves the average spectral efficiency, very close to the unlimited feedback case, with only a small feedback load. The simulation results also show that using flexible AMC tables, instead of using a fixed AMC table, significantly improves the average spectral efficiency of a wireless network. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT The system model considers a downlink channel, consisting of a BS with one transmit antenna and multiple users, each with one receive antenna. Although the channel between the BS and a user is frequency-selective, OFDMA divides the channel into multiple frequency-flat subchannels that are

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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 ICC 2008 proceedings.

T0 = 0

C1

C2

T1

T2

Fig. 1.

SNR

CN-1

T3

TN-1

TN =

8

outage C0 = 0

An example of the AMC table

A. Heuristic AMC-Table Design Algorithms

independent and wide-sense stationary. The signal-to-noisepower-ratio (SNR) distributions of these subchannels depend not only on short-term fading statistics, such as Rayleigh fading, but also on both channel environment and system parameters, e.g., a users’ geographical distribution, coverage radius, path loss, shadowing, and the BS’s transmit power. The BS uses the SNR distribution to optimize a flexible AMC table, which relates transmission rates Ci to the SNR intervals bounded by switching levels Ti and Ti+1 , as shown in Fig. 1. The number of intervals N is equal to 2b when the feedback load is b-bit per user. The first interval denotes the SNR range that causes an outage (C0 = 0), and the rate Ci increases with i. This paper assumes that each user has perfect knowledge of its SNR and feeds the corresponding interval index back to the BS through a zero-delay and error-free feedback channel. The BS sends a packet to a scheduled user using the transmission rate associated with the received index. The transmission is assumed to be performed in an orthogonal manner, i.e., only one user is assigned to each time/frequency slot. The maximum spectral efficiency C of an additive-whiteGaussian-noise channel is C = log2 (1+SN R) when the exact SNR is known to the transmitter [7]. However, the BS in this paper’s system model only knows the fed-back interval index of the AMC table; therefore, when the scheduled user’s SNR is Ti ≤ SN R < Ti+1 , the BS transmits to the user with a transmission rate Ci = log2 (1 + Ti ), which is the maximum spectral efficiency that guarantees error-free transmission for the given SNR range. As a result, the maximum average spectral efficiency for an adaptive-coded-modulation system (MASA) [3] with b-bit feedback load is, MASA

= =

N −1
i=0 N −1
i=0

Ci Pr{Ti ≤ SN R < Ti+1 }  log2 (1 + Ti )

Ti+1 Ti

fSN R (x)dx,

two heuristic AMC-table design algorithms are introduced. Next, the sufficient and necessary conditions for the optimal AMC table are shown, followed by a low-complexity iterative flexible AMC-table optimization (IFAO) algorithm based on the necessary conditions for the optimality.

(1)

where T0 = 0 < T1 < · · · < TN −1 < TN = ∞ and fSN R (·) denotes the average probability distribution function (PDF) of users’ SNRs. This PDF collectively represents the channel statistics of all users in a cell, thus is dependent on the channel characteristics of the cell. The objective of an optimization algorithm is to find the optimal set of switching levels that maximizes the MASA. III. D ESIGNING THE O PTIMAL AMC TABLE This section considers the design of the optimal AMC table that maximizes the MASA given a SNR distribution. First,

This subsection considers two simple heuristic AMC-table design algorithms: the equi-probable partition algorithm and the equi-rate partition algorithm [8]. First, the equi-probable partition algorithm divides the SNR to N intervals such that the probability of the SNR being in an interval Ti ≤ SN R < Ti+1 is equal for all i from 0 to N − 1, where T0 = 0 < T1 < · · · < TN −1 < TN = ∞: 1 for all i = 0, · · · , N − 1. (2) N This equi-probable partition algorithm finds the switching levels only based on the SNR distribution, ignoring its effect on the transmission rate. Second, the equi-rate partition algorithm [8] finds a switching level set such that the list of possible transmission rates {C1 , · · · , CN} is symmetric around the average transmission ∞ rate, µ = 0 log2 (1 + x)fSN R (x)dx, and has an equal granularity α:

Pr{Ti ≤ SN R < Ti+1 } =

{C1 , · · · , CN }

= {log2 (1 + T1 ), · · · , log2 (1 + TN −1 )} = {µ(1 − nα), · · · , µ(1 − α), µ, µ(1 + α), · · · , µ(1 + nα)},

(3)

where α is optimized to maximize the MASA. Contrary to the equi-probable partition algorithm, the equi-rate partition algorithm considers the switching levels’ effect on the transmission rate. B. The Sufficient and Necessary Conditions for the Optimal AMC Table When the feedback load is b-bit, the sufficient condition for the optimal AMC table is, −1 {Ti∗ }N i=1

=

arg max MASA

=

arg max

(4)

−1 {Ti }N i=1

N −1


−1 {Ti }N i=1 i=0

 log2 (1 + Ti )

Ti+1 Ti

fSN R (x)dx,

where T0 = 0 < T1 < · · · < TN −1 < TN = ∞. The −1 complexity of simultaneously finding the optimal {Ti }N i=1 using (5) is enormously high for large N , because the effect of Ti on the MASA is interwound with those of Tj,j=i . However, the necessary conditions for the optimal AMC table are relatively simple – the necessary conditions only depend on adjacent switching levels:   Ti ∗ fSN R (x)dx Ti = arg max log2 (1 + Ti−1 ) Ti

 + log2 (1 + Ti )

Ti−1

Ti+1 Ti



fSN R (x)dx ,

(5)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

T20

T30

T40

TN-10

TN =

T30

T40

TN-10

TN =

T40

TN-10

TN =

8

T10

8

T0 = 0

8

optimize T1 given T0 and T2

optimize T2 given T1 and T3

T11 T20

T0 = 0

optimize T3 given T2 and T4

T11

T0 = 0 Fig. 2.

T21

T30

Forward optimization of the switching levels

for i = 1, · · · , N − 1 where T0 = 0 < T1 < · · · < TN −1 < TN = ∞. The next subsection proposes the IFAO algorithm that uses these necessary conditions to find the optimal switching levels. C. Iterative Flexible AMC-Table Optimization Algorithm By using the necessary conditions (5), the proposed IFAO algorithm decomposes the complex problem into many simple sub-problems. Instead of simultaneously optimizing all switching levels, the IFAO algorithm optimizes only one switching level at each step. As a result, the IFAO algorithm’s complexity is low, regardless of the number of switching levels; on the contrary, simultaneous-optimization algorithms’ complexity prohibitively increases with the number of switching levels. Table 1. Iterative Flexible AMC-Table Optimization Initialization: initialize {Ti }N i=0 Recursion: Forward update: for i = 1 to N − 1 optimize Ti using Ti−1 and Ti+1 for (5) end Backward update: for i = N − 2 to 1 optimize Ti using Ti−1 and Ti+1 for (5) end repeat until M ASA converges or reaches the maximum number of iterations Result: N {Ti∗ }N i=0 = {Ti }i=0 ∗ M ASA = M ASA The IFAO algorithm starts by initializing Ti for i = 0, · · · , N to values obtained by a heuristic partition algorithm, such as the algorithms in subsection III-A. Then the IFAO

algorithm begins the forward optimization as shown in Fig. 2, where the superscript k of Tik denotes the number of iterations. The forward optimization starts by solving (5) using gradient method or grid search for the first switching level T1 . In the subsequent steps, the forward optimization similarly finds the optimal Ti using (5) while increasing i from 2 to N − 1. The backward optimization ensues by optimizing Ti using (5) while decreasing i from i = N − 2 to 1. The proposed IFAO algorithm iterates these forward and backward optimizations until the MASA converges within a prescribed threshold or until the number of iterations exceeds a prescribed number. This procedure is summarized in Table 1. Since the IFAO algorithm monotonically increases the MASA, which is upperbounded by the unlimited feedback case, the algorithm can be shown to converge. IV. S IMULATION R ESULTS Using computer simulations, this section compares the MASA of a wireless network achieved with AMC tables designed by the proposed IFAO algorithm and with AMC tables designed by other heuristic design algorithms. The simulation results also show how the MASA of a wireless network is affected by using flexible AMC tables versus by using a fixed AMC table. To simulate for different channel environments, a cellular system is considered. The cellular system consists of multiple round-shaped cells that have the same parameters with one of the four different cell scenarios defined in the 3GPP UMTS channel model [9]: suburban macrocell, urban macrocell, urban microcell-non-line-of-sight (NLOS), and urban microcellline-of-sight (LOS). Table 2 shows the parameters used for the channel scenarios, where fc , hBS , hM S , dBS−BS , and σSF respectively denote the carrier frequency, the BS’s antenna height, the user’s antenna height, the distance between BSs, and the standard deviation for log-normal shadow fading.

fc hBS hM S dBS−BS σSF

Table 2. Channel Scenarios suburban urban urban macrocell macrocell microcell 1900 MHz 1900 MHz 1900 MHz 32 m 32 m 12.5 m 1.5 m 1.5 m 1.5 m 3 Km 3 Km 1 Km 8 dB 8 dB NLOS: 10 dB LOS: 4 dB

The path-loss models for the channel scenarios are as follows:  31.5 + 35 log10 (d) suburban macrocell    urban macrocell 34.5 + 35 log10 (d) P L (dB) = (d) urban microcell-NLOS 34.53 + 38 log  10   30.18 + 26 log10 (d) urban microcell-LOS where P L denotes the path loss, and the distance d should be at least 35 m and 20 m for macrocells and microcells, respectively. For each cell scenario, the user’s empirical SNR

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

distribution is obtained by randomly generating the SNR, considering the path loss and the shadowing, at different locations within a cell. Although the simulations ignore inter-cell interference, the extension is straightforward by obtaining the distribution of users’ signal-to-interference-and-noise power ratio (SINR) instead of the SNR. It is assumed that the users are uniformly distributed in a cell, and a BS randomly schedules one user per subchannel, according to the transmission rate defined by the AMC table. A. The Maximum Average Spectral Efficiency

0.5

2−bit partition for urban macrocell 2 MASA (bit/s/Hz)

MASA (bit/s/Hz)

1−bit partition for urban macrocell 2 max IFAO 1.5 EQR EQP 1

0 20

MASA (bit/s/Hz)

0 20

MASA Gain (%)

Fig. 3.

25 30 TX power (dBm/MHz) 4−bit partition for urban macrocell 2

25 TX power (dBm/MHz)

1−bit partition for urban macrocell 150 IFAO over EQR IFAO over EQP 100

25 TX power (dBm/MHz)

30

50

2−bit partition for urban macrocell 80

20

0 20

25 TX power (dBm/MHz)

30

40 20

25 30 TX power (dBm/MHz) 4−bit partition for urban macrocell 60 MASA Gain (%)

40

60

0 20

25 30 TX power (dBm/MHz) 3−bit partition for urban macrocell 60 MASA Gain (%)

1 0.5

The MASA of an urban macrocell for the different algorithms

0 20

Fig. 4.

1.5

0 20

30

MASA Gain (%)

MASA (bit/s/Hz)

1 0.5

1 0.5 0 20

25 30 TX power (dBm/MHz) 3−bit partition for urban macrocell 2 1.5

1.5

40

20

0 20

25 TX power (dBm/MHz)

30

The MASA gain of the IFAO algorithm for an urban macrocell

Figure 3 shows the MASA for an urban macrocell for the feedback load ranging from 1-bit to 4-bit while varying the BS’s transmit power from 20 dBm/MHz to 30 dBm/MHz. The MASAs achieved with the IFAO algorithm, the equi-rate partition algorithm, the equi-probable partition algorithm, and unlimited feedback are denoted as ‘IFAO’, ‘EQR’, ‘EQP’, and ‘max’, respectively. The figure shows that the MASA achieved

by the IFAO algorithm approaches the MASA achieved by unlimited feedback with only a small feedback load, e.g., the IFAO algorithm achieves 70 % of the MASA achieved with unlimited feedback with only 3-bit feedback. Figure 4 shows that the proposed IFAO algorithm achieves significantly higher MASA over both the equi-rate partition algorithm and the equi-probable partition algorithm. For 1bit and 2-bit feedback cases, the equi-rate partition algorithm performs better than the equi-probable partition algorithm; however, the situation is opposite for 3-bit and 4-bit feedback cases. This transition is because the SNR distribution is skewed to the right. When the number of the switching levels is small, the equi-rate partition algorithm tends to set the switching levels in the high SNR region, resulting in high MASA. However, when the number of the switching levels is large, the equal granularity limits the equi-rate partition algorithm from setting the switching levels high enough. If the SNR is represented on x-axis with log scale, and the spectral efficiency is represented on y-axis, the plot follows an exponential curve for low SNR and linear curve for midto-high SNR. Therefore, to guarantee the equal granularity, the interval of the switching levels should be larger for lower SNR in dB. This imposes a limit on the optimal granularity size that the equi-rate partition algorithm can take. Therefore, the equi-rate partition algorithm has inherent limit on setting the switching levels, resulting in loss in the MASA. The IFAO algorithm, however, has no such limitation, thus, provides higher MASA than the heuristic partition algorithms. The IFAO algorithm’s relative MASA gain over the heuristic partition algorithms decreases when the BS’s transmit power is large. This is because the marginal transmission-rate gain diminishes with increasing SNR, which in turn makes MASA less sensitive to the choice of the switching levels. Although it is not shown here because of the lack of space, the IFAO algorithm converges fast, i.e., the IFAO algorithm converges after only one iteration of forward and backward optimization when the convergence threshold is 1 %. This fast convergence holds true not only for urban macrocell, but also for suburban macrocell, urban microcell-NLOS, and urban microcell-LOS. B. The Effect of Employing Flexible AMC Tables Figure 5 shows how using flexible AMC tables, each optimized for different cell scenarios, instead of using a fixed AMC table for all cell scenarios, affects the MASA of the cellular system. The system is assumed to have an equal number of cells for each of 4 cell scenarios. In the figure, ‘flexible’ denotes the case when each cell uses its own flexible AMC table, optimized for the user’s SNR distribution of that cell, and ‘fixed’ denotes the case when all the different cells share a single fixed AMC table, which is optimized for the average SNR probability distribution over four equallyweighted cell scenarios. The numbers in the legend denote the BS’s transmit power in dBm/MHz. As shown in the figure, employing flexible AMC tables improves the MASA by approximately 100 % for all feedback loads. Although in practice, the percentage of each cell scenario may be unequal,

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

table. As a result, among a large set of possible transmission rates for a wireless network, each flexible AMC table selects a subset of possible transmission rates according to the channel environment.

mixed channel scenario (1,1,1,1) 5.5 5 4.5

R EFERENCES

MASA (bit/s/Hz)

4 3.5 30.0−flexible 30.0−fixed 27.5−flexible 27.5−fixed 25.0−flexible 25.0−fixed 22.5−flexible 22.5−fixed 20.0−flexible 20.0−fixed

3 2.5 2 1.5 1

1

1.5

2

2.5 feedback load (bit)

3

3.5

Fig. 5.

The effect of employing flexible AMC tables

4

and the BS’s transmit power may vary for each cell scenario, the simulation result suggests that employing flexible AMC tables significantly improves the MASA of a wireless network. V. C ONCLUSION This paper proposes to use flexible AMC tables, instead of a fixed AMC table, for a wireless network. An efficient AMCtable optimization algorithm, necessary to employ flexible AMC tables, is also proposed. The proposed IFAO algorithm efficiently finds the optimal switching levels by the iterations of the forward and backward optimizations, which typically converge within 1 % in terms of the MASA only after one iteration. The IFAO algorithm has two major benefits: First, the IFAO algorithm finds an optimal AMC table for an arbitrary channel-state distribution, thereby, enabling the use of flexible AMC tables across the system, which is shown to significantly improve the MASA of a wireless network by computer simulations. Second, the IFAO algorithm requires low computational complexity; thus, the IFAO algorithm can be effectively used for a wireless network that frequently updates its operational parameters, such as the BS’s transmit power. For example, in CDMA systems with cell-breathing technique, an overpopulated cell decreases its BS’s transmit power, reducing its effective cell radius; as a result, the adjacent cells take over excessive users from the over-populated cell, balancing the users’ traffic load across cells. By using the IFAO algorithm, all these cells can quickly regenerate their own flexible AMC tables to maximize the MASA because of the IFAO algorithm’s low computational complexity. The proposed IFAO algorithm can be extended to the case when implementation issues limit the number of possible modulation and coding schemes. First, the IFAO scheme is used to find the flexible AMC table optimized for a channel environment. Then, each switching level is floored to the closest transmission rate possible. Finally, the converted set of switching levels is used to determine the flexible AMC

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