Variable Bit Allocation For FH-CDMA Wireless Communication Systems 1

Transcription

1 Variable Bit Allocation For FH-CDMA Wireless Communication Systems 1 Charles C. Wang 2 Gregory J. Pottie Jet Propulsion Laboratory Electrical Engineering Department Mail Stop University of California, Los Angeles 4800 Oak Grove Avenue 405 Hilgard Avenue Pasadena, California Los Angeles, CA Abstract In the wireless indoor environment, the channel may vary slowly as users and the interferers may move at slow speeds. A frequency hopped CDMA (FH-CDMA) system can adapt to the different interference levels in hopping patterns and assign the slots different bit rates to increase the system capacity. We show that the maximum throughput bit rate/channel assignment problem is NP-hard. Several practical ad hoc bit allocation algorithms are designed based on the insights derived from exhaustive searches. The algorithms that achieve the most system capacity perform interference avoidance. Users concentrate their throughput in a small fraction of the slots with low interference by transmitting large signal constellations, while avoiding the channels with large interference. Simulations show that the flexibility of users to adjust their bit rates to the interference environment can significantly increases the system capacity. 1. This work is funded by Motorola Inc., under the terms of a university partnerships in research grant. 2. Corresponding author. 1

2 1 Introduction With limited spectrum and increasing demand for services, the next generation of wireless communication systems need to utilize bandwidth more efficiently to support more users. Slow frequency hopped code-division multiple access (FH-CDMA) is proposed as it is capable of achieving high system capacity [1][2][3]. A slow hopped CDMA system divides the available spectrum and time into non-overlapped frequency bins and time slots. A user transmits in different frequency bins at different time slots, each containing several symbols. The sequence of timefrequency slots that a user transmits is called a hopping pattern. In this paper, we assume the hopping patterns of all users are periodic with the same period. Like most of the high capacity wireless systems where the available bandwidth is limited, an FH-CDMA system is interference limited. The transmitted power of all but the desired signal is seen as interference by the receivers. Channel assignment strategies that reduce the interference each user generates can directly increase the number of users that can communicate simultaneously. In an indoor wireless FH-CDMA system, the channels vary slowly as users and interferers move at low speeds. The system can take advantage of the different interference levels in a hopping pattern and assign the slots different bit rates to minimize the interference. Assuming all users desire the same data rate, our goal is to design bit allocation algorithms that can provide services to the most users. Rayleigh fading is not included in the simulations presented in this paper since it has already been shown that systems with Rayleigh fading can obtain capacities close to systems operating in additive white Gaussian channels [4], particularly when diversity techniques are employed. In addition, the use of adaptive antenna arrays has been shown to reduce the effects of fading greatly [5]. Here, we are more interested in showing the relative performances of the different proposed bit allocation algorithms. The effects of fading are, however, important in the implementation of an FH-CDMA system. We assume that the indoor wireless system with its high symbol rate can track the slowly changing fading environment. 2

3 We formulate the problem of bit allocation in an FH-CDMA system and present the simulation results in the context of the reverse link (users transmitting to base stations) of a cellular wireless system. The bit allocation algorithms can also be applied to the forward link of a cellular system or to a peer-to-peer wireless system. All of the variable bit algorithms described in this paper can be performed in a distributed fashion such that no centralized control is necessary. The simulation results presented here are for comparisons of different algorithms and are meant to be upperbounds for the simplified propagation models used. The results show that the interference avoidance strategies and the flexibility of users to adjust their bit rates to the interference environment can significantly increase the system capacity. This paper is divided into five sections. In Section 2, we study the problem of bit allocation in an indoor FH-CDMA wireless system. In Section 3, we prove that the problem of finding the channel allocations and bit assignments that maximizes the system throughput is NP-hard. The results from exhaustive searches of the bit allocations that maximize the system capacity are presented. In Section 4, several bit allocation algorithms developed from the insights obtained from the exhaustive searches are described. Simulation results of these algorithms are also presented. We then offer some concluding remarks in Section 5. 2 Variable Bit Allocation A frequency hopped CDMA (FH-CDMA) system can adapt to the different interference levels in a hopping pattern and assign to the slots different bit rates to maximize the system capacity. In an FH-CDMA system with variable bit allocation, the users can constantly monitor the interference statistics in all of the slots in their hopping patterns. They may then decide to transmit the larger signal constellations in the slots where there is little interference and transmit at a very low data rate or send no data at all in the slots where interference is already high. This flexibility allows the FH-CDMA system to achieve higher system capacity. At first glance, an FH-CDMA wireless system with variable bit allocation is similar to a 3

4 multitone wireline communication system [6][7]. In both of these systems, the users first probe the channels that are available to obtain channel statistics. There are many distributed channel probing algorithms proposed for wireless communication systems[8][9][10]. A new user transmits at several different power levels and monitors how the users sharing the same channels react to the interference the new user generates. The maximum achievable signal-to-interference ratio (SIR), the transmit power needed to achieve the desired SIR, and other statistics regarding power and interference can be estimated. The users in the multitone wireline system and the FH-CDMA system then allocate bits and power in the slots that have favorable channel conditions. In the multitone wireline communication system, however, the system has an average power constraint which limits the total transmit power of each link. In an FH-CDMA wireless system, there is usually no such limitation, although low transmit power is desirable as it generally leads to low interference. A maximum transmit power for every slot due to hardware limitations is more often the case. In addition, an FH-CDMA wireless system is almost always a multi-user system where the transmit power of one user interferes with the signals of the others. This leads to a coupling between power allocation and interference levels which complicates the bit allocation decisions. Thus, before we continue our discussion on finding the bit allocation algorithms that maximize the capacity of an FH-CDMA system and their complexities, we briefly describe the problem of power control. We assume there are B users in the system. In order to maintain the desired error performance, different SIR s are needed for different bit rates/signal constellation sizes. For now, we assume that the users know the bit rates for the slots and the SIR needed. If user i needs to obtain SIR greater than or equal to some SIR threshold γ ik to achieve the bit rate that it desires in the k-th slot, the user s transmit power needs to satisfy the following inequality: k P ik G ii γ k ik P jk G ij + n ik j i (1) where P jk is the transmit power of the j-th user in the k-th slot, n ik is equal to the noise floor at the 4

5 base station communicating with user i, and k G ij is the link gain from user j to the base station communicating with user i. The left hand side of (1) is simply the received power of user i divided by the sum of all of the interference and noise at the base station. We assume that and are independent of the frequency slots used and omit the superscript and subscript k in our notation. We then multiply the denominator of (1) on both sides and collect all of the terms associated with power to the left side of the inequality: k G ij n ik P ik G ii γ ik P jk G ij γ ik n i j i (2) There are B such linear inequalities, one for each of the users sharing the same slot, and B unknowns: P 1k, P 2k,, P Bk. Dividing G ii for each inequality, we can rewrite the B inequalities in matrix form, H( γ k )P k N( γ k ) (3) γ k is a B 1 vector where the i-th element is γ ik. γ k is the vector form of the B desired SIR s in the k-th slot. H( γ k ) is a B B matrix. The diagonal elements of H( γ k ) are equal to one and the off diagonal element (i, j) is equal to - γ ik G ij / G ii. P k = [ p ik ] is the B 1 non-negative power vector for γ in slot k. is a noise vector where i-th element is equal to γ ik n i /G ii. k N( γ k ) B 1 If there exists a non-negative power vector solution to (3), the B users can all achieve the SIR needed for their desired throughputs at this time-frequency slot. A set of linear equations needs to be solved for each time-frequency slot. This set of B B linear equations can be solved with power control algorithms with centralized control [11]. Distributed power control algorithms where no centralized control is needed can also be used [12][13]. These algorithms can find the same power solution as the centralized power control algorithms. If there are S slots in the system, the problem of variable bit allocation can be re-stated as finding the distributions of the SIR 5

6 thresholds, { γ i1, γ i2, γ, is }, hence bit allocations, for each user i such that the sum of the bit rates of each user is equal to the desired total throughput and that there exist feasible nonnegative power solutions in all of the S slots with the desired SIR distribution. In order to implement power control in a wireless system, the power amplifiers of the transmitters may not always operated in the saturation region where high amplifier efficiency can be obtained. However, with power control, a wireless system has been shown to be able to support more simultaneous links and the average power consumption of the transmitter may actually be lower than a system without power control as users with low path loss no longer needs to operate the power amplifier at the in the saturation region [13]. 3 Complexity of Optimal Bit Allocation Algorithms We now turn our attention to finding the channel and bit allocations for a given set of users that maximize the system capacity assuming all users must have the same throughput. We divide this section into two different parts. We first present the proof that the complexity of finding the optimal allocation is NP-hard. We then present the results of exhaustive searches which provide heuristics for the bit allocation algorithms discussed in Section Maximum Throughput Channel and Bit Assignments The optimization problem can be stated in the following compact matrix form: maximize B i = 1 X i (4) S X i log 2 ( 1 + γ ik ) k = 1 (5) X i = 1 i, j B X j 6

7 H( γ k )P k N( γ k ) (6) P k 0 γ k 0 1 k S We try to maximize the total throughput of the system with B users and S slots, where X i is the throughput of the i-th user. The logarithm function with base two in (5) is used to upper bound the achievable bit rate as a function of the desired SIR of γ ik [14]. All users have the same throughput. The relations in (6) are a set of non-linear power constraints described in (3). If the number of slots is greater than or equal to the number of users ( B S), the throughput of each user is infinite since we do not impose a maximum power constraint. The infinite throughput is achieved by assigning each user to a different slot. Since there is no interference, the maximum achievable SIR is unbounded. If there is only one slot ( S = 1), the maximum achievable SIR for these B users is equal to the reciprocal of the largest eigenvalue of a B B matrix F. F is closely related to the matrix H( γ 1 ) and has diagonal elements equal to zero and the off diagonal element (i, j) equal to G ij / G ii [11]. We will now show that the problem of finding the optimal bit allocation is NP-hard for B > S. Omitting the superscript k in (1) to simplify the notation, each user requires a different desired SIR, γ i, such that the inequality P i G ii γ i P j G ij + n i j i holds for 1 i B. Multiplying the denominator on both side and dividing both side by G ii, we have γ i j i P j Z ij P i + γ i ñ i 0 (7) where Z ij is equal to G ij / G ii and ñ i is equal to n i / G ii. G ii is path loss and is always greater than 7

8 zero. In the above equation, the transmit power, P j, and desired SIR, γ i, are the unknowns. The equation is thus a quadratic equation of the variables P j, 1 j B, and γ i. We now put the variables involved in the quadratic terms γ i P j in vector notation. Let x = [ P, 1 P, 2, P, i 1 P, i + 1, P, B γ i ] T We then put the sum of the quadratic terms γ i P j Z ij in matrix notation as x T Ax. A can be expressed as j i Z i Z i Z ib Z Z i1 i Z ib In other words, A is a symmetric B B matrix with all elements in the first ( B 1) rows and ( B 1) columns equal to zero. The j-th entry for the B-th row or column is equal to Z ij / 2 for 1 j < i and Z i(j +1) / 2 for i j < B. A B B = 0. To show that the problem is NP-hard, we need to show that the feasible region is nonconvex or equivalently, matrix A is not positive semi-definite [15]. The eigenvalues of the matrix A can be solved easily and they are ( B 2) zeros and ± ( Z ij 2) 2. With negative j i eigenvalues, matrix A is not positive semi-definite. This, along with the other ( B 1) non-convex feasible regions for different values of i in the inequality, makes the optimization problem NPhard. 8

9 3.2 Exhaustive Searches Since the optimization problem (4) is NP-hard, exhaustive searches are needed to find the optimum bit allocation. Searches were performed for the reverse link of an indoor cellular system. The cells are hexagonal in shape with users uniformly distributed in the coverage area. We assume propagation loss can be modeled by the third power attenuation model. Shadowing, which is caused by signals being blocked by objects or travelling through a room with the wave guide effect, can be modeled by a lognormal distribution with standard deviation of 10.0 db [16]. Quadrature-amplitude modulation (QAM) is used. The minimum constellation size is BPSK and requires an SIR of 10 db to maintain good error performance. We use the approximation that an additional 3 db is needed for every additional bit of information to be carried by a symbol. Let X denote the throughput of every user. If there are B users sharing S slots and the exhaustive search starts with one bit of throughput for all users (X=1), the search tries all possible bit allocations in these S slots for the total X bits for each user and all possible combinations of bit allocations among the B users. The exhaustive search does not stop until a feasible power vector for every slot is found such that every user can maintain X bits of throughput. We then increase the system throughput by allocating X+1 bits to all users and search for feasible power vectors again. This process is repeated until no more feasible power vectors can be found for the desired throughput. There are no maximum SIR or maximum power constraints. Unfortunately, the number of bit allocations that need to be searched is prohibitively large for a system with even a moderate number of slots and/or users. The number of allocations is ( X + S 1) C X closely upper bounded by [ ] B ( X + S 1) where C X is the combination of selecting X items out of (X+S-1). Table 1 shows the results of the exhaustive searches for systems with three to five users sharing two slots and a system with four users sharing three slots. One column tabulates the maximum throughput per slot for a bit allocation algorithm that assigns the same number of bits in 9

10 every slot for every user. This is computed using the single channel eigenvalue techniques described in [11]. The other column lists the results from the exhaustive searches. Table 1: Results of exhaustive searches for the maximum average bit allocation per slot. Equal Throughput Exhaustive Searches Difference [bits] [bits] [%] B = 3, S= B = 4, S= B = 5, S= B = 4, S= Table 1 shows that the equal throughput algorithm does not find the best allocation possible. As the number of the users increases, the difference in throughputs between the equal throughput algorithm and the exhaustive searches increases. Figure 1 shows the distribution of the maximum achievable bits for a system with four users and three slots. The equal throughput algorithm is not able to allocate throughput to the users if the users are close to each other in a large fraction of the cases. The distribution decreases rapidly as the maximum achievable throughput increases. The distribution from the results of exhaustive searches concentrates around the average maximum throughput and tapers off as the number of bits increases and decreases at about the same rate. These characteristics are also observed for the systems with a different number of users and slots tabulated in Table 1. 10

11 Solid -- Exhaustive search Dotted -- Equal throughput Distribution Throughput per pattern [bits] Figure 1 Distribution of total throughput for four users and three slots. From Table 1, for systems with four users, the exhaustive searches show that the average maximum achievable throughput per slot increases if the number of slots is increased from two to three for a system with four users. On the other hand, the throughput stays the same for the equal throughput system as the maximum throughput per slot is independent of the number of available slots. The increase in throughput per slot may seem similar to information theory in source coding which states that as the number of dimensions (slots in this case) increases, the average amount of information that can be represented by a symbol per dimension also increases. Figure 2, however, is more indicative of why the system throughput is much higher from exhaustive searches as the number of slots increases. Figure 2 shows that the normalized bit distributions among the S slots from the maximum throughput found by the exhaustive searches. It is normalized to the total maximum throughput of the S slots for each sample. Figure 2 also shows that the maximum throughput is usually achieved by concentrating the throughput of each user in one slot and transmitting no power in the rest. This can be viewed as a form of interference avoidance. The users transmit in the slots with low interference. They, on the average, do not generate too much interference to the system. The system throughput increases. In the variable bit allocation system, 11

12 users that are close to each other achieve most of their throughput from different slots. They are decoupled by avoiding transmitting in the same slots. The users arrange themselves so that they only share the transmit slots with whom they have very low mutual link gains. As the number of slots increases, the users have more freedom to avoid each other. The throughput thus increases. When the number of slots becomes as large as the number of users (B=S), the maximum throughput is infinite. This is achieved again by assigning each user to a different slot, thus avoiding any interference. Dotted line -- B = 4, S = 2 Solid line -- B = 4, S = 3 Distribution Normalized Throughput [Bit] Figure 2 Distribution of the bits allocated from the exhaustive searches. 4 Bit Allocation Algorithms As discussed in the last section, finding the optimum bit and channel allocation requires exhaustive searches which are impractical even for a moderate number of users and channels. We, thus, propose several practical ad hoc bit allocation algorithms that do not require centralized controls and compare their performances in dynamic simulations. The simulations are performed for the reverse link of an indoor cellular system. The system has nineteen hexagonal cells which 12

13 form three concentric circles with the base station located in the center of each cell. QAM is used. The minimum constellation is 4-QAM (QPSK) with desired SIR of 10 db, and we assume 3 db is needed for every additional bit. BPSK is not chosen since bandwidth is a premium in the wireless system and the required energy per bit for any bit error rate requirement is the same as 4-QAM (QPSK). We impose a maximum constellation constraint, 64-QAM, due to the linearity limitation of hardware. The users arrive at the system with nineteen-cell system with geographically uniform distribution. The user arrival process is Poisson distributed and the users have i.i.d. exponential service time. We assume that all users and surrounding objects are stationary so that channel conditions do not change for the duration of each call. The FH-CDMA system has 24 frequency bins. The period of all hopping patterns, N, is equal to 24 time slots and is conveniently defined as a frame. There are twenty four hopping patterns available in every cell. Each slot has C symbols, and we assume every user desires 48 C bits/frame. Channel probing as described in [10][17] is performed for every new user. It enables the system to estimate the maximum achievable throughput in every slot and predict the transmit power needed for any throughput less than or equal to the maximum for a new user. The system can also estimate the additional interference induced by the transmit power of the new user at the base station the new user is associated with as users already in the system adjust their transmit power to maintain satisfactory performance. All these statistics is used by the bit allocation algorithm to determine how slots and bits are assigned in hopping patterns. In our simulations, dropping an already existing link is assumed to be a lot more undesirable than denying newly arrived user service. No new links can be established unless the new links can share the same time-frequency slots with the users already communicating in the slots. We are, therefore, interested in finding the bit allocation algorithm that allows the most users to communicate simultaneously in the system while meeting the system blocking probability, P block, requirement. When a new user is arrived at the system, it first determines the base station to which it has the smallest path loss and associates itself with the cell covered by the base station. 13

14 The new user then probes all the unused hopping pattern in the cell to see if a hopping pattern can be found such that it can meet its throughput requirement without disruption of existing links. If the new user cannot share any of the unused hopping patterns with users already in the system or if all of the hopping patterns associated with the cell are already assigned, the newly arrived user is blocked and exits the system without attempting to establish links again. After a newly arrived user performs channel probing of all unused hopping patterns, the channel and bit allocation algorithm then determines how to assign the new user a hopping pattern. We compare six different heuristic bit allocation algorithms: 1. minimizing the total transmit power in the hopping pattern 2. minimizing the maximum transmit power in the hopping pattern in a slot 3. minimizing the total interference induced in the hopping pattern 4. minimizing the maximum of the interference induced in the hopping pattern 5. minimizing the total number of slots used in the hopping pattern 6. reverse water-filling of SIR Algorithms 1 and 2 are both transmit power based. These two algorithms seek to minimize the interference created by the newly arrived users by minimizing their total transmit power and the maximum transmit power in any slot of a hopping pattern. Since the smaller constellations require less energy per bit to meet the same bit-error-rate requirements, users in systems implementing these two algorithms are more likely to use the smallest constellations. Algorithms 3 and 4 are interference based. As a new user arrives at the system, the existing users need to increase their transmit power to maintain their desired SIR. Algorithms 3 and 4 seek to minimize the additional interference all the existing users need to transmit to maintain their current slot and bit assignment in their hopping patterns. Algorithm 5 performs interference avoidance by limiting the number of slots used. After probing all of the unused hopping patterns in the cell, the newly arrived user is assigned the hopping pattern where it will transmit in the fewest slots in a frame. The new user does 14

15 so usually by transmitting the largest constellations which is only possible when there are no interferers nearby. Algorithm 5 is designed to give the users that arrive later more freedom to perform interference avoidance. Algorithm 6 is based on reverse water-filling of SIR which has been shown to be the optimum algorithm in the single-user discrete multitone (DMT) system where the noise level is fixed at high SIR [7]. Reverse water-filling of SIR can be best described by turning Figure 3, which shows the maximum achievable SIR for all available slots, upside-down. A new user first determines the maximum achievable SIR, γ max, in each slot via channel probing. SIR is then poured into these slots with uneven depth until the desired throughput is achieved with SIR of γ. Algorithm 6 is, however, different from the reverse water-filling algorithm of the a DMT system as the interference level changes as more SIR is poured into the slots. γ max γ Frequency slots Figure 3 Reverse water-filling of SIR for a multislot system. We present the simulation results for indoor FH-CDMA wireless systems groupcoincidence hopping (GC) patterns. A new user shares every slot in its hopping pattern with one group of users. The group of potential interferers is the same in every hop. Depending on how the users in the system allocate their bit rates, a new user may not share every slot with all of the users of the same hopping pattern. There is no intra-cell interference as the users do not share any slot 15

16 in their hopping patterns with the other users in the same cells. Before presenting the simulation results, we define normalized offered load in units of Erlangs as the average arrival rate divided by the total number of hopping patterns available in the system which is equal to the number of base stations multiplied by the number of hopping patterns available in each cell. It is used as a measure of system capacity. We assume that once a user is admitted into the system, it keeps the bit assignment until the call is terminated. No users are dropped due to the arrivals of new users. It is up to the new users to adjust their channel and bit assignments to the different interference levels. Centralized power control is used for our simulations to reduce simulation time. Figure 4 shows the blocking probability as a function of the normalized offered load. Algorithms 2 and 4 are not shown in Figure 4 as Algorithm 2 has the same P block curve as Algorithm 1 and Algorithm 4 has the same P block curve as Algorithm 3. Figure 4 shows that the transmit power-based Algorithms 1 and 2 have the lowest system capacity for any given system P block requirement between 0.1% and 1% among six bit allocation algorithms. Algorithm 5, which minimizes the number of slots a new user uses, achieves the highest capacity among the six algorithms. The interference-based Algorithms 3 and 4 and the reverse water-filling of SIR, Algorithm 6, all have similar system capacities and achieve capacity close to Algorithm 5. The difference in system capacities for Algorithm 5 and Algorithm 1 for P block of 0.1% to 1% is approximately 25%. Figure 4 also shows the P block curve of a system without variable bit allocation. In this system, a user transmits 4-QAM in all of the twenty-four slots in its hopping pattern. The system capacity of this system is 25% to 55% lower than a system with any of the six bit allocation algorithms proposed for P block requirement of 1%. 16

17 P block Dashed line -- No bit allocation Dotted line -- Alg 1, min total power Solid line -- Alg 5, minimum slots x -- Alg 3, min total interference o -- Alg 6, reverse water-pouring Normalized Offered Load [Erlang] Figure 4 Bit allocation simulations for group-coincidence systems. Figure 5 shows the distributions of the slots used by the six algorithms as a function of the bits carried per symbol. Again, the distributions of Algorithms 2 and 4 are not shown as they are very similar to those of Algorithms 1 and 3 respectively. No slots carry just one bit of information as BPSK is not allowed. Algorithm 5, which achieves the highest system capacity, assigns the maximum constellation of 64-QAM to most slots among the six algorithms and at the same time leave most slots unused. It minimizes the number of slots a new user transmits and concentrates its throughput and power in these slots. The new user, at the same time, avoids the slots with large interference when possible. These are the characteristics associated with the maximum throughput bit assignments found by the exhaustive searches in Section 3.2. Minimizing the slots in which a user allocates power also gives future arrivals more flexibility to perform interference avoidance. 17

18 Distribution Solid -- Alg 5, minimum slots Dashed -- Alg 1, min in total power o -- Alg 3, min total interference + -- Alg 6, reverse SIR water-fill Normalized Throughput [Bit] Figure 5 Distribution of the bits allocated from simulations. The transmit power-based algorithms assign the smallest constellation, 4-QAM, to most of the slots as it often requires the least power. There are usually no slots left unused. These algorithms do not perform interference avoidance and are similar to the equal throughput algorithm in Section 3.2. The constellations assigned by these two algorithms are very similar to the system without variable bit allocation, but the additional flexibility that allows users to change bit assignments enables systems with Algorithms 1 or 2 to achieve 25% higher system capacity as shown in Figure 4 for P block of 0.1% to 1%. The interference-based algorithms, on the other hand, assign large constellations to the slots that have no interference or have interferers that are far away. In these slots, the new users can obtain the highest throughput without inducing much additional interference. The bit distributions of these algorithms are very similar to Algorithm 5. The interference-based algorithms better use the statistics derived from channel probing than the power-based algorithms. Instead of passively minimizing the interference generated by the new user as in Algorithms 1 and 2, Algorithms 3 and 4 estimate how the users already in the system would react to the interference generated by the new user. Algorithm 6, which performs reverse 18

19 water-filling of SIR, also obtains the most throughput desired in very few slots. These slots usually have very high achievable SIR and on the average, have no interferers or have interferers that have very low mutual link gain with the new user. The bit distribution of Algorithm 6 also takes on the general shape of Algorithm 5. We now compare the system capacity of the variable bit FH-CDMA systems with an FH- CDMA system deploying the interference avoidance strategy identical to that described for Algorithm 5 by transmitting in M out of every N slots (N = 24), but without the capability to vary the constellations among the slots used for communication [18]. The M/N system performs channel probing and then assigns a fixed constellation in the M most favorable slots. The receivers for this system is simplified as they only need to be able to demodulate one modulation. Power control is still implemented. Without power control, the system throughput would be much lower. We assume that the system operates in the same environment and that users have the same throughput requirement of 48 C bits/frame, where C is the number of symbols per slot. For comparison, the M/N system transmits eight out of every twenty-four slots using 64-QAM, the same largest constellation available for the variable bit allocation system, resulting in 48 C bits/ frame. The results are shown in Figure 6. Figure 6 shows that the M/N system can achieve higher system capacity for a given P block than a system transmitting QPSK in all twenty-four slots without bit allocation through interference avoidance. It also shows that the system capacity of the variable bit allocation system is 20%-25% better than that of the M/N= 8/23 system for P block of 0.1% to 1%. Even though both the M/N system and the bit allocation system use the largest constellation whenever possible, the bit allocation system has the additional flexibility to allocate bits according to the congestion condition in the hopping patterns. It can admit users that are unable to achieve the fixed constellation, 64-QAM, in every slot used in the M/N = 8/23 system. 19

20 No Bit Allocation P block Variable Bit Allocation M/N = 8/24 Normalized Offered Load [Erlang] Figure 6 Simulation results for variable bit allocation and M/N systems with all users having the same throughput. 5 Conclusion Finding the optimal bit allocation that maximizes the system throughput is an NP-hard problem. In an FH-CDMA system where every user requires the same bit rate, we propose several ad hoc algorithms that utilize the insights we have obtained from exhaustive searches. Interference avoidance algorithms which ask the users to avoid transmitting in the slots with large interferers nearby seems to be the best strategy. These bit allocation algorithms gives the system more flexibility to adapt to the interference statistics, and result in the highest system capacity. Under our simplified model, a system with variable bit allocation can achieve capacity more than 50% higher than a system with users transmitting the same bit rate in every slot of the hopping patterns for P block of 1%. In our simulations, we assume that power control and bit allocation can be performed instantly. We have not taken into account of the speed of convergence of various adaptive algorithms. The results presented here are meant to serve as an upperbound on how the various 20

21 proposed bit allocation algorithms can improve the capacity. We expect that high speed indoor systems can come close to these bounds, while quite different approaches will be required to cope with a combination of high mobility and low bit rate. The key factor is the ability to accurately estimate the channel and provide the communication partner with timely observations. With rapid channel and interference dynamics relative to the symbol rate, at some point interference averaging would be more attractive than interference avoidance. Development of an access strategy which effectively manages this transition would be an interesting research project. Appendix References [1] IEEE P , Draft Standard for Wireless Local-Area-Network Medium Access Control (MAC) and Physical Layer (PHY) Specifications. [2] T. Chebaro, Statistics of signal to interference plus noise ratio in a sectorized FH-TDMA system with a single cell frequency reuse pattern, IEEE Proc. of Int. Conf. on Comm., Seattle, WA, 1995, pp [3] A. M. C. Correia and A. A. Albuquerque, FH-SSMA with band-efficient modulations over cellular fading channels, European Transactions on Telecommunications, March- April 1996, vol.7, no.2, pp [4] A. Goldsmith and P. Varaiya, Increased spectral efficiency through power control, Proc. IEEE Int. Conf. on Comm, Geneva, May [5] E. Perahia and G.J. Pottie, Adaptive antenna arrays and equalization for indoor digital radio, IEEE Proc. of Int. Conf. on Comm., Dallas, TX, June 1996, vol. 1, pp

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