SUPERPOSITION CODING IN THE DOWNLINK OF CDMA CELLULAR SYSTEMS

Transcription

1 SUPERPOSITION ODING IN THE DOWNLINK OF DMA ELLULAR SYSTEMS Surendra Boppana, John M. Shea Wireless Information Networking Group Department of Electrical and omputer Engineering University of Florida 458 ENG Building #33 P.O. Box 630 Abstract Superposition coding techniques can be used to improve communications efficiency in a variety of scenarios. These techniques are particularly appropriate for applications in which there are spatial or temporal disparities in received signalto-noise ratios (SNRs). In this paper, we consider the application of superposition coding to increase the system capacity of the forward link in a DMA cellular system using orthogonal spreading sequences. By taking advantage of power disparities that are created by power control on the forward link, the system can transmit to a larger number of users by transmitting to multiple users on a single spreading sequence by using superposition coding. We use an abstraction of a cellular communication system in which Shannon capacity or the optimal capacity equations for broadcasting on an additive white Gaussian noise channel are used to determine the required transmit power to a user or users. We evaluate the user capacity achieved with the proposed superposition coding techniques under both average- and fixedpower constraints. We compare the performance of the proposed technique to that of a system that employs generalized Welch bound equality (GWBE) sequences. The results indicate that superposition coding can significantly increase the user capacity of a cellular system and that broadcasting is much more effective than using GWBE sequences. I. INTRODUTION In [], over introduced the broadcast channel (see also [2], [3]) and demonstrated that it is more efficient to simultaneously transmit to multiple users than to time-share or otherwise use orthogonal division of the channel resources among users. In [], over proposed the use of superposition coding to achieve better performance than orthogonal division of the channel and found the achievable rate region. Bergmans [4] showed that this rate region is the optimal capacity for additive white Gaussian noise broadcast channels. Motivated by this practical work, Pursley and Shea proposed practical schemes to exploit spatial differences among receivers in wireless networks in [5], [6]. Broadcast strategies for temporal fading were introduced by Shamai in [7] and extended to multiple-user and MIMO channels in in [8], [9], [0]. A summary of information-theoretic work focused on cellular communications is given in []. In this paper, we evaluate the use of superposition coding in a cellular DMA communication system that employs orthogonal spreading sequences and power control. In such systems, power control is applied to provide a target signalto-noise ratio for each user while minimizing multipath interference to users in that cell and minimizing interference to users in adjacent cells. Ideally, each user sees the same signal-to-noise ratio (SNR) for the signal on their designated spreading code. However, if a user despreads another user s signal, power control may result in vastly different SNRs. Superposition coding may offer some significant advantages in such scenarios by simultaneously transmitting messages to multiple users on a single spreading code. This paper generalizes and extends the earlier work in [2], in which nonuniform modulation is used as a way to achieve the effects of superposition. In this paper, we use an abstract model of a DMA system in which the required transmit powers are determined according to Shannon capacity or the capacity region of the additive white Gaussian noise (AWGN) broadcast channel. As in [2], our goal is to assess the number of users that can be supported at some target rate under a power constraint. Unlike other work on broadcasting in DMA cellular systems (cf. [9], []), we do not assume that sufficient channel knowledge is available to mitigate the interference among users through techniques such as linear precoding [3] or dirty paper coding [4], [9]. We compare the user capacity of a system employing superposition coding to that of a system employing GWBE sequences with the same power constraint. The use of Generalized Welch bound Equality(GWBE) sequences to accommodate more users than the processing gain of a cellular system was first considered in [5] for a synchronous AWGN channel with linear MMSE receivers. The design of spreading sequences to maximize user capacity for a DMA forward link in a fading channel was considered in [6]. The asynchronous DMA system was addressed in [7]. Iterative construction of the optimum sequences that maximize the user capacity with minimum total transmitted power was suggested in [8]. In [5], it is shown that for a DMA cellular system in which each user is assigned a unique spreading code and an MMSE receiver is employed, the user capacity K for a common target SNR of γ, is limited by K < N( + γ ), where N is the processing gain of the system. The remainder of the paper is organized as follows. In Section II, we describe the system model and introduce the notion of superposition coding. In Section III we construct an optimum pairing strategy that minimizes the total power transmitted for a given number of pairs. In Section IV we

2 2 compare the performance of a system employing superposition coding with a system employing GWBE sequences under an average power constraint and infinite user population. In Section V, we compare the user capacity of this broadcasting system to that of a system employing GWBE sequences under fixed total power constraint and fixed user population. The paper is concluded in Section VI. II. SYSTEM DESRIPTION In this section, we consider a cellular system that uses orthogonal spreading and power control on the forward link, which is typically seen in commercial systems, such as IS- 95, WDMA and DMA2000 [9]. onsider the base station at the center of a circular area of coverage. Without loss of generality, we assume that the circular region has unit radius and the base station is at position (0, 0). The users are uniformly distributed in the area of coverage. Let Z i denote the distance from the base station to mobile user/radio M i. The probability density function (pdf) of Z i is given by { 2z, 0 z f Zi (z) = () 0, otherwise. Another quantity we shall be referring to later is U i, the square of the distance from the base station to the mobile user M i. It is easy to see that U i is uniformly distributed in [0,]. We assume that the thermal noise, multiple-access interference and adjacent-cell interference can be modeled as a single additive white Gaussian noise (AWGN) source with two sided power spectral density N0 2 [20]. (The assumption is reasonable since the adjacent-cell interference is highest at the cell boundaries and multiple-access interference is highest in the interior of the cell). Let W be the bandwidth each user sees after despreading the received signal and N denote the number of orthogonal channels in the system. Power control is used to ensure that each mobile radio receives sufficient power to achieve the desired quality of service while minimizing the interference to other mobiles. We consider the case in which perfect power control is used to maintain a constant signal-to-noise ratio (SNR) at the mobile radio receiving information from the base station. In such a scenario, it is often possible to identify pairs of mobile radios such that the power received by one radio is much greater than the power received by the other radio. Fig. depicts such a scenario, where power control is used to achieve the same received power at each of the mobile radios M, M 2 and M 3. An exponential path loss model without the effects of fading and shadowing is assumed for the sake of exposition. In the figure, the abscissa denotes the distance of the mobile radios from the base station, and the ordinate denotes the power transmitted by the base station (in db). The radios are indexed in increasing order of their distance from the base station. It can be seen from the figure that when base station transmits information to M 3, maintaining the target SNR level, the SNR seen by both M and M 2 is much greater than their target SNR levels (by amounts A+B and respectively). Similarly, when base station transmits information to M 2, M receives an additional A db of power above its target SNR level. This implies that M has sufficient SNR to decode messages Power transmitted (db) > Fig.. B A M M2 M3 Distance from the base station > Power disparities in a cellular network. Target SNR level intended for both M 2 and M 3, and M 2 has sufficient SNR to decode the messages intended for M 3. The power disparities at radios M, M 2 and M 3 suggest that information for M 2 and M can be included in the transmission to M 3 through the use of superposition coding. Similarly, we can include information for M while transmitting information to M 2. The dotted line in Fig. indicates base station transmitting information to M 2 while transmitting to M 3 at the target SNR level by employing superposition coding. We can exploit such disparities to increase the user capacity of the system by employing superposition coding [], [2], [3]. Moreover we show that this additional system capacity comes at very little expense to the performance of the network. In our broadcasting strategy, the base station uses two-level superimposed codes to transmit to pairs of radios which are allocated the same spreading sequence. The superposition codes are composed of information at two different rates designed for two different SNR requirements for their accurate reception. The message with the lower SNR requirement for its accurate reception is known as the basic message, and the message with higher SNR requirement for its accurate reception is known as the additional message. The user capacity of such a system is dependent on the number of suitable pairs that exist and also on which radios pair. To analyze the performance of such a system, we index the mobile radios in decreasing order of their channel gains. We define a strategy function f(i) as a one-to-one function which associates/pairs radio M i with the radio M f(i), f(i) > i for i N. This means that radios M i and M f(i) share the same spreading code, and M i pairs with M f(i) to recover an additional stream of information that is superimposed on the message for M f(i) (Note that M i has a better channel than M f(i) ). The constraint on the domain of i indicates that the maximum number of pairs using our two-level superposition coding is equal to the number of orthogonal channels available. Here the one-to-one condition implies that no two radios pair with the same radio. This restriction is required by our use of two-level superposition coding. The fact that f(i) is a function restricts each radio to pair with at most one radio. Although these requirements are not necessary from theoretical standpoint, they represent a scenario that is of more practical interest.

3 3 An exponential path loss model with Rayleigh flat fading is assumed, where the power P r, received by radio M r, is related to the transmitted power P t by P r = d α r h r 2 P t, (2) where is a constant, d r is the distance of the mobile radio from the base station, α is the path loss exponent, and h r is the magnitude of the fading at the radio M r, which is assumed to be constant over many symbols. The information rates R bm and R am of the basic and the additional messages under twolevel superposition coding are given by [3] R bm = W log 2 ( R am = W log 2 ( + ( a)d α b h b 2 P t h b 2 P t + N 0 W ad α b + ad α a h a 2 P t N 0 W ) ), (3), (4) where P t is the power transmitted by the base station, d b and d a are the distances of the receivers receiving the basic message and additional message respectively, α is the path loss exponent, h a and h b are the magnitudes of the fading at the radios receiving additional and basic messages respectively, and 0 a is the ratio of the power utilized in the transmission of the additional message to the power utilized in the transmission of the basic message. Under an average power constraint, the transmission of an additional message while maintaining the same target SNR at the radio receiving the basic message results in an increase of the total transmitted power by the base station. In the context of a cellular network, this additional transmit power will result in increased interference to users in adjacent cells and to users in the current cell from multi path. Hence our focus is on increasing the capacity of the system, while maintaining the same average transmitted power at the base station. Such a throughput gain can achieved by decreasing the information rate or the target SNR of the radios. For the sake of simplicity, henceforth we assume that the target SNRs of both the additional and the basic messages are the same. III. MAXIMIZING THE USER APAITY In this section we derive an optimum pairing strategy for superposition coding that minimizes the total power transmitted by the base station for a given user capacity (We are interested in the case where the user capacity is greater than the number of orthogonal channels). First we need the following result to construct a pairing strategy that maximizes the user capacity. Proposition I: onsider a cellular network with K radios and N orthogonal channels such that N < K 2N. The total power transmitted by the base station using N orthogonal channels and two-level superposition coding is greater than that of direct transmission to the K radios through K orthogonal channels. Proof: Since we are considering only a two-level superposition code, it is sufficient to show that the total power transmitted to two radios with two orthogonal channels, maintaining a constant SNR at the radios is less than the power transmitted to the radios with one channel and a two-level superposition code and maintaining the same SNR at both the radios. onsider a network with two radios M and M 2 with channel gains z and z 2, z > z 2. Let P and P 2 be the powers transmitted by the base station to the radios M and M 2 respectively through two orthogonal channels, such that a target SNR of γ is maintained at the radios. Hence we have γ = zp N = z2p2 0W N 0W. Note that P 2 > P. In order for these radios to pair using a two-level superposition code with one channel such that both M and M 2 have a target SNR of γ, there should exist a pair of (a, P ) satisfying the following constraints. az P N 0 W = γ, (5) ( a)z 2 P az 2 P + N 0 W = γ, (6) 0 a, (7) 0 P P + P 2, where Z i = z i, i =, 2, α is the path loss exponent, N 0 is noise variance of the AWGN channel and W is the effective bandwidth seen by each user after despreading the orthogonal code. onstraints (5) and (6) state that the SNRs of basic and additional messages should satisfy the target SNR requirement. We show that no such pair (a, P ) exists that satisfy the above constraints. From (5) we have Substituting (8) in (6), a = γn 0W Z P a = P P ( γ = Z P N 0 W ) Z 2 (P P ) P Z 2 + N 0 W = γ Z 2 (P P ) = γ(p Z 2 + N 0 W ) Z 2 (P P ) = Z 2 (γp + P 2 ) ( γ = Z 2P 2 N 0 W ) P P = γp + P 2 P = P ( + γ) + P 2 > P + P 2 which violates constraint (8). orollary: The minimum additional power required for broadcasting to a pair of radios having the same spreading sequence is γp i, where γ is the target SNR and P i is the power required to maintain a constant SNR of γ at the radio M i with the better channel gain without employing broadcasting. A pairing strategy that minimizes the total power transmitted for a given number of pairs k N is f(i) = i + N, i k. (8)

4 4 Note that, even though the choice of the optimum pairing strategy is not unique, the minimum total transmitted power is unique. IV. USER APAITY UNDER AVERAGE POWER ONSTRAINT In this section, we compare the user capacity of a system employing superposition coding to that of a system employing GWBE sequences under the same average total power constraint. Since our focus is on increasing the capacity of a cellular network without increasing the total transmitted power, we derive the average power constraint from a cellular system that supports simultaneous transmission to N users through N orthogonal channels. We consider an infinite user population, but in each transmission period, the base station must transmit to a fixed number of arbitrary nodes that satisfy a path-loss constraint. We assume that all the radios are uniformly distributed in the circular area of coverage and have a target SNR requirement of γ. Let z denote the instantaneous channel gain between the base station and the radio M. The distribution of the channel gain z = d 2 h 2 (refer to (2)) is given by F Z (z) = + e z, z 0. z Under perfect power control, it is not possible to transmit to every node while maintaining finite transmit power because E[/z] does not converge on z [0, ). Thus, we assume that the system does not transmit to a set of overfaded users (cf. [6]) whose channel gain is such that F Z (z) ρ. Thus, the overfaded users experience outage during those intervals where their channel gains are too bad. Let Z ρ denote the maximum value of the channel gain for which a user will be classified as overfaded. A. ellular network without superposition coding onsider first a cellular system that allocates each of the N orthogonal spreading sequences to one user, such that it transmits to N users in each interval. Since we are assuming an infinite radio population, it is always possible to find N radios with channel gains z Z ρ to which the base station can transmit. This induces a conditional distribution for the channel gains (i.e. F z (z z Z ρ )). Thus, the average power transmitted by the base station to a user such that a target SNR of γ is maintained is given by γn0 W E{P T (Z ρ, γ)} = E z z Z ρ = γn 0W [ ] + Z 2 ρ Γ(0, Z ρ ) e Zρ ( + Z ρ ) 2Z ρ ( e Zρ ) where Γ(.) is the incomplete gamma function given by Γ(a, z) = z t a e t dt (0) Since the base station transmits to N users, the average total transmitted power is given by N E{P T (Z ρ, γ)}. (9) B. ellular network employing superposition coding onsider the cellular network employing two-level superposition coding, as described in Section II. Let K, N K 2N denote the number of mobile radios served in a symbol duration. We again assume that it is always possible to find K non-overfaded radios (i.e., these radios have channels gains greater than Z ρ ) for arbitrary K. In this section, we assume that the number of users K to which the base station transmits is constant across each transmission interval. The value of K is determined based on the power constraint from the last section. From the corollary to Proposition, the total power transmitted by the base station in a network with K users and N orthogonal channels under the pairing strategy of (8) is given by P bc T ( K ) PT bc (γ N ) = γ 0 W K N + γ, z > z 2 > > z K Z ρ where = PT nbc (γ ) + PT bc (γ ), PT nbc (γ N ) = γ 0 W P bc T (γ ) = γ 2 N 0 W K K N, and. In these expressions, γ is the common target SNR of all the radios, which is not necessarily equal to the target SNR γ for a cellular network without broadcasting. From Proposition, an increase in the user capacity under the same average power requires a decrease in the target SNR. The term PT nbc ) can (γ be interpreted as the total power required to transmit to K radios using K orthogonal channels and PT bc ) can be (γ interpreted as the increase in the transmitted power due to employing superposition coding to support these K radios over N orthogonal channels(refer to the orollary of Section III). The expected value of PT nbc is given by E{PT nbc (γ N )} = γ 0 W K { E } Z ρ N = Kγ 0 W E P T (Z ρ, γ ) where E P T (Z ρ, γ ) is given by (9). Similarly the expected value of P bc T (γ ) is given by E{ PT bc (γ )} = γ 2 K N N 0 W E, z > z 2 > > z K Z ρ The distribution of p k = z k can be evaluated from the principles of order statistics. Let P ρ = Zρ. The conditional

5 5 density of p i, i K is given by [2], [22] f pi (p p P ρ ) = where F p (P ρ ) is given by (K)! (p( e p )) i (i )!(K i)! F p (P ρ ) K [ ] K i F p (P ρ ) (p( e p )) ( e p ( + p )), Hence the expected value of P bc T E PT bc (γ ) = γ 2 N 0 W F p (P ρ ) = P ρ ( e Pρ ). () K N is Pρ 0 p k f pk (P ρ )dp k, (2) where f pk (P ρ ), k K is given by (). The expected value of PT bc nbc is the sum of the expected values of PT and PT bc. E { P bc T (γ ) } = E P bc T (γ ) + E PT bc (γ ). (3). ellular Network Employing GWBE Sequences onsider a cellular system with infinite radio population and (K g > N) supported users such that each radio is associated with a unique GWBE signature sequence [23]. As before, we assume that it is always possible to find K g radios with channels gains greater than Z ρ to which the base station will transmit. Let the target SNR of all the users be γ. Assuming that all the orthogonal channels are equally occupied by the users, the total transmitted power by the base station in a symbol duration is given by [6] P g T = Ng(γ )N 0 W/ N Kg(γ ) K g, (4) where g(γ ) = γ is the effective bandwidth of each user +γ and K g < N( + ) (5) γ and is the channel gain of user M k [5]. The average total power transmitted by the base station is given E{P g T (γ )} = NK gg(γ )N 0 W/ N Kg(γ ) E{P T (Z ρ, γ )} (6) where E{P T (Z ρ, γ )} is given by (9). We numerically evaluate the user capacities of systems employing superposition coding(sp) and GWBE sequences such that the average transmitted power in the both cases is equal to that of the system in Section IV-A. The results in Fig. 2 compare the user capacities of systems employing superposition coding and GWBE sequences for α = 2 and α = 4 as function of the degradation in the target SNR 0 log γ γ, compared to the target SNR γ of the system in Section IV- A. For these results, the number of orthogonal channels is N = 40, outage probability is ρ = 0.05, target SNR of the non-broadcasting system is γ = 0dB, power spectral density (PSD) of the AWGN channel is N 0 = 0 0, and bandwidth W = 0 6 Hz and = 0 2. Using superposition coding, the increase in the user capacity is about 20% for α = 4 and 5% for α = 2 for a degradation of db in the target SNR. The user capacity of the system employing GWBE sequences is 40 for both α = 2 and α = 4 and with a degradation of db in the target SNR. The maximum user capacity K g, of the system employing GWBE sequences in this case is 47 (Refer to (5), N = 40 and 5 < γ 0). For higher degradations, GWBE sequences can improve the user capacity, but superposition coding is always a much more effective strategy. The results indicate that significant increase in the user capacity can be achieved using superposition coding with little degradation in the target SNR. Number of users SP, α=4 SP, α=2 GWBE, α=2 GWBE, α= Decrease in the target SNR Fig. 2. User capacity of systems employing broadcasting and GWBE sequences under average power constraint and infinite radio assumption. N = 40, γ = 0dB, ρ = V. USER APAITY UNDER TOTAL POWER ONSTRAINT In this section, we evaluate the user capacity of a systems employing superposition coding and GWBE sequences under a finite radio population and a limit on the total transmitted power in any interval. For each set of radio channel gains, the base station chooses the set of radios to which it transmits in order to maximize the user capacity. In this case, we do not impose any constraint on the channel gains of the users. Since the radio population is finite, the number of radios that can be served during any transmission interval is random and depends on the channel gains of the radios to which the base station transmits in that interval. In order to maximize the user capacity, the base station chooses the nodes with the best channel gains using the strategy of (8), such that the total transmitted power to these nodes in that transmission interval doesn t exceed a predetermined value. We evaluate the average user capacity of the system using superposition coding and the system using GWBE sequences under the same total power constraint. We arbitrarily choose the total power constraint in a transmission interval equal to the average power constraint considered previously. (Refer to (9)).

6 6 The results in Fig. 3 illustrate the user capacity of systems employing superposition coding and GWBE sequences for α = 4, number of orthogonal channels N = 40, = 0 2, W = 0 6 Hz, and γ = γ = 0 db. It can be noted that there is no gain in the user capacity of the system employing GWBE sequences (the maximum user capacity is 43 in this scenario). However, the user capacity of the system employing superposition coding increases non-linearly with increasing radio population and reaches the theoretic maximum of 2N when the radio population is about 5 times the number of orthogonal channels available. With a sufficiently large population, it is always possible to find 2N radios with channel gains which satisfy the total power constraint under superposition coding. With fixed radio population and total power constraint, superposition coding can increase user capacity without any degradation in the target SNR by appropriately choosing the set of radios to which the base station transmits. Number of users supported by the base station Superposition oding GWBE Sequences User population Fig. 3. User capacity of systems employing superposition coding and GWBE sequences with fixed radio population and total power constraint. N = 40, γ = 0dB, α = 4. VI. ONLUSION We have evaluated the performance of superposition coding in increasing the user capacity of the forward link of DMA cellular systems. The proposed techniques exploit differences in power that exist in a network with power control to superimpose information to radios with better channel conditions. We have compared the performance of such a system employing superposition coding to that of a system employing GWBE sequences under both average- and fixed-power constraints. The results indicate that an average 20% increase in the capacity is possible for α = 4 under an average power constraint by employing superposition coding and a degradation of about db in the target SNR. With a fixed power constraint, the increase in the user capacity is far greater than that of a system employing GWBE sequences. These results suggest that superposition coding may be a useful technique to increase the number of users that can be supported in DMA systems. REFERENES [] T. M. over, Broadcast channels, IEEE Trans. Inform. Theory, vol. IT-8, pp. 2 4, Jan [2] P. P. Bergmans and T. M. over, ooperative broadcasting, IEEE Trans. Inform. Theory, vol. IT-20, pp , May 974. [3] T. M. over and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, New York, 99. [4] P. P. Bergmans, A simple converse for broadcast channels with additive white Gaussian noise, IEEE Trans. Inform. Theory, vol. IT-20, pp , Mar [5] M. B. Pursley and J. M. 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