Power Control Algorithms for MMSE Receivers in CDMA Systems

Transcription

1 Power Contro Agorithms for MMSE Receivers in CDMA Systems Yong Liu and Tan F. Wong Department of Eectrica & Computer Engineering University of Forida Gainesvie, Forida Emai: and Abstract In this paper, we introduce an adaptive and distribute power contro agorithm (PCA) for the MMSE receiver. This PCA utiizes the mean squared error (MSE) to adjust the transmission power of a user. We aso combine the proposed PCA with an existing PCA that empoys signato-interference ratio (SIR) in order to achieve the desired performance of suitabe convergence rate and stabiity. The proposed agorithms and the existing PCA based on SIR [7] are compared. We show that a these agorithms are derived from standard interference functions and so they converge to the same minimum power soution. Convergence rates and stabiity of these adaptive PCAs are aso compared by computer simuations. Keywords CDMA, MMSE receiver, power contro, mutiuser detection I. INTRODUCTION Mutiuser detection has received considerabe attention for over a decade as a promising technique to sove the near-far probem in CDMA systems. A number of mutiuser receivers have been proposed and studied, a survey of which can be found in [1]. Among many kinds of mutiuser detectors, the minimum mean squared error (MMSE) receiver [2] is a ikey candidate for practica appications of mutiuser detection in next generation CDMA systems because it can be impemented adaptivey and has reativey ow compexity. Power contro is another effective method to sove the nearfar probem and consequenty increases the user capacity of a CDMA system. The objective of power contro is to guarantee that certain signa-to-interference ratio (SIR) performance targets are satisfied at the receivers of a users by imiting the transmitted power and hence the eve of mutipe access interference (MAI). In [3] and [4], centraized power contro was considered and theoretica imits and anaytica approaches for the power contro probem were provided. Iterative and distributed power contro agorithms (DPCs), introduced in [5], update the transmitter power eve of each user using ony oca measurements. In [6], Yates gave a framework for upink power contro in ceuar radio systems. Convergence anaysis of a genera cass of PCAs, which are caed standard agorithms, was presented. This framework provides a simpe but effective guide to construct new PCAs. The appication of power contro to mutiuser receivers (the MMSE receiver in particuar) is proven to be usefu in further increasing the capacity of the system [7], [8]. Impementation of the PCA presented in [7] requires each user to estimate its own reverse-ink channe gain. Athough this can be achieved with the use of piot symbos, accurate estimation of the channe gain is difficut in the presence of MAI. We propose to use the estimates of the mean squared error (MSE) instead of the channe gain to update the power of a user. To increase the convergence rate, we aso construct a PCA by combining the proposed PCA and the PCA given in [7]. Adaptive fitering agorithms such as the RLS and LMS agorithms can be used for the practica impementations of both the MMSE receiver and the corresponding PCAs for the MMSE receiver. We prove the convergence of the newy proposed PCAs by showing that they converge to the same minimum power soution as the PCA in [7]. Convergence rates and stabiity of both the proposed PCAs and the PCA in [7] are aso examined and compared. The rest of the paper is organized in the foowing manner. In Section II, we describe the system mode. Power contro agorithms for the MMSE receiver are introduced in Section III. In Section IV, we give proofs of the convergence of these agorithms compare their convergence rates. In Section V, impementation issues are considered and stabiity of the PCAs is examined and compared. Concusions are drawn in Section VI. II. SYSTEM MODEL A synchronous DS-CDMA system over a nondispersive additive white Gaussian noise (AWGN) channe is considered. The compex baseband representation of the received signa in a bit interva is given by r(t) = pj h j b j s j (t)+n(t), (1) j=1 where p j and b j are the transmitted power and data bit of the jth user, respectivey, and h j is the channe gain from the jth user to the base station. The signature waveform of the jth user is s j (t) and n(t) represents the AWGN. We assume there are K users in the system. The received signa is synchronized, passed through a chip matched fiter (CMF), and samped at the chip rate. Hence the received vector formed by concatenating the chip-rate sampes can be expressed as r = pj h j b j a j + n, (2) j=1 where a j is the vector whose eements form a period of the spreading sequence of the jth user with a spreading gain N and n is a zero-mean Gaussian random vector with eementwise variance σ 2. The received vector r is fed into an MMSE fiter /02/$ IEEE. 1749

2 For the ith user, the SIR at the output of the MMSE fiter is given in [2]: γ i = p i h 2 i (ct i a i) 2 j =i p jh 2 j (ct i a j) 2 + σ 2 (c T i c i) 2, (3) where c i is the MMSE fiter coefficient vector empoyed by the ith user. For a given set of spreading sequences, the objective of the PCA is to choose the transmitted powers so that the tota transmitted power for a users to achieve the target SIR performance is minimized. Mathematicay, the power contro probem for the MMSE receiver can be expressed as: Minimize K i=1 p i (4) subject to γ i γ i, where γ i and γ i are the achieved SIR and the target SIR for the ith user, respectivey. III. POWER CONTROL ALGORITHMS FOR MMSE RECEIVER In this section, we first review the PCA for the MMSE receiver presented in [7]. Then we propose a new PCA that uses the mean-squared error (MSE) to update the transmission power of a user. We aso combine these two PCAs to obtain a new one. We use p =[p 1,,p K ] T 0 to denote the power vector in the foowing discussion. A. PCA based on signa-to-inference ratio (SIR) The PCA for the MMSE receiver given in [7] can be briefy described as T i (p(n), c i ) = γ i j =i p j(n)h 2 j (ct i a j) 2 + σ 2 (c T i c i) 2 h 2 i (c T i a, i) 2 p i (n +1) = I i (p(n)) = min T i (p(n), c i ). (5) c i Here we assume that the optima fiter weights can be obtained in each bock of transmission and the initia powers are not a zeros. From (3), it is straightforward to obtain the foowing aternative form for the PCA based on signa-to-interference ratio: p i (n +1)=I i (p(n)) = γ i γ i (n) p i(n), (6) p where γ i (n) = i(n)h 2 i (ct i j =i ai)2. For convenience, we ca this agorithm the PCA based on pj(n)h2 j (ct i aj)2 +σ 2 (c T ci)2 i SIR. B. PCA based on mean-squared error (MSE) We propose a new PCA based on mean squared error as foows: p i (n +1) = Īi(p(n)) = 1 MMSE i h 2 i at i ( j h2 j p j(n)a j a T j +, σ2 I N a i (7) where MMSE i is the desired MMSE of the ith user. Simiary to the PCA based on SIR, the PCA based on MSE can be written in an aternative form with the assumption that the optima fiter weights can be obtained in each bock of transmission and the initia powers are not a zero. For the ith user at the nth iteration, the MMSE is given by MMSE i (n) =1 p i (n)h 2 i at i Σ 1 (n)a i, where the Σ(n) is the autocorreation matrix and is defined as Σ(n) = E{rr T } = j h 2 j p j(n)a j a T j + σ2 I N. (8) Substituting the formua above into the PCA based on MSE (7) gives p i (n +1)= 1 MMSE i 1 MMSE i (n) p i(n). (9) In addition, since the MMSE and the SIR of the ith user are reated by 1 γ i (n) = MMSE i (n) 1, the PCA based on SIR can be transformed into the foowing form by substituting the above reation into (6): γ i p i (n +1)= 1 MMSE 1 p i(n). (10) i(n) C. Combined PCA We combine the above two different PCAs to form a new one: p i (n +1)=Îi(p(n)) = ɛi i (p(n)) + (1 ɛ)īi(p(n)), (11) where 0 ɛ 1 is a constant. Adjusting the vaue of ɛ heps achieve the desired performance of suitabe convergence rate and stabiity. When ɛ =1, the combined PCA reduces to the PCA based on SIR and when ɛ =0, it reduces to the PCA basedonmse. IV. CONVERGENCE OF POWER CONTROL ALGORITHMS We investigate the convergence of the proposed PCAs by foowing the genera approach presented in [6]. First, we wi revisit some important resuts from [6]: Definition 1: I(p) =[I 1 (p),,i K (p)] T is a standard interference function if the foowing three properties are satisfied for a power vectors p 0: I(p) > 0 (positivity). If p p,theni(p) I(p ) (monotonicity). If c>1,theci(p) > I(cp) (scaabiity). If I(p) is a standard interference function, the corresponding power contro agorithm p(n +1) = I(p(n)) is caed a standard PCA. From [6], the convergence property of a synchronous standard PCA can be described by the foowing resut: /02/$ IEEE. 1750

3 Theorem 1: If I(p) is feasibe, then for any initia power vector p, the standard PCA converges to a unique fix point p. It is shown in [7] that the PCA based on SIR is standard. The foowing resuts show that both the PCA based on MSE and the combined PCA are aso standard. Proposition 1: The interference function Ī(p) of the PCA based on MSE is a standard interference function. Proof: Positivity: From [2], we have MMSE i < 1. Therefore, 1 MMSE i > 0. Soifp > 0,thenĪi(p) > 0. Monotonicity: To prove monotonicity, we show that Īi(p) is increasing in p j,thatis,a T i ( j h2 j p ja j a T j + σ2 I N a i is decreasing in p j. To do so, we verify that the derivative is negative: d [ a T i dp j h 2 p a a T ai ] + σ 2 I [ d = a T i h 2 dp p a a T + σ 2 I N ]a i j { = a T i h 2 j p ja j a T j + σ2 I N j d ) h 2 p a a T + σ 2 I N dp j h 2 p a a T + σ 2 I N }a i, = h 2 j < 0. [ a T i Scaabiity: Note that and Ī i (cp) = cīi(p) = h 2 i at i h 2 i at i aj ] 2 h 2 p a a T + σ 2 I N c(1 MMSE i ) ai j h2 j p ja j a T j + 1 c σ2 I N c(1 MMSE i ) ai. j h2 j p ja j a T j + σ2 I N Assume that c>1. It is easy to see that j h2 j p ja j a T j + σ 2 I N and j h2 j p ja j a T j + 1 c σ2 I N have the same eigenvectors, and that each eigenvaue of the former matrix is smaer than that of the atter matrix corresponding to the same eigenvector. Thus j h2 j p ja j a T j + 1 c σ2 I N j h2 j p ja j a T j +σ2 I N is positive definite, which impies that cīi(p) > Īi(cp). Proposition 2: The interference function Î(p) of the combined PCA is a standard interference function. Tota Transmitted Power PCA based on MSE PCA based on SIR Combined PCA, ε =0.5 Combined PCA, ε = Iteration Fig. 1. Convergence of the tota transmit power given by different PCAs for the MMSE receiver. Proof: Since both I(p) of the PCA based on SIR [7] and Ī(p) of the PCA based on MSE are standard, it is straightforwardtoverifythatî(p) satisfies the three properties of a standard interference function. A the three PCAs considered are standard PCAs and they sove the same power contro probem described in (4). Therefore if there exists feasibe soution for the power contro probem, a the PCAs converge to the same unique fix point and have the same minimum tota transmitted power after convergence according to Theorem 1. We study the deterministic (assuming perfect estimation of a channe parameters) convergence of the power contro agorithms through computer simuations. A singe-ce synchronous MMSE receiver based DS-CDMA system is considered and the interce interference is ignored. We use a uniform channe gain of 1 and generate God spreading sequences with a processing gain of 31 for a users. The number of users in the system is set to be 15. We set the variance of the AWGN noise to be 0.2. Athough a the PCAs given in the paper permit heterogeneous target SIRs γ i for the users, we choose a common target SIR γ i =7dBto simpify the simuation. The simuation resut is shown in Fig. 1. We note that the tota transmitted powers of a the PCAs converge to the same vaue. The PCA based on SIR converges faster than the PCA based on MSE. The convergence rate of the combined PCA is between those of the above two PCAs. The smaer the vaue of ɛ, thesower the combined PCA converges. The difference between the convergence rates of the PCAs can be expained intuitivey in a simpe way. Comparing (9) and (10), we observe that the PCA based on SIR uses a arger scaing factor to update the user power eading to a faster convergence rate. Since the combined PCA is a tradeoff between the PCA based on SIR and the PCA based on MSE, adjusting /02/$ IEEE. 1751

4 the parameter ɛ can make the PCA achieve a convergence rate between that of the PCA based on SIR and that of the PCA basedonmse. V. IMPLEMENTATION AND STABILITY OF POWER CONTROL ALGORITHMS In [7], Uukus and Yates discussed some practica impementation consideration of the PCA based on SIR. We briefy reiterate the method here. The output of the receiver fiter for the ith user can be expressed as y i = p i h i (c T i a i)b i + pj h j (c T i a j)b j + z i, (12) j =i where z i is the corresponding Gaussian noise at the receiver output. Thus the MMSE fiter output of the ith user is E[yi 2 (n)] = p ih 2 i (c T i a i) 2 + p j h 2 j (ct i a j) 2 + σ 2 (c T i c i) 2. j =i (13) From (5), the power update step of the PCA based on SIR can then be written as p i (n +1)= γ i E[yi 2(n)] p ih 2 i (c T i a i) 2 h 2 i (c T i a. (14) i) 2 We note that E[yi 2 (n)] can be easiy estimated by using the time average of yi 2 (n) over mutipe bit intervas. However the impementation of the PCA based on SIR aso requires the desired user to estimate its own reverse ink channe gain. Reiabe estimation of the channe gain is hard to impement in the presence of MAI without the use of a arge number of piot symbos. This motivates the proposed power contro agorithm based on MSE that uses measurements of the mean squared error instead. Adaptive fitering agorithms such as the LMS and RLS agorithms are used for the practica impementation of the MMSE receiver. A training signa is needed for initia convergence (train period) and then the decisions made at the output of the decision device in the receiver are used instead of the true transmitted bits in the tracking period. For instance, the RLS agorithm is briefy described as foows: e i () = b i () c T i ()r(), K() = R 1 ()r() λ + r T ()R 1 ()r(), R 1 ( +1) = 1 [ R 1 () K()r T ()R 1 () ], λ c i ( +1) = c i ()+K()e i (). The RLS agorithm utiizes the error between the output of the receiver fiter and the true data bit to adjust the fiter weights adaptivey. We note that this side information e i () can be averaged over a bock of symbos to estimate the mean squared tota transmission power PCA based on SIR Combined PCA, ε =0.5 Combined PCA, ε =0.3 PCA based on MSE iteration Fig. 2. Convergence of the tota transmit power given by different PCAs using the estimated MSE vaues. error. The resuting time-averaged estimator of the MSE is MSE i = 1 L L e i () 2, (15) where L is ength of the bock. It is then used to update the power for both the PCA based on MSE, the PCA based on SIR and the combined PCA. In this approach, the estimated vaue of the MSE at the receiver is fed back to the transmitter. Using this vaue, the transmitter adjusts the power for the transmission of the next bock of data. We use MSE i in pace of MMSE i (n) in (9) and (10) for the PCA based on MSE and the PCA based on SIR, respectivey. The stabiity of the proposed PCAs is studied through simuation. In the simuation, the RLS agorithm is empoyed to adaptivey adjust the fiter weights of the MMSE receiver. At the same time, the side information e i () is used to approximate the mean squared error. Then we use this estimate to update the user power. The bockength is assumed to be 500 bits and other simuation conditions are set to be the same as those set for the simuation of the deterministic power contro agorthms in Section IV. The simuation resuts are shown in Fig. 2 and Tabe I. Fig. 2 shows typica reaizations of different PCAs. Tabe I shows the standard deviations of the SIRs and powers using different PCAs after convergence. They are normaized by the average SIRs and powers after convergence. From Fig. 2, we observe that the PCA based on SIR converges faster than the PCA based on MSE as in the deterministic case. The convergence rate of the combined PCA is between the above two PCAs. However the PCA based on SIR has a arge initia overshoot of the tota transmission power, consuming more power initiay. The overshoot is caused by errors in estimating the channe parameters. The PCA based on MSE is more robust against these estimation errors. From Tabe I, = /02/$ IEEE. 1752

5 it is seen that the PCA based on MSE is more stabe and has smaer osciations than the PCA based on SIR. The stabiity of the combined PCA is between the above two PCAs. This can be easiy expained by comparing (9) and (10). From (10), we see that a sma estimation error of the MSE can cause a arge estimation error of the SIR which eads to the arge osciations using the PCA based on SIR. At the beginning of power update process, the estimation error are usuay arge. This makes the PCA based on SIR more ikey to have overshoots. Hence there is a trade-off between the convergence rate and the stabiity of these PCAs for the MMSE receiver. Proper choices of ɛ in the combined PCA hep achieve a desired convergence rate and a desired degree of stabiity. For instance, with the choice of ɛ =0.3, the convergence rate of the combined PCA is amost the same as that of the PCA based on SIR and there is ony a sma overshoot at the beginning of the update process. VI. CONCLUSION In this paper, we propose a power contro agorithm for the MMSE receiver empoying the mean squared error (MSE) to update the power. We show that the proposed agorithm is a standard PCA so that it converges to a minimum power soution which guarantees that a users SIR performance requirements are satisfied. Compared with the PCA proposed by Uukus and Yates [7], the proposed agorithm updates the user power by using the estimates of the mean squared error instead of the estimated vaues of the channe gain. This makes the proposed agorithm more stabe. We aso note that the proposed agorithm has a drawback of sow convergence. To improve the convergence, we construct a new PCA by combining the two TABLE I THE STANDARD DEVIATIONS OF SIRS AND POWERS (NORMALIZED). σ SNR σ power PCA based on SIR 5.53% 8.22% Combined PCA (ɛ =0.5) 3.76% 5.14% Combined PCA (ɛ =0.3) 3.13% 3.97% PCA based on MSE 2.26% 1.93% PCA agorithms. Proper combinations of the two agorithms hep achieve a desired convergence rate and a desired degree of stabiity. REFERENCES [1] S. Verdú, Mutiuser Detection, New York: Cambridge Univ. Press, [2] U. Madhow and M. Honig, MMSE interference suppression for directsequence spread-spectrum CDMA, IEEE Trans. Commun., vo. 42, pp , Dec [3] J. Zander, Performance of optimum transmitter power contro in ceuar radio systems, IEEE Trans. Veh. Techno., vo. 41, pp , Feb [4] S. A. Grandhi, R. Vijayan and D. J. Goodman, Centraized power contro in ceuar radio systems, IEEE Trans. Veh. Techno., vo. 42, pp , Nov [5] G. J. Foschini and Z. Mijanic, A simpe distributed autonomous PCA and its convergence, IEEE Trans. Veh. Techno., vo. 42, pp , Nov [6] R. Yates, A framework for upink power contro in ceuar radio systems, IEEE J. Seected Areas Commun., vo. 13, pp , Sep [7] S. Uukus and R. Yates, Adaptive power contro and mmse interference suppression, Wireess Networks, vo. 4, no. 6, pp , [8] P. S. Kumar and J. Hotzman, Power contro for a spread spectrum system with mutiuser receivers, IEEE PIMRC 95, 1995, pp /02/$ IEEE. 1753