DOWNLINK TRANSMITTER ADAPTATION BASED ON GREEDY SINR MAXIMIZATION Dimitrie C Popescu, Shiny Abraham, and Otilia Popescu ECE Department Old Dominion University 231 Kaufman Hall Norfol, VA 23452, USA ABSTRACT In this paper we present algorithms for transmitter adaptation in downlin wireless systems based on maximization of the Signal-to-Interference plus Noise-Ratio (SINR) We investigate their fixed-point properties in various scenarios and their relationship with optimal points in dual uplin scenarios Index Terms CDMA, downlin, codeword adaptation, interference avoidance 1 INTRODUCTION Code Division Multiple Access (CDMA) enables multiuser communications along with efficient utilization of available spectrum and transmitter power in wireless systems and has been proposed for use in future generation wireless systems We note that, while the uplin CDMA scenario (dealing with transmissions from multiple users to a central receiver or base station) has been studied extensively over the past years and algorithms for uplin CDMA codeword adaptation are presented in [1, 2, 3, 4], the downlin scenario (which deals with transmission from a central base station to multiple receivers) has received less attention and only few wors discuss downlin transmitter adaptation [5, 6, 7] In this paper we discuss algorithms for downlin CDMA codeword adaptation that are based on greedy SINR maximization through interference avoidance [8] The first proposed algorithm is based on the collaborative approach introduced in [9] where the signals received by all active users are combined and used for joint decoding, and the proposed algorithm yields socially optimal codeword ensembles which maximize individual SINRs at the collaborative receiver as well as sum capacity Since collaboration can be difficult to achieve in the downlin scenario we discuss also two alternative algorithms for codeword adaptation where no collaboration among receivers is needed In the first one inverse channel observations similar to [3] are used to obtain the decision variables and decode received signals by users In this case the received signal by a given user is first equalized by multiplication with its inverse channel matrix followed by matched filter detection The second alternative algorithm is based on the interference avoidance procedure in [7] and uses matched filters and channel information to obtain the decision variable directly from the received signal The paper is organized as follows: we introduce the downlin CDMA system model and state our problem in Section II followed by presentation of the proposed algorithms in Sections III V In Section VI we discuss fixed-point properties of the proposed algorithms for various scenarios and conclude with final remars in Section VII 2 SYSTEM MODEL AND PROBLEM STATEMENT In the downlin of a CDMA system with signal space dimension N the base station transmits information to wireless terminals (or users) by means of distinct N-dimensional codewords Assuming that there are K active users in the system the signal transmitted by the base station is expressed as x = b p s = SP 1/2 b (1) =1 where S =[s 1 s K ] is the N K matrix of user codewords, P = diag[p 1,,p K ] is the K K diagonal matrix of transmitted powers, and b =[b 1,,b K ] > is the K- dimensional vector of transmitted symbols The transmitted signal x by the base station is received by the K user receivers through distinct vector channels characterized by channel matrices G 1,,G K of dimension N N, and is corrupted by additive Gaussian noise vectors n 1,,n K with zero-mean and covariance matrices W = E[n n > ], = 1,,K Thus, the received signal by a given user is given by r = G x + n = G SP 1/2 b + n =1,,K (2) The N N vector channel matrix G is the mathematical representation of the physical channel between the base station transmitter and user receiver and embeds channel characteristics lie attenuation, multipath, or multiple antennas [3, 4, 5] Channel matrices G 1,,G,,G K are assumed invertible and nown at the receiver as well as fixed for the entire duration of the transmission 978-1-4244-3508-1/09/$2500 2009 IEEE 94
In this setup we discuss algorithms that adapt the CDMA codewords {s 1,,s K } assigned to users through various approaches based on greedy SINR maximization and interference avoidance expression of the MMSE receiver for user is [9] c = q R 1 y y > R 2 y = q R 1 Gs s > G> R 2 Gs (8) 3 COLLABORATIVE CODEWORD ADAPTION Reference [9] considres a wireless system with multiple transmitters and receivers that collaborate and presents an algorithm for CDMA codeword adaptation based on greedy SINR maximization through interference avoidance This algorithm converges to a socially optimal ensemble of codewords which maximizes sum capacity We note that the downlin CDMA system considered in our paper and described in the previous section is a particular case of the general wireless scenario considered in [9] where the system has multiple receivers but only a single transmitter Following [9] and assuming collaboration among receivers we may form the KN-dimensional vector r by grouping the received signals by all users as shown below r 1 r K r or in compact form = G 1 G K G SP 1/2 b + n 1 n K n (3) r = GSP 1/2 b + n (4) The signal in equation (4) has correlation matrix R = E[rr > ]=GSPS > G > + W (5) where matrix W represents the NK NK correlation matrix of the aggregated noise vector n containing the noise vectors at each receiver and is expressed in terms of the covariance matrices W of individual noise vectors n, =1,,K,as W 1 W = E[nn > ]= W (6) W K Using the approach proposed for the multi-receiver system in [9] we rewrite (4) from a given user perspective as r = p Gs b desired signal + =1, 6= p Gs b + n interference+noise and assume that an MMSE receiver is used to decode user from the received signal r Denoting y = p Gs the (7) where R = R y y > is the correlation of the interference+noise corrupting the desired user signal The corresponding SINR expression for user is γ (col) = y > R 1 y = p s > G > R 1 Gs (9) The greedy SINR maximization procedure in [9] replaces user codeword s with a new one that maximizes the SINR expression (9) We note that the right-hand side of (9) contains the Rayleigh quotient of matrix G > R 1 G multiplied by the desired user power p and for fixed user power is maximized when s is replaced by the eigenvector corresponding to the maximum eigenvalue (also referred to as the maximum eigenvector) of G > R 1 G Based on this procedure the algorithm for downlin CDMA codeword adaptation can be formally stated as follows: Algorithm I 1 Initial data: All user codeword and power matrices S and P as well as all channel and noise covariance matrices G and W, 2 FOR each user =1,,K DO: (a) Calculate matrix G > R 1 G and determine its maximum eigenvector x (b) Replace user s current codeword s by x We note that Step 2) of the algorithm defines an ensemble iteration, in which all codewords are updated one time We also note that this algorithm monotonically increases the sum capacity of the multiple access channel (4) given by [9] C sum = 1 2 log R 1 log W (10) 2 and converges codeword ensembles that maximize C sum Algorithm I may be implemented at the base station transmitter which needs to now (in addition to the user codewords and powers which are already nown) also the downlin channel matrices, G, for all active users in the system along with their corresponding noise covariance matrices W We note that channel state information for the downlin can be made available at the transmitter by using either direct channel feedbac in the case of frequency-division duplex (FDD) systems or the reciprocal channel information from the uplin for time-division duplex (TDD) systems [10], while nowledge of noise covariance matrices at each receiver may be obtained over a dedicated feedbac channel However, in order 95
to tae advantage of the optimal codewords and to obtain the maximum SINR at the receiver collaborative decoding using all received signals is required which may not be practical Alternative algorithms in which users are not required to perform collaborative decoding to achieve maximum SINR are presented in the following sections 4 CODEWORD ADAPTATION BASED ON INVERSE CHANNEL OBSERVATIONS Assuming that the user channel matrices are invertible, in this approach the receiver at a given user uses an inversechannel observation to decode the transmitted information symbol by the base station The inverse-channel observation vector is obtained similar to [3] by equalizing the received signal through multiplication with the inverse of the given user channel matrix, r = G 1 r, and has the expression r = b p s + 6=1, =1 b p s + G 1 n (11) The inverse-channel observation is processed by a matched filter corresponding to user s codeword to obtain the decision variable for user, d = s > r, and is expresses as d = b p + =1, 6= b p s > s + s > G 1 n (12) which implies that the expression of the SINR for user is γ (ic) = s > G 1 p Q p G s s > G > G > Z s (13) where Q represents the correlation matrix of the received signal at user receiver in equation (2) In order to perform greedy maximization of the SINR in this case we note that the denominator in equation (13) contains the Rayleigh quotient of matrix Z = G 1 Q p G s s > G> G > which should be minimized in this case to achieve maximum SINRWenotethatZ represents the correlation matrix of the interference+noise that corrupts the desired signal from user in the inverse channel observation, and replacing the user codeword s by the minimum eigenvector of Z will ensure maximization of user SINR Based on this procedure we can define a second algorithm for downlin CDMA codeword adaptation which is formally stated as follows: Algorithm II 1 Initial data: User codeword and power matrices S, P, channel and noise covariance matrices G, W, 2 FOR each user =1,,K DO: (a) Determine minimum eigenvector u for matrix G > R 1 G (b) Replace user s current codeword s by u Similar to Algorithm I Step 2) defines an ensemble iteration in which all codewords are updated one time, but unlie Algorithm I for which convergence to a fixed point has been established analytically in [9] based on the monotonic increase of the sum capacity 1 C sum in (10), for Algorithm II we have only empirical evidence of convergence A fixed point of the algorithm is reached when the difference between two consecutive values of a stopping criterion is within a specified tolerance value ε, and we ran extensive simulations to assess the convergence of Algorithm II using the norm difference between a given codeword and its replacement as stopping criterion Numerical results have shown that codewords converge to within tight norm difference tolerances when starting with randomly initialized user codewords, and for ε =10 3 codeword convergence varied from tens of ensemble iterations for low values of the signal space dimension, to several hundred ensemble iterations for large values of the signal space dimensions We have also looed at the convergence of user SINRs which occurs much faster than codeword convergence and simulations have shown that typically this occurs in 10 15 ensemble iterations and does not depend on the dimension of the signal space Algorithm II may also be implemented at the base station and requires similar information as Algorithm I However, collaborative decoding using all received signals is not required and users decode their received signals independently in this case 5 CODEWORD ADAPTATION WITH MATCHED FILTER RECEIVERS This approach for downlin CDMA codeword adaptation has been proposed in [7] and in this case users employ linear filters matched to the expression of their corresponding received codeword to obtain the decision variable For a given user the receiver filter has the expression f = 1 qs > G> G s G s =1,,K (14) 1 Sum capacity is a global performance measure for the system in the collaborative approach and is maximized by Algorithm I 96
and its decision variable in this case d = f > r which implies that the SINR at user receiver is = f > p (f > G s ) 2 Q p G s s > G > Z f (15) and Z = Q p G s s > G> represents the correlation matrix of the interference+noise that corrupts the desired signal from user in this case Rewriting the SINR expression as = p f > (G s s > G> )f f > Z f (16) we note that for a given user codeword s, the user SINR is maximized by the choice of receiver filter f which implies maximization of the ratio of the two quadratic forms definedbymatricesg s s > G> and Z As discussed in [7] this ratio is maximized by the largest eigenvalue of the matrix pencil defined by the pair of matrices (G s s > G >, Z ) (17) and implies that the optimum receiver filter f is the eigenvector corresponding to the largest generalized eigenvalue of the matrix pair (17), that is (G s s > G > )f = λ Z f (18) Using the same assumption as in the previous section, namely that the user channel matrices are invertible, the codeword update for user is obtained using the expression of the SINR maximizing filter f as s = G 1 f (19) We note that this codeword update is similar to the uplin MMSE interference avoidance update [8] and based on it we define a third algorithm for downlin CDMA codeword adaptation using greedy SINR maximization which is formally stated below: Algorithm III 1 Initial data: User codeword and power matrices S, P, channel and noise covariance matrices G, W, 2 FOR each user =1,,K DO: (a) Determine the SINR maximizing filter f using equation (18) (b) Replace user s codeword with s obtained from equation (19) As in the case of the other two algorithms defined in previous sections, Step 2) defines an ensemble iteration in which all codewords are updated one time Convergence of Algorithm III to a fixed point was investigated in [7] where numerical results obtain from simulations are used to establish empirical convergence similar to Algorithm II Wenotethat no analytical convergence proof is available for Algorithms II and III yet, and this will be the object of future research We also note that Algorithm III may be implemented at the base station and requires similar information as Algorithms I and II with independent decoding of received signals by users and no need for collaborative decoding 6 FIXED-POINT PROPERTIES In order to study the fixed-points properties of the proposed algorithms we performed simulations for various scenarios and looed at the correlation properties of the resulting codeword ensembles given by the matrix SPS > In the case of ideal channels for all users, that is G = I N,, and white noise at all receviers, that is noise covariance matrices are expressed as W = σ I N, all algorithms resulted in codeword ensembles having the same correlation matrix SPS > = (K/N)I N This solution corresponds to Welch Bound Equality (WBE) codeword ensembles [1] for which the total squared correlation (TSC) of the ensemble, defined as the sum of squared correlations among all codewords in the system weighted by their corresponding powers, is minimized This result can be confirmed analytically as it can easily be shown that in the case of ideal channel matrices the codeword updates implied by the three algorithms are essentially the same: the codeword of a given user is replaced by the minimum eigenvector of matrix SPS > p s s > and iterating for all users leads to WBE codeword ensembles [8] We note that WBE ensembles maximize sum capacity of the multiple access channel corresponding to the dual uplin CDMA scenario with ideal channels and white noise at the receiver With ideal channel matrices and colored noise with the same covariance matrix at all user receivers, that is W = W, where W is no longer a scaled identity matrix, the three algorithms yield codeword ensembles with different correlation properties Among the three resulting codeword ensembles the one corresponding to Algorithm II has the same correlation properties SPS > as those of the dual uplin scenario with ideal user channels and same noise covariance matrix W at the receiver This result can also be confirmed analytically as one can easily note that with ideal channels and same noise covariance matrix at all receivers the codeword updates of Algorithm II are similar to those for the dual uplin scenario with ideal channels and colored noise with covariance W atthereceiver[8] Wenotethatwhennoise is colored but has different covariance matrices for different users, a meaningful comparison with a dual uplin scenario is difficult as the corresponding noise covariance matrix in the uplin scenario can not be clearly established 97
We have also looed at point-to-point communication scenarios where the channel and noise covariance matrices are the same for all users and observed that the algorithms yield in general codeword ensembles with different correlations, and that only the ensembles implied by Algorithm II have the same correlation as those of the dual uplin CDMA scenario This was also expected as in the point-to-point scenario the codeword updates implied by Algorithm II and those corresponding to the uplin scenario [3] are similar Simulations have also shown that in the case of interferencelimited systems, where the power of the noise corrupting the received signals corresponding to all users is small compared to the power of the transmitted signal, that is Trace [W ] Trace [P],, Algorithm I and Algorithm II yield very similar results with codeword correlations getting closer to those corresponding to WBE ensembles, that is SPS > =(K/N)I N, as the power of the noise gets closer to zero regardless of the user channel matrices We note that, while it is not obvious why WBE ensembles correspond to fixed points of Algorithm I for interference-limited scenarios, for Algorithm II this can be easily confirmed analytically as we note that in the case of very small noise power at the receiver the inversechannel observation in equation (11) is essentially identicalforallusers, =1,,K, and is the same as the uplin channel equation considered in [8] where it is shown that codeword adaptation based on greedy SINR maximization and minimum eigenvector replacement results in WBE codeword ensembles In the most general case of non-ideal channels where user channel matrices are only assumed to be invertible but can tae any form (diagonal, circulant, etc) and the noise covariance matrices W are different for distinct users, simulations have shown that the three proposed algorithms result in general in codeword ensembles that have different correlation properties In such scenarios additional criteria are needed to decide which of the resulting codeword ensembles is more desirable and this will be the object of future investigations 7 CONCLUSIONS In this paper we considered the downlin of a wireless system and presented three algorithms for CDMA codeword adaptation based on various greedy SINR maximization procedures Algorithm I uses a collaborative approach and yields codeword ensembles that maximize the sum capacity which is a global measure for the wireless system when collaboration among user receivers is assumed Since receiver collaboration may be difficult to achieve in the downlin scenario we discuss also two alternative algorithms where collaboration is not needed: Algorithm II is based on inverse channel observations and minimum eigenvector codeword replacement similar to the one proposed for uplin scenarios in [3], while Algorithm III uses matched filters and channel nowledge along with a maximum generalized eigenvector codeword replacement as outlined in [7] The proposed algorithms yield codeword ensembles that usually have different correlation properties and additional criteria should be specified in order to determine which of the ensembles is desirable in downlin scenarios 8 REFERENCES [1] P Viswanath and V Anantharam, Optimal Sequences for CDMA Under Colored Noise: A Schur-Saddle Function Property, IEEE Transactions on Information Theory, vol 48, no 6, pp 1295 1318, June 2002 [2] T Guess, Optimal Sequences for CDMA with Decision-Feedbac Receivers, IEEE Transactions on Information Theory, vol 49, no 4, pp 886 900, April 2003 [3] DCPopescuandCRose, CodewordOptimzation for Uplin CDMA Dispersive Channels, IEEE Transactions on Wireless Communications, vol 4, no 4, pp 1563 1574, July 2005 [4] GSRajappanandMLHonig, SignatureSequence Adaptation for DS-CDMA with Multipath, IEEE Journal on Selected Areas in Communications, vol 20, no 2, pp 384 395, February 2002 [5] H Bi and M L Honig, Power and Signature Optimization for Downlin CDMA, in Proceedings 2002 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