Gradient Descent Interference Avoidance with Target SIR Matching Dimitrie C. Popescu, Otilia Popescu, and Danda B. Rawat Department of Electrical and Computer Engineering Old Dominion University 231 Kaufman Hall, Norfolk, VA 23529 Contact e-mail:
[email protected] II. S YSTEM M ODEL AND P ROBLEM S TATEMENT We consider the uplink of a synchronous CDMA system with processing gain N and K active users, for which the Ndimensional received signal vector at the base station is given by the expression
Abstract—In this paper we present a new algorithm for joint codeword and power adaptation in the uplink of Code Division Multiple Access (CDMA) systems, in which codewords are adjusted using gradient-descent interference avoidance, and powers are updated to match target Signal-to-Interference plus NoiseRatio (SINR) values. The proposed algorithm is analyzed and illustrated with numerical examples obtained from simulations.
r=
I. I NTRODUCTION
K X
√ bk pk sk + n = SP1/2 b + n
(1)
k=1
Interference avoidance (IA) is a transmitter adaptation technique applicable to codeword optimization in CDMA systems, by which the transmitted signal energy is placed in the least occupied region of the signal space, such that the signal is received with minimum interference at the receiver [1]. IA was developed initially for codeword adaptation in CDMA systems with fixed user powers [2], [3], and was subsequently augmented with a power control mechanism [4] to allow more flexibility and enable users to achieve specified target SINRs. In this paper we present a new algorithm for joint IA and power adaptation, with gradient-based incremental codeword updates followed by power updates to match specified target SINRs. We note that gradient-based algorithms were used in the past for related IA algorithms [5] or for transmitter adaptation in multiple antenna systems [6]. The proposed algorithm is simple and suited for implementation in practical systems since it does not require complex computations like the minimum eigenvector or the MMSE updates for IA [2], [3]. In addition, empirical evidence shows that the proposed algorithm yields Generalized Welch Bound Equality (GWBE) ensembles of user codewords and powers [7]. When oversized users with disproportionate target SINR requirements relative to the other users in the system are present, the algorithm yields orthogonal codewords to oversized users without prior knowledge of their oversized status as in [7]. The paper is organized as follows: we describe the system model and formally state the problem in Section II. In Section III we present the expressions for codeword and power updates, and provide a formal statement of the proposed algorithm. In Section IV we present numerical results obtained from simulations illustrating convergence and fixedpoint properties of the algorithm vis-a-vis to those of the optimal GWBE codeword and power ensembles discussed in [7]. Final conclusions are presented in Section V.
where S = [s1 , . . . , sk , . . . , sK ] is the N × K codeword matrix having as columns the unit-norm codewords {sk } of active users in the system, P = diag[p1 , . . . , pk , . . . , pK ] is the K × K diagonal matrix containing received powers of active users, b = [b1 . . . bk . . . bK ]> is the K-dimensional vector containing the information symbols transmitted by users, and n is the additive white Gaussian noise (AWGN) that corrupts the received signal with zero-mean and positive definite covariance matrix equal to the scaled identity matrix W = E[nn> ] = σ2 IN . At the receiver a unit norm linear filter, ck , is used to obtain the decision variable dk for user k K X √ √ > > dk = c> b p s + n k r = bk pk ck sk + ck | {z } =1, 6=k desired signal | {z } interference + noise (2) The SINR for user k is the ratio of the desired signal corresponding to user k at the receiver to the power of interference and noise that affects user k’s signal at the receiver, and is expressed as γk =
2 pk (c> k sk ) K X
p
(c> ks
)
2
(3)
2 + E[(c> k n) ]
=1, 6=k
In the case of matched filter (MF) receivers ck = sk the SINR expression (3) becomes pk (4) γk = > sk Rk sk where Rk =
K X
=1, 6=k
p s s> + W = R − pk sk s> k
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1-4244-1457-1/08/$25.00 © IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE CCNC 2008 proceedings.
(5)
is the correlation matrix of the interference-plus-noise seen by user k and R = SPS> + W is the correlation matrix of the received signal in equation (1). We note that the presence of the positive definite noise covariance matrix W ensures that both Rk and R are also positive definite matrices. Our goal in this setup is to derive a distributed algorithm in which individual users adjust codewords and powers to achieve ∗ }. The proposed specified target SINRs {γ1∗ , . . . , γk∗ , . . . , γK algorithm will consist of two separate updates: a gradientbased codeword update designed to decrease the effective interference, followed by the power update designed to match the specified target SINR. We note that K users with specified SINR requirements, are admissible in the uplink of a CDMA system with processing gain N if and only if the sum of their effective bandwidths γk∗ k = 1, . . . , K (6) e(γk∗ ) = 1 + γk∗
does not require complex calculations like determining the minimum eigenvector or taking the inverse of Rk . After the gradient-descent codeword update in equation (11) user k SINR becomes pk (n) (12) γk0 (n + 1) = > sk (n + 1)Rk (n)sk (n) and may be matched to the specified target SINR γk∗ for user k by applying the power update pk (n + 1) = γk∗ s> k (n + 1)Rk (n)sk (n + 1)
(13)
(11)
We note that, when the SINR after codeword update γk0 (n+1) is above the specified target the power update equation (13) implies a decrease in user power, and when γk0 (n+1) is below the specified target the power update equation (13) implies an increase in user power. If the resulting power value implied by the power update equation (13) is above the maximum power by which user k is allowed to transmit, the new power value . will be updated with the maximum allowed power value pmax k Using the codeword and power updates in equations (11) and (13) the gradient-descent IA with SINR matching algorithm is formally stated as follows: 1) Input data: noise covariance matrix W, initial user codeword and power matrices S and P, desired tar∗ }, maximum user power matrix get SINRs {γ1∗ , . . . , γK Pmax , constant µ. 2) If admissibility condition in equation (7) is satisfied continue with Step 3, else STOP. 3) For each user k = 1, . . . , K do a) Compute Rk using equation (5). b) Update user k’s codeword to reduce effective interference using equation (11). c) Compute power required to match specified SINR target using equation (13). d) If pk (n + 1) implied by equation (13) is less update user k’s power with this value. than pmax k . Otherwise update power to pmax k 4) Iterate until a fixed point is reached. We note that Step 3) of the algorithm defines an ensemble iteration that consists of K individual user updates, in which all user codewords and powers are updated one time. A fixed point of the algorithm, is defined with respect to a stopping criterion, such that codeword and power updates at this point imply no change in the value of the stopping criterion. Numerically, we say that a fixed point is reached when the difference between two consecutive values of the stopping criterion is within a specified tolerance ε, and in the simulation studies performed to investigate convergence we used as stopping criterion the difference between the actual codeword SINR at the receiver and the desired target SINR, as well as the Euclidian distance between a given codeword and its corresponding replacement.
to ensure that it yields a unit norm codeword. We note that the proposed gradient-descent codeword update has a computational advantage over other IA updates like the minimum eigenvector update [3] or the MMSE update [2] since it
IV. S IMULATIONS , N UMERICAL E XAMPLES , AND D ISCUSSIONS In order to corroborate our theoretical findings and illustrate the proposed algorithm we performed extensive simulations
is less than the processing gain [7] K X
e(γk∗ ) < N
(7)
k=1
III. G RADIENT D ESCENT IA WITH TARGET SINR M ATCHING In order to obtain an IA codeword update equation we note that any codeword change which decreases the effective interference ik given by the denominator in the SINR expression (4), results in an increase of the SINR [1]. Thus, we formally write (8) ik = s> k Rk sk and note that this is a quadratic form implied by positive definite matrix Rk . As a consequence, ik is strictly convex over the N -dimensional unit sphere {sk |sk ∈ RN , ksk k = 1}, and has the global minimum point equal to the minimum eigenvalue of Rk achieved for codeword sk equal to the corresponding (minimum) eigenvector of Rk [8, Sec. 6.4]. The global minimum point can be approached using the gradientdescent update iteration ¯ ∂ik ¯¯ (9) sk (n + 1) = sk (n) − µ ∂sk ¯ instant n
with µ a suitably chosen constant, and the gradient of ik with respect to the codeword components given by ¯ ∂ik ¯¯ = 2Rk (n)sk (n) (10) ∂sk ¯ instant n
Using equations (9) and (10) we define the actual codeword update as sk (n + 1) =
[I − 2µRk (n)]sk (n) k[I − 2µRk (n)]sk (n)k
201 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE CCNC 2008 proceedings.
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Fig. 1. Codeword, respectively SINR convergence for 100 trials of the proposed algorithm for a system with K = 15 users in N = 10 dimensions, target SINRs for all users equal to 1.95, and gradient constant µ = 10−3 . Sum of effective bandwidths is 9.9153 – roughly 10% below the upper bound – and target SINRs can be achieved with arbitrary precision. 0
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for various scenarios, which we present in this section. Convergence of the algorithm is discussed in Section IV-A, followed by examples that show the variation in user power and SINR and illustrate fixed-point properties vis-a-vis to those of the optimal GWBE ensembles [7]. A. Algorithm Convergence Extensive simulations of the gradient descent IA with matching target SINRs algorithm have shown that, when initialized with random user codewords, random powers, and admissible target SINRs – as defined by equation (7), the algorithm reaches a GWBE ensemble of user codewords and powers [7] within some tolerance limits that can be adjusted through parameters µ and as it is the case in general with gradient-based algorithms. This behavior is consistent with that of related IA algorithms which always converge to GWBE ensembles of user codewords and powers from random initialization [1], and analytical approaches to establish convergence of the algorithm are currently being investigated. In the simulations investigating convergence we used both stopping criteria mentioned in Section III: the difference between the actual SINR and the desired target SINR, and the Euclidian distance between a given codeword and its corresponding replacement in two consecutive iterations. When the latter criterion is used the speed of convergence to tight norm difference tolerances (ksk (n + 1) − sk (n)k ≤ 10−6 , ∀k) depends, as it was expected, on the value of the µ constant, and the smaller this is, the slower the algorithm settles down. When the former criterion is used we note that that the precision with which the target SINRs are reached after the algorithm settles down and no more changes in user codeword and powers occur depends on the specified target SINR values: when these are such that the sum of user effective bandwidths in equation (7) is not close to N , then the specified targets can be achieved with arbitrary precision as shown in Fig. 1 which is typical for all the simulations we performed. However, when the sum of effective bandwidths corresponding to the specified targets is very close to the upper bound N , then the actual SINRs are reached only with limited precision of the order of 10−2 , as shown in Fig. 2, which is also typical for all simulations. Fig. 1 and 2 show also that, for the same value of the gradient constant µ, convergence occurs faster in both codewords and SINRs when the sum of user effective bandwidths is not very close to the upper bound N in equation (7), which is another characteristic observed in all simulations performed. Empirically we have noted that the number of ensemble iterations needed for convergence does not increase with increasing number of users and signal dimensions, which implies that the number of codeword updates per user needed for convergence stays approximately constant for increasing number of signal dimensions and users. We ran numerous simulations with N ranging from 5 to 50 and the ratio K/N ' 1.5 and observed that convergence was between 60 and 80 ensemble iterations when the sum of effective bandwidths was not close to its upper bound N in equation (7), and between 300 and 400 ensemble iterations when the sum of effective bandwidths was close to its upper bound.
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Fig. 2. Codeword, respectively SINR convergence for 100 trials of the proposed algorithm for a system with K = 15 users in N = 10 dimensions, target SINRs for all users equal to 1.99, and gradient constant µ = 10−3 . Sum of effective bandwidths is 9.9833 – only about 1% below the upper bound – and target SINRs are achieved with limited precision.
202 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE CCNC 2008 proceedings.
B. Variation of User Powers and SINRs, and Fixed-Point Properties
4 3.5
In this section we consider a CDMA system with K = 7 users in a signal space of dimension N = 5 and AWGN with σ 2 = 0.1. The tolerance for codeword convergence is = 10−6 . User codeword matrix is initialized randomly and initial user powers are also selected randomly between 0 and the maximum allowed power (taken to be 10 for all users). In a first example we set different target SINRs for users γ ∗ = {3.25, 3, 2.75, 2.5, 2.25, 2, 1.75}
SIR
User 7
(14)
pj ˜sj ˜s> j = 0.9I4
2.5
User 5 User 4 User 3 User 2
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1 0.5 0 0
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(a) µ = 10−2 4 3.5
User 7 User 6
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User 5 User 4
2.5 SIR
User 3 User 2
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(15)
1.5 1
which imply that user 1 is oversized according to [7]. In this case the algorithm yields codeword matrix S2 in equation (18) (shown on next page) and power matrix P2 = diag{1.5, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6}. such that oversized user 1 is orthogonal to all the other users as can be seen from the codeword correlation matrix in equation (19) (shown on next page). This is in agreement with properties of optimal codeword ensembles in [7] which show that oversized users must have private channels over which they transmit at the minimum power required to achieve their corresponding targets. We note that while the S2 matrix and the implied weighted correlation matrix S2 P2 S> 2 are not exactly in the form given in [7], they can be brought to the same form using the linear transformation implied by the left singular vectors [8, p. 443] of matrix S2 in (18) to align the signal space axes to the oversized user. This results in the transformed codeword ˜ 2 shown in equation (20) (shown on next page), which matrix S is in the form of [9] where oversized user 1 is using signal dimension 5 and non-oversized users 2–7 share dimensions 1– 4, and have the weighted correlation matrix [7] 7 X
User 6
1.5
which satisfy the admissibility condition in equation (7), and there are no oversized users (the sum of effective bandwidths is 4.9577 is less than the upper bound N = 5, and is not close to it). With a gradient constant µ = 0.01, convergence is achieved in approximately 30 ensemble iterations (corresponding to 210 individual codeword updates), and the SINR variation during the algorithm is shown in Fig. 3(a). Changing the gradient constant to µ = 0.001 slows down the convergence, as seen from Fig. 3(b), but it also reduces the range of SINR variations which may be desirable in practical implementations. Regardless of which gradient step µ is used, the algorithm yields final codeword matrix S1 in equation (17) (shown on next page) and power matrix P1 = diag{9.0345, 8.8605, 8.6634, 8.4384, 8.1788, 7.8758, 7.5179}, for which the weighted correlation matrix S1 P1 S> 1 + W = 11.814 I5 , and is within O(10−3 ) tolerance from the corresponding GWBE values implied by [7]. In the second example we set the target SINRs for users γ ∗ = {15, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5}
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(16)
Fig. 3. SINR variation for the system with K = 7 users in N = 5 dimensions, target SINRs in equation (14), for different gradient constants µ. Note: one ensemble iteration is equal to 7 codeword updates in this case.
j=2
We conclude presentation of this example by noting that in our case user 1 had no a priori knowledge of its oversized status in the system, and performed the same updates as the other users which resulted in the right ensemble of codewords and powers. This is unlike [7] where knowledge of the oversized status is necessary to obtain the right codewords and powers for oversized users. V. C ONCLUSIONS In this paper we presented a new algorithm for joint codeword and power adaptation in uplink CDMA systems, with codeword updates using gradient descent IA followed by power updates to match specified target SINRs. The proposed
203 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE CCNC 2008 proceedings.
Codeword matrix yielded by the algorithm for the first example in Section IV-B, with no oversized users: −0.2771 0.8654 −0.2527 −0.2680 0.4095 0.2838 −0.4029 −0.7015 0.2836 0.3208 0.6844 −0.4232 0.0921 0.1549 S1 = −0.0844 0.2588 −0.5990 0.1284 0.0325 −0.9008 0.4330 0.0864 0.1489 0.6783 0.1127 0.8062 −0.1877 0.4843 −0.6454 −0.2856 −0.1202 −0.6562 0.0466 0.2537 0.6258
Codeword matrix yielded by the algorithm for the second example in Section IV-B, with oversized user 1: −0.5468 0.6511 0.0681 0.1032 0.3167 0.3663 −0.6147 0.1139 0.6327 0.2169 0.6543 −0.5777 0.2072 0.4780 0.5482 0.1084 −0.7526 0.2409 S2 = −0.5551 −0.1594 −0.2750 0.0896 0.2679 0.8638 −0.1405 0.6293 −0.3753 0.3369 −0.6098 −0.2804 0.3558 −0.4908 −0.3978 0.3401 0.4714
Correlation matrix of codewords yielded by the algorithm for the second example in Section IV-B, user 1 is orthogonal to the other, non-oversized users: 1 0 0 0 0 0 0 0 1 0.3571 0.4938 0.1036 0.2936 −0.1782 0 0.3571 1 −0.2977 0.2685 0.0737 0.4543 > 0 0.4938 −0.2977 1 −0.1790 −0.3534 0.1027 S2 S2 = 0 0.1036 0.2685 −0.1790 1 −0.4568 −0.4201 0 0.2936 0.0737 −0.3534 −0.4568 1 −0.2736 0 −0.1782 0.4543 0.1027 −0.4201 −0.2736 1
Transformed codeword matrix for second example in Section IV-B, 0 −0.4839 −0.9595 0.0362 0 0.8151 −0.0334 0.3565 ˜2 = 0 0.3184 −0.2540 0.7068 S 0 0.0071 0.1173 −0.6099 1 0 0 0 algorithm is analyzed and accompanied by numerical examples obtained from simulations, which illustrate convergence and fixed-point properties. We note that analytical approaches to establish convergence of the proposed algorithm are currently being investigated. The gradient descent IA with target SIR matching algorithm uses simple codeword and power updates, which imply incremental changes in the system. This contrasts with related IA algorithms [2]–[4] which are more computationally intensive since they use more demanding updates based on computing the minimum eigenvector or the MMSE receiver filter. We note that in practice incremental adaptation is in general preferred, and from this perspective the proposed algorithm is more suited for implementation in practical systems. We also note, that the algorithm is amenable to distributed implementation in which individual users in the system independently adapt their codewords and powers employing gradient descent IA with target SIR matching updates, and using information about the received signal covariance matrix received on a feedback channel from the receiver [10]. R EFERENCES [1] D. C. Popescu and C. Rose, Interference Avoidance Methods for Wireless Systems. New York, NY: Kluwer Academic Publishers, 2004.
(17)
(18)
showing that oversized
(19)
with signal space axes aligned to the oversized user: −0.1317 −0.0183 −0.5712 0.3689 0.2707 −0.7057 −0.8097 0.1802 0.3827 (20) −0.4370 0.9455 −0.1713 0 0 0
[2] S. Ulukus and R. Yates, “Iterative Construction of Optimum Signature Sequence Sets in Synchronous CDMA Systems,” IEEE Transactions on Information Theory, vol. 47, no. 5, pp. 1989–1998, July 2001. [3] C. Rose, S. Ulukus, and R. Yates, “Wireless Systems and Interference Avoidance,” IEEE Transactions on Wireless Communications, vol. 1, no. 3, pp. 415–428, July 2002. [4] D. C. Popescu and C. Rose, “Interference Avoidance and Power Control for Uplink CDMA Systems,” in Proceedings 58th IEEE Vehicular Technology Conference – VTC 2003 Fall, vol. 3, Orlando, FL, October 2003, pp. 1473–1477. [5] J. Singh and C. Rose, “Distributed Incremental Interference Avoidance,” in Proceedings 2003 IEEE Global Telecommunications Conference GLOBECOM ’03, vol. 1, San Francisco, CA, December 2003, pp. 415– 419. [6] B. C. Banister and J. R. Zeidler, “Feedback Assisted Stochastic Gradient Adaptation of Multiantenna Transmission,” IEEE Transactions on Wireless Communications, vol. 4, no. 3, pp. 1121–1135, May 2005. [7] 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. [8] G. Strang, Linear Algebra and Its Applications. San Diego, CA: Harcourt Brace Jovanovich College Publishers, 1988. [9] P. Viswanath, V. Anantharam, and D. Tse, “Optimal Sequences, Power Control and Capacity of Spread Spectrum Systems with Multiuser Linear Receivers,” IEEE Transactions on Information Theory, vol. 45, no. 6, pp. 1968–1983, September 1999. [10] W. Santipach and M. L. Honig, “Signature Optimization for CDMA with Limited Feedback,” IEEE Transactions on Information Theory, vol. 51, no. 10, pp. 3475–3492, October 2005.
204 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE CCNC 2008 proceedings.
