Gradient Descent Interference Avoidance for. Uplink CDMA Systems with Multipath. Dimitrie C. Popescu, Danda B. Rawat, and Otilia Popescu. Department of ...
Gradient Descent Interference Avoidance for Uplink CDMA Systems with Multipath Dimitrie C. Popescu, Danda B. Rawat, and Otilia Popescu Department of Electrical and Computer Engineering Old Dominion University Norfolk, VA 23529 Contact e-mail: {dpopescu, drawa001, opopescu}@odu.edu Abstract— In this paper we present an algorithm for joint codeword and power adaptation in uplink Code Division Multiple Access (CDMA) systems with multipath channels between users and the base station. The algorithm is based on gradient descent (GD) interference avoidance (IA) and adapts user codewords and powers incrementally to meet specified targets for the signalto-interference+noise-ratio (SINR). The proposed algorithm is illustrated with numerical examples obtained from simulations.
I. I NTRODUCTION IA is a transmitter adaptation technique applicable to codeword optimization in CDMA systems, by which the transmitted signal energy is placed in the signal subspace with minimum interference power [1]. IA algorithms for CDMA systems with multipath proposed so far use codeword updates based on the minimum eigenvector of the interference matrix [2], [3] or on the minimum mean squared error (MMSE) filter [4], [5]. These updates may result in abrupt codeword changes that can be difficult to be tracked by the receiver and can disrupt the transmitted data streams leading to increased probability of error and outages. In addition, direct combination of these IA codeword updates with a power control mechanism to meet specified target values for the SINR does not always converge to a fixed point [6]. In order to enable smooth adaptation of the system and to allow variable number of active users and/or quality of service (QoS) requirements adaptive algorithms for joint IA and power control were proposed recently [7], [8]. We note that the algorithm in [7] employs incremental codeword updates based on the minimum eigenvector of the interference matrix while the one in [8] uses gradient-based updates that do not require the computation of the minimum eigenvector. We also note that both algorithms [7], [8] move the system in the direction of the optimum configuration for which the interference at the receiver is minimized and specified target SINRs are achieved with minimum user powers. In this paper we discuss extension of the gradient-based adaptive algorithm in [8] from the basic scenario with ideal user channels to a more practical scenario in which multipath channels between mobile transmitters and the base station are explicitly considered. The paper is organized as follows: in Section II we present the system model and formally state the problem. In Section III we discuss application of GD and IA to joint codeword and power adaptation for the system under
978-1-4244-2110-7/08/$25.00 ©2007 IEEE
consideration. In Section IV we formally state the proposed algorithm and discuss its convergence to a fixed point. We illustrate the algorithm with numerical examples obtained from simulations in Section V, and present final conclusions in Section VI. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT We consider the uplink of a synchronous CDMA system with K active users in a signal space of dimension N implied by finite common bandwidth and signaling interval constraints [9], for which the N -dimensional received signal vector at the base station corresponding to one signaling interval is given by the expression r=
K
√ bk pk Hk sk + n
(1)
k=1
where {s1 , . . . , sk , . . . , sK } are the N -dimensional user codewords assumed to have unit norm, {p1 , . . . , pk , . . . , pK } are the received powers at the base station, {b1 . . . bk . . . bK } are the information symbols transmitted by users, and n is the additive Gaussian noise that corrupts the received signal with zero-mean and positive definite covariance matrix W = E[nn ]. The multipath channels between users and the base station are described by the N × N user channel matrices H1 , . . . , Hk , . . . , HK , which may be either diagonal [3] or circulant [4], and are assumed invertible. In addition, we assume also that they are fixed and do not change during the entire duration of the transmission, and that they are known at the receiver. In order to decode the information transmitted by a given user k the receiver uses an “inverse-channel” observation obtained by equalizing the received signal with the given user k channel matrix as in [3] rk
= H−1 k r
interference + noise ⎛ ⎞ K √ √ ⎝ b p H s + n⎠ = bk pk sk + H−1 k k=1,=1 desired signal
(2)
The “inverse-channel” observation in equation (2) is processed by a matched filter corresponding to user k’s codeword to
1060
obtain the decision variable for user k dk
=
⎛ ⎞ K √ √ −1 ⎝ = bk pk + s b p H s + n⎠ k Hk
(3)
=1,=k
which implies that the expression of the SINR for user k is pk ⎛ ⎞ γk = (4) K −1 ⎝ ⎠ H− sk s p s H H s + W k Hk k =1,=k
We note that the matrix ⎛ ⎞ K ⎝ ⎠ H− p s H H Rk = H−1 s + W k k
(5)
=1,=k
in the denominator of the SINR expression (4) is the correlation matrix of the interference+noise that affects user k’s symbol in the “inverse-channel” observation and is related to the correlation matrix of the received signal in equation (1) R=
K
p H s s H + W
instant n
(6)
with μs a suitably chosen constant and the gradient of ik with respect to the codeword components given by
∂ik
= 2Rk (n)sk (n) (11) ∂sk instant n
(7)
Using equations (10) and (11) we define the actual codeword update as
=1
by the expression Rk
− R − pk Hk sk s = H−1 k Hk Hk k − = H−1 − pk sk s k k RHk
If the SINR after codeword update is below γk∗ then the power update must increase user k power. ∗ • If the SINR after codeword update is above γk then the power update must decrease user k power. We note that in practical systems the codeword and power should be changed in small increments allowing corresponding incremental changes in the receiver matched filter. These will avoid steep changes that may not be tracked by the receiver and could result in increased probability of error at the receiver or even connection loss between transmitter and receiver. We will derive such incremental updates using GD-based adaptation for the user codeword and “lagged” adaptation for the user power. In order to obtain a GD-based codeword update equation we note that the effective interference function ik in equation (9) is a quadratic form implied by positive semidefinite matrix Rk . Thus, ik is a convex function over the N -dimensional unit sphere {sk |sk ∈ RN , sk = 1}, and is decreased by the GD update iteration
∂ik
(10) sk (n + 1) = sk (n) − μs ∂sk •
s k rk
The expression of user k SINR can then be rewritten in the simpler form pk γk = (8) sk Rk sk and note that the denominator term represents the effective interference+noise power that is present in user k’s decision variable ik = s (9) k Rk sk We also 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 their corresponding codewords and powers using gradient-based updates to meet a specified ∗ }. set of target SINRs {γ1∗ , . . . , γk∗ , . . . , γK III. GD-BASED C ODEWORD AND P OWER U PDATES From the perspective of a given user k the transmitted power is a valuable resource, and user k is interested in transmitting with the minimum power that meets its specified target SINR. In order to achieve this goal, user k will first apply a codeword update designed to reduce the effective interference that corrupts its transmitted symbol at the receiver, and which results in an increase of its SINR. This will be followed by a power update which adjusts the transmitted power such that the specified target γk∗ is approached:
sk (n + 1) =
[I − 2μs Rk (n)]sk (n) [I − 2μs Rk (n)]sk (n)
(12)
to ensure that it yields a unit norm codeword. The GD-based codeword update in equation (12) has a numerical advantage over the minimum eigenvector or the MMSE updates for IA [3]–[5] since it does not involve computationally intensive calculations like finding the minimum eigenvector or matrix inversions needed by the MMSE update. The value of the effective interference function after the codeword update ik (n) = sk (n + 1) Rk (n)sk (n + 1)
(13)
requires that user k power be pk (n) = γk∗ ik (n)
(14)
in order for its SINR to meet the specified target value γk∗ . However, the value pk (n) may not be close to pk (n), and in order to avoid abrupt variations we use the “lagged” update pk (n + 1) = (1 − μp )pk (n) + μp pk (n)
(15)
with 0 < μp < 1 a suitably chosen constant, which adapts user k power to a new value that is a combination of the current power pk (n) and the power pk (n) needed to meet the specified target SINR after the effective interference function has been reduced by the incremental codeword update (12). We note that the smaller the μp constant is the more pronounced the
1061
lag in the power update is, and the smaller the incremental power change will be.
3.5
3
IV. T HE GD IA A LGORITHM FOR M ULTIPATH C HANNELS
User 1 2.5 User 2 2 SIR
The proposed algorithm for GD IA for multipath channels uses the codeword and power updates defined in the previous section and consists of two distinct stages performed sequentially by active users in the system: one in which users perform incremental adaptation of their codeword followed by “lagged” power adaptation. The algorithm is formally stated below: 1) Initial Data: • Codewords sk , powers pk , channel matrices Hk , and target SINRs γk∗ for active users k = 1, . . . , K. • Noise covariance matrix W • Constants μs , μp , and tolerance . 2) FOR each user k = 1, . . . , K DO a) Compute corresponding Rk using equation (7). b) Update user k’s codeword using equation (12). c) Update user k’s power using equation (15). 3) REPEAT Step 2 until all user SINRs are within specified tolerance of their corresponding target value.
User 4 1 User 5 0.5
0
0
500
1000 1500 codeword updates
2000
(a) SINR variation 40 user 1 35 user 2 30
user 3 user 4
25 Power
Extensive simulations of the GD IA algorithm have shown that, when initialized with random user codewords and powers, and admissible target SINRs, the algorithm converges to a fixed point where the specified target SINR values are met for all users. We note that unlike the ideal channel scenario where a formal admissibility condition for the SINRs is defined in terms of the sum of user effective bandwidth requirements implied by their corresponding target SINR values [10], no such formal admissibility condition is available for the multipath channel scenario and admissible target SINRs have been identified empirically in this case. We also note that, provided that the specified target SINRs are admissible, an analytical convergence proof for the algorithm may be obtained by using the various approaches to convergence of gradient-based algorithms discussed in [11].
User 3 1.5
user 5 20 15 10 5 0
V. S IMULATIONS AND N UMERICAL E XAMPLES
0
500
1000 1500 codeword updates
2000
(b) Power variation
In this section we present numerical results obtained from simulations that illustrate convergence and tracking properties of the proposed algorithm, as well as the structure of the codeword matrix and received signal correlation matrix at a fixed point of the algorithm. We performed extensive simulations for various scenarios with both diagonal and circulant channel matrices. In a first experiment we illustrate convergence of the algorithm from random initial codewords and powers, and plot the variation of user SINRs and powers during the algorithm in Figure 1. We considered a CDMA system with K = 5 users in a signal space dimension N = 3 and white noise with covariance matrix W = 0.1I3 . The algorithm constants were chosen to be μs = 0.1, μp = 0.001, and tolerance = 0.001. User channel matrices were initialized randomly to
1062
Fig. 1.
Variation of user SINRs and powers from random initialization.
H1 = diag{0.9501, 0.8913, 0.8214} H2 = diag{0.4447, 0.7382, 0.9169} H3 = diag{0.4103, 0.8132, 0.1987} H4 = diag{0.6038, 0.7468, 0.4186} H5 = diag{0.8462, 0.8381, 0.8318} and the target SINRs were selected γ ∗ = {2.5, 2.0, 1.5, 1.0, 0.5}
The user codewords yielded by the algorithm for this scenario are the columns of the codeword matrix ⎡ ⎤ −0.9707 0.0440 −0.1134 0.5861 −0.4745 S1 = ⎣ −0.2086 0.0182 −0.9935 −0.8091 −0.0598 ⎦ −0.1197 0.9989 −0.0059 0.0423 0.8782
7 user 1 user 2
6
user 3 user 4
5
and the resulting user powers are
old user 5 new user 6
p1 = 1.1554, p2 = 1.2586, p3 = 2.1335
4 SIR
p4 = 2.3425, p5 = 0.7889
3
We note that these values imply user SINRs that are within O(10−3 ) of the specified target SINRs, and the correlation matrix of the received signal is
2
R1 = diag{1.5083, 2.3900, 1.5887}
1
The variation of the user SINRs and powers for this example plotted in Figure 1 shows a sharp increase at first followed by and gradual adaptation and convergence to the specified target SINR values. In a second experiment we illustrate the tracking ability of the algorithm for variable number of active users in the system, and plot the variation of user SINRs and powers during the algorithm in Figure 2. The signal space dimension is the same as in the previous example N = 3, and we start the simulation with the same number of users K = 5, same user channels, and same target SINRs
0
0
1000
2000 3000 codeword updates
(a) SINR variation 40 35
user 1 user 2
30
user 3
γ ∗ = {2.5, 2.0, 1.5, 1.0, 0.5}
user 4 25 Power
such that when the algorithm settles down yields the user codewords and powers mentioned before. Once the system reached the fixed point user 5 was dropped from the system leaving the total number of active users K = 4 with codewords equal to the first 4 columns of matrix S1 and powers p1 , . . . , p4 above, and same target SINRS γ1∗ , . . . , γ1∗ as before. The algorithm was applied to the new system configuration and yielded the new codeword matrix ⎡ ⎤ −0.9753 −0.0250 −0.1391 0.6124 S2 = ⎣ −0.1949 −0.0041 −0.9899 −0.7896 ⎦ −0.1041 0.9997 0.0282 −0.0390
old user 5 new user 6
20 15 10 5 0
0
1000
and the new user powers matrix
2000 3000 codeword updates
4000
(b) Power variation
p1 = 0.6771, p2 = 0.2443, p3 = 1.2545, p4 = 1.3940 that imply the new correlation matrix of the received signal R2 = diag{0.8763, 1.4183, 0.3106}
Fig. 2. Variation of user SINRs and powers for the tracking example where one user is dropped from the system followed by subsequent addition of another user.
and new user powers
After the system reached this new fixed point a new user was added to the system with randomly generated codeword, power p5 , corresponding channel matrix H5
4000
= diag{0.5028, 0.1897, 0.5417}
p1 = 0.7944, p2 = 1.0717, p3 = 1.5270 p4 = 1.6899, p5 = 1.3367
which imply the new correlation matrix of the received signal and target SINR γ5∗ = 0.4. For this new system configuration R3 = diag{1.0422, 1.7299, 1.3537} the algorithm results in new user codewords represented as columns of matrix S3 below The variation of the user SINRs and powers for this example ⎡ ⎤ plotted in Figure 2 shows sudden sharp changes in user SINRs −0.9827 −0.0352 −0.1438 0.5827 0.3215 S3 = ⎣ −0.1748 −0.0144 −0.9896 −0.8122 0.0139 ⎦ and powers when one user is dropped from the system as well as when the new user is added to the system, that are gradually 0.0609 0.9993 0.0023 −0.0300 0.9468
1063
compensated by the algorithm which brings the users SINRs to the specified target SINR values.
[3]
VI. C ONCLUSION In this paper we proposed a new algorithm for joint codeword and power adaptation in uplink CDMA systems in which multipath channels between users and the base station are explicitly considered. The algorithm is based on GD IA and adapts user codewords and powers incrementally to meet specified targets for user SINRs. Extensive simulations of the proposed algorithm have shown that, provided that the specified target SINR values are admissible, the algorithm converges to a fixed point where the user SINRs are within specified tolerance of their desired target values. However, unlike the ideal channel scenario where a formal admissibility condition for the SINRs is available, admissible target SINRs have been identified empirically in this case. Future work related to the proposed algorithm will focus on establishing an analytical condition that defines admissible target SINRs which generalizes the one applicable to ideal user channel scenarios in [10], and on establishing formal properties of the fixed point of the algorithm along with an analytical convergence proof.
[4] [5]
[6]
[7]
[8]
[9]
[10]
R EFERENCES [1] D. C. Popescu and C. Rose, Interference Avoidance Methods for Wireless Systems. New York, NY: Kluwer Academic Publishers, 2004. [2] D. C. Popescu, O. Popescu, and C. Rose, “Interference Avoidance and
[11]
1064
Multiaccess Vector Channels,” IEEE Transactions on Communications, vol. 55, no. 8, pp. 1466–1471, August 2007. D. C. Popescu and C. Rose, “Codeword Optimzation for Uplink CDMA Dispersive Channels,” IEEE Transactions on Wireless Communications, vol. 4, no. 4, pp. 1563–1574, July 2005. G. S. Rajappan and M. L. Honig, “Signature Sequence Adaptation for DS-CDMA with Multipath,” IEEE Journal on Selected Areas in Communications, vol. 20, no. 2, pp. 384–395, February 2002. J. I. Concha and S. Ulukus, “Optimization of CDMA Signature Sequences in Multipath Channels,” in Proceedings 53rd IEEE Vehicular Technology Conference – VTC’01 Spring, vol. 3, Rhodes, Greece, May 2001, pp. 1227–1239. D. C. Popescu, “Interference Avoidance for Wireless Systems,” Ph.D. dissertation, Rutgers University, Department of Electrical and Computer Engineering, 2002, thesis Director: Prof. C. Rose. Available online at http://www.lions.odu.edu/∼dpopescu/thesis.pdf. C. L˘ac˘atus¸ and D. C. Popescu, “Adaptive Interference Avoidance for Dynamic Wireless Systems: A Game-Theoretic Approach.” IEEE Journal on Selected Topics in Signal Processing, vol. 1, no. 1, pp. 189–202, June 2007, special issue on adaptive waveform design for agile sensing and communications. D. C. Popescu, O. Popescu, and D. B. Rawat, “Gradient Descent Interference Avoidance with Target SIR Matching,” in Proceedings 5th Annual IEEE Consumer Communications and Networking Conference – CCNC 2008, Las Vegas, NV, January 2008, accepted for presentation, to appear. H. J. Landau and H. O. Pollack, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – III: The Dimension of the Space of Essentially Time- and Band-Limited Signals,” The Bell System Technical Journal, vol. 41, no. 4, pp. 1295–1335, July 1962. 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. D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods. Belmont, MA: Athena Scientific, 1997.
