Iterative Power Control for Imperfect Successive ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 3, MAY 2005

Iterative Power Control for Imperfect Successive Interference Cancellation Avneesh Agrawal, Jeffrey G. Andrews, Member, IEEE, John M. Cioffi, and Teresa Meng, Fellow, IEEE

Abstract—Successive interference cancellation (SIC) is a technique for increasing the capacity of cellular code-division multipleaccess (CDMA) systems. To be successful, SIC systems require a specific distribution of the users’ received powers, especially in the inevitable event of imperfect interference cancellation. This apparent complication of standard CDMA power control has been frequently cited as a major drawback of SIC. In this paper, it is shown that surprisingly, these “complications” come with no additional complexity. It is shown that 1-bit UP/DOWN power control—like that used in commercial systems—monotonically converges to the optimal power distribution for SIC with cancellation error. The convergence is proven to within a discrete step-size in both signal-to-noise plus interference ratio and power. Additionally, the algorithm is applicable to multipath and fading channels and can overcome channel estimation error with a standard outer power control loop. Index Terms—Code division multiple access (CDMA), interference cancellation, multiuser receivers, power control.

I. INTRODUCTION

S

UCCESSIVE interference cancellation (SIC) shows promise as a practical approach toward multiuser detection [18] for code-division multiple-access (CDMA) systems. Its complexity is linear in the number of users, it does not require dimensional separation of users using short signature sequences, and it can work equally well in an asynchronous environment. These desirable properties derive from the fact that SIC systems typically use the proven matched-filter receiver and strong error correction codes. However, SIC has its own set of challenges that need to be overcome. Most importantly, the received power distribution for all the users needs to be set properly. For a conventional matched-filter receiver, an equal decision-time signal-to-noise plus interference ratio (SINR) requirement for all the users corresponds to the familiar requirement that all users be received with the same power. For SIC, the situation is a bit more complicated, as users are decoded successively and have a large portion of their interference subtracted from the composite signal prior to the decoding of the next user. It is thus necessary for earlier users to have higher received powers, and later users to have lower received powers, in order to satisfy the equal SINR requirement. The optimum power distributions have been studied previously under the

Manuscript received May 1, 2003; revised November 3, 2003, February 10, 2004; accepted March 1, 2004. The editor coordinating the review of this paper and approving it for publication is D. I. Kim. A. Agrawal is with QUALCOMM, San Diego, CA 92121 USA (e-mail: [email protected]). J. G. Andrews is with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78701 USA (e-mail: [email protected]). J. M. Cioffi and T. Meng are with Department of Electrical Engineering, Stanford University, Standford, CA 94305 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TWC.2005.846996

assumption of perfect interference cancellation [19], [21], and more generally when the cancellation is not perfect [4], [13]. Although previous work [4], [13] successfully addressed the power inequality required for SIC, the result was a more complicated power control distribution than required in conventional CDMA. This apparently complex power distribution has been frequently cited as a major disadvantage of SIC [17], [22], suggesting the need for an iterative power control scheme that is able to maintain the optimum power distribution for SIC as the channel varies. Further, this iterative scheme will need to have low overhead, low complexity, and be able to track a fading channel. Ideally, it would be based on or even compatible with proven second-generation (2G) and third-generation (3G) cellular power control schemes, which send 1-bit UP/DOWN commands at a rate on the order of 1 kb/s [10]. There has been significant prior work on iterative power control for CDMA systems, but not for SIC systems. In [7], a distributed power control scheme is proposed that converges to the feasible solution whenever such a feasible solution exists. In [23], Yates extended the work of [7] and derived general conditions for the convergence of a power control algorithm to the optimal solution, if it exists. The amount of feedback required for both of these techniques is fairly large. In [16], a fixed step iterative power control scheme was studied in which the receiver sends a two bit command (up, down, or no change) every power control cycle. Herdtner and Chong [9] provided a more comprehensive analysis of 1-bit (UP/DOWN only) power control and proved convergence for all Standard interference functions with asynchronous power control. All of these focused on a matched-filter receiver in which all other users are treated as interference. Recent work has attempted to apply some of the well-known results on iterative CDMA power control to SIC. An iterative scheme for SIC was developed in [6], and the ternary (up, down, no change) power control algorithm was shown to converge in [2]. We will show that the interference function for SIC receivers can be modeled as a Standard interference function, so it follows that the results of [9] are directly applicable to SIC. The contribution of this paper is to show, summarizing from our recent results [1], [3] and applying [9] and [23], that single-bit iterative power control for SIC is indeed possible. Also, we derive important additional properties: monotonic convergence, robustness to multipath, active link protection, and convergence in power with maximum power constraints. II. OPTIMAL POWER CONTROL FOR SIC In this section, we introduce the SIC system model and derive the optimum power control distribution for a SIC system with imperfect interference cancellation. We show that seminal work on iterative power control by Yates [23] includes SIC as a special

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Fig. 1. SIC using a standard CDMA matched filter receiver.

case. Finally, we show that although the discussion in this paper is for either a static or flat-fading channel, the results generalize to a multipath channel. A. System Model Fig. 1 shows a high level diagram of a matched-filter receiver with SIC. The receiver for the th user uses a standard matched-filter receiver to demodulate its waveform. The decoded signal is re-encoded, remodulated, and by performing a post-decoding channel estimate, the received signal for user can be estimated. Using this estimated received signal , user ’s interference can be cancelled prior to demodulation of subsequent users. This process is continued until all users are decoded. Since the channel estimation is imperfect, and because , the interference it is possible that the decoded bits cancellation will never succeed in perfectly cancelling out user ’s interference. Hence, is defined as the fractional error in cancelling the th user’s signal. For a sufficiently large number of interfering users and/or a large processing gain, the residual interference can be accurately approximated using a Gaussian distribution regardless of the channel estimation algorithm, due to the Central Limit Theorem [8]. In an additive Gaussian noise channel, the receiver performance is determined by the SINR. Hence, only user powers are considered in the analysis, and actual transmitted bits and other subtleties affecting power control are ignored. For a matched-filter receiver with SIC (after hard decisions), the received interference and noise power vector can be expressed as (1)

is the normalized channel impulse response for the where th user (i.e., ), is the fractional cancelis the vector lation error for user as discussed earlier, and . For typical bit-error rates norm operation, i.e., (BERs) achieved with error correction coding (less than 10 , i.e., 0.01%), error propagation has a very minor effect on rela, tive to channel estimation error (on the order of 10%). As , the rethe receiver approaches ideal SIC [21], and as ceiver approaches a conventional matched filter receiver as discussed in [5] and [7]. B. Feasible Solution The goal of the power control algorithm is to make sure each user can meet its target SINR, . To find a set of powers that can achieve the target SINRs, the parameters of the system must allow for a feasible solution. The following lemma describes the necessary and sufficient conditions for the existence of a feasible solution. exists iff Lemma 1: A feasible power vector where , is the spectral radius (magnitude of maximum eigenvalue) of , and is a diagonal matrix of target SINRs for the users, i.e., (3) Proof: This is a known result for irreducible, nonnegative [5], [14]. It is true that in most cases, is irreducible;1 however, there are some special conditions under which may not be irreducible. For instance, perfect cancellation of the first user would result in a reducible . Assuming the existence of a feasible solution, the optimal power vector can be derived as (4)

is the vector of receiver noise and other cell where is the vector of received powers, interference power, and is a nonnegative matrix that accounts for interference from other users in the same cell. Each vector is dimensional. For the case of SIC, can be represented as

(2)

C. Calculation of Feasible Solution This section presents optimal power distributions for the special case of SIC with equal target SINRs and fractional cancellation errors, i.e., and . The assumption of equal fractional cancellation error is never precisely true; however, assuming the same quality-of-service (QoS) requirement for each user and post-decoding channel estimation (as in Fig. 1), the BER and errors in channel estimation should be statistically 1A

nonnegative matrix, A

, is irreducible iff (I + A)

> 0 [11].

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similar for all the users. With the given assumptions, represented as


can be

1) 2)

(5) The optimal power distribution can be derived using the following result for irreducible, nonnegative matrices [11], [14]. Lemma 2 (Perron–Frobenius Theorem): If is an irreducible nonnegative matrix, then: 1) the maximum eigenvalue of is positive; 2) the eigenvector corresponding to the maximum eigenvalue is nonnegative. Using this Lemma, it can be shown [1] that the total number of supportable users in a CDMA system using SIC is bounded by (6) , (with perfect cancellation we can where as always add more users if we are willing to use more power), , (an intuitive result since the and as ). pre-despreading SINR is proportional to The optimal power vector can now be derived directly from , with each element of equal to the condition . After some algebra, the following optimal transmit powers for the users are found [1]:

, then that fixed If there exists a fixed point, point is unique. If the interference function is feasible, then for any inithe following iterative algorithm altial vector ways converges to the fixed point: (10)

It is straightforward to show that the interference function for SIC in (1) satisfies the conditions of positivity, monotonicity, and scalability, and, hence, is a standard interference function. Then, using similar reasoning as in [24], the iterative algorithm in (10) may be used to determine the optimum power allocation. E. Effect of Multipath The interference matrix in (2) implicitly assumed that there is no multipath. Multipath can be included in the analysis without modifying any of the general conclusions by including the selfinterference. The resulting feasibility equation can be written as (11) is a nonnegative, diagonal matrix with entries , where is the post-despreading intersymbol interference where (ISI) power remaining at detection for user . Thus (12) (13)

(7) (8)

The special case of perfect interference cancellation is for , and the resulting optimal power distribution is the same as the solution derived in [21] (9) As , it can be seen that all the received power levels should be the same, which is the familiar result for no interference cancellation. More generally, it has also been shown in [3] that (7) and (8) are consistent with the previous SINR-equalizing power distribution for SIC derived in [4]. D. Standard Interference Function Yates [23] has developed an elegant theory of standard interference function. According to [23], an interference function is standard if it satisfies the following properties: 1) positivity— ; , then ; 2) monotonicity—if 3) scalability— , ; where all the relationships are component-wise relationships. Yates also proves the following theorems for standard interference functions.

where . Hence, multipath can be incorporated in the analysis by replacing the previous target SINR ma. Note that exists and is trix with . positive only if III. ITERATIVE POWER CONTROL Wireless channels generally change rapidly as users enter, move within, and leave the system. It is impractical for the base station to update all the users’ powers constantly based on a complex relation such as that in (7). However, it is important to maintain this power distribution or performance suffers dramatically [4]. This suggests the need for an iterative power control scheme that achieves very near the optimum distribution, while requiring minimal feedback between the receiver and transmitter. Previously, Foschini and Miljanic [7] and Yates [23] have proposed distributed iterative power control for the matched-filter receiver, and Yates [23] derived general conditions for the convergence of a power control algorithm to the optimal solution. However, the amount of feedback required for both schemes is quite large. Actual implementation would require much lower feedback rates. The 3G CDMA systems [10] use a fixed-step, binary-feedback power control algorithm that is described as if if (14)


where is the power control step size. The scheme in (14) requires minimal feedback (only a 1-bit UP/DOWN command every power control cycle), and is easy to implement as each receiver independently determines the power control command based on a single comparison. Commercial implementations of this scheme for conventional matched-filter receivers have shown to be very robust under different channel conditions [20]. We propose applying this scheme, (14), directly to a SIC-based system. In [12], Kim proved that for a matched-filter receiver, the algorithm in (14) eventually converges such that the actual SINR of each user is within of the target SINR, . Herdtner and Chong [9] provided a more comprehensive analysis of (14) and proved convergence of (14) for all standard interference functions with asynchronous power control, and different step sizes for power increment and decrement commands. Since the interference function for SIC receivers is a standard interference function, it follows that the results of [9] should be directly applicable here. This section shows that even with fixed power control step ), convergence to the optimum power sizes (i.e. control vector of (7) is achievable in both SINR and in power, even with a maximum power constraint. While the SINR convergence follows naturally from [23] and [9], the power result is to the best of our knowledge original. A. SINR Convergence Theorem 1: If a standard interference function is feasible with an SINR margin of , the fixed-step binary feedback algorithm in (14) converges monotonically to a power distribution such that at time (15) Proof: The proof relies on the following propositions. The proofs of the propositions are omitted here for brevity, but are similar to those presented in [1], [3]. , decreases monoProposition 1: While tonically every power control update until . Proposition 2: While , increases monotonically every power control update. Proposition 3: If the interference function is feasible with , then increases an SINR margin of , and monotonically every power control update until . Proposition 4: If , then , . Thus, irrespective of the initial condition, each user’s SINR of the target would eventually be bounded within SINR. Note that Propositions 1, 2, and 4 do not require the interference function to be feasible. Proposition 3 requires feasibility with an SINR margin of . This means that if the target SINR matrix is , then should also be a feasible SINR matrix.

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B. Power Convergence With a Maximum Power Constraint While Theorem 1 showed convergence in SINR with fixedstep, binary feedback power control, it did not actually show that the power levels converged to the minimum required for feasibility, i.e., . This section shows that (14) also converges in terms of received powers. Theorem 2: If a standard interference function is feasible with a feasibility margin of , then (14) converges monotonically to a power distribution at time such that , where is the optimal power vector for a feasibility margin of . Proof: Define (16)

Clearly, , and by the scalability and monotonicity is also property of the standard interference function, a feasible power vector with a SINR margin of , and it can be shown that (17) . For both downwards with a strict inequality if and upwards convergence, it can be shown that , with . Hence, and convergence equality only if in power is attained. For an alternate, more lengthy proof, we refer the interested reader to [1]. Thus, Theorem 2 states that for a feasible interference function the received powers are eventually upper bounded by . As long as the maximum transmitter power and channel gain , then the maximum power constraints will allow not prevent convergence since the received powers for all users lie in a convergence region upper bounded by . IV. PERFORMANCE OF ITERATIVE POWER CONTROL In this section, the convergence of the power control algois simulated for both static rithm to the optimum solution and fading channels. The convergence is quantified using the normalized mean square error (NMSE) of the received power versus the optimum power vector, . The NMSE vector at any time is defined as

(18) Important assumptions made in this section include the following. 1) It is assumed for simplicity that all users have the same amount of estimation error . for each user is 2) The target SINR is the spreading factor. known, where 3) The receiver is assumed to have perfect estimates of . the power and interference vectors and

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Fig. 2. Power control convergence for different step sizes in a static channel. It can be seen that for = 0:1 dB, the NMSE is about 24 dB, implying that p(n) = p for large n and a small enough . It is also seen that the matched filter receiver approaches steady-state at the same rate as an SIC receiver.

0

4)

Fig. 3. Power control convergence for different number of users K . This implies that one factor in reduced performance at high system loads is a degradation in the power control accuracy.

For the fading simulations, time variations in the received power are simulated using Clarke’s model for Rayleigh fading [15]. The PC update rate (PCGs/s) is 1500 b/s, equal to wideband CDMA (WCDMA) [10].

A. Static Channels and the Outer Power Control Loop First, to demonstrate the steady-state convergence, we focus on a static channel where each user has an i.i.d. channel gain and initial power level. Fig. 2 shows the convergence of the 1-bit power control algorithm as a function of time, parameterized by the step size . As expected, larger values of allow faster convergence, but smaller values allow finer resolution, and ultimately lower steady-state NMSE. Fig. 3 shows power control convergence for different system loads. For the parameters used, a feasible solution exists for a maximum of 60 users. It is seen system, the convergence that for this fully loaded is not accurate, as the NMSE is greater than 0 dB. The jagged steady-state behavior results from the 1-bit UP/DOWN commands which cause the received powers to oscillate about the optimum value. SIC relies on the ability of the system to accurately estimate the received signal of each user and subtract it from the composite received signal. Inevitably, the channel estimation—and, hence, interference cancellation—is imperfect, and it is important that the power control algorithm adapt to this. In general, the amount of cancellation error, , for each user is not known. In order to generate correct UP/DOWN commands, the received must be estimated. From our stated assumptions, any SINR is a result of error in estimating . An imerror in estimating perfect estimate of causes the power control loop to converge to a nonoptimal power distribution. In order to mitigate this problem while closely modeling a practical system, an outer power control loop is introduced. An outer power control loop is employed in commercial systems to adjust the target SINR based on the achieved frame-error rate (FER) [10]. Typically, the outer loop tries to maintain the target FER for each user by increasing or decreasing its target SINR. A

Fig. 4. Steady-state NMSE versus " for different " values. The outer power control loop allows similar steady-state convergence even for (initially) inaccurate estimates.

similar approach is adopted in this section. The th user’s target SINR is increased by if , otherwise it is . This section uses a slightly simplified decreased by model in which the receiver is assumed to have an accurate estifor the purposes of outer power control mate of loop, and does not have to rely on the imperfect estimates used for the inner loop. Fig. 4 shows the steady-state NMSE for the system with both inner and outer loop power control, as a function of , used in the inner power control loop. the estimate Steady state means that the power control distribution has of the optimum, as guaranteed by converged to within Theorems 1 and 2. The outer power control step-size is set at dB. The striking feature of this plot is that the NMSE is roughly similar for various . Hence, it can be inferred that if the outer values of and power control loop is enabled, then errors in the receiver’s


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Fig. 5 shows the variations in NMSE for different system as a function of the Doppler frequency . As loading and expected, the NMSE increases with higher Doppler higher system loading. Fig. 6 shows the effect of the step-size on NMSE for Hz. These results are in contrast with those presented in Fig. 2 which showed that in a static channel, the final NMSE is smaller for smaller values of . As the channel varies with time, the increase in tracking ability offsets the poorer convergence properties of larger step-sizes. V. CONCLUSION

Fig. 5. Average steady-state NMSE as a function of Doppler f averaged over 1000 runs. NMSE