Optimal Power Allocation over Fading Channels with Stringent Delay Constraints

Transcription

1 1 Optimal Power Allocation over Fading Channels with Stringent Delay Constraints Xiangheng Liu Andrea Goldsmith Dept. of Electrical Engineering, Stanford University Abstract We study the optimal power allocation scheme for i.i.d. block-fading channels [1] with a strict transmission delay constraint. In particular, we consider the maximization of the total throughput within a finite interval under a short-term average power constraint. When all the channel gains are known apriori, we demonstrate the mapping between the delayconstrained channel and the corresponding parallel channel for which both the single-user and multi-user channel results are well-known. The maximization problem becomes more complicated when only causal channel side information (CSI) is available. We show that under this assumption, constant power transmission is optimal in the limit of high signal-to-noise ratio (SNR). We also examine the optimal power policy and show that a simple linear power control scheme has near optimal performance. We next consider a two-user broadcast channel with a stringent delay constraint and causal feedback. We solve the optimal power control problem for this channel via dynamic programming. We discuss the solutions in the limit of both low and high SNR. Numerical results show the optimal scheme is approximately piece-wise linear for general SNRs. I. INTRODUCTION The wireless communication channel is a highly constrained and unpredictable medium with throughput severely limited by the scarce radio spectrum. The wireless channel is time-varying due to shadowing, fading and user mobility. Failure to adapt to the time variations of the channel results in decreased throughput. However, there exist various adaptive techniques that increase throughput in a time-varying channel. Power control is one of the key techniques for increasing channel capacity. Shannon capacity defines the maximum transmission rate of a channel. It was shown in [3] that a single Gaussian codebook scaled by the optimal power allocation achieves the Shannon capacity under the assumption of perfect causal CSI in i.i.d. fading channels. The optimal power control policy without a delay constraint is known to be water-filling [2]. Under water-filling, the amount of transmit power allocated to a channel increases with channel quality. Shannon capacity, also called ergodic capacity, requires the codeword length to be long enough to average over all channel states. Thus, the transmission delay is infinite. The assumption of infinite delay is unrealistic for applications with bounded delay constraints. Different notions of capacity have been proposed to account for delay constraints. Two such definitions are zero-outage capacity and outage capacity [6], which are both defined as the minimum achievable rate over every channel state. Outage capacity further allows for some small nonzero outage probability. The optimal power allocation scheme for both capacity definitions inverts the channel and transmits at a constant rate over every channel state. The optimal scheme for outage capacity will not allocate power when the channel gain is below a threshold. Both schemes are very power inefficient because more power is needed when the channel is bad. Although the zero-outage and outage capacity are designed to account for delay constraints, their optimal power allocation schemes do not maximize throughput within a finite delay. In this paper, we investigate the optimal power allocation scheme to maximize the throughput with a finite delay constraint and a short-term average power constraint. We consider both single-user and multiuser channels under two different assumptions: known channel states and causal CSI feedback. We also analyze the optimal power scheme under the limit of low and high SNR for causal feedback. The rest of the paper is organized as follows. In Section 2 we formulate the problem with certain assumptions on the channel model. In Section 3 we present the mapping between a delay-constrained channel with known channel states and a corresponding parallel channel model. Section 4 examines the optimal power allocation scheme under causal feedback in a singleuser channel. We also propose a near optimal linear power control algorithm. In Section 5 we solve the optimal power control problem for a two-user broadcast channel with causal feedback via dynamic programming. We conclude in Section 6. II. PROBLEM FORMULATION A. Channel Model and Assumptions We assume an i.i.d. Block Fading - Additive White Gaussian Noise (BF-AWGN) channel model [1]. The i.i.d. assumption is valid in frequency hopping systems when the random hopping rate is sufficiently high. In the BF-AWGN model, blocks of Æ symbols experience the same fading gain. The fading gain is often referred to as the channel state and is random but constant over the whole block. Different blocks usually experience different fading gains. The block size, Æ, depends on the channel coherence time. We assume a slow fading channel so that Æ is large. It is reasonable to study the performance limits of a BF- AWGN channel as Æ because there exist practical codes with finite codeword length and data rates close to capacity. For multi-user channels, we assume different users have independent channels. The delay constraint is imposed by maximizing the throughput within a finite time interval. We say the problem has a Ã- block delay constraint if the maximum number of blocks within the delay constraint is Ã. In ergodic capacity, à approaches while in the delay constrained case, à is fixed and finite. We consider two different assumptions on channel side information. In the case of known channel states, we assume that all the channel states within the à blocks are known apriori. In

2 2 the case of causal CSI, the transmitter has perfect knowledge of the current and previous fading blocks but does not have any knowledge of the future channel gains. B. Problem Setup Without loss of generality, we incorporate the receiver noise power into the channel gain,, of the -th block. Let denote the power allocation to the -th block. We impose a stringent Ã-block delay constraint and a short-term average power constraint which requires à Ã. As Æ, a Gaussian codebook achieves the capacity in each block, thus the maximum achievable rate in the -th block is ¾ ÐÓ µ. In this paper, all the capacities are measured in nats and we ignore the constant factor of in all capacity expressions. ¾ We consider the following optimization problem for the single-user channel. Maximize Subject to à ÐÓ µ à à When all channel states are known apriori, we do not need to take expectations on the throughput. With known channel states, the power allocation depends on all the à channel gains within the delay constraint whereas with causal CSI, only depends on the current and previous channel states µ. In a two-user broadcast channel (BC), we consider the following optimization problem. Maximize Subject to à µ à à ¾µ µ where µ is the channel gain of the -th user in the -th block and is the total power allocated to both users in the -th block. The capacity of an AWGN broadcast channel with channel gains ¾µ ¾µ µ and total power is denoted as µ µ. This capacity is the total throughput of both users weighted by their priorities. Since successive interference cancellation decoding is used to achieve capacity, the throughput of a particular user depends on his decoding order. In an AWGN broadcast channel, the optimal decoding order is solely determined by the noise levels of different users where the user with the best channel is decoded last. III. KNOWN CHANNEL STATES In this section, we study the optimal power allocation of a Ã-block delay constrained system and we assume the receiver and the transmitter have apriori knowledge of all the channel states within the à blocks. Although this assumption is not valid for some practical systems, it is a good model for multicarrier systems when the channel is known [5]. There exists a direct mapping between a Ã-block delay constrained channel with known channel states and a Ã-component parallel channel. Each block in the delay-constrained channel can be viewed as a component in the parallel channel. This mapping is valid for multi-user channels as well. Each joint (1) (2) channel state becomes a multi-user component channel in the corresponding parallel channel. The optimal power allocation schemes for single-user and multi-user parallel channels are well-known for various different capacity definitions and power constraints [7] [8] [9] [11]. Therefore, we apply these known results to the delay-constrained channel with known channel states. Caire et. al. [5] studied the single-user optimal power allocation problem under a stringent delay constraint with various power constraints and objective functions. Caire s solution was based on Lagrangian techniques. By applying the known results from parallel channels, we can obtain the same conclusions as in [5]. We can also solve new problems. For example, in a two-user multiple access channel(mac), the optimal power scheme with the stringent delay constraint is also a two level water-filling scheme as in the corresponding parallel channel. We also note that the solutions to these problems are similar to their ergodic counterparts but with different water-fill levels. The similarity follows from the fact that for ergodic channel capacity, knowing the statistics is equivalent to knowing the channel states over the long-term average. IV. CAUSAL CHANNEL SIDE INFORMATION - SINGLE USER We now assume the channel states are only known causally. With causal CSI, the transmitter and the receiver have perfect channel side information of the current and previous channel states. The mapping between a delay-constrained channel and a parallel channel does not hold because of the unknown future channel states. Thus, we now examine the optimization problem (1) with the constraint that the power policy only depends on causal information. The optimal solution can be obtained via the dynamic programming algorithm ÔØ ÔØ as briefly presented in [4]. The algorithm is essentially a backward exhaustive search. Due to its complexity the authors in [4] do not further investigate the ÔØ ÔØ algorithm, but instead, they consider a threshold policy. This threshold power policy was shown to be optimal for low SNR. In this section, we show that constant power transmission is optimal under the limit of high SNR. We then propose a simple near optimal power allocation scheme for general SNRs to approximate the optimal solution. A. Limiting Optimal Power Control Low SNR: (Threshold Policy) When the SNR is low, the throughput of the -th block can be approximated using the first-order Taylor expansion ÐÓ µ. This approximation greatly simplifies the optimization problem. The optimal transmission scheme under this approximation, as shown in [4], is to transmit all the power in one block chosen according to a threshold policy. All power à µ is allocated to a given block if the gain in that block exceeds a threshold, otherwise no power is allocated. The threshold decreases as the delay deadline approaches. The transmission threshold is zero for the last block. High SNR: (Constant Power Transmission) We now consider the case of high SNR. When the SNR is high,, the capacity can be approximated as ÐÓ µ ÐÓ µ. The optimality of constant power

3 3 8 7 Optimal Power Allocation Average SNR = 1dB K=1 K=8 K=6 K=4 K=2 Ergodic 8 7 Optimal Power Allocation Average SNR = 1dB K=1 K=8 K=6 K=4 K=2 Ergodic Power 4 Power Channel Gain (g) Noise (1/g) Fig. 1. Power Allocation versus Channel Gain (single user) Fig. 2. Power Allocation versus Equivalent Noise (single user) transmission under this approximation can be easily proven by induction. Therefore, no power adaptation is necessary when the SNR is sufficiently high. This explains the observation in [4] that optimal power control has diminishing improvement over the constant power policy as the average SNR increases. The optimal power allocation for low and high SNR can be explained from an information-theoretical point of view. When SNR is low, we only have enough power to water-fill one block. In order to maximize capacity, we water-fill the first block that exceeds our threshold since we do not know if the channel will exhibit a higher gain in the future. For high SNR, we have plenty of power to water-fill all the blocks. Since the power level is high, the amount of power we use in different blocks is approximately equal. B. Numerical Results for Single User Channels Although the dynamic program ÔØ ÔØ was given in [4], the optimal power control policy in the general case was not discussed. We implemented the algorithm for i.i.d. Rayleigh fading under SNRs of -2, -1,, 1, and 2 db. We show the numerical results of SNR db below as an example. The plots for other SNRs are similar. Fig. 1 shows the optimal power allocation as a function of the channel gain in the first block for different delay constraints, Ã, under the same average power constraint. The thick line is the optimal water-filling solution with no delay constraint. The rest of the curves are the power allocation of the first block for different delay constraints, Ã, and ¾. We see a monotonic increase in the cutoff channel gain for transmission as à increases. This is because the more delay we can tolerate, the more likely that a good channel will appear later within the delay constraint, and therefore, we can be more selective about which channel gains are acceptable for transmission. However, if à is small, we are more willing to transmit on not-so-good channels due to the stringent delay constraint. The curve flattens out for à ¾ as the channel gain increases since the maximum total power is reached. Also, the power allocation schemes for delay-constrained channels approach the optimal water-filling solution for ergodic capacity as à increases. Fig. 2 is essentially the same plot as Fig. 1 except we use as the x-axis instead of. The term can be viewed as the equiv- alent noise floor at the receiver. Therefore, this figure shows the optimal power allocation as a function of noise. Although it appears that the optimal power scheme is a linearly decreasing function of the equivalent noise, a more careful examination of the numerical results indicate otherwise. However, the linear appearance of the optimal power policy in Fig. 2 indicates that a linear approximation to this policy may lead to near optimal performance. This is appealing because the ÔØ ÔØ algorithm is both time-consuming and memory-consuming. Since we generally do not have perfect statistics of the channel and the channel statistics may be varying with time, we cannot assume that the algorithm is executed once and results stored for the future. Therefore, a real-time algorithm, such as our linear allocation scheme described below, is very desirable. C. Optimal Power Allocation The optimal power policy is the solution of the optimization problem (1) with the constraint that only depends on causal CSI. Let denote the remaining power for blocks through Ã. Thus, Ã. Let µ be the optimal power allocation to the -th block when the channel gain is For the last block, à µ à µ because no more delay can be tolerated. At block, the total remaining power is Ã. In order to determine as a function and, we need to solve the following optimization problem. Maximize Subject to à ÐÓ µ ÐÓ µ à Let Ë µ be the expected maximum throughput from blocks through à when the total available power is. Thus, Ë µ can be found by taking the expected value over of the maximized objective function in (3). We can then rewrite (3) as a single-variable optimization problem (3) Ñ ÐÓ µ Ë µ (4)

4 4 Taking the first derivative of the objective function with respect to in Eqn. (4), we have Ë µ (5) Setting the first derivative to zero, we get the optimal power allocation. However, the analytical form is very difficult to derive even for the first block when à ¾. Thus, in the next section, we will derive a linear approximation to the optimal allocation using the results from this section. D. Near Optimal Linear Power Allocation Scheme In this section, we present a simple linear power allocation scheme and show the performance is near optimal. Our approach is to find two characteristic points on the optimal policy and use the line through these two points as the suboptimal power control algorithm. The two points we use are where the optimal power allocation scheme begins to allocate zero power,, and where the optimal scheme starts to allocate full power, (see Fig. 2). Since Eqn. (4) is a single-variable optimization problem and the objective function is strictly concave in for, we can use the first derivate (Eqn. 5) for our analysis. Due to the strict concavity, if Eqn. (5) is negative when, then the first derivative remains negative for all. Thus, it is optimal to allocate zero power to the -th block:. Similarly, if the first derivative is positive when, then it is positive for all. Thus,. Specifically, if Ë µ, then. When Ë µ Ë µ, then. Denoting and Ë µ, our linear approximation has the form where Ì if if Ì,, if Ì Ì, for à and à Ã. Both and are functions of and. Different delay constraints and power constraints yield different slopes,, and different water-levels,. For the ergodic water-filling solution, for all. Finding is difficult because Ë µ is hard to calculate for most when. For and, we can determine exactly but require an approximation for. Since and Ë µ, we have Ì Ë µ Ì Ë Ë µ Ë µ ÐÑ ÐÑ (6) (7) Ë µ For, we are in the low SNR regime and the optimal power control is a threshold policy [4]. All power,, is transmitted in the very first block which exceeds its threshold. The index of this block is denoted as õ. The threshold for the -th block, õ, depends on the number of blocks remaining, i.e. Ã. In the following derivation, we will assume the inherent dependence on à and simply denote õ as and õ as. Thus, denotes the gain of the channel we used under the threshold policy during blocks à in the limit of low SNR. The low SNR approximation yields Ë µ. Therefore,. We can calculate recursively as Pr µ Pr µ Pr µ Pr µì where Ìà Ã. Thus, we have determined the full power allocation cutoff point for every within the delay constraint Ã. In order to calculate, we need to evaluate Ë µ at for any Ã. It can be shown that Ì Ë µ Ñ Ã where à is the optimal power allocation and. Eqn. (8) can be calculated by the following recursion. Pr µ Pr µ Ë µ Pr µ Pr µì with initial condition Ì Ã Ë Ã µ à for all. This recursion algorithm requires the knowledge of, the optimal power allocation to block when the total power available to the remaining blocks à is. We approximate using the same linear technique,, where and are determined by and. Since Eqn. (9) is a backward recursion, is already known. We also know since the recursion of can be calculated exactly without the knowledge of previous power allocations. Error propagation may be a problem because we use the linear approximation instead of the optimal solution in our recursion for. However, numerical results shown in Fig. 3 indicate that error propagation is not a concern, as the suboptimal scheme performs about the same as the optimal scheme. As an example, for à ¾, we have Ì ¾ and Ì ¾ ¾. The zero power cutoff noise floor ¾ and à ¾ the full power allocation cutoff noise level ¾. As, ¾ and ¾, which shows the optimal power allocation scheme converges to the threshold policy in the limit of low SNR. As, and, which are the same limiting values for and as in the constant power transmission which is optimal in the high SNR regime. Fig. 3 compares the expected throughput within the Ã-block delay constraint of the linear power control with the optimal power control and the constant power transmission scheme for i.i.d. Rayleigh fading with SNR db and an average (8) (9)

5 Optimal Constant Power Linear Approximation Expected Capacity vs. Delay Average SNR = 1dB available to blocks Ã. The optimal power allocation to each user at block is uniquely determined by µ, and ¾µ. The quantity, Ê µ is the expected throughput of blocks à where the total power of these blocks is where Ã. Expected Capacity # blocks Fig. 3. Comparison of Optimal, Linear and Constant Power Control (single user) power constraint. The linear scheme achieves approximately the same rate as the optimal power control. Both of these schemes have a substantial improvement over constant power transmission. This improvement increases with à because more degrees of freedom are available for adaptation to channel variations. The plots are similar for other SNR values. The linear power scheme is near optimal for all SNRs. For Ã, the linear power control scheme achieve 62% higher throughput than constant power allocation for SNR db. For SNR db, the performance improves by 15%, while for SNR db, the performance only improves by 2%. The gains through power control diminish as SNR increases because constant power transmission is optimal in the limit of high SNR. V. TWO-USER BROADCAST CHANNEL A broadcast channel has one transmitter sending information to many receivers (down-link in a cellular system). The capacity region of the BC channel is known to be convex and the optimal coding scheme is superposition coding with interference cancellation [9]. The decoding order only depends on the noise level at receivers. The user with the best channel is decoded last. Since the base station is the only transmitter in a broadcast channel, there is only one power constraint. We solve the one-dimensional optimization problem (2) via dynamic programming. The algorithm Optimal Broadcast below calculates the optimal power allocation for each block under a delay constraint of à blocks and an average power constraint of per block. Algorithm 1: (Optimal Broadcast) For à Ã, choose Ö Ñ µ ¾µ µ Ê µ and Ê µ µ ¾µ µ Ê µ with initial conditions Ê Ã µ for any and Ê Ã µ for any. The optimal power allocation in the -th block,, is a function of the joint channel state and the total power remaining A. Low SNR In the limit of low SNR, we assume the channel gains for both users, µ and ¾µ, are small for all. Under this assumption, the total throughput of two users in block can be approximated as µ µ ¾ ¾µ ¾µ. We first determine the power allocation among the two users. If the total power allocated to block is, i.e., µ ¾µ, the total throughput is µ ¾µ ¾ µ µ ¾µ ¾. The optimal power allocation among the two users is µ if µ ¾µ ¾ and otherwise. Thus the throughput of the - th block is Ñ µ ¾µ ¾ µ. This throughput expression is very similar to the single-user channel throughput approximation under the limit of low SNR where Ñ µ ¾µ ¾ µ replaces. It can be shown that the optimal power policy between blocks in the two-user broadcast channel is a threshold policy. If Ñ µ ¾µ ¾ µ exceeds a threshold, all the power is allocated to this block. Again, the threshold depends on the the total number of blocks left to go and the total remaining power. B. High SNR In the high SNR regime we assume the channel gains for both users, µ and ¾µ, are large for all. Under this assumption, the power allocation among users is completely determined by the priorities of the users. The user with highest priority transmits all the time. Thus, the two-user broadcast channel degrades to a single-user channel. The total throughput of each block is approximately Ð ÐÓ Ðµ µ, where Ð refers to the index of the highest priority user. The optimal power policy between blocks is a constant power scheme. This can be proven by induction. C. Numerical Results The numerical results in this section are for a two-user broadcast channel. Both User 1 and User 2 have an i.i.d. BF-AWGN channel. The channel gain has a Rayleigh distribution with ËÆÊ db. The average power is.2 per block. The user priorities are ¾ ¾. Fig. 4 shows the power allocated to User 1 when User 2 sees a fixed channel ( ¾ ). We see two segments for each delay constraint Ã. When User 1 has a good channel, the power is only allocated to encode User 1 s information since User 1 has a higher priority. When User 1 s channel is not as good, power is allocated to encode information for both User 1 and User 2. Since the transmitter is also sending information to User 2, as the noise level at User 1 increases, the power allocation to User 1 drops faster than when only User 1 s information is transmitted. This result also agrees with the two-level water-filling in the ergodic broadcast channel capacity results. We see an approximately piecewise linear relationship between the power and noise with different slopes similar to the single-user case.

6 User 1 Power Allocation vs. Noise n 1 with fixed n 2 = [1] T. Cover, J. Thomas, Elements of Information Theory, John Wiley & Sons, New York, NY, USA, [11] D. Tse, Optimal Power Allocation over Parallel Gaussian Broadcast Channels, IEEE Transactions on Information Theory, October P n 1 (1/g 1 ) Fig. 4. Two-user Broadcast Channel: User 1 Power allocation versus channel state ( ¾ ) The power allocation for User 2 is similar to User 1 s allocation. However, less power is allocated to User 2 because User 2 has a lower priority. VI. CONCLUSIONS The problem of optimal power allocation with a stringent delay constraint for i.i.d. BF-AWGN channels is studied with both causal and non-causal channel side information. We show that when all the channel gains are known apriori, the problem is equivalent to the power allocation over a corresponding parallel channel for which the single-user, MAC and BC channel solutions are known. Under causal feedback, the optimal power allocation is much more complicated and requires a dynamic programming algorithm. A simple linear power allocation algorithm is proposed and numerical results show the performance of this linear scheme is near optimal with much lower complexity than the optimal scheme. We extend the results to a two-user broadcast channel. We give the dynamic program to solve the power allocation problem and numerical results show that the optimal solution is approximately piece-wise linear in the equivalent noise level. The optimal power allocation under the limit of low and high SNR is also derived. REFERENCES [1] E. Biglieri, J. Proakis, S. Shamai, Fading Channels: Information-Theoretic and Communication Aspects, IEEE Transactions on Information Theory Vol. 44, October [2] A. Goldsmith, P. Varaiya, Capacity of Fading Channels with Channel Side Information, IEEE Trans Inform Theory, vol. 43, November [3] G. Caire, S. Shamai, On the Capacity of Some Channels with Channel Side Information, IEEE Trans Inform Theory, vol. 45, September [4] R. Negi, J. Cioffi, Transmission Optimization with Delay Constraint and Causal Feedback, Preprint. [5] G. Caire, G. Taricco, E. Biglieri, Optimum Power Control Over Fading Channels, IEEE Transactions On Information Theory, Vol. 41, July [6] V. Hanly, D. Tse, Multiaccess Fading Channels. Part II: Delay-limited Capacities, IEEE Trans Inform Theory, vol. 44, November 1998 [7] V. Hanly, D. Tse, Multiaccess Fading Channels. Part I: Polymatroid structure, optimal resource allocation and throughput capacities, IEEE Trans Inform Theory, vol. 44, November 1998 [8] S. Vishwanath, A. Goldsmith, Optimum Resource Allocation for Multiple Access Fading Channels, Preprint [9] L. Li, A. Goldsmith, Capacity and Optimal Resource Allocation for Fading Broadcast Channels: Part I: Ergodic Capacity, IEEE Trans on Information Theory, Jan 2.