Capacity Expression and Power Allocation for Arbitrary Modulation and Coding Rates

Transcription

1 IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE (WCNC), 23 Capacity Expression and Power Allocation for Arbitrary Modulation and Coding Rates Weisi Guo, Siyi Wang, Xiaoli Chu University of Warwick, University of Sheffield, United Kingdom {siyi.wang, Abstract This paper addresses the dichotomy that exists in the analysis of wireless channels, which exists between employing: the tractable Shannon capacity bound and the accurate simulated capacity of modulation and coding schemes (MCSs). By considering the effects of mutual information saturation in modulation schemes and the efficiency of error correction codes, the paper proposes a new tractable capacity expression that can accurately represent any MCS and adaptive combinations. To demonstrate this methodology, the paper considers the adaptive MCS capacity for the Long-Term-Evolution (LTE) system. The potential benefit is that system-level optimization can employ the more accurate capacity expression. In the second part of the paper, the proposed capacity expression is applied to optimize power allocation to the parallel channels of a MIMO system. The results show that when mutual information saturation occurs, neither the water-filling nor channel inversion schemes are optimal. In fact the optimal power allocation is a combination of the two aforementioned schemes, switching adaptively between their relative merits. A. Challenges I. INTRODUCTION The Shannon capacity theorem has presented the community with a tractable deterministic capacity expression based on the maximum achievable mutual information of a Gaussiannoise channel []. This has been extensively used in literature to derive optimisation algorithms for various wireless systems under the assumption that the adaptive modulation and coding scheme (MCS) can closely match the performance of the Shannon bound. Whilst, the bound has been demonstrated as being theoretically possible with advances in multiple-antenna and forward-error-correction (FEC) techniques, a realistic system to this date is far away from achieving the Shannon bound. For example, the current proposal for the 3GPP Long- Term-Evolution (LTE) cellular network s physical layer can achieve a maximum capacity of bit/s/hz for different propagation environments [2]. Therefore, in the analysis of wireless communication channels, there is a methodology dichotomy between employing the tractable Shannon bound and using a simulated MCS performance. Whilst the Shannon theory yields an insightful theoretical bound, the MCS simulation results produce a more accurate performance prediction. The key differences between the two approaches are: Mutual Information Saturation: this is determined by the modulation scheme s minimum-mean-square-error (MMSE) performance [3]; FEC Code Efficiency: this is determined by the FEC coding scheme and code rate. B. Contribution This paper proposes a novel and tractable capacity expression, which can yield the capacity of any PSK and QAM modulation scheme, along with any turbo/convolution coding rate. The paper presents a set of tractable expressions that yields the capacity of each individual modulation-coding scheme, as well as the combined adaptive MCS. The investigation methodology employed in this paper involves: ) Simulated Link-Layer Capacity: 27 different MCSs for an LTE physical-layer using an established simulator [4] in a 3GPP channel [2]. 2) Proposed Capacity Expression: curve fit a function to the simulation data for each individual MCSs and the combined adaptive MCS. 3) Proposed Optimal Power Allocation: using this closedform capacity expression, derive the optimal power allocation for parallel channels and present a comparison with existing schemes. The paper will show that the proposed scheme accurately match a large sample of different MCSs with a mean capacity error of bit/s/hz, and a variance of Furthermore, the resulting power allocation scheme for MIMO yields a capacity gain over both channel inversion and waterfilling schemes. It is worth noting that the proposed Arctangent capacity expression is common to all transmission schemes, but the specific parameter values in the function is specific to the MCS and the channel. A. Deterministic Approach II. REVIEW Existing research conducted by [3] has attempted to derive a deterministic expression for the capacity of arbitrary modulation schemes with perfect FEC coding. To that end, they have succeeded in formulating a relationship between capacity (C) and the MMSE of the modulation scheme: dc(γ) dγ = MMSE(γ), () where γ is the signal-to-noise-ratio (SNR). The disadvantage of this approach is that the resulting expressions for different

2 IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE (WCNC), Special efficiency (bit/s/hz) SNR QPSK, R=/9 QPSK, R=/6 QPSK, R=.2 QPSK, R=/4 QPSK, R=/3 QPSK, R=.42 QPSK, R=/2 QPSK, R=.58 QPSK, R=2/3 QPSK, R=.73 6QAM, R=.43 6QAM, R=.46 6QAM, R=/2 6QAM, R=.54 6QAM, R=.58 6QAM, R=.6 6QAM, R=2/3 6QAM, R=.73 6QAM, R=4/5 64QAM, R=.58 64QAM, R=.62 64QAM, R=2/3 64QAM, R=.7 64QAM, R=.74 64QAM, R=4/5 64QAM, R=.85 64QAM, R=.9 Special efficiency (bit/s/hz) SNR QPSK, R=/9 QPSK, R=/6 QPSK, R=.2 QPSK, R=/4 QPSK, R=/3 QPSK, R=.42 QPSK, R=/2 QPSK, R=.58 QPSK, R=2/3 QPSK, R=.73 6QAM, R=.43 6QAM, R=.46 6QAM, R=/2 6QAM, R=.54 6QAM, R=.58 6QAM, R=.6 6QAM, R=2/3 6QAM, R=.73 6QAM, R=4/5 64QAM, R=.58 64QAM, R=.62 64QAM, R=2/3 64QAM, R=.7 64QAM, R=.74 64QAM, R=4/5 64QAM, R=.85 64QAM, R=.9 Combined MCS Fig.. Simulated adaptive MCS capacity of LTE physical layer using the simulator in [4] and 3GPP indoor channel [2], with SNR values is in linear. Fig. 2. Theoretical adaptive MCS capacity of LTE physical layer using expression (4), with SNR values is in linear. m-psk and m-qam modulation schemes do not take into account the important role played by adaptive FEC coding. Furthermore, the expressions are only tractable at high SNRs and do not scale easily to higher order modulation schemes. Nonetheless, work conducted in [3] [5] has shown that optimization results using realistic MCS expressions can yield very different wireless system optimisation results. B. Empirical Approach An alternative empirical approach is to modify the Shannon expression, so that the effects of imperfect coding and capacity saturation (C s ) are considered [6]. The resulting link-level capacity is of the form: ( C(γ) min [a log 2 + γ ) ], C s, (2) b where a and b are adjustment factors that depend on the combined MCS. The disadvantage of this approach is that the resulting expression is not continuous and cannot be employed in analysis that requires continuous tractable functions, such as formulations for the the MIMO power allocation and stochastic-geometry [7]. Furthermore, the expression is not a function of individual MCS s modulation and coding rates. Whilst there is also research on approximating the error rate performance of modulation [8] and coding schemes [9], the research community is still yet to find a tractable expression that captures the dynamic nature of adaptive MCS. Alternative empirical expressions that modify the Shannon expression to take into account of mutual information saturation and coding rates are typically break functions [6], which are not tractable for many optimisation approaches. C. Power Allocation for Parallel Channels For optimal power allocation in independent and parallel channels, the Water-filling (WF) policy was derived using the Shannon bound [] and then rigorously formalized for dispersive channels. Whilst Gaussian inputs yield the greatest mutual information in a Gaussian noise channel, they are not feasible in reality. Realistic communication systems employ inputs from discrete constellations. A relationship between the mutual information and the MMSE of a modulation scheme was derived in [3], and the associated power allocation for parallel channels is known as Mercury-filling (MF). This yielded an insightful allocation scheme, whereby MF agreed with WF at low SNRs, but deviated away at high SNRs. The rationale with MF is that at high SNRs, the mutual information of channel is saturated and further power allocation to high SNR channels (WF) is wasteful. Instead, the remaining power should be allocated to the unsaturated low SNR channels. However, the work in MF assumed perfect forward-error-correction (FEC) code rates, and closed form expressions were only available for high SNRs and PSK modulation schemes. III. APPROXIMATE CAPACITY EXPRESSION A. LTE Physical Layer The LTE physical layer consists of 27 different modulation and coding combinations [2], and the combined MCS has a significantly lower performance than the Shannon bound [6]. Each MCS yields the highest capacity for a given SNR range, which ranges from approximately -6 db to capacity saturation at 4 db [2]. The MCS results from the physical layer simulator can be seen in Fig.. It is worth noting that a similar set of data can be obtained for any MCS for systems such as Wi-Fi, Wi-Max, HSPA, etc. B. Arctangent Function Based Capacity The Arctangent-function has been chosen to characterize the capacity of arbitrary discrete modulation inputs and FEC coding. The rationale for the heuristic approach is that typically, a MCS will achieve a non-zero and monotonically increasing capacity in an SNR region of γ to γ 2. For low-snr regimes

3 IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE (WCNC), 23 3 of below γ, the capacity is zero, and for high-snr regimes of above γ 2, the capacity saturates at C sat.. The Arctangent function is selected because it has the aforementioned characteristics, which are common to all capacity profiles of any MCS in any transmission environment: γ γ arctan (γ) = C γ < γ < γ 2, (3) C sat. γ γ 2 where the channel SNR is defined as: γ = h 2 PtΛ N. The parameter h is the magnitude of the complex fading coefficient h, P t is the transmitted signal power, Λ is the pathloss, and N is the additive white Gaussian noise power. The proposed capacity expression relates the capacity (C, bit/s/hz) as a function of the channel SNR (γ): ( ) γ + b C(γ) a arctan (γ, b, c) = a arctan. (4) c The parameters a, b and c are adjustment factors, whereby through curve-fitting, an empirical relationship can be established with the modulation bits/symbol rate (N) and FEC coding rate (R) of any particular MCS. To demonstrate this concept, the LTE physical layer with a 3GPP channel model yields: a =.3 +.5N +.4NR.N R 2 b = 2..N 2.6R +.5NR +.5N 2.5R 2 c = N 25.3R + 4.5NR +.54N 2 3.3R 2, Given that the proposed capacity expression is a function of the modulation bits/symbol rate and the coding rate, it allows inference to other unconsidered MCS schemes. Having addressed individual MCSs, the combined adaptive MCS for LTE has a unique set of adjustment parameters: a = 2.27, b = 3, and c = 4. A similar set of parameter values also exist for other communication systems, which is beyond the scope of this paper. The derivative of the capacity function with respect to γ is: dc(γ) dγ = (5) AC (γ + B) 2 + C 2 (6) which is a very tractable function. This identity is useful for the optimal power allocation derivation. The antiderivative of the capacity function is composed of an arctangent function and a logarithm function, which is also tractable. C. Validation Results In Fig., the simulated adaptive MCS of LTE physical layer is presented. The theoretical expression (4) with different modulation and coding rates is presented in Fig. 2, along with a combined MCS plot. The approximation accuracy is very high: a maximum error of.53 bit/s/hz on any MCS, a mean error of bit/s/hz across all MCSs, and a variance of It is worth noting that the proposed arctangent capacity expression is common to all transmission schemes, but the specific parameter values in the function is specific to the MCS and the channel. This section has merely used LTE physical layer to demonstrate the accuracy and applicability. D. Application and Benefit The application of capacity expressions that include the effects of capacity saturation has already been demonstrated for Water-Filling (WF) power control in MIMO channels. The resulting effect is that by considering capacity saturation, the opposite conclusion of WF is found at medium-high SNRs. Therefore, the usage of accurate capacity expressions in performance characterization and optimisation is of profound importance to the wireless industry. To that end, this paper has so far offered a tractable capacity expression for a wide range of individual MCSs and their combined MCS. Furthermore, the expression can infer the performance of other MCSs, before extensive simulations are performed. For a given SNR (γ) and modulation scheme (N), the proposed expression in (4) can find the theoretical coding rate (R) that maximizes the capacity. The paper now considers an optimisation application using the proposed capacity expression. Existing research examples include optimising the energy-capacity trade-off in wireless networks [] and outage-throughput tradeoffs in cooperative networks []. For the purpose of demonstration, the paper now examines the detailed impact of applying the proposed expression onto optimizing power allocation to parallel Gaussian-noise fast-fading channels [2]. IV. OPTIMAL POWER ALLOCATION A. Water-filling, Mercury-filling Many wired and wireless systems employ a transmitter configuration consisting of a bank of independent parallel channels, such as multi-tone transmission and multi-antenna communication. For N c parallel channels with Gaussian inputs and perfect coding, the existing WF algorithm [3] is: P WF,n = ( λ N ) + h n 2, for: N c P n = P, (7) where λ is the Lagrangian constant and P is the total power budget. Figure 3a shows WF power allocation, whereby each channel with N h n is supplemented with a power allocation 2 up to the λ threshold. Any channel that exceeds the threshold (poor channel SNR), receives no power allocation. The maximum capacity is achieved by allocating the most power to the best SNR channels. When the effects of capacity saturation are considered, the existing WF algorithm is modified to become the MF algorithm [2]: ( PMF,n = [ MMSE n min, η ] ) N c, for: Pn = P, γ n γ i (8)

4 IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE (WCNC), 23 4 /γ /λ γ P* CI P* WF a) Max. Long Term Capacity: Water-filling (WF) c) Max. Delay-Limited Capacity: Channel Inversion (CI) /γ saturated P* /η MF /γ /κ WF regime CI regime G(η/γ)/γ P WF /P CI =(B/C) 2 P* proposed saturated b) Max. Long Term Capacity: Mercury Filling (MF) d) Max. Long Term Capacity: Proposed Scheme Fig. 3. Optimal power allocation algorithms for independent parallel Gaussian-noise channels: a) Water-filling (WF) [3], b) Mercury-filling (MF) [2], c) Channel-Inversion (CI) [3], d) Proposed Scheme (2). where η is the Lagrangian constant. Figure 3b shows MF power allocation, whereby each channel with γ is supplemented with a mercury-filling modifier (G(η/γ)/γ) to take into account the capacity saturation effects [2]. Compared to WF, MF allocates power to the best SNR channels up to capacity saturation, and then allocating any remaining power to poor SNR channels. However, MF only considers specific modulation schemes with perfect coding, as opposed to the completely adaptive MCS (Shannon bound) in WF. B. Channel Inversion One can interpret WF/MF as maximising the long-term average rate flow of information in a fast fading channel. Channel inversion (CI) attempts to achieve a constant flow of information rate by allocating power such that: PCI,n = γ n. Whilst the CI algorithm yields a steadier flow of information on an instantaneous basis, it causes a high demand for peakto-average ratio in transmit power. The CI algorithm shown in Figure 3c ensures a delay-limited capacity [3], whereby for every channel SNR of γ, the corresponding power allocated is the inverse of the channel. Note that Figure 3c is for illustrative purposes only. C. Optimal Power Allocation For N c parallel channels, the paper attempts to maximize the long-term throughput for a non-delay-limited channel. P n * (W) Spectral Efficiency (bit/s/hz) A A B B h n 2 /N (db) Proposed Channel Inversion Water filling h n 2 /N (db) C C Combined MCS Shannon Fig. 4. a) Simulated power allocation values for N c independent channels. b) Combined MCS and Shannon capacity results. The capacity expression in (4) is employed, whereby the Lagrangian (L) is: N c ( ) γn + B N c L(ρ, P n ) = A arctan ρ P n P, C (9)

5 IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE (WCNC), 23 5 where ρ is the Lagrangian multiplier. The resulting partial differentiation yields: L P n = N c [ AC (γ n + B) 2 + C 2 ( hn 2 N ) ] ρ. () The Kuhn-Tucker condition for power allocation optimality is: L P n { = if Pn > if P n =. () Therefore, the optimal power allocation policy is (Proof given in Appendix): P proposed,n = [ + C P CI,n P WF,n CI,n] (κ) BP, (2) for: N c P n = P. From (2), it can be observed that the proposed scheme is a combined CI and WF scheme, whereby the term P CI,n P WF,n (κ) is equivalent to the average of the CI and WF power allocation values in the dbm scale. The modified Lagrangian constant is κ = ρc A. It is worth mentioning that no approximations were made during the power allocation derivation process. An explanation on the power allocation mechanism and its rationale is given below. D. Low SNR Regime: Water-filling Channels with extremely low SNRs will require a large amount of power to achieve a non-zero capacity. That is to say, in order to maximise long-term capacity of multiple parallel channels, they should be allocated zero power. This is reflected in the WF and MF solutions, which has been proven to be near-optimal [2]. The same is also reflected in the proposed solution (2), whereby at low SNRs: Pproposed,n = for: P WF,n =. (3) This is represented as Region A Fig. 4. E. High SNR Regime: Channel Inversion At high SNR regimes, the WF solution allocates the most power to the strongest channel, whereas the MF solution deviates from this [2]. The proposed solution is actually a modified CI solution, where a decreasing amount of power is P CI,n P WF,n allocated to increased channel SNR. The term is equivalent to the geometric mean of the CI and WF power allocation values in the dbm scale. This is shown as Region B to C in Fig. 4, whereby the spike in power allocation corresponds to the medium SNR range of the combined MCS. The power allocation decreases with increased SNR in order not to waste resources on saturated channels (Region C). F(Capacity) Fig Empirical CDF.2 Proposed. Channel Inversion Water filling Capacity, bit/s/hz Simulated capacity CDF for N c independent channels. F. Capacity Improvement Results The investigation considers N c = independent channels with a mean power constraint of W. Each channel is random and uniformly distributed across a SNR range of - and 5 db. From the capacity cumulative distribution function (CDF) results presented in Fig. 5, the proposed scheme yields a mean capacity improvement of 8% over CI and 7% over WF. Figure 5 shows that the proposed scheme primarily improves medium SNR channels to the point of capacity saturation, whilst sacrificing the performance of low SNR channels. Due to the fact that the MF algorithm in [2] only considers separate modulation schemes (not a combined MCS), it was not possible to draw a direct comparison between MF and the proposed scheme in this paper. V. CONCLUSIONS In this paper, the analytical dichotomy of employing the tractable Shannon bound and using simulated physical layer data is consolidated. The paper proposes a novel tractable expression that can yield accurate capacity expressions for a wide range of specific modulation and coding schemes (MCSs). The proposed capacity expression is a function of the modulation bits/symbol rate and the coding rate, which allows inference to other unconsidered MCSs. The proposed capacity expression is then applied to find the optimal power allocation for multiple parallel channels. The resulting power allocation solution is effectively a combination of the channel inversion and water-filling schemes. The results prove that the proposed scheme yields a better capacity compared to both existing schemes.

6 IEEE WIRELESS COMMUNICATIONS AND NETWORKING CONFERENCE (WCNC), 23 6 APPENDIX By differentiating the Lagrangian in (9), the following is L yielded for = : P n + (AC hn ) γ n = 2 + C N ρ 2 B, (4) [3] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, UK: Cambridge University Press, 25. where γ n = h 2 P proposed,n N. Solving for the optimal power allocation with respect to Pproposed,n, yields: Pproposed,n = C = N h n 2 ( κ N ) + h n 2 B N h n 2 [ + C P CI,n P WF,n CI,n] (κ) BP, + (5) where κ = ρc A. The notation P indicates that the power allocation has the same form as the isolated case, but not the same numerical value due to the overall power constraint, which is determined by the factor κ. The Lagrangian factor κ needs to be found iteratively, as is the case for WF/MF. Note, the proposed power allocation solution is a function of the combined MCS, and no approximations were used during the derivation beyond the proposed capacity expression itself. REFERENCES [] C. E. Shannon, Communication in the presence of noise, in Proc. of the IRE, 949. [2] 3GPP, TR36.84 V9..: Further Advancements for E-UTRA Physical Layer Aspects (Release 9), 3GPP, Technical Report, Mar. 2. [3] D. Guo, S. Shamai, and S. Verdu, Mutual information and minimum mean-square error in gaussian channels, in Info. Theory, IEEE Trans. on, Apr. 25. [4] C. Mehlfuhrer, M. Wrulich, J. Ikuno, D. Bosanska, and M. Rupp, Simulating the long term evolution physical layer, in European Signal Processing Conference, EURASIP, Aug. 29, pp [5] W. Guo, I. Chatzigeorgiou, I. J. Wassell, and R. Carrasco, Partner selection and power control for asymmetrical collaborative networks, in Vehicular Technology Conference, IEEE, Taipei, Taiwan, May 2. [6] P. Mogensen, W. Na, I. Kovacs, F. Frederiksen, A. Pokhariyal, K. Pedersen, T. Kolding, K. Hugl, and M. Kuusela, LTE Capacity Compared to the Shannon Bound, in Vehicular Technology Conference, 27. VTC27-Spring. IEEE, Apr. 27, pp [7] M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, Stochastic geometry and random graphs for the analysis and design of wireless networks, Selected Areas in Communications, IEEE Journal on, vol. 27, no. 7, pp , September 29. [8] O. Afelumo, A. Awoseyila, and B. Evans, Simplified evaluation of apsk error performance, Electronics Letters, vol. 48, no. 4, pp , [9] H. El Gamal and A. Hammons Jr, Analyzing the turbo decoder using the Gaussian approximation, Information Theory, IEEE Transactions on, vol. 47, no. 2, pp , 2. [] W. Guo and T. O Farrell, Capacity-Energy-Cost Tradeoff for Small- Cell Networks, in Vehicular Technology Conference, IEEE, Yokohama, Japan, May 22. [] W. Guo and I. J. Wassell, Capacity-outage-tradeoff for cooperative networks, in Selected Areas in Communications (JSAC), IEEE Journal on, vol. 3, Oct. 22, pp [2] A. Lozano, A. M. Tulino, and S. Verdu, Optimal power allocation for parallel gaussian channels with arbitrary input distributions, in Info. Theory, IEEE Trans. on, vol. 52, Jul. 26, pp