Degrees of Freedom in Adaptive Modulation: A Unified View

Degrees of Freedom in Adaptive Modulation: A Unified View Seong Taek Chung and Andrea Goldsmith Stanford University Wireless System Laboratory David Packard Building Stanford, CA, U.S.A. taek,andrea @systems.stanford.edu Abstract We examine adaptive modulation schemes for flat-fading channels where the data rate, transmit power, and instantaneous BER are varied to maximize spectral efficiency, subject to an average power and BER constraint. Both continuous-rate and discrete-rate adaptation are considered, as well as average and instantaneous BER constraints. We find the general form of power, BER, and data rate adaptation that maximizes spectral efficiency for a large class of modulation techniques and fading distributions. The optimal adaptation of these parameters is to increase the power and data rate and decrease the BER as the channel quality improves. Surprisingly, little spectral efficiency is lost when the power or rate is constrained to be constant. Hence, the spectral efficiency of adaptive modulation is relatively insensitive to which degrees of freedom are adapted. Introduction Adaptive modulation is a promising technique to increase the data rate that can be reliably transmitted over fading channels. For this reason some form of adaptive modulation is being proposed or implemented in many next generation wireless systems. The basic premise of adaptive modulation is a realtime balancing of the link budget in flat fading through adaptive variation of the transmitted power level, symbol transmission rate, constellation size, BER, coding rate/scheme, or any combination of these parameters [, ]. The question therefore arises as to which of these parameters should be adapted to obtain the best performance. In particular, [] investigates adapting power and rate, and [] investigates adapting power, rate, and instantaneous BER. However, no unified study on the tradeoffs in adapting all combinations of different modulation parameters has been previously undertaken. In this paper we provide a systematic study on the increase in spectral efficiency obtained by optimally varying combinations of the transmission rate, power, and instantaneous BER. We assume that the resulting adaptive modulation schemes are subject to an average power and BER constraint. We first analyze adaptive modulation with continuous rate adaptation (C- Rate) where the set of signal constellations is unrestricted, and then consider the more practical scenario, discrete rate adaptation (D-Rate) where only a discrete finite set of constellations is available. Analysis is done for both an average BER constraint (A-BER) and an instantaneous BER constraint (I-BER). Our goal is to determine the impact on spectral efficiency of adapting various modulation parameters under different constellation restrictions and BER constraints, for a large class of modulation techniques and fading distributions. The remainder of this paper is organized as follows. The next section describes the system model, including the average power and BER constraints. Section 3 presents the BER approximations used to derive the optimal adaptive modulation scheme. We derive the optimal rate, power, and BER adaptation strategies under different constellation restrictions and BER constraints in Section. Numerical results and plots of spectral efficiency, optimal power, BER, and rate adaptation are presented in Section 5. We examine constant power and constant rate adaptation in Section. Conclusions will be given in Section 7. System Model In this section we present our system model and notation, following that of []. The system model is illustrated in Figure. We assume a discrete-time Ô channel with stationary and ergodic time-varying gain and additive white Gaussian noise Ò. Let denote the average transmit signal power, denote the variance of Ò, denote the received signal bandwidth, and denote the average channel power gain. With appropriate scaling of, we can assume that =. There is an instantaneous and error free feedback path from the receiver to the transmitter for sending channel estimates, so the channel gain estimate. For a constant transmit power, the instantaneous received SNR is. We denote the transmit power at time, which is a function of, by µ. The received SNR at time is then µ. Since is stationary, the distribution of is independent of, and we denote this distribution by Ô µ. When the context is clear, we will omit the time reference relative to and µ. We also assume ideal coherent phase detection. The parameters that we can adapt at the transmitter include the transmission rate, power, and instantaneous BER. For the D-Rate case we assign one signal constellation and a corresponding data rate of bits/symbol to each rate region µ Æ, where Æ. No signal is transmitted if. Thus, serves as a cutoff SNR be-

TRANSMITTER CHANNEL RECEIVER r[i] Adaptive Power Modulation Control and Coding S[i] x[i] g[i] n[i] y[i] Figure. System Model Demodulation and Decoding Channel Estimator low which the channel is not used. We will find that in the C-Rate case there is also an optimized cutoff value,, below which the channel is not used. Thus, for both continuous and discrete rate adaptation, when the channel quality is significantly degraded, the channel should not be used. The spectral efficiency of our modulation scheme equals its average data rate per unit bandwidth (Ê ). When we send µ ÐÓ Å µ (bits/symbol), the instantaneous data rate is µ Ì (bps), where Ì is the symbol time. Assuming Nyquist data pulses ( Ì ), for continuous rate adaptation the spectral efficiency is given by Ê and for discrete rate adaptation it is given by Ê Æ ^r[i] µô µ bps/hz () Ô µ bps/hz () The rate adaptation µ is typically parameterized by the average transmit power and the BER of the modulation technique, as discussed in more detail in Section. We assume an average transmit power constraint given by µô µ (3) For the BER, we assume either an average (A-BER) or an instantaneous (I-BER) constraint. The instantaneous BER constraint implies that the system must maintain a constant probability of bit error for each fading value (BER( )=BER). This is more restrictive than the average constraint, which is BER Ê for continuous rate adaptation and BER È Æ for discrete rate adaptation. BER µ µô µ Ê µô µ Ê È Æ BER µô µ Ê Ô µ 3 BER Approximations In order to derive the optimal power and rate adaptation we need to invert and differentiate BER approximations with respect to their arguments. However, for most nonbinary modulation techniques (e.g. MQAM and MPSK) BER expressions with such properties are hard to find. Therefore, we () (5) Modulation MQAM MPSK (Model ) MPSK (Model ) MPSK (Model 3) Table. Constants in BER approximations () now find new tight BER approximations for several modulation techniques in AWGN that can be easily differentiated and inverted. All BER approximations can be written in the following generic form: BER µ ÜÔ µ () µµ where µµ µ.,, and are positive fixed constants, and is a real constant. Table shows one approximation for MQAM and three approximations for MPSK. All these approximations are valid to within.5 db of error for µ and BER. More details on these approximations can be found in []. Optimal Rate, Power, and BER Adaptation In this section we determine the optimal rate, power, and BER adaptation for maximizing spectral efficiency in the following four cases: C-Rate with an A-BER constraint, C-Rate with an I-BER constraint, D-Rate with an A-BER constraint, and D-Rate with an I-BER constraint. Our analysis here applies for any fading distribution. Derivations are outlined below: details can be found in [].. Continuous Rate and Average BER (C-Rate A-BER) The optimal continuous rate, power, and BER adaptation to maximize spectral efficiency () subject to the average power constraint (3) and the average BER constraint () are obtained via a Lagrangian optimization []. The resulting optimal power adaptation is µ µµ µµ µ µµ µµ µ where µµ µµ, for nonnegative µ and the constants and. The corresponding optimal BER adaptation µ is (7) BER µ BER µµ µ () The optimal rate adaptation µ is either zero or the nonnegative solution of µµ µµ µ µ ÐÒ (9) BER µµ The values of µ and the constants and are found through a numerical search such that the average power (3) and BER () constraints are satisfied. More details on the numerical search process are given in [].

. Continuous Rate and Instantaneous BER (C-Rate I- BER) The optimal continuous rate and power adaptation to maximize spectral efficiency () subject to the average power constraint (3) and an instantaneous BER constraint BER µ BER are also obtained using the Lagrangian method []. It turns out that the optimal power adaptation completely changes for different values []. Moreover, the cutoff value below which the channel is not used is based on the positivity constraint on power when and the positivity constraint on rate when or. We show the optimal power and rate adaptation scheme for each in Table, where à and à ÐÒ BER µ ÐÒ BER µ. The numerical value of the cutoff depth is obtained from the average power constraint (3). The optimal power adaptation is a waterfilling (more power as the channel improves above ) for, an inverse-waterfilling (less power as the channel improves above ) for, and an on-off power transmission (either power is zero or a constant nonzero value) for. Thus, although all three BER approximations for MPSK (Table ) are tight, they lead to very different optimal adaptive power policies (Table ). µ µ à à õ õ ÐÓ µ ÐÓ µ à ÐÓ µ Table. Optimal µ and µ in C-Rate I-BER for The optimal power adaptations of Table are plotted in Figure for Rayleigh fading with BER and db. This figure clearly shows the water-filling, inverse waterfilling, and on-off behavior of the different schemes. Note that the cutoff for all these schemes is roughly the same. From the cumulative density function of also shown in Figure, the probability that is less than is.. Thus, although the optimal power adaptation corresponding to low SNRs is very different for the different BER approximations, this behavior has little impact on spectral efficiency since the probability of being at those low SNRs is quite small..3 Discrete Rate and Average BER (D-Rate A-BER) In the discrete rate case, the rate is varied within a fixed set Æ, and we assign rate to the rate region µ. Under this fixed rate assignment we wish to maximize spectral efficiency () through optimal rate, power, and BER adaptation subject to an average power (3) and BER constraint (5). Since the set of possible rates and their corresponding rate region assignments are fixed, the optimal rate adaptation corresponds to finding the optimal rate region boundaries Æ. Necessary conditions on the optimal power, BER adaptation and boundaries are found in []. However, these conditions are very difficult to solve for the optimal boundary points Æ. By assuming that µ is continuous at each bound-........ Model Model Model 3 cdf of γ 5 5 5 5 3 35 Figure. µ in Model,,3 for MPSK (BER db). ary, we obtain the suboptimal rate region boundaries and the corresponding suboptimal power adaptation []. We will see in Section 5 that these suboptimal boundaries and suboptimal power adaptations yield a spectral efficiency close to that of the optimal C-Rate A-BER case.. Discrete Rate and Instantaneous BER (D-Rate I- BER) With the same discrete rate set Æ and the same rate assignment to region µ as in the previous section, we now assume an instantaneous BER constraint, so that BER µ BER. Under these constraints the optimal power adaptation is given as µ µ () BER µ. The optimal rate region where µ ÐÒ boundaries that maximize spectral efficiency must satisfy and µ µ µ Æ () where is determined by the average power constraint (3). 5 Numerical Results Although our derivations are for general fading distributions, modulations, and BER approximations, we compute our numerical results for adaptive MQAM in Rayleigh fading based on the BER () with parameters given in Table. We assume a BER requirement of either or. For the discrete rate cases we assume that different MQAM signal constellations ( QAM, QAM, QAM, 5 QAM, QAM, and 9 QAM) are available. The spectral efficiencies for the four adaptation policies (C-Rate A-BER, C-Rate I-BER, D-Rate A-BER, and D-Rate I-BER) are plotted in Figure 3. The spectral efficiencies of all four policies under the 3

C Rate A BER C Rate I BER D Rate A BER (suboptimal) D Rate I BER numerical calculations we use the same MQAM constellations as in Section 5, with a target BER of either or under Rayleigh fading. BER= 3 BER= 7. Constant Power In this section we maximize spectral efficiency assuming a constant transmit power with a cutoff threshold below which the channel is not used. We first consider the C-Rate I-BER policy. The constant transmit power µ satisfies Ê Ô µ () 5 5 3 35 Figure 3. Spectral Efficiency for MQAM same BER constraints are very close to each other. The spectral efficiency of D-Rate I-BER is slightly higher than that of D-Rate A-BER since the latter is calculated with suboptimal rate region boundaries. The optimal power control scheme, µ, for BER is given in Figure. We see from these figures that the optimal transmit power follows a smooth water-filling with respect to under the C-Rate A-BER and C- Rate I-BER policies while the optimal power adaptation curve is quite steep under the D-Rate A-BER and D-Rate I-BER policies. Specifically, the peak-to-average-power-ratio (PAPR) in the D-Rate case is about db. The optimal BER and rate adaptation are not shown here due to space constraints. These adaptation policies, shown in [], indicate that BER is always within an order of magnitude of its target value and rate increases with, as expected..5 (a) C Rate A BER.5 (b) C Rate I BER from the average power constraint (3). The optimal rate µ corresponding to () can be derived by inverting the BER approximation (). Then we can find the maximum value of spectral efficiency () by optimizing it with respect to. Numerical values of this maximum spectral efficiency are given in Figure 5. The spectral efficiency loss is less than % when compared with the maximum possible spectral efficiency obtained using the adaptive power C-Rate A-BER policy (Section.), so the two curves are indistinguishable in Figure 5. If we assume the threshold, then the constant transmit power is µ. Numerical values of the spectral efficiency () in this case are also given in Figure 5. We see that optimizing the threshold results in little performance improvement relative to the zero threshold ( ) case, especially at high SNRs. We do not analyze the spectral efficiency of the C-Rate A-BER policy with constant transmit power, since its efficiency will lie between the efficiency of the C-Rate I-BER policy with constant power and that of the adaptive power C-Rate A-BER policy. C Rate A BER (Section.) C Rate I BER with threshold C Rate I BER w/o threshold D Rate A BER (suboptimal) D Rate I BER.5 3.5 (c) D Rate A BER (suboptimal).5 3.5 (d) D Rate I BER BER= 3 BER= 7.5 3.5 3 Figure. µ for MQAM (BER db). Constant Power and Rate We now restrict our system to have either constant transmit power or constant rate, which simplifies system design. For our 5 5 3 35 Figure 5. Spectral Efficiency for Adaptive MQAM with Constant Transmit Power. We now consider the D-Rate I-BER policy with constant transmit power. The constant transmit power µ satisfies () and the optimal rate region boundaries Æ are found from the instantaneous BER constraint. Numerical values for the spectral efficiency () in this case are given in Figure 5. The spectral efficiency of this scheme is between 7% and

9% of the C-Rate A-BER policy with optimal power adaptation. This penalty is predictable since we have removed two degrees of freedom with respect to the adaptive power C-Rate A-BER policy: power and BER adaptation. For this policy, the instantaneous BER is often lower than the target BER. This is due to the rate discretization and constant power restriction. We now consider the D-Rate A-BER policy with constant transmit power. The optimal solution is hard to find as was also the case in Section.3. As a suboptimal solution, we will use the discrete rate region boundaries of the D-Rate I-BER with constant power. We scale each value of these discrete rate regions equally such that the A-BER constraint (5) is satisfied exactly. Numerical values of the spectral efficiency () with this scaled discrete rate region boundaries are given in Figure 5. Even though we obtain the spectral efficiency of this D- Rate A-BER policy using suboptimal rate regions, the spectral efficiency is between 75% and 95% that of the optimal power C-Rate A-BER policy.. Constant Rate We now consider constant rate policies ( µ ) with a cutoff threshold. Let us first assume an I-BER constraint. From (), the optimal power control scheme to maintain the BER target over all for rate is µ µ else (3) where the threshold value is obtained from the power constraint (3). We obtain Ê the optimum value of to maximize spectral efficiency ( Ô µ ) at each average SNR value by numerical search. The resulting spectral efficiency values are given in Figure. Here the penalty in spectral efficiency is about % with respect to the adaptive power C-Rate A-BER policy (Section.) if we don t restrict the fixed rate to be an integer. However, as we see in Figure, the restriction of to integer values does not significantly decrease spectral efficiency. Now consider the A-BER constraint. Assume a threshold below which no data is sent. Then when the optimal power control is µ and the optimal BER adaptation is µ µ ÐÒ () BER µ µ (5) where the constant and threshold are obtained from (3) and (5). Using numerical search techniques, the optimal rate and threshold are found. Details are Ê described in []. The corresponding spectral efficiency Ô µ µ is plotted in Figure. As the figure shows, the spectral efficiency in this case is quite close to that of a constant rate policy with an I-BER constraint, so we do not get much gain by relaxing the I-BER constraint. Figure also shows the case when the constant rate is restricted to integer values. We see that all constant rate policies yield almost the same spectral efficiency regardless of differences in the rate restriction (integer or noninteger) and BER constraint (average or instantaneous). All of these policies show a penalty of about db relative to the optimal adaptive power C-Rate A-BER policy of Section.. In the constant rate case, an optimal transmit power level could approach infinity when the channel gain is very small, which results in a bad PAPR. In [] it is shown that the optimal transmit power under the constant rate transmission strategy in Rayleigh fading with BER and = 3 db has a PAPR of 9 db for the I-BER constraint and 7 db for the A-BER constraint. These PAPRs are much worse than in schemes where both power and rate are adapted, e.g. Figure, where PAPR is around db. C Rate A BER (Section.) Integer Rate I BER Noninteger Rate I BER Integer Rate A BER Noninteger Rate A BER BER= 3 BER= 7 5 5 3 35 Figure. Spectral Efficiency for Adaptive MQAM with Constant Rate. 7 Conclusion We have shown that the maximum spectral efficiency of adaptive modulation is nearly the same under continuous and discrete rate adaptation as well as under an instantaneous or average BER constraint. We have also derived the optimal power, rate, and BER adaptation for these schemes for a large class of modulation techniques and general fading distributions. Restricting the power or rate of the adaptive modulation to be constant achieves near optimal performance in most cases. Therefore, the parameters to adapt should be chosen based on implementation considerations. References [] A. J. Goldsmith and S. Chua, Variable-rate variable-power MQAM for fading channels, IEEE Trans. on Commun., Vol. 5, pp. 3, Oct. 997. [] C. Kose and D. L. Goeckel, On power adaptation in adaptive signaling systems,, IEEE Trans. on Commun., Vol., pp. 79 773, Nov.. [3] J. G. Proakis, Digital Communications, nd ed. New York:McGrawHill,99 [] S. T. Chung and A. J. Goldsmith, Degrees of Freedom in Adaptive Modulation: A Unified View, To appear in IEEE Trans. on Commun. 5