ADAPTIVE modulation is a promising technique that is

716 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004 Impact of Channel Estimation Error on Adaptive Modulation Performance in Flat Fading José F. Paris, M. Carmen Aguayo-Torres, and José T. Entrambasaguas Abstract Presented here is an approach to analyzing the effect of imperfect channel estimation on adaptive modulation. The sensitivity of the main performance parameters to short-term and long-term estimation error sources is summarized in a set of new formulas that are either closed-form expressions or simple to compute numerically. Index Terms Adaptive modulation, channel estimation, flat-fading channels. I. INTRODUCTION ADAPTIVE modulation is a promising technique that is being progressively incorporated into wireless communication systems [1] [6]. One important premise for adaptive modulation performance is the availability of an accurate knowledge of the channel fading. In a likely scenario, the time-varying frequency-flat slow fading consists of short-term variations, due to multipath propagation, and long-term variations as a consequence of path loss/shadowing. Typically, short-term variations are modeled by a Rayleigh distribution, and long-term variations by a log-normal distribution. Information concerning both phenomena is needed to implement adaptive modulation schemes. For this task, the instantaneous signal-to-noise ratio (SNR) as the short-term descriptor, and the average SNR as the long-term descriptor, are usually considered. Pilot symbols can be inserted into the data stream to obtain this information. Finally, the transmitter is dynamically configured according to these estimates to optimize the system spectral efficiency. Any uncertainty about the channel fading descriptors may cause significant degradation of adaptive modulation performance. Several papers have addressed the usage of imperfect channel estimates in adaptive modulation [1], [7], [8]. In [1], a simple approximation to evaluate the effect of delayed instantaneous SNR estimates on the bit-error rate (BER) is presented. The characterization of the channel fading variations is considered in [7], to design effective adaptive modulation schemes for systems using outdated fading estimates. Finally, in [8], the impact of the estimation delay on the BER is analyzed for adaptive quadrature amplitude modulation (AQAM). The motivation of this letter is providing an approach for the analysis of the combined effect of the different error sources related to the channel estimation for AQAM. The remainder of this letter is organized as follows. Section II briefly describes Paper approved by M.-S. Alouini, the Editor for Modulation and Diversity Systems of the IEEE Communications Society. Manuscript received April 25, 2003; revised September 9, 2003 and October 19, 2003. This work was supported in part by the Spain CICYT under project TIC2003-07819. The authors are with the Department of Ingeniería de Comunicaciones, University of Málaga, Málaga E-29071, Spain (e-mail: paris@ic.uma.es; aguayo@ic.uma.es; jtem@ic.uma.es). Digital Object Identifier 10.1109/TCOMM.2004.826254 the adaptive modulation scheme and the channel estimation subsystem considered here. In Section III, close approximations are found for the cutoff SNR to simplify the performance analysis carried out in Section IV. Finally, conclusions about the presented approach are provided in Section V. II. SYSTEM MODEL Fig. 1 shows a block diagram of the adaptive modulation system. It is assumed flat fading, thus, the channel model consists of the baseband complex envelope plus additive white Gaussian noise (AWGN). The received power follows a multiplicative model, where is the power fluctuation due to path loss/shadowing, and is the power fluctuation due to multipath with. Although is time dependent, its variation is assumed very slow, compared with the temporal variations of characterized by the Doppler spread. From the point of view of short-term temporal variations, exhibits Rayleigh fading with average power gain. When long-term variations are also considered, is modeled as a random variable (RV) that is widely accepted as being log-normal. The aforementioned random processes are ergodic, stationary, and statistically independent, thus, temporal references can be dropped for simplicity of notation. At the transmitter, fixed pilot symbols are periodically inserted into the data symbols. Both types of symbols have the same average power 1 and are assembled into frames that consist of one pilot symbol followed by data symbols. Among all possible AQAM schemes, continuous-rate continuous-power adaptation with instantaneous BER constraint is chosen, as in [8], for the analysis performed in next sections. This scheme preserves all degrees of freedom concerning rate-power adaptation, while achieving the target BER in any channel state. The rate adaptation (bits/symbol) and power adaptation that maximize the spectral efficiency subject to the instantaneous BER constraint are [2] where, the average transmitted power, and is the unit-step function, i.e., power transmission is disabled below the cutoff SNR. The instantaneous BER is kept constant above the cutoff SNR, thus, it is fixed to a target value 1 The choice of equal data and pilot symbol power is not necessarily optimal. (1) (2) 0090-6778/04$20.00 2004 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004 717 Fig. 1. AQAM system model, including the channel estimation subsystem. At the receiver, channel estimation is performed by the symbol pilots inserted at the transmitter [9], [10]. As shown in Fig. 1, the channel estimation subsystem required by the AQAM algorithm can be split into two parts, which provide estimates for the instantaneous SNR ( ) and average SNR ( ). The estimate can be obtained from a finite-impulse response (FIR) filtered version of samples of the channel complex envelope followed by a squared norm operation. 2 Following the analysis presented in [10], the conditional probability density function (PDF) is then derived, assuming the filter coefficients are chosen under the minimum mean-square error (MMSE) criterion Rayleigh fading can be determined by solving [2] (5) where and are the normalized instantaneous SNR and cutoff SNR, respectively. Although can be bounded and numerically calculated [13], it is not possible to find an exact closed-form solution. The cutoff SNR can be expressed as. An initial approximation for the cutoff function, within 1 db for the usual range of values of and, is given in [14] by (3) the zeroth-order modified Bessel function. The pa- represents the MMSE of (4) with rameter (6) where tighter closed-form approximations are also provided. The function is strictly monotonically increasing, and when, then. and depends on, the number of filter taps, and the normalized frame interval ( is the symbol interval). Note that is the normalized covariance matrix given by, is the -dimensional normalized covariance row vector where, and is the zeroth-order Bessel function. With regard to the estimate of the average SNR, [12] provides a simple model to characterize its statistics for different average power measurement methods. More specifically, the relative error is nearly Gauss distributed, with its mean easily driven to zero and its standard deviation within the 2 4 db range. III. APPROXIMATIONS FOR THE CUTOFF SNR All adaptive modulation performance parameters depend on the cutoff SNR. For each given value of, the cutoff SNR for 2 This estimator could be improved by considering the bias in the squared norm channel gain [11]. IV. PERFORMANCE ANALYSIS In this section, several expressions are obtained to evaluate the impact of the imperfect channel estimates and on the more relevant AQAM performance parameters: BER, spectral efficiency, average transmitted power, and outage probability. For each parameter, the analysis is split into two steps. First, the short-term error only is considered, and later the long-term error is included. Finally, numerical and simulation results show the relevance of the different error sources. A. Averaged BER Analytical results for the BER, shown in (7) (8) at the bottom of the next page, are derived as follows. Let us suppose that and. The instantaneous BER is no longer kept constant and equal to above the cutoff SNR. According to Fig. 1, two different impairments can be identified: rate-power selection is performed by using the noisy estimate and, when the transmitter starts to send data with this selection, the channel

718 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004 has evolved to a new state. Therefore, from (2), the instantaneous BER is now given by When the long-term error is included in the analysis, i.e.,, the cutoff SNR becomes where and represent the noisy and delayed instantaneous SNR ( ) and the current instantaneous SNR ( ), respectively. Note that when because there is no data transmission. From the standpoint of performance evaluation, the averaged BER above the cutoff SNR is used to quantify the variation of the instantaneous BER (9) (10) Consistently, when channel estimation is perfect, then. Evaluation of (10) only requires the identification of the joint PDF, since can be derived from it. In [8, p. 136, eq. (41)], the conditional PDF is obtained as a function of two parameters: the adaptation delay normalized to the Doppler shift, and the channel gain correlation coefficient. Hence, the joint PDF can be obtained if the following conditional probability chain is regarded: (13) and expectation over must be applied to (12a). Then, after considering the change of variable, we obtain (7), where the former is redefined as (14) and and as previously defined in (12b). Integration over and in (7) can be accurately approximated by the Gauss Laguerre and Gauss Hermite quadrature formulas, respectively. By this method, (8) is obtained, where and are the order of the Laguerre and Hermite polynomials, and the zeros of such polynomials, and and the associated weight factors [16, p. 223 224]. B. Spectral Efficiency, Average Transmitted Power, and Outage Probability A similar analysis can be carried out for these performance parameters. By integrating (11) with [15, p. 1182, eq. (109)], it is shown that the marginal PDF is exponential distributed with mean. Consequently, considering (1), it is straightforward to evaluate how the short-term error affects the spectral efficiency 3, average transmitted power, and outage probability (15) (11) Substituting (3) and [8, p. 136, eq. (41)] into (11), then (11) in (10), using twice the Laplace transform given in [15, p. 1182, eq. (109)] and applying the change of variable allows us to obtain (12a) (12b) (16) (17) where is the first-order exponential-integral function classically defined as in [17, p. xxxiv]. 3 The spectral efficiency penalty factor (L 0 1)=L caused by the insertion of pilot symbols is not included. (7) (8)

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004 719 Fig. 2. Impact of channel estimation error on the AQAM system performance parameters as a function of the MSE db, for different channel conditions and values of the long-term error standard deviation 1 db with BER =10 and T =5 1 10. Simulation results are also shown for M = 1, 2, 4, 8, and 16 filter taps. (a) Averaged BER in terms of the normalized adaptation delay for any 1 2 [0; 4]. (b) Relative variation of the spectral efficiency < (%) = 100 2 (hi 0)=. (c) Relative variation of the average transmitted power < (db) = 10 1 log hsi= S. (d) Relative variation of the outage probability < (%) = 100 2 (Pr f < g0pr f < g). Inclusion of the long-term error in (15) (17) yields (18) (19) (20) where,,, and are previously defined in Section IV-A. C. Numerical Results Summarized in Fig. 2 is the numerical evaluation of the set of analytical formulas (8), (18) (20) as a function of the MSE for different values of,, and. It is assumed that and. Simulation results are also superimposed in Fig. 2 for two purposes, to check the validity of these analytical results and to display the effect of the estimation filter complexity through the number of taps. To estimate, we consider the use of 1, 2, 4, 8, and 16 taps. The averaged BER plotted in Fig. 2(a) is only affected by the short-term error. It suffices to use and in the Gauss Laguerre and Gauss Hermite quadrature formulas to achieve an acceptable accuracy. For favorable channel conditions, e.g.,, it is observed how the BER does not tolerate a normalized delay greater than or an MSE above db if is considered as the criterion for the maximum admissible degradation. It can be pointed out that in the adverse channel conditions case, e.g.,, the system becomes less sensitive, thus, these limits relax to approximately for the normalized delay, and db for the MSE. As shown in Fig. 2(b) (d), the remainder performance parameters are mainly affected by the MSE and exhibit appreciable variations when the estimation noise is above db. To quantify these variations, the following metrics are defined: (21) (22) (23) In average terms, the estimation filter tends to underestimate the channel state, because, leading

720 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004 us to employ more robust constellations and lower transmitted power. Basically, the effect of the long-term error is only noticeable in the spectral efficiency and the average transmitted power when the channel conditions are adverse. The insensitivity of all performance parameters to the long-term error when is high can be easily explained through the properties of the cutoff function. When, from (14), it follows that independently of the long-term error standard deviation db. V. CONCLUSIONS In this letter, the impact of imperfect channel estimation on variable-rate variable-power QAM performance is studied. A set of new analytical expressions are derived that show the high sensitivity of the BER to both the estimation MSE and the system adaptation delay. The rest of performance parameters mainly depend on the estimation MSE. The long-term error associated with the average SNR estimation is also included in the analysis. Its impact is negligible on the BER and the outage probability, and moderate on the spectral efficiency and the average transmitted power. According to [8], similar results should be expected for practical discrete AQAM schemes [3], [4]. ACKNOWLEDGMENT The authors would like to thank Dr. E. Martos-Naya, Prof. M.-S. Alouini, and the anonymous reviewers for their useful comments. REFERENCES [1] A. J. Goldsmith and S.-G. Chua, Variable-rate variable-power MQAM for fading channels, IEEE Trans. Commun., vol. 45, pp. 1218 1230, Oct. 1997. [2] S. T. Chung and A. J. Goldsmith, Degrees of freedom in adaptive modulation: A unified view, IEEE Trans. Commun., vol. 49, pp. 1561 1571, Sept. 2001. [3] C. Kose and D. Goeckel, On power adaptation in adaptive signaling systems, IEEE Trans. Commun., vol. 48, pp. 1769 1773, Nov. 2000. [4] J. F. Paris, M. C. Aguayo-Torres, and J. T. Entrambasaguas, Optimum discrete-power adaptive QAM scheme for Rayleigh fading channels, IEEE Commun. Lett., vol. 5, pp. 281 283, July 2001. [5] B. Classon, K. Blankenship, and y. V. Desai, Channel coding for 4G systems with adaptive modulation and coding, IEEE Personal Commun. Mag., pp. 8 13, Apr. 2002. [6] H. Holm, Adaptive coded modulation performance and channel estimation tools for flat fading channels, Ph.D. dissertation, Norwegian Univ. Sci. and Technol. (NTNU), Dept. Telecommun., Trondheim, Norway, 2002. [7] H. Shengquan and A. Duel-Hallen, Combined adaptive modulation and transmitter diversity using long-range prediction for flat-fading mobile radio channels, in Proc. Global Telecomm. Conf., vol. 2, 2001, pp. 1256 1261. [8] M.-S. Alouini and A. J. Goldsmith, Adaptive modulation over Nakagami fading channels, Wireless Pers. Commun., vol. 13, pp. 119 143, May 2000. [9] J. K. Cavers, An analysis of pilot-symbol assisted modulation for Rayleigh fading channels, IEEE Trans. Veh. Technol., vol. 40, pp. 686 693, Nov. 1991. [10] X. Tang, M.-S. Alouini, and A. J. Goldsmith, Effect of channel estimation error on M-QAM BER performance in Rayleigh fading, IEEE Trans. Veh. Technol., vol. 47, pp. 1856 1864, Dec. 1999. [11] T. Ekman, Prediction of mobile radio channels: Modeling and design, Ph.D. dissertation, Uppsala Univ., Uppsala, Sweden, 2002. [12] A. J. Goldsmith, L. Greenstein, and G. Foschini, Error statistics of real-time power measurements in cellular channels with multipath and shadowing, IEEE Trans. Veh. Technol., vol. 43, pp. 439 446, Aug. 1994. [13] M.-S. Alouini and A. J. Goldsmith, Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques, IEEE Trans. Veh. Technol., vol. 48, pp. 1165 1181, July 1999. [14] J. F. Paris, M. C. Aguayo-Torres, and J. T. Entrambasaguas, Closed-form approximations for the cutoff SNR of variable-rate variable-power QAM in Rayleigh fading channels, Tech. Rep., Dept. Ing. de Comun., Univ. Málaga, Málaga, Spain, 2003. [15] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. New York: Dover, 1970. [16] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. San Diego, CA: Academic, 1984. [17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic, 1994.