2003 Conference on Information Sciences and Systems, The Johns Hopkins University, March 12{14, 2003 Adaptive Modulation with Adaptive Pilot Symbol Assisted Estimation and Prediction of Rapidly Fading Channels Xiaodong Cai and Georgios B. Giannakis1 Dept. of Electrical and Computer Engineering University of Minnesota 200 Union Street SE Minneapolis, MN 55455 e-mail:fcaixd,[email protected] Abstract | Adaptive modulation requires channel performance and flexibility when operating in changing en- state information (CSI) at both transmitter and re- vironments. Adaptive modulation for multi-antenna systems ceiver. In practice, CSI can be acquired at the re- with channel mean feedback was designed in [12], where it ceiver by inserting pilot symbols in the transmitted was shown that multi-antennae transmissions increase spec- signal. In this paper, we ¯rst analyze the e®ect chan- tral e±ciency, as well as relax the requirement on the channel prediction quality. While the e®ect of channel prediction error nel estimation and prediction errors have on bit er- has been considered in [10] and [12], perfect CSI is assumed ror rate (BER). Based on this analysis, we develop at the receiver. adaptive pilot symbol assisted modulation (PSAM) In this paper, we deal with adaptive pilot symbol assisted schemes that account for both channel estimation and modulation (PSAM) that accounts for both channel estima- prediction errors to meet the target BER. While pilot tion and prediction errors. As advocated in [4, 9], pilot sym- symbols facilitate channel acquisition, they consume bols are periodically inserted in the transmitted signal to fa- part of transmitted power and bandwidth, which in cilitate channel estimation and prediction at the receiver. Dif- turn reduces spectral e±ciency. With imperfect (and ferent from these non-adaptive PSAM schemes, we will adjust thus partial) CSI available at the transmitter and transmission parameters to maximize spectral e±ciency while receiver, two questions arise naturally: how often adhering to a prescribed (target) bit error rate (BER). Our should pilot symbols be transmitted? and how much goals are: i) to design adaptive PSAM schemes that take into account both channel estimation and prediction errors to meet power should be allocated to pilot symbols? We ad- the target BER; and ii) to optimize the spacing and power of dress these two questions by optimizing pilot param- pilot symbols to maximize spectral e±ciency. eters to maximize spectral e±ciency. The rest of this paper is organized as follows. Section II describes the system model, and studies the e®ect channel es- I. Introduction timation and prediction errors have on BER. In Section III, Channel-adaptive modulation is a promising technique to en- adaptive PSAM schemes are developed, and their average hance spectral e±ciency of wireless transmissions over fading BER performance is analyzed. Numerical results are pre- channels [3, 8, 11]. In adaptive systems, certain transmission sented in Section IV, and conclusions are drawn in Section V. T ¤ H parameters such as constellation size, transmitted power, and Notation: Superscripts , and stand for transpose, conjugate, Hermitian, respectively; E[ ] denotes expectation code rate, are dynamically adjusted according to the channel ¢ with the random variable within the brackets; and x repre- quality, which increases the average spectral e±ciency without b c wasting power or sacri¯cing error probability performance. sents the smallest integer less than x. Column vectors (ma- Channel state information (CSI) is required for the trans- trices) are denoted by boldface lower (upper) case letters; IN represents the N N identity matrix; and D(x) stands for mitter to adapt its parameters, and for the receiver to perform £ coherent demodulation. It has been shown that adaptive mod- the diagonal matrix with x on its diagonal. ulation with perfect CSI o®ers performance gains relative to non-adaptive transmissions [5, 8]. However, adaptive modu- II. System Model and Bit Error Rate lation schemes designed based on perfect CSI work well only A. System Model when CSI imperfections induced by channel estimation error The adaptive system under consideration is outlined in the and/or feedback delays are su±ciently small [1, 8]. For exam- block diagram of Fig. 1. A pilot symbol is inserted every L 1 ple, an adaptive transmitter relying on an error-free channel ¡ information bearing symbols, which results in the transmitted estimate to predict future channel values requires feedback frame structure shown in Fig. 2, where P and D denote pilot delay ¿ < 0:01=f , where f is the Doppler frequency [1]. d d and data (information) symbols, respectively. The discrete- The impact of channel prediction error on the performance of time equivalent baseband channel includes transmitter and re- adaptive coded modulation was studied in [10]. While it is ceiver ¯lters, time-selective frequency-flat fading e®ects, and necessary to employ reliable channel estimators and predic- additive white Gaussian noise (AWGN). At the receiver, a tors to minimize the e®ect of imperfect CSI, adaptive trans- channel estimator extracts the pilot signal, and estimates the mitters that account for CSI errors explicitly may have better channel periodically. Using the estimated channel supplied 1This work was supported by the ARL/CTA grant no. by the channel estimator, the demodulator performs coher- DAAD19-01-2-011. ent detection of the data symbols. The pilot symbols are also Adaptive Power Pilot Fading power allocation between pilot and data symbols, as well as Control Insertion AWGN Demodulation Modulation the pilot spacing L, will be optimized to maximize spectral ef- ¯ciency with a target BER. Since both channel estimation and Power Channel Selection Estimator prediction errors a®ect BER, to design the adaptive PSAM, Feedback we ¯rst need to analyze the BER in the presence of channel Delay Constellation Channel estimation and prediction errors. Selection Predictor B. BER in the Presence of Channel Estimation Errors Let h^(n; l) be the estimator of h(n; l), and ²(n; l) := h(n; l) Figure 1: Adaptive PSAM System Model ¡ h^(n; l) denote the channel estimation error. The quality of channel estimation is measured by the channel mean square 2 2 error (MSE) which is de¯ned as σ² (l) := E[ ²(n; l) ]. Given . D P D D . D D P D D . D D P D D . 2 j j σ² (l) and a realization of the channel estimator h^(n; l) = L L h^0(n; l), our goal in this subsection is to derive the condi- tional BER P (e h^0(n; l)) for BPSK and square M-QAM. In j Figure 2: Transmitted frame structure Section II-C, based on P (e h^0(n; l)), we will derive the BER j in the presence of both channel estimation and prediction er- rors, which will be used later in Section III in adapting PSAM used by the channel predictor to estimate the channel ¿ sec- to meet the target BER. onds ahead, where ¿ is the feedback delay that accounts for We consider the linear MMSE channel estimator, and refer both actual transmission delay, and processing time at the re- the reader to [4] for the detailed derivation. This estimator ceiver and transmitter. Based on the predicted channel and uses Ke pilot samples, yp(n Ke=2 ); : : : ; yp(n+ (Ke 1)=2 , the quality of channel estimation and prediction, a constella- L¡1 ¡b c b ¡ c to estimate h(n; l) l=1 . De¯ning s := [sp(n Ke=2 ); : : : ; f g T ¡ b c tion size is selected and fed back to the transmitter. There, sp(n + (Ke 1)=2 )] , h := [h[(n Ke=2 ); 0]; : : : ; h[(n + b ¡ T c H ¡ b c ¤ the adaptive modulator maps incoming binary symbols to the (Ke 1)=2 ); 0]] , R := E[hh ], rl := E[hh (n; l)], the b ¡ c selected constellation. If the instantaneous transmit power is linear MMSE channel estimator for h(n; l) is given by wl = allowed to vary, the transmit power level is determined adap- ¤ ¡1 p pD(s)RD (s) + N0IKe D(s)rl [4], which does not tively along with the constellation size, and is also fed back to E E depend on n. The estimated channel is obtained as h^(n; l) = the transmitter. If the transmit power is constant, the system pH ¡ ¢ wl y(n), where y(n) := [yp(n Ke=2 ); : : : ; yp(n + (Ke does not include the dashed line blocks of power selection and T ¡ b c b ¡ 1)=2 ] ; and the channel MMSE can be written as [4] power control depicted in Fig. 1. c 2 H ¤ Let r(n; l) denote the received signal sampled in the lth σ² (l) = 1 prl D (s)wl; (3) symbol period of the nth frame, or equivalently at t = (nL + ¡ E p2 l)T , where T is the symbol (equal to the sampling) period. which con¯rms that indeed σ² (l) does not depend on n. The received samples corresponding to the pilot symbols can We are interested in the BER of square M-QAM and BPSK, ^ ^ be written as given h(n; l) = h0(n; l). If the channel were known per- fectly at the receiver, the decision variable for s(n; l) with yp(n) := r(n; 0) = ph(n; 0)sp(n) + ´(n; 0); (1) symbol-by-symbol maximum likelihood detection would be: E z(n; l) = yd(n; l)=(p dh(n; l)). With the estimated channel, p E and similarly, those corresponding to the data symbols are: we can replace h(n; l) with h^(n; l) in the decision variable, although this detection rule is no longer optimum. Since yd(n; l) = p dh(n; l)s(n; l) + ´(n; l); l [1; L 1]; (2) E 2 ¡ the channel is a Gaussian random process with zero-mean, h^(n; l) and ²(n; l) are zero-mean Gaussian random variables.