Adaptive Modulation with Adaptive Pilot Symbol Assisted Estimation and Prediction of Rapidly Fading Channels

2003 Conference on Information Sciences and Systems, The Johns Hopkins University, March 12–14, 2003

Xiaodong Cai and Georgios B. Giannakis1 Dept. of Electrical and Computer Engineering University of Minnesota 200 Union Street SE Minneapolis, MN 55455 e-mail:{caixd,georgios}@ece.umn.edu

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 first analyze the effect chan- tral efficiency, as well as relax the requirement on the channel prediction quality. While the effect 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 efficiency. 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 efficiency 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 efficiency. eters to maximize spectral efficiency. The rest of this paper is organized as follows. Section II describes the system model, and studies the effect 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 efficiency 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 efficiency without b c wasting power or sacrificing 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 offers 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 sufficiently 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 τ < 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 filters, time-selective frequency-flat fading effects, and necessary to employ reliable channel estimators and predic- additive white Gaussian noise (AWGN). At the receiver, a tors to minimize the effect 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 efficiency with a target BER. Since both channel estimation and Power Channel Selection Estimator prediction errors affect BER, to design the adaptive PSAM, Feedback we first 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 hˆ(n; l) be the estimator of h(n; l), and ²(n; l) := h(n; l) Figure 1: Adaptive PSAM System Model − hˆ(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 defined as σ² (l) := E[ ²(n; l) ]. Given . . . D P D D . . . D D P D D . . . D D P D D . . . 2 | | σ² (l) and a realization of the channel estimator hˆ(n; l) = L L hˆ0(n; l), our goal in this subsection is to derive the conditional BER P (e hˆ0(n; l)) for BPSK and square M-QAM. In | Figure 2: Transmitted frame structure Section II-C, based on P (e hˆ0(n; l)), we will derive the BER | in the presence of both channel estimation and prediction errors, which will be used later in Section III in adapting PSAM used by the channel predictor to estimate the channel τ sec- to meet the target BER. onds ahead, where τ 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 . Defining s := [sp(n Ke/2 ), . . . , { } 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 hˆ(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 confirms 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, ˆ ˆ 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) + η(n; 0), (1) symbol-by-symbol maximum likelihood detection would be: E z(n; l) = yd(n; l)/(√ 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 hˆ(n; l) in the decision variable, although this detection rule is no longer optimum. Since yd(n; l) = √ dh(n; l)s(n; l) + η(n; l), l [1, L 1], (2) E ∈ − the channel is a Gaussian random process with zero-mean, hˆ(n; l) and ²(n; l) are zero-mean Gaussian random variables. where p and d represent, respectively, the power per pilot E E L−1 Because the orthogonality principle renders ²(n; l) uncorre- and data symbol; sp(n) is the pilot symbol and s(n; l) { }l=1 ˆ ˆ ˆ are the data symbols of the nth frame. We assume that lated with h(n; l), given h(n; l) = h0(n; l), the actual chan- 2 2 nel h˘(n; l) = hˆ0(n; l) + ²(n; l), is Gaussian distributed with E[ s(n; l) ] = sp(n) = 1; η( ) is complex AWGN with | | | | · ˆ 2 zero-mean, and variance N /2 per dimension; and the chan- mean h0(n; l), and variance σ² (l). Hence, given the realiza- 0 ˆ nel h(n; l) is a stationary complex Gaussian random random tion h0(n; l) of the channel estimator, the decision variable for 2 s(n; l) can be written as process with zero-mean, and variance σh = 1. Hence, the average signal-to-noise ratio (SNR) per pilot (data) symbol is √ dh˘(n; l)s + η s² η p/N0 ( d/N0). The autocorrelation function of the channel z(n; l) = E = s + + , (4) E E √ dhˆ0 hˆ0 √ d hˆ0 is determined by the Doppler spectrum. Unlike the PSAM E E in [4], where pilot and data symbols have the same transmit where we omitted the time indices of s(n; l), ²(n; l), hˆ (n; l), power, and similar to [9], we allow the pilot and data sym- 0 and η(n; l), for notational brevity. Since ² and η are indepen- bols to be transmitted with different power. The PSAM in [4] dent, the SNR of (4) can be expressed as uses pilot symbols to enable coherent demodulation; and the ergodic capacity of time-selective fading channels with PSAM 2 2 s d hˆ0(n; l) is maximized in [9], without CSI available at the transmitter. γ1(n; l) = | | E | 2 | 2 . (5) N0 + dσ² (l) s The adaptive PSAM we will introduce here uses the es- E | | timated channel to perform coherent demodulation, and the For BPSK and 4-QAM, we have s = 1; and thus, BER can | | predicted channel to adjust the constellation size and trans- be calculated using the SNR in (5). For M-QAM with M > 4, mitted power, i.e., adaptive PSAM exploits partial CSI at we have s = 1; and the SNR in (5) can not be directly used to | | 6 the transmitter. Given a total transmitted power budget, the calculate BER. However, since E[ s 2] = 1 and s 2 < 2.65 for | | | | 0 ˇ 2 10 BER for BPSK and M-QAM, as a function of h0(n; l), σ² (l), Approximate BER 2 and σε (l). Using this BER expression, the adaptive PSAM in Simulated BER Section III will choose transmission parameters to maximize −1 10 spectral efficiency, while satisfying the target BER. T Defining sp := [sp(n D), . . . , sp(n D Kp + 1)] , − T− − H −2 hp := [h(n D; 0), . . . , h(n D Kp+1; 0)] , Rhp := E[hphp ], 10 − ∗ − − rp(l) = E[hph (n; l)], the MMSE of the predicted channel is 2 H ∗ ∗ written as σε (l) = 1 prp (l)D (sp) pD(sp)Rhp D (sp) + −3 −1 − E E 2 2 10 N0IKp D(sp)rp(l), where similar to σ² (l), σε (l) does not

BER ¡ depend on n. As shown in [10], the length of the channel M=4 ¢ −4 predictor, Kp, should be chosen large enough to achieve a 10 suﬃciently small MSE with a moderate delay. Similar to M=2 M=16 M=64 M=256 the estimated channel hˆ(n; l), the predicted channel hˇ(n; l) 2 −5 is Gaussian distributed with zero-mean, and variance σ (l) = 10 hˇ σ2 σ2(l) = 1 σ2(l). h − ε − ε Considering the relationship between the estimated and −6 10 predicted channels, we have

0 5 10 15 20 25 30 35

¢¡¢£ ¤ £ ¦¨§ © ¥ hˆ(n; l) = h(n; l) ²(n; l) − (9) ˇ 2 = h(n; l) + ε(n; l) ²(n; l). Figure 3: Conditional BER, σ² (l) = N0/Ed = 0.01 − By the orthogonality principle, ε(n; l) and hˇ(n; l) are uncorrelated. The correlation between ²(n; l) and hˇ(n; l) is M 256, our simulations show that the BER of M-QAM (4 < ˇ∗ ∗ ∗ 2 ≤ E[²(n; l)h (n; l)] = E[²(n; l)(h (n; l) ε (n; l))] = σ² (l) M 256) with channel estimation error is well approximated ∗ ∗ − − ≤ E[²(n; l)ε (n; l)]. Since E[²(n; l)ε (n; l)] σ²(l)σε(l), we by the BER formula using an average SNR given by | | ≤ have ˆ 2 d h0(n; l) ˇ∗ γ (n; l) = . (6) σ²(l) σε(l) E[²(n; l)h (n; l)] σ²(l) + σε(l) 2 E | |2 . (10) N0 + 1.3 dσ² (l) − E σhˇ (l) ≤ σ²(l)σhˇ (l) ≤ σhˇ (l) Considering both BPSK and M-QAM, we will henceforth use In practice, σ (l) and σ (l) are very small. And because the SNR ² ε ˆ 2 σhˇ (l) is close to one, (10) implies that the correlation coef- d h0(n; l) ˇ γ(n; l) := E | 2| , (7) ficient between ²(n; l) and h(n; l) is very small. Hence, we N0 + g dσ (l) E ² can assume that ²(n; l) and hˇ(n; l) are uncorrelated. Given where g = 1 for BPSK and 4-QAM, and g = 1.3 for M-QAM hˇ(n; l) = hˇ0(n; l), hˆ(n; l) is Gaussian distributed with mean with M > 4. ˇ ˇ N h0(n; l). Since h(n; l) is uncorrelated with ε(n; l) and ²(n; l), Suppose that N different constellations of size Mi i=1 are the conditional variance of hˆ(n; l) is σ˜2 := E[ ε(n; l) ²(n; l) 2]. { } hˆ | − | available, and Gray coding is used to map information bits to Based on the definition of ε(n; l) and ²(n; l), σ˜2 can be cal- ˆ ˆ hˆ QAM symbols. Then, given h(n; l) = h0(n; l), the BER of the culated using the channel autocorrelation function. However, ith constellation can be approximately calculated by [5] 2 2 2 since (σ²(l) σε(l)) σ˜hˆ (σ²(l) + σε(l)) , we will use 2 2 − 2 ≤ ≤ cγ(n; l) σ˜hˆ = σ² (l) + σε (l) in calculating the BER, which amounts to P (ei hˆ0(n; l)) 0.2 exp , (8) | ≈ − Mi 1 assuming that ²(n; l) and ε(n; l) are uncorrelated. µ − ¶ Given hˇ(n; l) = hˇ0(n; l), the amplitude of hˆ(n; l) follows where γ(n; l) is given in (7), c = 3.2/3 for BPSK, and c = ˇ 2 2 a Rice distribution with Rician factor K = h0(n; l) /σ˜hˆ . 1.6 for square M-QAM. Fig. 3 compares the simulated BER | | Letting p( hˆ0 hˇ0) denote this conditional probability density 2 | | with the approximate BER calculated from (8), when σ² (l) = function (pdf), the conditional BER of s(n; l) can be expressed ¯ N0/ d = 0.01. The approximation is within 2dB for M-QAM, as E ¯ ∞ when the target BER < 10−2. P (ei hˇ0) = P (ei hˆ0 )p( hˆ0 hˇ0)d hˆ0 , (11) | | | | | | | C. BER in the Presence of Channel Estimation and Prediction Z0 ¯ ˆ¯ ˇ Errors where we omitted the time indices¯ of h0¯(n; l) and h0(n; l). From (8), it is seen that p(ei hˆ0(n; l) ) = p(ei hˆ0(n; l)); thus, Suppose that the length of the linear MMSE channel pre- | | | we substitute (8) into (11), and obtain [6] diction filter is Kp, and the feedback delay is τ = DLT , ¯ ¯ where D is a positive integer, and we assume that the de- 2 ai(l)Ed|hˇ0(n;l)| 0.2 exp 2 ai(l)E σ˜ +1 lay is a multiple of LT for notational simplicity. The channel − d hˆ L ˇ −1 P (ei h0(n; l)) 2 , (12) predictor estimates h(n; l) l=1 , using Kp past pilot samples | ≈ a¡i(l) dσ˜ + 1 ¢ { } hˆ Kp−1 ˇ E yp(n D k) k=0 . Letting h(n; l) be the predicted chan- { − − } 2 2 2 nel, the prediction error is given by ε(n; l) = h(n; l) hˇ(n; l), where ai(l) := c/[(Mi 1)(N0 + g dσ² (l))]. If σε (l) = σ² (l) = 2 − 2 − E and the MSE of the predicted channel is σ (l) := E[ ε(n; l) ]. 0, then hˇ0(n; l) = hˆ0(n; l) = h0(n; l); and it is seen from (8) ε | | Given the predicted channel hˇ(n; l) = hˇ0(n; l), our adaptive and (12) that P (ei hˇ0(n; l)) = P (ei h0(n; l)), as expected. The | | PSAM will adjust the constellation size of s(n; l) based on BER P (ei hˇ0(n; l)) is mainly determined by the exponential | hˇ0(n; l), and the quality (MSE) of channel estimation and pre- term in (12). To see how the channel prediction and estima- diction. To this end, in this subsection, we will express the tion errors affect the BER, we express the exponent in (12) as in a frame, this also reduces the feedback rate, since hˇ0(n; l) ˇ 2 ˇ 2 ai(l) d h0(n; l) c d h0(n; l) is almost identical l, and the same constellation can be used E | 2 | = E | | G, (13) L−1 ∀ ai(l) dσ˜ + 1 (Mi 1)N hˆ 0 × for s(n; l) l=1 . We will henceforth omit the time indices in E − { ¯ } where γˇi(l), γˇ(l), ai(l), bi(l), and di(l), for notational simplicity.

N0 A. Constant Power G = 2 2 N0 + dσ² (l)(g + c/(Mi 1)) + c dσε (l)/(Mi 1) Let γˇ denote the switching threshold for the ith constella- E − E − i (14) tion, which will be found later to meet a target BER. In this reflects the performance loss caused by channel estimation subsection, we assume that the transmit power is constant, and prediction. From (14), we observe that the same chan- and no data is transmitted when γˇ < γˇ1. Then, the actual nel prediction error incurs less performance loss for large transmit power is [5] 2 constellation sizes, because the term c dσ (l)/(Mi 1) re- E ε − ¯ duces as Mi increases. On the other hand, the same chan- d d = E = ¯d exp(γˇ1/γˇ¯). (18) E ∞ p(γˇ)dγˇ E nel estimation error causes almost the same performance loss γˇ1 for all M-QAM and BPSK constellations, since the term 2 Suppose that the targetR BER is equal to B. Using (15) and dσ² (l)(g+c/(Mi 1)) changes very slowly when Mi increases E − letting P (e γˇi) = B, we can find the switching thresholds beyond a moderately large value. | Let ¯d denote the average transmit power of the data ln(5biB) E ¯ ˇ 2 γˇi = , i = 1, . . . , N. (19) symbols. Defining γˇ(l) := d h0(n; l) /N0, bi(l) := − di 2 E | | ¯ ai(l) dσ˜hˆ (l) + 1, and di(l) = (ai(l) dN0)/( dbi(l)), we can E E E Letting ki := log2(Mi), and P (ki) := P (γˇi < γˇ < γˇi+1) = also write P (ei hˇ0(n; l)) in (12) as | exp( γˇi/γˇ¯) exp( γˇi+1/γˇ¯), where γˇN+1 := , the spectral − − − ∞ 0.2 exp( di(l)γˇ(l)) efficiency can be expressed as P (ei γˇ(l)) − , (15) | ≈ bi(l) N 1 S = (1 ) kiP (ki) − L where we omitted the frame index n for notational brevity. i=1 2 2 Letting γˇ¯(l) := ¯dσ (l)/N0 = ¯d(1 σ (l))/N0, the pdf of X (20) E hˇ E − ε N γˇ(l) can be expressed as 1 γˇ1 γˇi = (1 )[k1 exp( ) + (ki ki−1) exp( )]. − L − γˇ¯ − − γˇ¯ i=2 exp( γˇ(l)/γˇ¯(l)) X p(γˇ(l)) = − . (16) γˇ¯(l) It is evident from (17) that the MSE of the estimated channel depends on L and α. Similarly, the MSE of the predicted channel is also a function of L and α. Hence, the switching III. Adaptive PSAM thresholds, as well as the spectral efficiency depend on L and Let be the total average power including both pilot and data α. Numerical techniques can be used to find the optimum E power. Clearly, ¯d = αL /(L 1), and p = (1 α)L , where L and α to maximize S. Specifically, the maximum value E E − E − E of L can be found as Lmax = 1/(2fdT ) [4, 9]. For each α determines the power allocation between data and pilot b c L [2, Lmax], we have the following optimization problem to symbols. Note that α = (L 1)/L corresponds to equal power ∈ − be solved for the optimum α and d: allocation. The channel MSE can be explicitly expressed as E a function of α. Let ui be the ith eigenvector of R, and λi max S(α, d) be the corresponding eigenvalue. Using the constant modulus α,Ed E 2 property of pilot symbols, we can write σ² (l) in (3) as 0 < α < 1 (21) subject to ln(5b1B) K K (L 1) d = αL exp( ¯ ), e H 2 e H 2 ( d1γˇ 2 p ui rl ui rl (1 α)Lγ¯ − E E − σ² (l) = 1 E | | = 1 | | − , (17) − pλi + N0 − (1 α)Lγ¯λi + 1 where the last constraint follows from (18), and we explicitly i=1 E i=1 − X X write S as a function of α, and d. Numerical search based E where γ¯ := /N0 is the average SNR. Since R depends on L, on e.g., sequential quadratic programming (SQP) [7], can be 2 E 2 so does σ² (l). The MSE of the predicted channel, σε (l), can used to solve this optimization problem with starting point also be written in a similar formula. α = (L 1)/L, d = ¯d. After solving (21) for each L, the − E E Based on the feedback, our goal in this section is to adapt optimum L and the maximum spectral efficiency can be found the constellation size Mi, and possibly transmitted power per by searching over all possible values of L. If the pilot and data constellation, as well as the spacing L, and the power level (α) symbols have the same transmit power, we let α = (L 1)/L; − of PSAM, so as to maximize spectral efficiency, while adhering and the same procedure can be used to find the maximum to the target BER. The BER in (15) depends on the predicted spectral efficiency, as well as the optimum L. channel value hˇ0(n; l), as well as the channel MSE σ²(l) and σε(l). As we showed in [2], when Ke 20, the MSE of the B. Discrete Power ≥ estimated channel is almost identical for any l. The MSE of Here, we allow symbols chosen from different constellation the predicted channel hˇ(n; l) is largest when l = L 1. Hence, sizes to be transmitted with different power levels. For the − the receiver can use σ²(L 1) and σε(L 1) in calculating the perfect CSI case, it has been shown that the spectral effi- − L−1 − switching thresholds for s(n; l) . With a small penalty ciency of discrete power transmissions is within 2dB of those { }l=1 in spectral efficiency, this reduces computational complexity with continuous power adaptation [8], and is higher than that to 1/L relative to the case where the switching thresholds of of constant power transmission [5]. Here, the advantage of dis- L−1 L−1 s(n; l) l=1 are, respectively, calculated using σ²(L 1) l=1 crete power adaptation relative to constant power transmis- { } L−1 { − } and σε(L 1) . When the channel changes very slowly sion may be even larger, because discrete power adaptation { − }l=1 6 6 Perfect CSI Perfect CSI Optimal power and L Optimal power and L Equal power, optimal L Equal power, optimal L 5 Equal power, L=10 5 Equal power, L=10

4 4

3 3 Capacity Capacity

2 2 Spectral Efficiency (bps/Hz) Spectral Efficiency (bps/Hz) Adaptive PSAM Adaptive PSAM 1 1

0 0 5 10 15 20 25 30 5 10 15 20 25 30 SNR (dB) SNR (dB)

Figure 4: Spectral eﬃciency of constant power adaptive Figure 5: Spectral eﬃciency of discrete power adaptive modulation modulation increases the probability of using large constellations, and as The average BER for the ith constellation can be written corroborated by (14), the same channel prediction error incurs as γˇ smaller performance loss for larger constellations. i+1 P (ei) = P (ei γˇ)p(γˇ)dγˇ. (24) Letting di be the transmit power of the ith constellation, γˇ | E Z i the switching threshold γˇi can be calculated from (19) with The overall average BER is then computed as the ratio of the d being replaced by di. Since the average transmit power of average number of bits in error over the total average number E ¯E the data symbols is d, we must have of transmitted bits [1, 5] E N N i=1 kiP (ei) diP (ki) = ¯d. (22) P (e) = N . (25) E E kiP (ki) i=1 Pi=1 X Based on the conditionalPBER in (15) and the pdf of the For each L, we can formulate the following optimization prob- N predicted SNR in (16), we can approximately calculate P (ei) lem to be solved for the optimum α and di : {E }i=1 from (24) as

max S(α, d1, . . . dN ) N 0.2 βiγˇi βiγˇi+1 α,{Edi}i=1 E E P (ei) exp exp , (26) ≈ biβi − γˇ¯ − − γˇ¯ 0 < α < 1 · µ ¶ µ ¶¸ (23) di > 0, i = 1, . . . , N where βi := γˇ¯di + 1. Then, the average BER in (25) can be subject to  E N di[exp( γˇi/γˇ¯) exp( γˇi+1/γˇ¯)] approximately calculated by substituting (26) into (25).  i=1 E − − −  = αL /(L 1), P E − IV. Numerical Results  where the last equation in the constraints follows from (22), We consider an adaptive modulation system which employs N and we explicitly write S as a function of α, and di . This 4 different square M-QAM constellations corresponding to {E }i=1 optimization problem can also be solved by numerical search 2 (4-QAM), 4 (16-QAM), 6 (64-QAM), and 8 (256-QAM) based on e.g., sequential quadratic programming (SQP) [7], bits/symbol, in addition to BPSK. The carrier frequency is using α = (L 1)/L and di = , i, as the starting point. f = 2GHZ, and the mobile velocity is v = 108km/hr. This − E E ∀ c Similar to the constant power case, the optimum L and the results in a Doppler spread fd = 200Hz. We assume that the maximum spectral efficiency can be found by searching over channel has a Jakes’ Doppler spectrum. The symbol rate is all possible values of L, using the solution to the optimization 200ksps, which corresponds to a symbol period of 5µs. The −3 problem (23). normalized Doppler spread is thus fdT = 10 . We consider a delay τ = 0.2/fd, and choose the length of the channel estima- C. Average BER Performance Analysis tion (prediction) filter to be Ke = 20 (Kp = 250). The spec- Since the switching threshold γˇi is chosen such that the BER tral efficiency of constant power adaptive PSAM is depicted of the ith constellation meets the target BER, and the same in Fig. 4. It is seen that the scheme with optimum power allo- constellation is used in the interval [γˇi, γˇi+1), the actual av- cation has about 1dB advantage over the equal power scheme erage BER is expected to be less than the target BER. In at SNR = 15dB. We also show the channel capacity with this subsection, we will analyze the average BER performance perfect CSI, as well as lower bounds on channel capacity with of the adaptive PSAM schemes developed in Sections III-A estimated channels, which were derived in [9]. With optimum and III-B. training, channel capacity is very close to that with perfect −5 10 [2] X. Cai and G. B. Giannakis, “Adaptive PSAM account- Constant power IEEE Discrete power ing for channel estimation and prediction errors,” Trans. Wireless Commun., submitted, Feb. 2003. [3] J. K. Cavers, “Variable-rate transmission for Rayleigh fading channels,” IEEE Trans. Commun., vol. 20, pp. 15– 22, Feb. 1972. [4] J. K. Cavers, “An analysis of pilot symbol assisted mod- −6 10 ulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, pp. 686–693, Nov. 1991.

Average BER [5] 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. [6] D. Divsalar and M. K. Simon, “The design of trellis coded MPSK for fading channels: performance criteria,” IEEE

−7 Trans. Commun., vol. 36, pp. 1004–1012, Sept. 1988. 10 5 10 15 20 25 30 SNR (dB) [7] R. Fletcher, Practical Methods of Optimization, Vol. 2, Constrained Optimization. John Wiley and Sons, 1980. Figure 6: Average BER [8] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable- power MQAM for fading channels,” IEEE Trans. Com- mun., vol. 45, pp. 1218–1230, Oct. 1997. CSI, because only channel estimation error affects capacity. [9] S. Ohno and G. B. Giannakis, “Average-rate optimal On the other hand, both channel prediction and estimation er- PSAM transmissions over time-selective fading chan- rors degrade the performance of adaptive PSAM. Hence, adap- nels,” IEEE Trans. Wireless Commun., vol. 1, pp. 712– tive PSAM with optimally estimated channels loses consider- 720, Oct. 2002. ably spectral efficiency compared to that with perfect CSI. Fig. 5 shows the spectral efficiency of discrete power adaptive [10] G. E. Øien, H. Holm, and K. J. Hole, “Impact of im- PSAM. Comparing Fig. 4 with Fig. 5, we infer that the dis- perfect channel prediction on adaptive coded modula- crete power adaptive PSAM has larger spectral efficiency than tion performance,” IEEE Trans. Veh. Technol., submit- the constant power scheme, as expected. From Fig. 5, we see ted June 2002; see also Proc. of Int. Symp. Advances in that the optimum power allocation has higher spectral effi- Wireless Commun., Victoria, BC Canada, Sept. 2002, pp. ciency than the equal power allocation scheme. Fig. 6 shows 19–20. the average BER of both constant and discrete power adaptive [11] W. T. Webb and R. Steele, “Variable rate QAM for mo- PSAM. We observe that the average BER is much lower than bile radio,” IEEE Trans. Commun., vol. 43, pp. 2223– the target BER, even though the feedback delay, τ = 0.2/fd, 2230, July 1995. is relatively large. On the other hand, the adaptive modu- [12] S. Zhou and G. B. Giannakis, “Adaptive modulation lation schemes treating the predicted channel as perfect CSI for multi-antenna transmissions with channel mean feed- may not be able to meet the target BER with such a large back,” in Proc. of ICC, Anchorage, Alaska, May 2003. feedback delay. This clearly shows the advantage of our adaptive PSAM that accounts for both channel estimation and prediction errors.

V. Conclusions We have studied adaptive transmission systems with pilot symbol assisted estimation and prediction of rapidly fading channels. The eﬀect of channel estimation and prediction errors on BER was investigated for BPSK and square M-QAM. Adaptive pilot symbol assisted modulation schemes that account for both channel estimation and prediction errors were developed; and their BER performance was analyzed. The spacing between two consecutive pilot symbols and the power allocation between pilot and data symbols were optimized to maximize spectral eﬃciency.

References

[1] M. S. Alouini and A. J. Goldsmith, “Adaptive modulation over Nakagami fading channels,” Kluwer Journal on Wireless Commun., vol. 13, pp. 119–143, May 2000.