Adaptive Modulation for MIMO Beamforming under ... - IEEE Xplore

Adaptive Modulation for MIMO Beamforming under Average BER Constraints and Imperfect CSI José F. Paris

Andrea J. Goldsmith

Dept. of Ingenier´ıa de Comunicaciones University of Málaga, SPAIN Campus de Teatinos, 29071 [email protected]

Dept. of Electrical Engineering Stanford University Stanford CA, 94305 [email protected]

Abstract— Adaptive modulation schemes for fading channels are usually required to fulfill certain long-term average BER targets. However, for simplicity and mathematical tractability, these schemes are often designed by fixing the short-term instantaneous BER to the target value. In this paper, the analysis and design of variable-rate variable-power QAM schemes with average BER constraints are tackled for a MIMO beamforming system with MRC and imperfect CSI. The SISO system is considered as a special case of these more general results. Approximate closed-form policies are derived for continuous rate and power adaptation which are compared to fully discrete policies designed numerically. Closed-form expressions for the ASE of the proposed policies are derived and evaluated. Our results indicate that imperfect CSI and discrete rate and power constraints do not significantly degrade performance.

I. I NTRODUCTION Wireless communication systems require higher data rates to support new data-intensive services. Many techniques have been proposed to increase data rates in wireless systems without requiring additional power or bandwidth. Within this context, two of the most promising and powerful techniques are adaptive modulation and multiple-input multiple-output (MIMO) systems [1]. Among the many possible degrees of freedom for adaptive modulation, in this work we focus on uncoded QAM schemes that can adapt their rate and power to the channel state in order to maximize average spectral efficiency (ASE) [2]. We assume these schemes can have an average or instantaneous BER constraint and a discrete or continuous rate and power adaptation associated with each constellation. We will call this wide group of schemes adaptive QAM (A-QAM) with the following nomenclature. We say an A-QAM scheme is XY-Z-L for X and Y representing the type of variation for rate (equivalently, constellation size) and power, respectively. Three options are possible for the variation denoted by X and Y: ’C’ (Continuous), ’D’ (Discrete) and ’K’ (Constant). The Z corresponds to the type of BER constraint, which can be ’I’ (Instantaneous) or ’A’ (Average). Finally, for discretepower schemes, L is the allowed number of power levels per constellation. Imperfect channel state information (CSI) can adversely impact adaptive modulation performance [3]-[7]. In [4] the impact of noisy and delayed channel estimates on the performance of continuous A-QAM (CC-I) is evaluated. In [5] the impact of noisy channel prediction on the BER of a coded

discrete A-QAM scheme (DK-I) with antenna diversity is analyzed. In [6] three types of A-QAM schemes (DC-I, DK-I and DK-A) are designed to account for prediction errors. A similar design philosophy to deal with prediction errors is used in [7] for discrete A-QAM schemes (DK-I and DD-I-1). Most of these schemes are designed for an instantaneous BER constraint (I-BER) for simplicity and mathematical tractability, leading to a final average ASE performance inferior to that achievable when long-term BER (A-BER) targets are required [8]. It is well-known that MIMO systems, where there are multiple antennas at the transmitter and the receiver, can dramatically increase ASE [9]. Moreover, adaptive modulation and MIMO can be combined to leverage both of their potentials [10]-[15]. Although sub-optimal from a MIMO capacity point of view, MIMO beamforming with maximal ratio combining (MRC) provides increased robustness through diversity in exchange for a lower ASE over MIMO multiplexing techniques [16]. In addition, MIMO beamforming can be easily combined with adaptive modulation since it can be reduced to an equivalent SISO channel [15]. The focus of this paper is on adaptive modulation for MIMO beamforming under different constraints and with imperfect CSI. Adaptive modulation for MIMO multiplexing under imperfect CSI is addressed in a companion paper [17]. For MIMO beamforming we determine adaptation policies under a broad set of constraints and evaluate their ASE in closed form. These results indicate that imperfect CSI does not lead to significant degradation in ASE, even with low complexity estimation techniques. In addition, practical discrete-rate discrete-power adaptation is within 1 dB of a fully continuous scheme. Thus, adaptive modulation in a MIMO beamforming system achieves near-optimal performance under practical design constraints. The remainder of this paper is organized as follows. Section II describes the system model. In Section III, the continuous and discrete rate-power adaptation policies are obtained, with their performance analyzed in Section IV. Finally, conclusions are provided in Section V. II. S YSTEM M ODEL The system model for MIMO beamforming with MRC is shown in Fig. 1. The following channel model is assumed. We consider NT ≥ 1 transmit antennas and NR ≥ 1 receive antennas. With only a single antenna at the transmitter

1312 1-4244-0355-3/06/$20.00 (c) 2006 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

x

z (t )

Input Data

QAM MAPPING

H

y

TX

RX MRC

B EAMFORMING PILOT INSERTION

r (t )

Flat Fading Channel

X

Transmitter

Fig. 1.

Output Data

DETECTION

AGC Pilot Extraction

CHANNEL PREDICTION

Hˆ

vˆ Chosen Rate-Power

QAM

ADAPTATION PROCESS Feedback Channel

λˆ

Hˆ → λ,ˆ vˆ Receiver

System model for MIMO Beamforming with MRC and imperfect CSI.

and receiver, this reduces to a SISO channel. Channel gain is modeled by the NR × NT complex matrix H, so that each entry Hij denotes the channel gain between the jth transmit and the ith receive antennae. All channels exhibit frequency-flat slowly time-varying fading. The entries Hij are assumed independent and identically distributed (i.i.d.), zeromean, unity-variance, complex circularly symmetric Gaussian random variables (RVs), i.e. Hij ∼ CN (0, 1) where the symbol ∼ means statistically distributed as. Channel noise is modeled by the NR -dimensional complex vector n that is additive, Gaussian and white both in space and time, i.e. the entries ni are i.i.d. RVs ∼ CN (0, σn2 ). The received signal is expressed as y = Hx + n,

(1)

where y is an NR -dimensional complex vector and x is the transmitted NT -dimensional complex vector. ˆ is obtained by a MIMO exThe predicted channel matrix H tension of classical pilot symbol assisted modulation (PSAM) with optimal Wiener filtering. A detailed description of this technique can be found in [15]. In short, the data stream is parsed into blocks of length P , and NT known pilot symbols are inserted per block. Thus, the ASE penalty factor for MIMO channel prediction is (P − NT )/P . Orthogonal signatures of length NT decouple the MIMO estimation into NT single transmit-antenna problems, followed by separate channel estimation at each receiver branch. This initial channel estimate is improved by further optimal Wiener FIR filtering ˆ The prediction process can be expressed by the to obtain H. following imperfect CSI model ˆ i,j i.i.d. RVs ∼ CN (0, 1 − χ) H ˆ + Ξ, ˆ (2) H=H with ˆ i,j i.i.d. RVs ∼ CN (0, χ) Ξ

where the minimum mean square error (MMSE) χ includes the global effect of the PSAM prediction subsystem. The MMSE ˆ ij and orthogonality principle guarantees that matrix entries H ˆ Ξij are uncorrelated. The optimal Wiener FIR prediction of time-correlated fading according to Jakes’ model yields −1 1 T χ = χ(γ P ; P) = 1 − w(P) W(P) + I w(P), (3) γP where (Wi,j ) = J0 (2π(TD |i − j|)) (F × F -dimensional) and (wi ) = J0 (2π(TD (i − 1) + τD )) (F -dimensional). Note that χ depends on the pilot symbols SNR γ P and a certain set P of other prediction parameters. Constant power is usually employed for the pilot symbols, thus, their SNR γ P is strongly linked to the average channel SNR. The remaining prediction parameters are P = {F, TD , τD } where F is the number of filter taps, TD = fD TS P is the normalized signaling interval (fD is the Doppler spread, TS the symbol interval and P the pilot insertion interval) and τD = fD τ is the normalized adaptation delay to be predicted (τ is the absolute adaptation delay which must be known along with fD for Wiener filtering). Transmit and receiver MIMO processing are as follows1 [15]. The input data stream is mapped onto a single signal z(t) at the transmitter. The NT -dimensional vector ˆ is the x=v ˆ · z(t) is then sent across NT antennas where v beam-steering vector with v ˆH v ˆ = 1. To maximize the received SNR, v ˆ is chosen as the eigenvector corresponding to the ˆ of H ˆ At the receiver, MRC results in ˆ H H. largest eigenvalue λ a single signal r(t) to be detected. Pilot symbols can be reused to perform very accurate noncausal channel estimation for both the MRC and the automatic gain control (AGC). Thus, we 1 Note that in [15] the channel matrix H is defined as N × N T R which leads to some minor differences in the system model formulation.

1313 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.

assume perfect CSI for such signal processing at the receiver. Adaptation of the transmit signal rate and power is based on ˆ The adaptation policies considered in the predicted CSI H. this paper are designed in the next section and are based on long-term constraints, i.e. the target average power (S¯T ) and BER (BERT ) constraints must be satisfied when averaged over all channel realizations. III. A DAPTIVE M ODULATION A. SISO Channels For SISO channels the matrix H reduces to a scalar h. Thus, . it is natural to define the instantaneous SNR as γ = |h|2 γ 2 where the instantaneous power gain |h| is unity-mean expo. nentially distributed random variable and γ = E[γ] = S¯T /σn2 . Then, we can deduce from (2) that the predicted instantaneous . ˆ ˆ = |h| ˆ2 = with the eigenvalue λ SNR is given by γˆ = λγ 2 (1 − χ)ξ and ξ statistically distributed as |h| , i.e. ξ ∼ |h|2 . Consequently, the probability density function (pdf) pγ ( γ ) is exponentially distributed with mean (1 − χ)γ. Let us revisit the original CC-A problem, as stated in [2, sec. IV.A], with an extension to the imperfect CSI case. To obtain the optimum adaptation policy for this scheme we have to tackle a calculus of variations problem with two isoperimetric γ ) and fS ( γ ), respectively, constraints [18]. We denote by fR ( any rate and power candidate adaptation laws for our problem and by R( γ ) and S( γ ), respectively, the optimum laws i.e. those that maximize the ASE. The power law is normalized to the target average power S¯T . Mathematically, the SISO design problem is expressed as follows.

In [2, sec. IV.A] the particular case of perfect CSI ( γ = γ) was addressed. By setting the gradient of the Lagrange equation associated with the functional of (4) to zero, a nonlinear functional equation is obtained which is satisfied for every γ by the optimum adaptation laws R( γ ) and S( γ ) [2, eq. 20]. However, an involved numerical search procedure is needed to obtain R( γ ) and S( γ ). For the perfect CSI case it is possible to express the solution of (4) in closed-form using a tight approximation which establishes an elegant parallelism between the CC-I and CC-A adaptation strategies. In particular, consider the following approximation: Approximation I: The shape of the optimum rate adaptation law R( γ ) for our CC-A scheme is the same as that of the CC-I scheme [2], i.e. the instantaneous rate grows one bit per 3 dB of instantaneous SNR increment. An alternative interpretation of this approximation is that the optimum rate slope is the same as that of the rate slope to achieve the SISO Shannon capacity [1, eq. 4.13]. by

Under this approximation the optimum rate policy is given γ (7) · u( γ − γ0 ), R( γ ) ≈ log2 α0

where u(·) is the unitary step function, α0 is a constant that depends on the power and BER constraints, and γ0 is the cutoff SNR that is typically different from the cutoff SNR γ0 of the CC-I case. Applying again the first-order extremum condition to the Lagrange functional associated with (4) and using (7), it is straightforward to obtain the CC-A adaptation policy that appears in Table I along with the CC-I max Eγ [fR ( γ )] (ASE) fR ,fS case for comparison. As in the CC-I case, the mathematical form of the adaptation laws are independent of the channel subject to γ ) and its influence is included prediction distribution pγ ( −1 + Eγ [fS ( γ )] = 0 (Average Power) through the constants α0 , β0 and γ0 . Given a certain cut-off value γ0 , the constants α0 and β0 are fixed to exactly fulfill the Eγ [fR ( γ )(1 − B(ˆ γ , fR , fS ))] = 0 (Average BER) (4) long-term constraints given in (4). The parameter γ0 is then chosen to maximize the ASE above a certain minimum value which guarantees the nonnegativity of R( γ ) and S( γ ). This With regard to the conditional BER B (normalized to the procedure is carried out by standard numerical search methods target BER BERT ) for QAM under imperfect CSI, from such as those used in [2],[8]. For the usual system parameters [15, eq. 38] we have that the CC-A and CC-I ASE achieved with the policies in Table I are nearly the same [2]. The power adaptation of the CC-A 1 . B (ˆ γ , fR (ˆ γ ), fS (ˆ γ )) = Eγ [BER (γ, γˆ ) | γˆ ] scheme is quasi water-filling in time with the constant factor BERT . K = − log(5BERT ) of the CC-I scheme replaced by T 8 γˆ fS (ˆ γ) 1 1 exp − ≈ , log γ /α0 )/(1/α0 − 1/ γ )). This term is responsible log(β 0 2 ( Φ (ˆ γ , fR , fS ) 5 2fR (ˆγ ) − 1 5BERT Φ (ˆ γ , fR , fS ) for saving some power with adverse channel conditions to (5) be reused when the instantaneous SNR γ is close to or above with the diversity loss factor defined as its mean. This implies that the following instantaneous BER γ) 8 fS (ˆ . γ f (ˆγ ) ≥ 1. (6) curve exactly satisfies the A-BER constraint: Φ (ˆ γ , fR , fS ) = 1 + χ¯ 52 R −1 1 1 1 α0 − γ (8) u( γ − γ0 ). BER( γ) = Note that Φ (ˆ γ , fR , fS ) = 1 when χ = 0 in which case (5) 5β0 log2 ( γ ) α0 reduces to the same BER approximation used in [2]. The diversity loss factor Φ within the exponential in (5) dominates All these results are consistent with those given in [2] obtained the logarithmic slope growth of the conditional BER B due to using numerical search procedures. However, our results are in closed form, assuming that Approximation I is adopted. imperfect CSI.


TABLE I SISO A DAPTATION P OLICIES WITH P ERFECT CSI.

β0 and γ0 are obtained by standard numerical methods as in the perfect CSI case. Obviously, the CC-A closed-form expressions in (11) reduce to those of Table I when χ = 0 (perfect CSI).

CC-I A-QAM γ R( γ ) = log2 u( γ − γ0 ) γ0 1 1 5KT u( γ − γ0 ) − S( γ) = 8 γ0 γ CC-A A-QAM γ R( γ ) ≈ log2 u( γ − γ0 ) α0   γ log β 0 2 α0 1 1 5  S( γ ) ≈ log  − u( γ − γ0 ) 1 8 α0 γ −1 α0

γ

Our next step is to extend these closed-form results to the imperfect CSI case (ˆ γ = γ). Even with Approximation I, we reach an analytical dead end without some further assumptions. Fortunately, there is a reasonable assumption on the diversity loss factor Φ that allows us to obtain a closed-form adaptation policy that includes the effect of imperfect CSI. This assumption is captured as follows: Approximation II: The effect of the instantaneous BER variations around the target BER on the diversity loss factor Φ is negligible. This approximation is justified by plotting (8) for the usual range of system parameters and confirming that the average and instantaneous BERs do not differ significantly. We have generated such plots and confirmed that the instantaneous BER law does not deviate much from its target value for χ ≈ 0. Approximation II can be applied in the conditional BER function (5), in which case Φ becomes a function of γ independent of the rate and power laws:

B. MIMO Beamforming with MRC We now extend the SISO analysis of the previous section to MIMO beamforming. This is facilitated by recognizing that finding the optimum rate and power for MIMO beamforming with the same long-term constraints as in the previous section is equivalent to finding the optimal adaptation for a generalized SISO channel. Thus, this optimization problem is also expressed by (4). The key difference with the SISO case is that we have to obtain appropriate expressions for both the pdf of the predicted instantaneous SNR and the conditional BER for MIMO beamforming. The predicted instantaneous SNR, written in a form equivalent to the SISO case, is ¯ ˆ ST = (1 − χ)ξγ, (12) γ =λ σn2 . ˆ where ξ = λ/(1 − χ) is defined as the largest eigenvalue ˆ HH ˆ ∼ HH H. Note that for the particular case of (1 − χ)−1 H NT = NR = 1, the instantaneous power gain ξ is simply characterized by the unity-mean exponential distribution exhibited in the SISO case. The pdf of the largest eigenvalue ξ of complex Wishart matrices can be expressed as a sum of elementary Gamma pdfs [16]. Therefore, the pdf of the predicted instantaneous SNR γ = (1 − χ)ξγ is given by m (n+m−2k)k

ak,l γˆ l kˆ γ ) exp(− ((1 − χ)¯ γ )l+1 (1 − χ)¯ γ k=1 l=n−m (13) . . where m = min{NT , NR },n = max{NT , NR }, the constant m m Km,n = ( i=1 (m − i)! i=1 (n − i)!)−1 and the coefficients ak,l determined by the algorithm proposed in [16]. The conditional BER function is again given by [15, eq. 38] γ) 8 fS (ˆ KT . 1 γ) 1 8 γˆ fS (ˆ = Φ( γ) γ f (ˆγ ) ≈ 1 + χ¯ γ Φ (ˆ γ , fR , fS ) = 1 + χ¯ R B= exp − , 52 γˆ −1 N Φ 5 2fR (ˆγ ) − 1 5BERT Φ (ˆ γ , fR , fS ) R (9) (14) assuming in Φ that with the diversity loss factor Φ defined as in the SISO case (6). 1 8 γˆ fS (ˆ γ) exp( (10) When NT = NR = 1, equations (13) and (14) reduce to the ) ≈ BERT , f (ˆ γ ) R 5 52 −1 expressions for the SISO case. The same process employed in Section III-A, based on Apas in the CC-I case. Applying the first-order extremum condition to the La- proximations I and II, can be repeated for MIMO beamforming grange functional associated with (4) and using Approxima- with MRC. Consequently, the following adaptation policy is obtained tions I and II, we obtain the following adaptation policy  γ   R( γ ) ≈ log2 u( γ − γ0 )    α 0   γ      R( γ ) ≈ log2 u( γ − γ0 )   S(  γ) ≈  α0     γ γ   log log  2 α0 2 α0   1 1 5Φ( γ) β0 β0 1 1 5         γ ) log log − − S( γ ) ≈ Φ( u( γ − γ0 )  u( γ − γ0 )    2 1 − 1 NR +1 1 − 1 8 α0 γ 8 α0 γ   Φ Φ α0 γ α0 γ (11) (15) which is close to the optimum solution of the design problem where the constants α0 , β0 and γ0 can be found numerically (4) as long as the approximations are accurate. Again, α0 , as previously explained. γ ) = Km,n pγˆ (ˆ


C. Discrete Adaptation In this section a discrete rate and power DD-A-1 MIMO beamforming scheme under imperfect CSI is designed. Our objective is to show that a practical scheme based on discrete adaptation achieves near-optimal results. The adaptation policy of the DD-A-1 AQAM scheme emF −1 (including R0 = 0 ploys a set of NF prefixed rates {Ri }N i=0 as NOTX) and a fixed power level for each constellation, thus, this scheme considers NF fading regions. Within the fading region R(i) that spans from the lower SNR switching i , the rate Ri and threshold γ i−1 to the upper threshold γ the normalized power level Si are employed. For notation . . NF −1 = ∞. simplicity we assume that γ −1 = 0 and γ To design the DD-A-1 scheme we have to determine the optimum sets { γi } and {Si }. Mathematically, this is a constrained nonlinear programming problem with 2(NF − 1) variables which can be stated as follows. γi N F −1 max Ri pγ ( γ ) d γ (ASE) { γi },{Si }

subject to  γi N F −1     −1 + S pγ ( γ ) d γ=0 i   γ i−1

(Average Power)

. with ak,l = ak,l Km,n l!/k l+1 and Γ(0, ·) = E1 (·) where, as expressed in [20, eq. 6.5.15], E1 (·) is the first-order exponential integral function. The ASE for the discrete DD-A-1 scheme in MIMO beamforming must be calculated according to Section III-C. Applying again (19) to (13) we obtain l m (n+m−2k)k ak,l w! i=1 k=1 l=n−m w=0 w w kˆ γi−1 kˆ γi kˆ γi−1 kˆ γi − (1−χ)¯ − (1−χ)¯ γ γ e − e × . (1 − χ)¯ γ (1 − χ)¯ γ (22)

ν¯ =

γ i

γ i−1

N F −1

Ri

Note that the ASE penalty factor due to MIMO pilot insertion (P − NT )/P is not included in (21) and (22).

i=1

i=1

m (n+m−2k)k l ak,l w! k=1 l=n−m w=0 w kγ0 kγ0 γ0 kγ0 − (1−χ)¯ γ e × Γ w, , + log (1 − χ)¯ γ α0 (1 − χ)¯ γ (21)

ν¯ = log2 (e)

γ i−1

i=1

N F −1     R  i 

Introducing (20) in (17) yields

(1 − B( γ , Ri , Si )) pγ ( γ ) d γ = 0 (Av. BER)

B. Numerical Results

(16) Using pγ ( γ ) and B given in (13) and (14), the integrals in (16) can be computed in closed form [19]. Finally, applying standard numerical methods as in [8] the optimum sets { γi } and {Si } are obtained.

Figure 2 shows the ASE of adaptive rate and power modulation with average BER and power constraints, channel prediction, and beamforming in a Rayleigh MIMO channel. The target BER is BERT = 10−3 and the predicted adaptation delay is τD = 4TD . The prediction MMSE χ is calculated with γ P = γ, fD TS = 1/4000, and pilot insertion interval IV. P ERFORMANCE A NALYSIS P = 32, i.e. TD = 8·10−3 . For example, this set of parameters A. Average Spectral Efficiency corresponds to a 3 GHz wireless system with 1/TS = 500 The ASE for the proposed MIMO beamforming policies KHz, terminal speed v = 45 Km/h, and adaptation delay can be calculated in closed form. From (13) and (15) it is τ = 256 µs. Results are shown for both F = 1 and F = 16 straightforward to express the ASE for the CC-A scheme as prediction filter taps and several antenna configurations. m (n+m−2k)k The excellent performance of adaptive modulation and ak,l γ )] = log2 (e)Km,n Ik,l , ν¯ = Eγ [R( MIMO is shown in Fig. 2-(a). Roughly speaking, the NT = k l+1 k=1 l=m−n NR = 2 configuration yields a 7 dB average SNR improve(17) . ment relative to the SISO case, while improvements for the where Ik,l is given by (setting ψ0 (k) = kα0 /((1 − χ)¯ γ ) and . NT = 4, NR = 2 and NT = NR = 4 configurations are γ )) θ0 (k) = kγ0 /((1 − χ)¯ 10 and 12 dB, respectively. These results also show that ∞ x )xl e−x dx. Ik,l = log( (18) the performance degradation associated with imperfect CSI ψ0 (k) x=θ0 (k) on the ASE can be minimized if the number of prediction filter taps is appropriately chosen, taking into account the Integrating by parts in (18) and taking into account that normalized signaling interval TD = fD TS P and the nor ∞ i i! xw , ti e−t dt = e−x (19) malized adaptation delay τD = fD τ to be predicted. In Γ(i + 1, x) = w! x Fig. 2-(b) the ASE performance of a DD-A-1 AQAM scheme w=0 where Γ(·, ·) is the incomplete Gamma function defined in is plotted for F = 1, with the corresponding CC-A scheme for comparison. This discrete AQAM scheme employs square [20, eq. 6.5.3] and i is a nonnegative integer, we obtain QAM constellations (from 4-QAM to 256-QAM) with a single l θ0 (k) l! Ik,l = Γ (w, θ0 (k)) + log e−θ0 (k) θ0 (k)w . power level per constellation. This discrete rate and power w! ψ (k) scheme achieves an ASE performance close to that of the 0 w=0 (20) continuously varying schemes.


12

)z H/ sp10 b( υ yc 8 nei cif El 6 ar tc ep 4 S eg ar 2 ev A 0 5

CC-

12

A

4

N R= 4,

Perfect CSI

N T=

F=1 F=16

2

N R= 2 N R= 2,

4, =

NT N T=

1, =

NT

10

)z H s/p10 b( υ yc 8 nei cif El 6 ar tc ep 4 S eg rae 2 vA

15

Average SNR (a)

20

γ (dB)

25

1 =

NR

0 5

30

A DD-A-1

4

CC-

4, =

NT

N R=

2

4,

N T=

N R=

2

2, =

NT

Densest Constellation

N R= 1

1,

N T=

10

15

20

Average SNR γ (dB)

25

N R=

30

(b)

Fig. 2. ASE of adaptive modulation with average BER and power constraints, channel prediction and beamforming in Rayleigh MIMO channels. The target BER is BERT = 10−3 and the predicted adaptation delay τD = 4TD . The prediction MMSE χ is calculated with γ P = γ, fD TS = 1/4000 and pilot insertion interval P = 32, i.e. TD = 8 · 10−3 . Different curves are shown for F = 1 and F = 16 filter taps and several antenna configurations. (a) CC-A AQAM scheme. (b) DD-A-1 AQAM scheme for F = 1 compared to the corresponding CC-A scheme.

V. C ONCLUSIONS We obtain a closed-form adaptation policy to perform variable-rate variable-power QAM with MIMO beamforming under MRC with average power and BER constraints. The policy considers continuous rate and power adaptation, taking into account imperfect CSI due to MIMO channel prediction. Results for the special case of a SISO channel establish an elegant parallelism between the adaptation policies designed under instantaneous and average BER constraints. We also design fully discrete schemes using numerical methods to show that these schemes have almost the same performance as the continuously adaptive schemes. Closed-form expressions for the ASE achieved by both continuous or discrete rate and power adaptation policies are derived and evaluated. These results indicate that practical constraints on channel prediction and discrete adaptation do not significantly degrade ASE performance. ACKNOWLEDGMENTS The work of A. J. Goldsmith is supported in part by the U.S. Army MURI award W911NF-05-1-0246. The work of J. F. Paris is partially supported by the ’Plan Propio de Investigación’ of the University of Málaga and by the Spanish Government and the European Union under project TIC200307819 (FEDER). R EFERENCES [1] A. J. Goldsmith, Wireless Communications, New York: Cambridge University Press, 2005. [2] S. T. Chung and A. J. Goldsmith, “Degrees of freedom in adaptive modulation: a unified view,” IEEE Trans. Commun., vol. 49, no. 9, pp. 1561–1571, Sept. 2001. [3] M-S. Alouni and A. J. Goldsmith, “Adaptive modulation over Nakagami fading channels,” Kluwer Wireless Personal Commun., vol. 13, pp. 119– 143, May 2000. [4] J. F. Paris, M. C. Aguayo-Torres and J. T. Entrambasaguas, “Impact of imperfect channel estimation on adaptive modulation performance in flat fading,” IEEE Trans. Commun., vol. 52, no. 5, pp. 716-720, May 2004.

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