Adaptive Modulation for MIMO Multiplexing under ... - Semantic Scholar

Adaptive Modulation for MIMO Multiplexing 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 MIMO multiplexing with 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 the fully discrete policies. Our results indicate that MIMO multiplexing is much more sensitive to imperfect CSI than MIMO beamforming due to the self-interference that arises from channel coupling. In particular, if interference between eigenchannels is large, MIMO multiplexing should utilize only one of its eigenchannels, in which case all multiplexing gain is lost.

I. I NTRODUCTION Multiple input multiple output (MIMO) techniques provide dramatic capacity gain in wireless communication systems without requiring additional power or bandwidth. Adaptive modulation is another powerful technique, which can be used to increase the data rate that can be reliably transmitted over fading MIMO channels [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 refer to this class of schemes as 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 this variation: ’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 discrete-power 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 ASE performance inferior to that achievable when long-term BER (A-BER) targets are required [8]. MIMO systems where there are multiple antennas at the transmitter and the receiver allow to achieve a very high ASE [9]. Moreover, adaptive modulation and MIMO can be combined to leverage both of their potentials [10]-[14]. In this paper we investigate the performance of adaptive modulation for MIMO multiplexing under different constraints and with imperfect CSI. Specifically, we determine adaptation policies under a broad set of constraints and derive closed-form expressions to evaluate their ASE. An alternative to MIMO multiplexing is MIMO beamforming, which trades ASE for robustness through diversity. The performance of MIMO beamforming with adaptive modulation under imperfect CSI was investigated in a companion paper [15]. In this work, we will compare the performance of MIMO multiplexing and beamforming with adaptive modulation under imperfect CSI. Our results show that imperfect CSI can lead to significant degradation in the ASE of MIMO multiplexing, as compared with an equivalent MIMO beamforming system. This indicates that MIMO beamforming has greater ASE robustness under imperfect CSI. 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 multiplexing 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 and receiver, this reduces to a SISO channel. Channel gain is modeled

1318 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

Input

H

y

r

Output Data

Data

QAM MAPPING

TX PRECODING PILOT INSERTION

RX SHAPING

Flat Fading Channel

Chosen Rate-Power

Fig. 1.

Feedback Channel

(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 [12]. 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 + Ξ, ˆ H=H

with

CHANNEL PREDICTION

ADAPTATION PROCESS

Hˆ SVD

λˆ

Receiver

System model for MIMO multiplexing with imperfect CSI.

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.), zero-mean, unityvariance, 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,

Pilot

Uˆ

V

QAM DETECTION

AGC Extraction

ˆ

Transmitter

X

ˆ i,j i.i.d. RVs ∼ CN (0, 1 − χ) H (2) 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 based on singular value decomposition (SVD) are briefly described as follows [1]. The predicted channel matrix is decomposed ˆ is the NT × NT unitary matrix ˆ =U ˆΣ ˆV ˆ H where V as H ˆ is the NR × NR unitary used for transmit precoding, U ˆ is an NR × NT matrix required for receiver shapingand Σ ˆ i }. The MIMO multidiagonal matrix of singular values { λ plexing system is characterized by the m-dimensional vecˆ = (λ ˆ i ) of unordered eigenvalues of H ˆH ˆ H with tor λ m = min{NT , NR } representing the equivalent number of parallel single-input single-output (SISO) channels. The uncorrelated binary data from the input is multiplexed and mapped onto an NT -dimensional vector signal z at the transmitter, ˆ is sent across NT then the NT -dimensional vector x = Vz

1319 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.

γ f Si ∞ ˆ i 8¯ λ 1 + χ¯ γ fSj t 1 5(2fRi −1) ˆ ˆ Bi = Λj (fS1 (λ), . . . , fSm (λ)) exp(− )e−t dt 8fSi 8fSi 5BERT j=i 1 + χ¯ γ (fSj t + 5(2fRi −1) ) 1 + χ¯ γ (fSj t + 5(2fRi −1) ) t=0

1 ˆ . . . , fS (λ)) ˆ ≈ Λj (fS1 (λ), m 5BERT j=i

NP k=1

Lxk

1 + χ¯ γ fSj xk 1 + χ¯ γ (fSj xk +

ˆ H y is detected antennas. At the receiver, the vector r = U and demultiplexed to deliver the output data stream. Pilot symbols can be reused to perform very accurate noncausal channel estimation for automatic gain control (AGC) prior to detection, thus, we assume perfect CSI for such signal processing at the receiver. Separate adaptation of the transmit signal rate and power of each eigenchannel is performed using ˆ The adaptation policies considered in the predicted gains λ. 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 To obtain the optimum CC-A adaptation policy for MIMO multiplexing we have to tackle a calculus of variations problem with two isoperimetric constraints [16]. We denote by ˆ fS (λ) ˆ : Rm → R any nonnegative rate and power fR i (λ), i candidate adaptation laws for the ith eigenchannel and by ˆ and Si (λ) ˆ the optimum laws i.e. those that maximize Ri (λ) the ASE. The power laws are normalized to the target average power S¯T . Mathematically, the MIMO multiplexing design problem is expressed as follows. m ˆ fR (λ) (ASE) max E ˆ λ

i

i=1

exp(−

1 + χ¯ γ (fSj xk +

8fSi ) 5(2fRi −1)

(8)

)

(SINR), thus, using the usual exponential expression for MQAM [2], ˆ H) 1 8 SINRi (λ, ) . (6) EH exp(− Bi ≈ ˆ 5 2fRi (λ) 5BERT −1 It will be shown at the end of this section that this Gaussian approximation is quite accurate due to the particular form of the optimum adaptive policy. Introducing (2) in (1) and after some algebra (see [14]) it is straightforward to obtain 2 ˆ ˆ ˆ ii fS (λ) λi + Υ i ˆ H) = SINRi (λ, , (7) ˆ 2 ˆ + 1/¯ γ Υij fSj (λ) j=i

A. Continuous Policy

{fR i },{fS i }

8fSi ) 5(2fRi −1)

γ f Si ˆ i 8¯ λ 5(2fRi −1)

. ¯ ˆ ∼ Ξ ˆ and the average SNR defined as γ¯ = with Υ ST /σn2 . According to Appendix A, the conditional BER expression in (6) can be accurately computed by (8) at the top of the page, where xk are the zeros of the NP th-order Laguerre polynomial and Lxk the associated weight factors [17, p. 223-224] used for Gauss-Laguerre quadrature integration. Specifically, in expression (8) Λj = 1 for m = 2 and for m > 2 Λj =

m l=1,l=i,j

ˆ fSj (λ) , ˆ − fS (λ) ˆ fSj (λ) l

(9)

which must be interpreted as a limit1 when fSl = fSj . ˆ To perform expectation over the predicted channel gain λ, ˆ note that λ = (1 − χ)ξ with ξ the m-dimensional vector ˆH ˆH ∼ ξ = (ξi ) of unordered eigenvalues of 1/(1 − χ)H H HH . Consequently, the joint probability density function ˆ is easily obtained from the Wishart pdf pξ (ξ) (pdf) pλˆ (λ) given in [19] which can be expressed as

subject to  m   ˆ =0  fS i (λ) (Average Power)  −1 + Eλˆ    i=1   m  ˆ ˆ fR , . . . , fR , fS , . . . , fS ))  Eˆ fR i (λ)(1 − Bi (λ,  1 m 1 m λ    i=1    =0 (Average BER) (4) where the conditional BER (normalized to the target BER BERT ) for the ith eigenchannel is defined as

1 . ˆ H) | λ ˆ , Bi = EH BERi (λ, (5) BERT ˆ H) the instantaneous BER given the with BERi (λ, predicted and the true CSI. Under the Gaussian approximation [1, pag. 195] the conditional BER can be computed from the signal-to-noise plus interference ratio

pξ (ξ) =

1 (−1)per(a)+per(b) m! i a,b

ai +b i −2+d

Aj (ai , bi )ξij e−ξi ,

j=d

(10) . where d = |NT − NR |, a = (ai ) and b = (bi ) represent permutation vectors of {1, . . . , m}, the function per(·) is 0 or 1, respectively, depending on whether the permutation is even or odd, and Aj (ai , bi ) is defined as the (j + 1)th coefficient of 1 This limit is easily circumvented when this expression is evaluated numerically. An alternative expression without this exception can be obtained following Appendix A and using the pdf representation in [18, eq. 14], however, this requires defining equivalence classes formed by equal-power ˆ which makes the computation of (8) very cumbersome. interferers for each λ


the following polynomial (ai − 1)! (bi − 1)! xd Ld (x)Ldbi −1 (x), (ai − 1 + d)! (bi − 1 + d)! ai −1

(11)

where Ldn (x) is the generalized Laguerre polynomial [17]. It is shown in [19] that the marginal pdf pξ (ξ) can be represented as follows: 2(m−1)+d Bj ξ j , (12) pξ (ξ) = e−ξ j=d

with Bj defined as the (j + 1)th coefficient of the following polynomial 2 1 d (i − 1)! d Li−1 (x) . x m i=1 (i − 1 + d)! m

(13)

As will be shown later, the pdf representations given in (10) and (12) enable us to obtain closed-form expressions. In general, solving the problem stated in (4) is hard due to the coupling between eigenchannels introduced through both the conditional BER (imperfect CSI induced interference) ˆ and the statistical dependence between the components of λ. However, under certain approximations it is possible to find an accurate closed-form adaptation policy for MIMO multiplexing with an average BER constraint and imperfect CSI. To analyze the behavior of optimum A-QAM MIMO multiplexing we distinguish two scenarios according to the quality of the available CSI: good quality (χ relatively small) and bad quality (χ relatively high). An expression to perform such classification will be provided later. We first assume good CSI quality. Approximation I: Under relatively good CSI quality the design problem is essentially decoupled, i.e. to control the rate and power of the ith eigenchannel we can just observe î. its corresponding instantaneous power gain λ Applying this approximation and considering the symmetry ˆ and Bi it follows that fR (λ) ˆ = fR (λ ˆ i ) and of pλˆ (λ) i ˆ = fS (λ ˆ i ) for all i = 1, . . . , m. Consequently, the fSi (λ) following decoupled approximation for the conditional BER at the optimum is obtained ˆ i γ¯ fS (λ î) (1 + χ¯ γ m−1 1 1 8 λ m ) exp(− ), B˜i ≈ ˆ ˜ i 5 2fR (λi ) − 1 î) ˜ i (λ 5BERT Φ Φ (14) m − 1 KT ˆ ˜ + γ with Φi (λi ) = 1 + χ¯ , ˆ i γ¯ m λ . ˜ i ≥ 1 representing KT = − log(5BERT ), and the function Φ the diversity loss due to imperfect CSI (which is 1 under ˜i perfect CSI, i.e. χ = 0). Note that the diversity loss factor Φ within the exponential in (14) dominates the logarithmic slope growth of the conditional BER due to imperfect CSI. The steps to obtain (14) are briefly sketched. First, we approximate the random interference term in (7) by mean values 2 ˆj ) ≈ χ m − 1 , ˆ ij fS (λ (15) Υ m j=i

then (6) is recalculated (by a procedure similar to that of [12, sect. IV]) to obtain a new expression for Bi where we finally assume that, within the terms multiplied by χ (which is assumed small because the channel quality is good), the instantaneous BER is approximately constant and equal to the target BER BERT , ˆ i S(λ î) 8 γ¯ λ ≈ KT , ˆ 5 2R(λi ) − 1

(16)

as happens for the policy in the interference-free case [15]. Under Approximation I the decoupled formulation of the original problem (4) becomes

ˆ (ASE) max Eλˆ fR (λ) fR ,fS

subject to

 ˆ  f −1 + mE ( λ) =0 (Average Power) ˆ S  λ

  E f (λ)(1 ˆ ˆ fR , fS )) = 0 (Average BER) ˜ λ, − B( ˆ R λ (17) ˆ corresponds to any eigenchannel, B˜ is given in (14) where λ ˆ = 1/(1 − χ)pξ (ξ/(1 − χ)) with and the marginal pdf pλˆ (λ) pξ (ξ) in (12). Approximation II: The shape of the optimum rate adaptation ˆ for our CC-A scheme with imperfect CSI is the law R(λ) same as that of the CC-I scheme with perfect CSI [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 under perfect CSI [1, eq. 4.13]. Under this approximation (that accurately holds for the equivalent MIMO beamforming problem [15]) and applying the first-order extremum condition to the Lagrange functional associated with (17), it is straightforward to obtain the following policy for MIMO multiplexing with imperfect CSI:  ˆ  λ   R(λ) ˆ ≈ log ˆ − λ0 ) u(λ  2   α0      ˆ β0 log2 αλ0  ˆ λ)  1 5 Φ( 1  ˆ ˆ − λ0 )    log S( λ) ≈ − u(λ   ˆ 1 1 8γ α 2  0 λ Φ α0 − λˆ   (18) where u(·) is the unitary step function and {α0 , β0 , λ0 } are three constants. Given a cut-off value λ0 , the constants α0 and β0 are fixed to exactly fulfill the long-term constraints given in (4). The parameter λ0 is then chosen to maximize the ASE subject to a certain minimum value which guaranties ˆ and S(λ). ˆ This procedure is carried the nonnegativity of R(λ) out by standard numerical search methods such as those used in [2],[8]. The same continuous policy is applied to every ˆ i . For the SISO eigenchannel using its corresponding gain λ case (m = 1) the interference term χ¯ γ (m − 1)/m within (14)


vanishes and the adaptation policy in (18) coincides with the SISO policy obtained in [15]. Approximation I is valid as long as the diversity loss in (14) remains close to unity. Specifically, we establish the following criterion to describe the CSI quality. The prediction MMSE threshold χ0 is defined as the value above which the error term in the diversity loss (14) is greater than the unity term ˆ = (1 − χ)ξ¯ for the average channel state E[λ] 2(m−1)+d χ0 K T m−1 ¯= + χ0 γ¯ = 1 with ξ (j + 1)!Bj . ¯ m (1 − χ0 )ξ j=d (19) Solving (19) we obtain 2 χ0 = . 2 KT m−1 m−1 1 + 1 + Kξ¯T + γ¯ m−1 + + γ ¯ − 4¯ γ m m m ξ¯ (20) Hence, we consider that the decoupled solution deduced from Approximations I and II is valid when CSI quality satisfies χ < χ0 . When CSI quality is bad, i.e. χ ≥ χ0 , the behavior of the optimum A-QAM MIMO multiplexing is described as follows.

Approximation III: Under poor CSI quality the shape of the optimum adaptation policy given in (18) is still valid. However, to avoid performance degradation due to the severity of interference, optimum adaptation only allows transmission along the best eigenchannel at every instant. Therefore, using Approximations I, II and III we formulate the following closed-form policy for continuous adaptation in MIMO multiplexing: For eigenchannels i = 1, . . . , m, χ < χ0 ⇒

ˆ = R(λ î) Ri (λ) ˆ = S(λ î) Si (λ)

  ˆ i = max λ ˆk ˆ i ) if λ  R(λ    k = 1,...,m ˆ  R ( λ) = i   0  otherwise  χ χ0 ⇒  ˆ i ) if λ ˆ i = max λ ˆk  S(λ    k = 1,...,m ˆ  ( λ) = S   i 0 otherwise

on both available power as well as the potential interference between eigenchannels. The accuracy of Approximations I, II and III that lead to the formulation of (21) will be analyzed in Section IV.

B. Discrete Policy In this section a discrete rate and power DD-A-1 MIMO multiplexing scheme under imperfect CSI is designed. Our objective is to show that a practical scheme based on discrete adaptation achieves near-optimal results. We show a rectangular partition of the fading regions for a bidimensional MIMO system (m = 2) with NF = 4 fading states per dimension in Fig. 2. In the figure the external regions Rlk which are interference-free have a darker shading than the internal regions Rij with simultaneous transmission along both eigenchannels, where i, j, k = 1, . . . , NF − 1 and l = 1, . . . , m. For the general m-dimensional case, we define a set of fading regions {R} associated with the rectangular partition of the eigenchannels space. Within a specific region R we employ a constant rate for the lth eigenchannel Rl) (R) chosen from a certain set Q of possible QAM constellations. In addition, we employ a constant power level S l) (R) for the lth eigenchannel within the region R. The design problem is . to obtain the optimum switching thresholds U = {ulh } with h = 1, . . . , NF − 1, which fix the optimum fading regions l) partition {R } and the optimum sets of rates {R (R )} and l) power levels {S (R )} to be employed within each specific fading region R . For notational simplicity we assume that ul0 = 0 and ulNF = ∞ for every l. The probability of staying within the region R is denoted by PR (U) which is determined by the set U and the joint ˆ Every fading region R is a polyhedron defined as pdf pλˆ (λ). l) ˆ l ≤ ul)+ }l=1,...,m with h− , h+ ∈ {0, 1, . . . , NF } {uh− ≤ λ h + and h = h− + 1. Using (10) and taking into account that for any nonnegative integer i:

(21)

Γ(i + 1, x) =

∞

ti e−t dt = e−x

x

i i! w x , w! w=0

(22)

where Γ(·, ·) is the incomplete Gamma function defined in [20, eq. (6.5.3)], it is straightforward to obtain

with R(·) and S(·) given in (18). Note that the Gaussian approximation in (6) holds regardless of χ; if χ < χ0 the AWGN noise term dominates over interference and if χ ≥ χ0 interference is avoided. We see from (21) that when interference due to the MMSE would be large (χ ≥ χ0 ) we only use the eigenchannel associated with the largest singular value to avoid interference. A similar phenomenon happens in single-user MIMO systems when the power is not sufficient to water-fill over more than one eigenchannel. In our case, however, using more than one eigenchannel depends

PR (U) =

m al +b l −2+d

1 (−1)per(a)+per(b) m! a,b

n n! × w! w=0

l=1

l)

uh− 1−χ

w

l) u −

h − 1−χ

e

−

An (al , bl )

n=d l)

uh+ 1−χ

w

l) u +

h − 1−χ

e

. (23)

Therefore, the discrete MIMO multiplexing problem equiv-


R

R

2)

u

3

R

2)

u

Cl = 1 + (1 − χ)

R

2, 2

2, 3

2)

2

R

R

2)

R

R

1,2

1,1

1

u

R

2,1

2

1, 3

2)

1

R

R

1)

1

u

1)

1

R

1)

1)

2

λˆ1

1)

3

2

u

u

1) 3

Fig. 2. Bidimensional MIMO multiplexing DD-A-1 scheme (m = 2) with NF × NF = 16 fading states.

alent to (4) is expressed as max {Rl) },{S l) },U

m

Rl) (R)PR (U)

(ASE)

l=1 {R}

subject to  m   −1 + S l) (R)PR (U) = 0 (Average Power)     l=1 {R}      m Rl) (R)(1 − B l (R, Rl) , S 1) , . . . , S m) ))PR (U) = 0    l=1 {R}   (Average BER)       l) R (R) ∈ Q (Constellations Set) (24) where B l is the expectation of Bl conditioned on a certain fading region:

ˆ ∈R Eλˆ Bl , λ . . Bl (R, Rl) , S 1) , . . . , S m) ) = Eλˆ [Bl |R] = PR (U )

×

1 Λj (S 1) , . . . , S m) ) PR (U)5BERT j=l

NP k=1

Lxk

1 + χ¯ γ S j) xk 1 + χ¯ γ (S j) xk +

8S l) ) l) 5(2R −1 )

m ai +b i −2+d 1 An (ai , bi ) (−1)per(a)+per(b) m! Cin+1 i=1 a,b n=d w w i) i) i) i) n Ci u Ci u Ci uh+ Ci uh− n! h− h+ − 1−χ − 1−χ × e − e w! 1−χ 1−χ w=0 (25)

×

1 + χ¯ γ S j) xk +

8S l) l) 5(2R −1 )

.

(26)

In general, solving (24) by standard numerical methods is hard if the set of possible constellations Q is restricted to be discrete, e.g. the set of rates corresponding to square QAM S = {2, 4, 6, 8, . . .} bits per symbol. The reason is that in contrast to the SISO case, where rate allocation is based only on the fading level, the optimum rate allocation for the internal regions, where eigenchannel interference is possible, will depend on the rate allocation of the interfering channels. Hence, to analyze the achievable performance of the DD-A-1 scheme we assume that Q = R+ and that optimum rate allocation is unknown a priori. Obviously, the ASE for Q = R+ upper bounds the ASE for Q = S and optimization can be performed by extending the method used in [8] to the multidimensional problem stated in (24). IV. P ERFORMANCE A NALYSIS A. Average Spectral Efficiency In this section the ASE of the proposed adaptation policies is derived in closed form. For the continuous CC-A AQAM scheme in the MIMO multiplexing case the ASE when χ < χ0 is given by ˆ λ ˆ − λ0 ) ν¯ = mEλˆ log2 u(λ α0 2(m−1)+d

= mlog2(e)

log

∞

Bj

ξ=λ0 /(1−χ)

j=d

ξ α0 /(1 − χ)

ξ j e−ξ dξ, (27)

Integrating by parts (27) and taking into account (22) yields 2(m−1)+d

j j! w! w=0 j=d w λ0 λ0 λ0 λ0 e− 1−χ . × Γ w, + log 1−χ α0 1−χ (28)

ν¯ = mlog2(e)

Using (10), (8) and (22), for the fading region R we obtain Bl =

8¯ γ S l) l) 5(2R −1 )

3, 3

3,2

2)

R ˆλ2

with Ci = 1 for i = l and for i = l

R

R

3,1

3

Bj

For χ ≥ χ0 the ASE of the continuous CC-A AQAM scheme is given by ξ1 ξ1 ∞ ξ1 ν¯ = m log2 . . . pξ (ξ)dξ1 . . . dξm . ξ2 =0 ξm =0 ξ1 =λ0 /(1−χ) α0 /(1 − χ) (29) Using the joint pdf representation (10) in (29), rearranging terms and considering the simple identity Q k=1

(1 + Xk ) =

X1c1 X2c2 · ·XQcQ ,

(30)

c∈{0,1}Q

with c = (ci ) representing all possible Q-dimensional binary vectors, the multidimensional integral (29) can be reduced to


a form equivalent to (27). Consequently, the ASE for χ ≥ χ0 can be expressed as ν¯ =

log2 (e) (−1)per(a)+per(b) (m − 1)! a,b

×

a1 +b 1 −2+d

am +b m −2+d

··

l1 =d

×

l2

lm

··

w2 =0

lm =d

z=0

+ log

(−1)Σ(c)

λ0 α0

(l1 + Ω(c, w))! z!

m li +1 ci−1 wi ! i=2 1+l1 +Ω(c,w)

(1 + Σ(c))

wm =0 c∈{0,1}m−1

l1 +Ω(c,w)

×

m li !Al i Al1 l +1 i=2 i

λ0 (1 + Σ(c)) 1−χ

λ0 (1 + Σ(c)) Γ z, 1−χ

z

λ0 (1 + Σ(c)) exp − 1−χ

,

(31) where Σ(c) and Ω(c, w) are nonnegative integers given by  m−1    Σ(c) = cr   r=1 (32) m−1     Ω(c, w) = cr wr+1  r=1

The ASE for the discrete DD-A-1 scheme in MIMO multiplexing must be calculated according to Section III-B ν¯ =

m

l)

R (R ) PR (U ),

(33)

l=1 {R } l)

with {R (R )}l=1,.,m the optimum rate allocation for the fading region R , U the optimum set of switching thresholds and PR (U ) the probability of staying in fading region R when the set U is chosen. The ASE penalty factor due to MIMO pilot insertion (P − NT )/P is not included in (28), (31) and (33). B. Numerical Results Fig. 3-(a) shows the ASE of adaptive rate and power modulation with average BER and power constraints, channel prediction and multiplexing in Rayleigh 2×2 MIMO channels. The target BER is BERT = 10−3 and the number of prediction filter taps is F = 16. 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 . For example, this set of parameters corresponds to a 3 GHz wireless system with 1/TS = 500 KHz and terminal speed v = 45 Km/h. The predicted adaptation delays are τD = 0, 4TD , and 16TD , which corresponds to the adaptation delays τ = 0, 256, and 1024 µs for our example wireless system. Fig. 3-(b) shows the prediction MMSE corresponding to the ASE curves of

Fig. 3-(a). For comparison the ASE of the corresponding MIMO beamforming system obtained by the methods developed in [15] is also displayed. Note that in all cases the parameters α0 and β0 are obtained numerically to accurately satisfy the average power and BER constraints. Fig. 3-(a) shows the accuracy of the proposed continuous policy for MIMO multiplexing with respect to the numerically optimized discrete policy. It is also clear that for relatively bad CSI MIMO multiplexing gain is lost, i.e. for exactly the same performance (average BER and power) the achieved ASE of MIMO multiplexing can be less than that of MIMO beamforming when CSI results in a poor channel estimate. This is not surprising, since a poor channel estimate leads to interference between eigenchannels in MIMO systems. The behavior model for MIMO multiplexing collected in Approximation III can be explained by Fig. 3-(b), where the prediction MMSE χ is plotted versus the average SNR γ¯ . When χ(γ) < χ0 (γ) diversity loss is relatively small and the impact on ASE is not too important. However, when χ(γ) ≥ χ0 (γ) the impact of imperfect CSI on the ASE of MIMO multiplexing is quite important. A simple rule for a MIMO system designer is keeping the curve χ(¯ γ ) below the curve χ0 (γ). To achieve this goal, different prediction subsystem parameters can be employed according to the average SNR γ¯ , e.g. a variable number of filter taps or unequal pilot symbol power allocation. V. C ONCLUSIONS We obtain a closed-form adaptation policy to perform variable-rate variable-power QAM in MIMO multiplexing with average power and BER constraints. The policy considers continuous rate and power adaptation, taking into account imperfect CSI due to MIMO channel prediction. 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 are derived for the ASE of the proposed adaptation policies. Our results also show that adaptive modulation for MIMO multiplexing is much more sensitive to imperfect CSI that MIMO beamforming. We obtain design guidelines as to how good channel estimation must be for multiplexing to outperform beamforming under imperfect CSI. We show that MIMO multiplexing should utilize only the best channel when interference between eigenchannels is too large due to imperfect CSI. A PPENDIX A From a statistical point of view, the SINR in (7) is equal to SINRi (x, y) =

γ¯ fSi x with y = yj , 1+y

(34)

j=i

where x has noncentral chi-squared pdf with two degrees of ˆ i (see [21]) and freedom and parameters 2σ 2 = χ and s2 = λ each yj for j = i has exponential pdf with mean γ¯ χfSj . All these RVs are mutually independent. Using the well-known


12

)z H/ sp10 b( υ cyn 8 eic if 6 El ar tc ep 4 S eg ar 2 ev A

-5

D=0

y cit pa Ca . lt u M

D=4TD

I S -C ct fe r e P

I t-CS rfec e P m. Bea

-10

) B d (

χ

E-15 S M M n io t ic d e r P

Mult. CC-A Mult. DD-A-1 Beam. CC-A 10

15

20

Average SNR γ (dB)

TD

-20

D=16TD

0 5

D=16

χ( 0 γ )

25

D= 4T D

-25

-30 5

30

(a)

D= 0

10

15 20 Average SNR γ (dB)

25

30

(b)

Fig. 3. Performance of MIMO multiplexing with imperfect CSI compared with MIMO beamforming for a 2 × 2 system. The prediction MMSE χ is calculated with γ P = γ, F = 16 filter taps, fD TS = 1/4000 and pilot insertion interval P = 32, i.e. TD = 8 · 10−3 . The predicted adaptation delays are τD = 0, 4TD and 16TD . (a) ASE. (b) Prediction MMSE.

moment generating function method, the pdf of y is expressed as y − γ¯ χf 1 Sj py (y) = Λj e , (35) γ¯ χfSj j=i

with Λj given in (9). The conditional BER is given by 1 Bi = 5BERT

∞ ∞

−

e

8SINRi (x,y) f −1 5(2 Ri )

px (x)py (y)dxdy.

(36)

x=0 y=0

Using the bidimensional RV transformation u = 1 + y, v = γ¯ fSi x/(1 + y) and the Laplace transform given in [20, p. 1182, eq. (109)] the expression in (8) is obtained. ACKNOWLEDGMENTS The work of A. J. Goldsmith is supported in part by the U.S. Army under 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, Cambridge University Press, New York, 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. [5] G. E. Oien, H. Holm and K. J. Hole “Impact of channel prediction on adaptive coded modulation performance in Rayleigh fading,” IEEE Trans. Veh. Techn., vol. 52, no. 5, pp. 758-769, May 2004.

[6] S. Falahati, A. Svensson, M. Sternad y T. Ekman, “Adaptive modulation systems for predicted wireless channels,” IEEE Trans. Commun., vol.52, no. 2, pp. 307-316, Feb. 2004. [7] X. Cai and G. B. Giannakis, “Adaptive PSAM accounting for channel estimation and prediction errors,” IEEE Trans. Wireless Commun., vol. 4, no. 1, pp. 246-256, Jan. 2005. [8] J. F. Paris, M. C. Aguayo-Torres and J. T. Entrambasaguas, “Nonideal adaptive modulation: bounded feedback information and imperfect channel estimation,” Proc. of IEEE Global Commun. Conf., Dallas, Nov. 2004. [9] A. J. Goldsmith, S. A. Jayar, N. Jindal and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Select. Areas Commun., vol. 21, pp. 684-702, June 2003. [10] A. Maaref and S. Aissa, “Rate-adaptive M-QAM in MIMO diversity systems using space-time block codes,” Proc. of IEEE Personal, Indoor and Mobile Radio Commun. Conf., vol. 4, pp. 2294 - 2298, Sept. 2004. [11] B. Holter, G. E. Oien, K. J. Hole and H. Holm, “Limitations in spectral efficiency of a rate-adaptive MIMO system utilizing pilot-aided channel prediction,” Proc. of IEEE Veh. Techn. Conf., VTC 2003, pp. 282-286, April 2003. [12] S. Zhou and G. B. Giannakis, “How accurate channel prediciton needs to be for transmit-beamforming with adaptive modulation over Rayleigh MIMO Channels,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1285-1294, July 2004. [13] Z. Zhou and B. Vucetic, “MIMO systems with adaptive modulation,” Proc. of IEEE Veh. Techn. Conf., VTC 2004, pp. 765-769, May 2004. [14] Z. Zhou and B. Vucetic, “Design of adaptive modulation using imperfect CSI in MIMO systems,” Elec. Lett., vol. 40, no. 17, Aug. 2004. [15] J. F. Paris and A. J. Goldsmith, “Adaptive Modulation for MIMO beamforming under average BER constraints and imperfect CSI,” Proc. IEEE Internat. Conf. Commun., June 2006. [16] R. Weinstock, Calculus of Variations, 1st ed., McGraw-Hill, New York, 1952. [17] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, San Diego, 1984. [18] X. W. Cui, Q. T. Zhang, and Z. M. Feng, “Outage performance for maximal ratio combiner in the presence of unequal-power co-cochannel interferers,” IEEE Comm. Lett., vol. 8, n. 5, May 2004. [19] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecom. ETT, vol. 10, n. 6, pp. 585-596, Nov. 1999. [20] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed., Dover, New York, 1970. [21] J. G. Proakis, Digital Communications, 8th ed., McGraw-Hill, Singapur, 1995.
