Performance of Adaptive Modulation with STBC ... - Semantic Scholar

JOURNAL OF NETWORKS, VOL. 6, NO. 1, JANUARY 2011

137

Performance of Adaptive Modulation with STBC and Imperfect Feedback Information over Rayleigh Fading Channel XiangBin Yu1, 2, YanFeng Li1, TingTing Zhou1, DaZhuan Xu1 1

College of Information Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China 2

National Mobile Communications Research Laboratory, Southeast University, Nanjing, China Email: [email protected]

Abstract—Based on imperfect channel information feedback, the performance of space-time coded multi-antenna systems with adaptive modulation and feedback delay is investigated in Rayleigh fading channel, and the corresponding theoretical analysis for spectrum efficiency (SE) and bit error rate (BER) is presented. The fading gain value is partitioned into a number of regions by which the modulation is adapted in terms of the region the fading gain falls in. The fading gain switching thresholds for attaining maximum spectrum efficiency under a target BER and fixed power constraint are achieved. Based on these results, we derive the theoretical calculation formulae for the system spectrum efficiency and average BER in detail. As a result, the closed-form expressions for the spectrum efficiency and average BER are respectively obtained for perfect feedback. Besides, the system performance with delayed feedback information is analyzed, and the accurate and approximate BER expressions of the systems with imperfect feedback are derived, respectively. The obtained theoretical formulae will have more accuracy than the existing formulae. Simulation results show that the theoretical SE and BER are in good agreement with the corresponding simulated values, and the system has slight BER performance degradation for the normalized time delay less than 0.01. Index Terms—Adaptive modulation; Space-time coding; feedback delay; Spectrum efficiency; Bit error rate; MIMO

I. INTRODUCTION With the fast development of modern communication techniques, the demand for high data rate service is grown increasingly in the limited radio spectrum. For this reason, the future wireless communication system will require spectrally efficient techniques to increase the system capacity. To satisfy this requirement, adaptive modulation (AM), as a powerful technique for improving the spectrum efficiency (SE), has obtained fast development recently. It can take advantages of the timevarying nature of wireless channels to transmit higher

Manuscript received January 1, 2010; revised June 1, 2010; accepted July 1 2010. Corresponding author: Xiangbin Yu

© 2011 ACADEMY PUBLISHER doi:10.4304/jnw.6.1.137-144

data rate under favorable channel conditions and to reduce throughput as the channel degrades by varying the transmit power, symbol rate, code rate, and their combination. Thus it can provide much higher spectrum efficiency without sacrificing bit error rate (BER) [1-5]. Moreover, multiple antennas technique is well known to offer improvements in spectrum efficiency along with diversity and coding benefits over fading channels. Especially, space-time coding in multiple input multiple output (MIMO) system provides effective transmit diversity for combating fading effects [6-7]. Hence, the effective combination of adaptive modulation and spacetime coding techniques has received much attention in the literatures [8-12]. The performance of single-and multicarrier adaptive modulation with constant power and single antenna is analyzed in [8]. A performance analysis of space-time block coding (STBC) with fixed and variable rate Quadrature Amplitude Modulation (QAM) is given in [9]. Using multi-antenna space-time coding scheme, a discrete-rate adaptive QAM scheme with constant power over flat Rayleigh fading is studied in [10]. An adaptive coded modulation scheme with 2antennas space-time coding is considered in [11] that make use of turbo code to improve the system performance. The impact of feedback delay on adaptive modulation with constant power transmission is analyzed in [3, 12]. Variable-power adaptive modulation are investigated in Rayleigh fading channel in [4-5], where AM schemes are both designed for single antenna systems with perfect channel state information. However, the above AM schemes are basically limited in perfect feedback information and two-antenna STBC scheme for analysis simplicity, whereas in practice, the feedback information is imperfect due to the feedback delay or estimation error. Moreover, the above schemes usually adopt the approximate BER formula to compute the average BER of the system. As a result, the obtained BER expressions are not accurate to reflect the actual values. Motivated by the reasons above, we will present the performance analysis of adaptive modulation with multi-antenna space-time coding schemes and imperfect feedback information in Rayleigh fading channel. By

138


f ( g ) = g NK −1 e − g / Γ ( NK )

utilizing the exact BER formula of each modulation constellation and the related mathematical calculation, the closed-form expressions of SE and average BER are derived for both perfect and imperfect feedback information. It is shown that the obtained expressions under imperfect feedback include perfect feedback case as a special case. Besides, by utilizing Taylor's series expansion, a closed-form approximate BER formula is derived to simplify the calculation of the theoretical BER effectively, and this approximate BER can obtain the performance close to the actual BER. The results will show that the derived theoretical expressions have better agreement with simulation results than the existing expressions, and that the system has BER performance degradation for imperfect feedback information.

(3) Using transformation of random variable, the pdf of ρ can be obtained as follows f ( ρ ) = ( RN ρ / ρ ) NK exp(−RN ρ / ρ ) [ ρ ⋅Γ( NK )] (4)

II. SYSTEM MODEL

In this paper, a space-time block coded MIMO system with adaptive modulation is referred as AM-STBC, and the adaptive modulator in the system employs square MQAM because of its inherent SE and ease of implementation. For discrete-rate MQAM, the constellation size Ml is defined as {M0=0, M1=2, and Ml=22l-2, l=2,…,q-1}, where M0 means no data transmission. The instantaneous SNR range is divided into q fading regions with switching thresholds {ρ0, ρ1,…, ρq-1, ρq; ρ0=0, ρq=+∞}. The MQAM of constellation size Ml is used for modulation when ρ falls in the l-th region [ρl, ρl+1). Consequently, the data rate is bl=log2Ml bits/symbol with b0=0. According to [4] and [5], the BER of square MQAM with Gray code over additive white Gaussian noise (AWGN) channel for the received SNR ρ and constellation size Ml is approximately given by

In this section, a wireless communication system with N antennas at the transmitter and K antennas at the receiver are considered; and the system operates over a flat and quasi-static Rayleigh fading channel. Given that H={hnk} is N ×K fading channel matrix, where hnk denotes the complex channel gain from transmit antenna n to receive antenna k. Considering quasi-static channel, the corresponding channel gains are constant over a frame (P symbols) and vary from one frame to another. The channel gains are modeled as samples of independent complex Gaussian random variables with zero-mean and variance 0.5 per real dimension. A complex orthogonal STBC, which is represented by an L×N transmission matrix D, is used to encode P input symbols into an N dimensional vector sequence of L time slots. The matrix D is a linear combination of P symbols satisfying the complex orthogonality: DHD=a(|d1|2+…+|dP|2)IN, where IN is the N×N identity matrix, {di}i=1,…,P are the P input symbols, and a is a constant which depends on the STBC transmission matrix [7]. Accordingly, the transmission rate of the STBC is R =P/L. Utilizing the complex orthogonality of STBC, the decoded signal vector y after space-time block decoding can be written as [6, 7]

y = a || H ||2F d + n

(1)

where d=[d1, …, dP]T, and each symbol has an average energy LEs/(aNK), where Es is the average transmitted power radiated from the Nt transmit antennae. || H ||2F is the squared Frobenius norm defined as

|| H ||2F = ∑n=1 ∑k =1| hnk |2 . n is a P×1 noise vector, N

K

whose elements are independent, identically distributed (i.i.d) complex Gaussian random variables with mean zero and variance a || H ||2F N 0 . Thus the effective SNR per symbol after STBC decoding is expressed as

ρ = gEs /( RNN 0 )

(2)

where g =|| H || . For Rayleigh fading channel, g is a central χ2 distributed random variable with 2NK degrees of freedom. From (2-1-110) in [13], the probability density function (pdf) of g is given by 2 F

© 2011 ACADEMY PUBLISHER

where

ρ = Es / N 0

is the average SNR per receive ∞

antenna

and

Γ( z ) = ∫ t z −1e− t dt is the Gamma 0

function. III. PERFORMANCE OF ADAPTIVE MODULATION WITH SPACE-TIME BLOCK CODING AND PERFECT FEEDBACK A. AM with STBC

BERρ ≈ 0.2exp{−1.6ρ /(Ml −1)} , ρ ∈[ρl , ρl +1)

(5)

This approximate formula is slightly tighter than the existing approximate BER formula which has a similar form with 1.5 (not 1.6) in the exponent [3-4, 10, 12] for Ml ≥ 4. Namely, the exact BER is tightly bounded from above by this approximated BER. In spite of that, the above formula can not still reflect the actual BER values accurately. Suppose we set a target BER equal to BER0. Then the region boundaries (or switching thresholds) can be set to the required SNR to achieve the target BER0 over an AWGN channel as follows:

ρ1 = [erfc −1 (2 BER0 )]2

(6)

ρl = −(1/1.6)(Ml −1)ln(5BER0 ) = (Ml −1)/ A , l=2,…q-1. (7) where A=-1.6/ln(5BER0), erfc-1(.) denotes the inverse complementary error function, which can be evaluated by table look-up. When the switching thresholds are chosen according to (6) and (7), the system will operate with a BER below target BER0, as will be confirmed in the following simulation results in section IV. B. Performance analysis with perfect feedback We now derive closed-form expressions for the spectrum efficiency and average BER of the system,


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which will be employed to evaluate the performance of adaptive MQAM with space-time block coding. Based on the switching thresholds described in (6) and (7), using (4) and the incomplete Gamma function

Γ ( k ,υ ) ∆

+∞

∫υ

Pl = ∫

ρl

+∞

+∞

ρl

ρl+1

f (ρ)d ρ = ∫ f (ρ)d ρ − ∫

f (ρ)d ρ

=[Γ(NK, ρl / ρ′) −Γ(NK, ρl+1 / ρ′)]/ Γ(NK) where ρ ′ = ρ /( NR) .

q −1

η = R∑ bl ∫

ρl +1

ρl

l =1

(8)

(9)

l =1

ρ ρ R bl [Γ( NK, l ) −Γ( NK, l +1 )] (10) ∑ Γ( NK ) l =1 ρ′ ρ′ q−1

Eq.(10) is the average SE of the adaptive MQAM scheme with STBC. We define ensemble average BER for multi-antenna system combined with adaptive MQAM and space-time block coding as ρl +1 ⎡ q −1 ⎤ ⎢ ∑ Rbl ∫ρl f ( ρ )d ρ ⎥ ⎣ l =1 ⎦

q −1

BER = ∑ Rbl BERl l =1

= ∑ bl ∫ l =1

ρl +1

ρl

BERl f ( ρ )d ρ

⎡ ⎤ ⎢ ∑ bl Pl ⎥ ⎣ l =1 ⎦ q −1

BERl = ∑ ζ lj erfc{ κ lj ρ }

(11)

(12)

j

where erfc{.} is complementary error function, ζlj and κlj are constants which depend on the constellation size Ml. The higher order terms (HOTs) of (12) are found negligible at high SNR, but first few HOTs can not be neglected at low SNR. The values of the constant sets {ζlj ,κlj} for MQAM can be found in [15-16]. With (4), we can evaluate the cumulative distribution function (cdf) of ρ as follows:

F ( ρ ) = 1 − ∑i =0

NK −1

( ρ / ρ ′) exp ( −ρ / ρ ′) / i ! i

(13)

Using (13) and (120, the following equation can be obtained:

∫ρ

l

ρl+1

BERl f (ρ)d ρ = ∑ζ lj ∫ erfc{ κlj ρ} f (ρ)d ρ j

ρl

Substituting (13) into (14) yields ρl +1

∫ρ

BERl f ( ρ )d ρ

j

{

κlj π

ρ′

NK−1

∑ i!(ρ′κ +1)

(15)

i+0.5

i=0

lj

1 1 ⎡ ⎤⎫ ⋅ ⎢Γ(i + ,(κlj + ρ′−1)ρl+1) −Γ(i + ,(κlj + ρ′−1)ρl )⎥⎬ 2 2 ⎣ ⎦⎭ Substituting (8), (15) and (13) into (11), the average BER of the system can be evaluated as follows: q −1 ⎪⎧ κ BER = ∑ bl ∑ ζ lj ⎨ lj l =1 j ⎪⎩ π

ρ′

NK −1

∑ i !( ρ ′κ i =0

lj

+ 1)i + 0.5

⎡⎣Γ(i + 0.5,(κlj + ρ′−1)ρl +1) −Γ(i + 0.5,(κlj + ρ′−1)ρl )⎤ ⎦

+erfc{ κlj ρl }e− ρl / ρ′ ∑i=0 (i!)−1 ( ρl / ρ′) NK −1

i

−erfc{ κlj ρl +1}e− ρl+1 / ρ′ ∑i=0 (i!)−1 ( ρl +1 / ρ′) NK −1

i

∑ [Γ( NK , ρ / ρ′) − Γ( NK , ρ

/ ρ ′)]/ Γ( NK )

q −1

l +1

l

(16)

}

Eq.(16) is a closed-form exact expression for the average BER of the multi-antenna system with AM and spacetime block coding. It is in good agreement with the corresponding simulation. To simplify the calculation of the above exact BER expression effectively, in what follows, we will derive an approximate closed-form expression of average BER by using the complementary error function erfc{.} and Taylor's series. In Appendix, we give an approximate expression of erfc{ z } by means of Taylor's series expansion. Namely, erfc{ z } can be approximated as

erfc{ z } ≅ µ exp(−ν z )

(17) where the values of the parameter µ and ν depend on the expansion point z0. More details on (17) can refer to the Appendix. Substituting (17), (12), (4) and (10) into (11) gives q −1

BER = ∑ bl ∑ζ lj {µ /(vκlj ρ ′ + 1) NK l =1

j

}

⋅ ⎡⎣Γ(NK,(νκlj + ρ′−1)ρl ) −Γ(NK,(νκlj + ρ′−1)ρl +1)⎤⎦ q−1

∑[Γ(NK, ρ / ρ′) −Γ(NK, ρ l =1


ρ

l =1

where BERl is the BER of MQAM with constellation size Ml and Gray coding and its exact expression is given by [13,15]

ρl+1

ρl

(14)

κlj / π exp(−κlj ρ)F(ρ)d ρ ⎤ ⎥⎦

−1/ 2

+erfc{ κlj ρl }−erfc{ κlj ρl+1}+

q −1

f ( ρ )d ρ = R∑ bl Pl

ρl+1

= ∑ζlj erfc{ κlj ρl+1}F(ρl+1) −erfc{ κlj ρl }F(ρl )

Substituting (8) into (9) gives:

q −1

+∫

l

For discrete-rate adaptive modulation scheme, the spectrum efficiency is defined as the ensemble average of effective transmission rate. Hence, the spectrum efficiency of the adaptive MQAM scheme with spacetime block coding can be given by

η=

j

t k −1 e − t dt [14], we can calculate the

probability of the effective SNR ρ falls in the lth region [ρl, ρl+1), denoted as Pl. ρl+1

= ∑ζ lj ⎡⎣erfc{ κlj ρl +1 }F(ρl +1) − erfc{ κlj ρl }F(ρl )

l

l +1

/ ρ′)]

(18)

140


Eq.(18) is a closed-form approximate expression of average BER of the AM-STBC system, it obviously reduces the calculation of exact BER (16), and may obtain the performance similar to the latter. Moreover, the above theoretical equations (i.e., (16) and (18)) are shown to provide better agreement with simulation results than the existing closed-form expressions in [3] and [10]. IV. PERFORMANCE ANALYSIS OF ADAPTIVE MODULATION WITH STBC AND IMPERFECT FEEDBACK In section III, we give the performance analysis of AM with STBC in the presence of perfect feedback information. However, due to the feedback delay, the available information will be imperfect in practice. Based on the above reason, the AM-STBC system performance with imperfect feedback will be investigated in this section. With imperfect feedback information, the transmitter will use the delayed channel state information, ρˆ , to adjust the modulation modes, where ρˆ is written as

ˆ ||2 = ρ ′gˆ ρˆ = ρ ′ || H

Since the delayed ρˆ has the same pdf as the original ρ, the optimal switching thresholds for the AM-STBC with delayed CSI are thus identical to those of the AMSTBC system with ρ as given by Eqs.(6)-(7). Since the switching thresholds and the pdf in (21) are the same as the perfect case, the spectrum efficiency of the system with ρˆ is thus given by Eq.(10). In other words, the time delay does not affect the system SE. But the time delay can affect the BER performance of AM-STBC system. In what follows, we will give the derivation of average BER. The average BER of the AM-STBC with the delayed feedback information can be expressed as

BER =

R q−1

ρl+1

∑b ∫ρ ∫ η l =1

l

l

(19)

ˆ is the τ time-delayed version of H, gˆ =|| H ˆ ||2 . H The entries {hˆ } are correlated with {hnk} with

where the instantaneous BERρ |ρˆ is given by

BERρ |ρˆ = ∑ j ζ lj erfc

J 0 (2π f dτ ) , where J0(·) is

the zero-order Bessel function of the first kind [14], and fd is the maximum Doppler frequency [13]. Thus according

ˆ are jointly complex to [17], the true channel H and H Gaussian with the correlation c, and correspondingly, their relation can be repressed as ˆ +E H = cH

(20) where each element of E has zero mean and variance

ˆ , we can obtain σe =1-c . With (20), conditioned on H ˆ , σ 2I ) , the probability distribution of H as CN (cH e K 2

where ‘CN’ denotes the complex Gaussian distribution. For flat Rayleigh fading channels, with (19), the pdf of ρˆ is given as

f ( ρˆ ) = (ρˆ / ρ ′)

NK

exp(−ρˆ / ρ ′) [ρˆ ⋅Γ( NK )]

(21) According to the above analysis, using (20) and [13, eq.(2-1-118)], the conditional pdf of g given gˆ , is expressed as NK−1 2

1 ⎛ g⎞ f (g | gˆ) = ⎜ ⎟ (1−c2) ⎝ c2gˆ ⎠

⎛ 2c ⎞ ⎛ g +c2gˆ ⎞ (22) INK−1 ⎜ ggˆ ⎟ ⎟exp⎜− (1−c2) ⎠ ⎝ 1−c2 ⎠ ⎝

Using transformation of random variable and (22), the conditional pdf of ρ given ρˆ can be obtained as follows ( NK −1)/ 2

⎛ ρ ⎞ 1 f (ρ | ρˆ ) = ⎜ ⎟ 2 (1 − c )ρ′ ⎝ c2 ρˆ ⎠

⎛ g + c2 gˆ ⎞ exp ⎜ − ⎟ 2 ⎝ (1− c )ρ′ ⎠ (23) ⎛ 2c ⎞ ⋅ I NK −1 ⎜ ρρˆ ⎟ c2 )ρ′ ⎠ (1 − ⎝


2

(

π/2

π∫

0

κ lj ρ

)

(25)

exp(-x2 /sin2 φ)dφ [18],

(25) can be changed into

BERρ|ρˆ = (2/ π)∑ ζ lj ∫ exp ( -κlj ρ / sin2 φ )dφ (26) π/2

0

j

nk

2

BERρ|ρˆ f (ρ | ρˆ )d ρ f (ρˆ )d ρˆ (24)

Using the equality erfc(x) =

and

correlation coefficient c =

∞

0

Substituting (26), (23) and (21) into (24), we can evaluate the average BER of the system.

BER =

π /2 ρl+1

2R q−1

(sin2 φ)NK ρˆ NK −1 NK 2 2 lj + sin φ ]

∑b ∑ζ ∫ ∫ [(1-c )ρ′κ ηπ ρ l =1

l

lj

j

0

l

⎛ κlj c2 ρˆ (ρ′)−NK ρˆ ⎞ exp ⎜ − ⋅ − d ρˆ dφ ⎜ (1-c2 )γ ′κ + sin2 φ ρ′ ⎟⎟ Γ(NK) lj ⎝ ⎠

(27)

When the feedback information is perfect, i.e., time delay τ is equal to zero, and c will be one. Using c2=1, (27) will be changed to (16) under perfect feedback. The above equation can be further simplified as q−1

BER = ∑bl ∑ζ lj l =1

⋅

j

π /2

∫ ( sin φ 2

(ρ′κlj + sin2 φ ))

NK

0

κlj c2 ρl ρ 2R ⎡ ( , NK Γ + l ) (28) ⎢ 2 2 (1-c )ρ′κlj + sin φ ρ′ ηΓ(NK )π ⎢⎣ κlj c2 ρl +1 ρl +1 ⎤ )⎥ dφ −Γ( NK, + (1-c2 )ρ′κlj + sin2 φ ρ′ ⎥⎦

Considering the above equation needs the integral, we employ the (17) to simplify the calculation of the theoretical average BER. Substituting (17) into (25) yields (29) BER ρ | ρˆ ≅ ζ lj µ exp( −ν κ lj ρ )

∑

j

Substituting (29), (23) and (21) into (24) gives

BER ≅

ζ lj µ R q−1 bl ∑ ∑ ηΓ(NK ) l =1 j [ρ ′κlj v + 1]NK


⋅{Γ(NK,

κljc2vρˆl κljc2vρˆl+1 ρˆl ρˆ ) ( , NK + −Γ + l+1 )} 2 2 ′ ljv +1 ρ′ ′ ljv +1 ρ′ (1-c )ρκ (1-c )ρκ

(30) Eq.(30) is a closed-form approximate expression of the average BER of the AM-STBC, and can effectively reduce the calculation complexity of the theoretical BER from (27) or (28). It is shown that the (28) and (30) have better agreement with the actual simulation values than the existing theoretical expression in [12]. With perfect feedback, c2=1, (30) is reduced to (18), that is, it includes perfect feedback as its special case. V. SIMULATION RESULTS In this section, we will use the derived theoretical formulae and computer simulation to evaluate the SE and average BER of the MIMO system with adaptive MQAM and space-time coding in Rayleigh fading channel. The system is referred to as AM-STBC. In simulation, the channel is assumed to be quasi-static flat fading. Gray code is employed to map the data bits to MQAM constellations. The set of MQAM constellations is {M l }l =0,1,…,5 ={0, 2, 4, 16, 64, 256}. The target BER is

141

In Figure 1, we plot the theoretical SE and corresponding simulation of the AM-STBC system at the target BER0=10-3 for different transmit antennae and one receive antennae. The theoretical SE is calculated by (10) with the switching thresholds defined by (6) and (7). As shown in Fig.1, the theoretical analysis of SE is in good agreement with the simulation result, that is, the theoretical curves coincide with the corresponding simulated curves. It means that the derived theoretical formula is valid. We see from this figure that the SE of adaptive MQAM with G2 code is higher than that of the other three adaptive schemes. It is because the G2 code is a full rate code while G4 and H4 codes are having code rate less than one. Due to the same reason, the systems using H4 have larger SE than that of using G4 code. Besides, the SE of adaptive MQAM with G2 code is also higher than that with one transmit antenna because more transmit antennae are used for G2 code.

set as BER0=0.001. Different space-time block codes, such as G2, G3, G4, H3 and H4, are adopted for evaluation and comparison. Eq.(16) and (18) are used for the theoretical calculation of average BER with perfect feedback, while Eq.(28) and (30) are employed for the BER performance evaluation with imperfect feedback. In the following figures, xTyR denotes a multi-antenna system with x transmit antennae and y receive antennae.

Average BER of AM-STBC system with perfect feedback for three transmit antennae and two receive antennae.

Figure 1. Spectrum efficiency of AM-STBC system with multiple transmit antennae and one receive antenna.


In Figure 2, we plot the theoretical average BER and simulation of AM-STBC system with three transmit antennae and two receive antennae for perfect feedback, G3 code is used. Three theoretical average BERs are considered for the comparison. The average BER described by (16) is referred to as ‘theory 1’, the approximate BER described by (18) is referred to as ‘theory 2’, and the one described by [3, 10] is referred to as ‘theory 3’. In this figure and the following Fig.3, z0 may be set as to four. From Fig.2, it is observed that the average BER of the AM-STBC systems is always less than the target BER0 while the SE effectively increases with SNR. This result verifies the effectiveness of the AM-STBC system on SE. Moreover, we can see that the theoretical analysis of (16) is in good agreement with the

142

simulation result, whereas the theoretical evaluation of using Refs.[3,10] is very different from the simulation result because of using looser approximation. Besides, the derived ‘theory 2’ is also very close to the simulated values due to the close approximation. These results show that our derived BER expressions are effective. Besides, it is observed that the average BER curves have ripples phenomenon. This is because the BER of adaptive modulation scheme will rise up when the constellation is switched to one with larger constellation size as SNR increases and falls in the next fading region. The BER will decrease when the SNR keeps increasing within the region boundaries.


However, the existing theoretical expression [12] is different with the simulation, and they have obvious difference. In Figure 4, we plot the average BER versus the normalized time delay (fdτ) for the AM-STBC with 1T1R, 2T1R, 3T1R and 3T2R. The G2 code is used for 2T1R, G3 code is used for 3T1R and 3T2R. The average SNR is set as to 15 dB. The results show that the AM-STBC can tolerate the normalized time delay up to about 0.01 (corresponding to the time delay τ=0.13ms for 90km/hr) with a slight degradation in the average BER. But when fdτ increases beyond 0.01, the BER performance will degrade gradually, especially for single antenna 1T1R system, the degradation is more significant. Namely, for 1T1R system, it fails to meet the target BER at fdτ =0.05, 2T1R system with G2 code fails at fdτ =0.09, while for 3T1R and 3T2R systems with G3 code, they fail at fdτ beyond 0.1. Moreover, the 3T2R system can tolerate more time delay than 3T1R system because of using multiple receive antennae. The BER performance of the 1T1R system is more sensitive to time delay than the 2T1R and 3T1R system. This is because the latter employs multiple transmit antennae and has greater diversity than the former. Based on the above results, the application of space-time coding can reduce the effect of the time delay on AM performance effectively.

Figure 3. Average BER of AM-STBC system with imperfect feedback for three transmit antennae and one receive antenna.

In Figure 3, we plot the theoretical average BER and corresponding simulation of the AM-STBC system at the target BER0=10-3 for three transmit antennae and one receive antennae, where H3 code and imperfect feedback is considered, the normalized time delay (fdτ) is equal to 0.08. Three theoretical average BERs are considered, that is, the average BER described by (28) is referred to as ‘theory 1’ , the approximate BER described by (30) is referred to as ‘theory 2’, and the one described by [12] is referred to as ‘theory 3’. It is shown in Fig.3 that the BER performance with imperfect feedback is worsen than that with perfect feedback because of the influence of feedback delay. Moreover, due to the same reason, the average BER of AM-STBC system is beyond the target BER at low SNR, but with the increase of average SNR, the BER performance will be improved effectively. Besides, the derived two theoretical BER expressions (i.e., Eqs.(28) and (30)) can match the simulation well, and they are basically identical at different SNR region. © 2011 ACADEMY PUBLISHER

Figure 4. Effect of time delay on the average BER of AM-STBC system with different transmit antenna and receive antenna

V. CONCLUSIONS We have investigated the performance of MIMO system with adaptive modulation and space-time block coding over Rayleigh fading channel. Subject to the


143

target BER and fixed power constraint, the switching thresholds of the fading gain are attained. Based on the obtained switching thresholds, utilizing the probability density function of effective SNR and exact BER formula of MQAM, the closed-form expressions of the SE and average BER for AM-STBC systems are derived in detail. Besides, the impact of the delayed feedback information on the system performance is analyzed. The analysis shows that the AM-STBC system is insensitive to the time delay lower than 0.01. For performance evaluation, the theoretical BER expression is derived for imperfect feedback. Moreover, the closed-form approximate BER expressions are also derived for both perfect and imperfect feedback by using mathematical calculation and a tight approximation of the error function. With these approximate expressions, the computation complexity of the theoretical BER expressions is reduced effectively. Simulation results show that the derived theoretical expressions are more accurate than those in the existing adaptive schemes, and can match the simulation well. APPENDIX In this appendix, we will give the derivation of (17). Let function

I ( z ) = ln(erfc{ z }) ,

then using the

Taylor's series expansion, the I(z) can be approximately written as:

I (z) ≅ ln(erfc{ z0 }) + (erfc′{ z}/ erfc{ z}) z (z − z0 ) 0

exp(−z0 ) ⎛ z0 ⎞ exp(−z0 ) z ⎜ ⎟ − erfc{ z0 } π z0 erfc{ z0 } ⎝ π ⎠ 1/2

= ln(erfc{ z0 }) +

According to the definition of I(z), the function

erfc{ z } can be expressed as: erfc{ z} = exp(I (z)) ⎛ exp(−z0 ) z0 ⎞ ⎛ exp(−z0 ) ≅ erfc{ z0 }exp⎜ ⎟ exp⎜ − ⎜ erfc{ z } π ⎟ ⎜ erfc{ z } π z 0 0 0 ⎝ ⎠ ⎝ = µ exp(−ν z) where

⎞ z⎟ ⎟ ⎠

v = e− z0 /[ π z0 erfc{ z0 }] , z0 is the expansion

point and

⎛ exp(−z0) z0 ⎞ ⎟ = erfc( z0 )exp(ν z0 ) . ⎜ erfc{ z } π ⎟ 0 ⎝ ⎠

µ = erfc{ z0 }exp⎜

This approximate expression may implement better approximation with the function erfc{ z } , and its value is close to the actual value. ACKNOWLEDGMENT The authors would like to thank the editors and reviewers of NSWCTC2010 for useful suggestions and comments. This work is supported by the Doctoral Fund of Ministry of Education of China © 2011 ACADEMY PUBLISHER

(No.20093218120021), the Open Research Fund of National Mobile Communications Research Laboratory (No.N200904), Southeast University, and NUAA Research Funding (No. NS2010113). REFERENCES [1] A.Svensson, G.E. Øien, M.S. Alouini, S.Sampei. “Special issue on adaptive modulation and transmission in wireless systems,” Proceedings of the IEEE, vol.95, no.12, pp. 2269-2273, 2007. [2] T. Kwon, D-H Cho, “Adaptive-Modulation-and-Codingbased transmission of control messages for resource allocation in mobile communication systems,” IEEE Trans.Veh.Technol., vol. 58, no. 6, pp. 2769-2782, July 2009. [3] M.-S.Alouini and A.J.Goldsmith, “Adaptive Modulation over Nakagami Fading Channels,” Wireless Personal Communications, vol.13, pp.120-143, May. 2000. [4] A.J.Goldsmith and S.G.Chua. “Variable-rate variablepower MQAM for fading channel,” IEEE Trans. Commun., vol.45, no.10, pp. 1218-1230, 1997. [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, Sep. 2001. [6] V.Tarokh, H.Jafarkhani, and R.Calderbank, “Space-time block coding for wireless communications: performance results”, IEEE J. Select. Areas Commun., Vol.17, pp. 451460, 1999. [7] H. Shin and JH Lee, “Performance analysis of space-time block codes over keyhole Nakagami-m fading channels,” IEEE Trans.Veh.Technol., vol. 53, no. 2, pp. 351-362, Mar. 2004. [8] B.Choi and L.Hanzo,“Optimum mode-switching-assisted constant-power single-and multicarrier adaptive modulation,” IEEE Trans.Veh.Technol.,vol.52, pp.536559, 2003. [9] H.M.Carraro, R.J.Fonollosa and A.J.Delgado-Penin, “Performance analysis of space-time block coding with adaptive modulation,” in Proc. IEEE PIMRC'2004, pp.493-497, 2004. [10] A.Maaref and S.Aissa, “Rate-adaptive M-QAM in MIMO diversity systems using space-time block codes,” in Proc. IEEE PIMRC'2004, pp.2294-2298, 2004. [11] Jiangbo Dong, Yongzhou Zhou and Daoben Li, “Combined adaptive modulation & coding with space-time block code for high data transmission,” in Proc. IEEE ICCT'03, pp.1476-1479, 2003. [12] Y. Ko and C. Tepedelenlioglu, “Orthogonal space-time block coded rate-adaptive modulation with outdated feedback,” IEEE Trans. Wireless Commun., vol.5, no.2, pp.290-295, Feb. 2006. [13] J. G. Proakis, Digital communications, 4th ed. New York: McGraw-Hill, 2001. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic, 2000. [15] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques Signal Design and Detection, Englewood Cliffs: NJ: Prentice-Hall, 1995. [16] L. Hanzo, W. T. Eebb, and T. Keller, Single-and multicarrier quadrature amplitude modulation. New York: IEEE Press, Wiley, 2000. [17] V. Sharma, K. Premkumar, and R. N. Swamy, “Exponential diversity achieving spatio-temporal power allocation scheme for fading channels,” IEEE Trans. Inform. Theory, vol. 54, no. 1, pp. 188–208, Jan. 2008. [18] A.Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005.

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XiangBin Yu received his Ph.D. in Communication and Information Systems in 2004 from National Mobile Communications Research Laboratory at Southeast University, China. From 2004 to 2006, he worked as a Postdoctoral Researcher in the Information and Communication Engineering Postdoctoral Research Station at Nanjing University of Aeronautics and Astronautics, Nanjing, China. He has been an Associate Professor with the Nanjing University of Aeronautics and Astronautics since May 2006. From February 2007 to June 2007, he was a Postdoctoral Research Associate in the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong. Currently, he also works as a Research Fellow in the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. He has served as a technical program committee of Globecom 2006, International conference on communications systems 2008 (ICCS’08), International Conference on Communications and Networking in China (Chinacom 2010), and ICCS’10. He has been a member of IEEE ComSoc Radio Communications Committee (RCC) since May 2007. His research interests include Multi-carrier CDMA, multiple antennae technique, space-time coding, adaptive modulation and space-time signal processing.



YanFeng Li received the B.S. degree in Electronic Information Engineering from Nanjing University of Aeronautics and Astronautics in 2009. She is pursuing M.S. degree at Nanjing University of Aeronautics and Astronautics.

TingTing Zhou received the B.S. degree in Electronic Information Engineering from Nanjing University of Aeronautics and Astronautics in 2009. She is pursuing M.S. degree at Nanjing University of Aeronautics and Astronautics.

DaZhuan Xu was graduated from Nanjing Institute of Technology, Nanjing, China, in 1983. He received the M.S. degree and the Ph.D. in communication and information systems from Nanjing University of Aeronautics and Astronautics in 1986 and 2001, respectively. He is now a Full Professor in the College of Information Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include digital communications, software radio and coding theory