Adaptive modulation for space–frequency block coded OFDM systems

Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533 www.elsevier.de/aeue

Adaptive modulation for space–frequency block coded OFDM systems Mohammad Torabi∗ Département de génie électrique, École Polytechnique de Montréal, Pavillon Lassonde, 2500 Chemin de Polytechnique, Montréal, Québec, Canada H3T 1J4 Received 19 February 2007; accepted 3 July 2007

Abstract The growing popularity of both multiple-input multiple-output (MIMO) and orthogonal frequency division multiplexing (OFDM) systems has created the need for adaptive modulation to integrate temporal, spatial and spectral components together. In this article, an overview of some adaptive modulation schemes for OFDM is presented. Then a new scheme consisting of a combination of adaptive modulation, OFDM, high-order space-frequency block codes (SFBC), and antenna selection is presented. The proposed scheme exploits the benefits of space–frequency block codes, OFDM, adaptive modulation and antenna selection to provide high-quality transmission for broadband wireless communications. The spectral efficiency advantage of the proposed system is examined. It is shown that antenna selection with adaptive modulation can greatly improve the performance of the conventional SFBC–OFDM systems. 䉷 2007 Elsevier GmbH. All rights reserved. Keywords: OFDM; Adaptive modulation; MIMO; Space–frequency block coding

1. Introduction A major limitation of wireless communication systems is the result of the channel fading. Conventional fixed-mode modulation schemes suffer from errors caused by deep fades. An effective approach to mitigating these detrimental effects is to adaptively adjust the modulation and/or channel coding format as well as a range of other system parameters based on the near-instantaneous channel quality information perceived by the receiver, which is fed back to the transmitter with the aid of a feedback channel. This plausible principle was recognized by Hayes [1] as early as 1968. Hayes [1], proposed an adaptive modulation scheme based on power adaptation. Several years later, Goldsmith et al. [2–4] proposed variable-rate, variable-power adaptive schemes. They found that “the extra throughput achieved by the additional

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E-mail address: [email protected]. 1434-8411/$ - see front matter 䉷 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2007.07.008

variable-power assisted adaptation over the constant-power, variable-rate scheme is marginal” for most types of fading channels [2–4]. Hanzo et al. [5–8], proposed an adaptive scheme based on a set of mode switching levels designed for achieving a high average bits per second (BPS) throughput, while keeping the target BER of the system at a desired level. They showed that adaptive modulation provides promising advantages compared to the fixed-mode schemes in terms of BER performance, spectral efficiency, etc. The associated principles can also be applied in multicarrier systems such as discrete multi-tone (DMT) and OFDM systems. The adaptive modulation scheme for multicarrier systems, the so called Adaptive OFDM (AOFDM), was first proposed by Kalet [9] and was developed by Cioffi et al. [10] and by Czylwik [11], and some other schemes have been studied in [12–16]. In [11] Czylwik showed that from the optimum power distribution, only a gain in order of 1 dB is obtained. Therefore, it is recommended not to optimize the power distribution and to use a constant power spectrum in order to save computational complexity. AOFDM exploits

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M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

the variation of the signal quality due to the variation of the channel in the time and frequency domains. The goal of adaptive modulation in an OFDM system is to allocate (according to the instantaneous condition of a subchannel) an appropriate number of bits and to choose the suitable modulation mode for transmission in each subcarrier, in order to improve the system performance or to keep the overall bit error rate (BER) performance at a certain desired level. The performance improvement offered by the adaptive modulation over non-adaptive systems is remarkable. Furthermore, other dimensions such as frequency and space may yield further gains by providing additional degrees of freedom that can be exploited by adaptive modulation [17,18]. To this end, a combination of space–time block coding (STBC) and AOFDM has been considered in [19]. In [19], it is shown that the full benefit of AOFDM and STBC cannot be exploited at the same time. Considering the popular space–frequency block coded OFDM (SFBC–OFDM), we presented some SFBC–OFDM schemes in conjunction with transmit antenna selection and adaptive modulation that can improve the overall system performance [20,21]. In this article, we generalize these schemes with highorder space–time block codes such as G2 , G3 , G4 , H3 and H4 provided in [22–24], and propose a subcarrier-by-subcarrier basis antenna selection scheme for non-flat fading channels that can provide superior system performance. In this study, the term adaptive modulation refers to bit rate adaptation, and therefore, it is the variable-rate case. The performance of the proposed system called A-SFBC–OFDM, has been analytically evaluated and has been compared with that of the non-adaptive SFBC–OFDM system. It is shown that using the antenna selection scheme at the transmitter in conjunction with adaptive modulation can improve the performance of the SFBC–OFDM system. The rest of this article is organized as follows. Section 2 presents an overview of adaptive OFDM systems. The proposed A-SFBC–OFDM scheme is presented in Section 3. In Section 3, both BER and spectral efficiency of the adaptive and non-adaptive SFBC–OFDM systems are analytically evaluated. Numerical results for the performance evaluation are presented in Section 4. Finally, Section 5 concludes this paper.

2. Adaptive OFDM The allocation of power and bits to the OFDM subcarriers can be uniform or non-uniform. In the uniform case, each subcarrier carries the same number of bits with equal power. On the other hand, in the non-uniform case, bits and power can be allocated in a way to maximize the number of transmitted bits (spectral efficiency) or to minimize the overall probability of error. It has been shown that the bit error probability of OFDM subcarriers over frequency-selective fading channels depends on the frequency response of the channel. In a

frequency-selective fading channel, some subchannels have a deep fade while others have relatively negligible attenuation. The occurrence of bit errors is mostly caused by a set of severely faded subcarriers, while in the rest of the subcarriers, bit errors are observed less often. If the subcarriers having a high BER can be identified and excluded from transmission, the overall BER can be improved at the price of a slight loss of throughput. Since the fading only deteriorates the SNR of certain subcarriers but improves others’, the potential loss of throughput due to the exclusion of faded subcarriers can be mitigated by employing higher-order modulation modes on the subcarriers having high-SNR values. This is the principal idea of AOFDM. Several AOFDM schemes have been proposed in the past. In this section, we review some of the most effective schemes, including AOFDM based on water-filling, AOFDM based on mode switching levels (discrete-rate) [16] and a continuous-rate AOFDM scheme. The term continuous-rate means that the number of bits per symbol is not restricted to integer values. While continuous-rate adaptive modulation is possible [25], discrete-rate adaptive modulation is more practical.

2.1. Adaptation based on water-filling The concept of water-filling can be used for power allocation and then for bit allocation. It says that the optimal power allocation maximizes the channel capacity and it is given by [26,27]  P (f ) =

n (f ) K − |H (f )|2 0

+

f ∈ W, f∈ / W,

(1)

where P (f ) is the signal power density function, H (f ) is the channel frequency response with bandwidth W , and n (f ) is the power spectral density function of the AWGN. K is a constant, determining the total power to be allocated, and [.]+ sets all negative values to zero. K is chosen such  that P (f ) satisfies W P (f ) df Pmax , where Pmax is the maximum power allowed. From (1), more power is allocated at the subchannels with greater channel magnitude or lower noise power. This approach is known as water-filling. A sample output of the bit and power allocation algorithm is shown in Fig. 1 for a frequency-selective fading channel. As can be observed, no bit is allocated to the subchannels corresponding to a deep fade, while more bits are allocated to the subchannels with strong channel gain. Also, it can be seen that subchannel power varies as a function of channel gain and bit allocation. It can be observed that adaptive modulation adapts the transmission parameters to take advantage of prevailing channel conditions. It aims to exploit the variation of the wireless channel by dynamically adjusting certain transmission parameters to changing environmental conditions.

Channel Amplitude

M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

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2 1.5 1 0.5 0 1

8

16

24 32 40 Number of Sub-Carriers

48

56

64

1

8

16

24 32 40 Number of Sub-Carriers

48

56

64

1

8

16

24 32 40 Number of Sub-Carriers

48

56

64

6

Bits

4 2 0

Energy

0.3 0.2 0.1 0

Fig. 1. An example of bit and power allocation.

2.2. Adaptation based on mode-switching level (discrete-rate adaptation) The general idea of adaptive modulation is to choose a set of suitable modulation parameters based on the channel state information (CSI) known at the transmitter. The signal-tonoise ratio (SNR) can be considered as a proper metric that can be obtained from the CSI. In this case, we determine the switching level, which is the lowest required SNR for a given mode corresponding to a given target BER. Then, based on the fading channel condition, i.e., instantaneous SNR range, we select the modulation mode. This provides the largest throughput while maintaining the target BER. For example, consider Fig. 2, that represents a set of BER curves for BPSK, QPSK, 16-QAM, and 64-QAM constellations. The set of adaptation/switching thresholds can be obtained from the closed-form BER expression as a function of SNR or simply by reading the SNR points corresponding to a target BER. For example, for the target BER of 10−4 , the thresholds will be 8.4, 11.4, 18.2, and 24.3 dB, respectively. The goal of this algorithm for adaptive modulation is to ensure that the most efficient mode is always used over varying channel conditions based on a mode selection criterion. In contrast, systems with fixed-mode modulation are designed for the worst-case channel conditions, resulting in insufficient utilization of the full channel capacity.

In Fig. 3, we present the spectral efficiency performance (bits/s/Hz) vs. SNR (dB) for four different uncoded modulation levels referred to as BPSK, QPSK, 16-QAM, and 64QAM. The spectral efficiency was obtained for each modulation scheme by taking into account the need to maintain the target BER of 10−4 . It can be seen that each modulation is optimal for use in different quality regions, and adaptive modulation selects the mode with the highest spectral efficiency for each link.

2.3. Adaptation based on target BER (continuous-rate adaptation) In this section, we review a continuous-rate adaptive bit allocation scheme for the OFDM systems. The goal is to allocate an appropriate number of bits and to choose the suitable modulation mode for transmission in each subcarrier, in order to keep the overall BER performance at a desired level, i.e. target BER. An M-QAM is employed for each subcarrier and [n, k] bits/symbol are assigned for the kth subcarrier in the nth block, where M = 2[n,k] . Also the negligible degradation due to the cyclic prefix in the OFDM has not been considered. The expression for the instantaneous BER of the kth subcarrier in the nth block of the OFDM (square M-QAM with Gray bit mapping on each subcarrier), over a frequency

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100 BPSK QPSK 16-QAM 64-QAM 10–1

BER

10–2

10–3

10–4

10–5

0

5

10 8.4

15 11.4

20

25

30

24.3

18.2 SNR (dB)

Fig. 2. BER for various modulation schemes.

7 64-QAM

Spectral Efficiency (Bits/Sec/Hz)

6

5 16-QAM 4

3 QPSK 2 BPSK 1

0 0

5

10

15

20

25

30

SNR (dB)

Fig. 3. Spectral efficiency for various modulation schemes.

35

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selective-fading channel and with an additive white Gaussian noise (AWGN), can be written as   1 2 1− √ BER[n, k] = [n, k] 2[n,k] ⎞ ⎛

1.5s |H [n, k]|2 ⎠ · erfc ⎝ , (2) 2[n,k] − 1 where s = Es /N0 , Es is the symbol energy at the transmitter and N0 /2 is the variance of the real/imaginary part of the AWGN. The  ∞ erfc(x) is the complementary error function: erfc(x) = x exp(−t 2 ) dt and H [n, k] is the frequency response of the fading channel. As stated before, the goal is to find [n, k] as a function of target BER. As the use of exact erfc(.) function results in complex solutions, in this paper we use the exponential approximation [14,15,28] to express the BER in an “invertible” form, in the sense that it provides a simple expression as a function of the [n, k] and the SNR. An approximate expression for (2) is given by   1.6s |H [n, k]|2 BER[n, k] = 0.2 exp − . (3) 2[n,k] − 1 By inverting (3), the maximum instantaneous data rate [n, k] that can be transmitted under a target BER (BERt ) for a given instantaneous SNR can be represented as ⎛ ⎞ ⎜ 1.6s |H [n, k]|2 ⎟  ⎟  [n, k] = log2 ⎜ 1 + ⎝ ⎠, 0.2 ln BERt

written as BER = EH [n,k] {BER[n, k]},  ∞ BER = BER[n, k]p() d,

(6) (7)

0

where p() is the probability density function of  = s |H [n, k]|2 . Since |H [n, k]| is Rayleigh-distributed, |H [n, k]|2 has a chi-square probability distribution with two degrees of freedom. Consequently,  is also chi-square-distributed according to p() = 1¯ exp(− ¯ ),  0 where ¯ is de-

fined as a function of the average of |H [n, k]|2 , i.e., ¯ = s EH [n,k] {|H [n, k]|2 } = s . Then by substituting (3) into (7) we can obtain [14]   1.6s −1 BER = 0.2 1 +  . (8) 2 −1 We can invert (8) to express  as a function of s and the BERt as ⎛ ⎞

⎜  = log2 ⎜ ⎝1 + 

⎟ 1.6s ⎟ ⎠. 0.2 −1 BER t

(9)

The average spectral efficiency, in this case is equal to .

3. Adaptive modulation for SFBC–OFDM (4)

which is a simple closed-form expression and provides insights that are unattainable with more complicated BER expressions. The term continuous-rate adaptive modulation which means that the number of bits per symbol is not restricted to integer values, responds to the instantaneous channel fluctuation by varying the number of allocated bits according to (4). The average spectral efficiency (number of bits per second per Hz) can be written as R = EH [n,k] {[n, k]},

525

(5)

where the term EH [n,k] {.} denotes the average operator and the expectation is taken over all possible channel realizations.

2.4. Non-adaptive OFDM In order to compare the spectral efficiency of AOFDM (the scheme explained in subsection 2.3) with that of nonadaptive OFDM, we evaluate the spectral efficiency of non-adaptive OFDM systems. In the case of non-adaptive modulation for OFDM, the same number of bits is allocated to each subcarrier, i.e., [n, k] = . The average BER can be

As stated earlier, adaptive modulation is a well-known and powerful technique for increasing the spectral efficiency and improving the system performance of the OFDM systems. The performance improvement offered by adaptive modulation over non-adaptive systems is remarkable. Furthermore, other dimensions such as frequency and space may yield further gains by providing additional degrees of freedom that can be exploited by adaptive modulation. Antenna selection has been considered and studied as an efficient technique for improving performance and for reducing the complexity. Recently, there has been a growing interest in applying the antenna-selection technique to multiple input/multiple output (MIMO) systems [29–34]. In [30] an interesting antenna-selection technique is presented for the application of MIMO systems over flat fading channels, when space–time coding is used. In this article, we propose a new space–frequency block coded OFDM scheme called A-SFBC–OFDM, in conjunction with transmit antenna selection and adaptive modulation that can improve the overall system performance.

3.1. System model Fig. 4 shows a block diagram of the proposed system. The channel state information can be estimated at the receiver

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Fig. 4. Block diagram of the proposed system.

and then fed back to the transmitter. We assume that the estimation process and feedback channel are perfect. We also assume that the subcarrier allocation information is sent to the receiver via a dedicated control channel. Antenna selection maximizes the signal-to-noise ratio and, therefore, minimizes the probability of error. The basic concept of antenna selection is to transmit each subcarrier using the antenna which has the smallest attenuation for that subcarrier. Given that KT transmit antennae are available, we select MT out of KT antennae (MT KT ), whose subchannel amplitudes are larger than the rest. Considering a multiple input/single output (MISO) system with KT transmit antennae and one receive antenna, the kth subchannel response in the nth block, i.e., H[n, k] is a KT × 1 vector, given by H[n, k] = (H1 [n, k] H2 [n, k] · · · HKT [n, k])T , where Hj [n, k] is the DFT (size N) of the channel response between the jth transmit and the receive antennae, hj (n), j = 1, 2, . . . , KT . The superscript T in (.)T denotes the transpose operator. In order to minimize the average BER, we can maximize the average SNR that is equivalent to selecting MT out of KT rows of H[n, k]. Therefore, the optimal transmission is to transmit on the transmit antennae corresponding to the MT rows with the highest Frobenius norms [30]. Thus, the equivalent selected subchannel vector will be a MT × 1 vector,

 [n, k])T . given by H [n, k] = (H1 [n, k] H2 [n, k] · · · HM T Note that Hi [n, k] is equal to some Hj [n, k], for each i ∈ {1, 2, . . . , MT } and for some j ∈ {1, 2, . . . , KT }. Based on the characteristics of the selected subchannels, i.e., Hi [n, k], the suitable modulation scheme and the corresponding number of bits can be determined. This bit allocation can be done using water-filling. Here, we consider a simpler method of adaptive bit allocation where an average target BER is set. The size of the constellation for each subcarrier is determined based on the target BER. Therefore, a signal vector S(n) = (s[n, 0], s[n, 1], s[n, 2], . . . , s[n, Nt − 1])T is provided as the input for the SFBC system, where Nt is equal to N multiplied by the code rate of SFBC system (Rc ). A space–time block code is defined by a (q × MT ) transmission matrix G given by ⎞ ⎛ g1,1 g1,2 · · · g1,MT ⎜ g2,1 g2,2 · · · g2,M ⎟ T ⎟ ⎜ , (10) G=⎜ . .. .. ⎟ ⎝ .. . ··· . ⎠ gq,1 gq,2 · · · gq,MT

where each element gj,i is a proper linear combination of a subset of elements of S(n) and their conjugates, as G is an orthogonal design.

M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

In order to utilize the space–frequency diversity, the input blocks for OFDM at each transmit antenna should be of length N. SFBC provides MT blocks, S1 (n), S2 (n), . . . , SMT (n) each of length N and consisting of N/q sub-blocks, i.e., Si (n)=(si [n, 0] si [n, 1] · · · si [n, Nq −

1])T for i = 1, 2, . . . , MT , where each sub-block si [n, k  ], (k  = 0, 1, . . . , Nq − 1), of length q, is simply equal to the transpose of the ith column of the transmission matrix G for different values of k .1 Then the OFDM modulators generate blocks X1 (n), X2 (n), . . . , XMT (n) that are transmitted by the first, second, . . ., and MT th transmit antennae simultaneously. In order to avoid ISI, the guard time interval is chosen to be longer than the largest delay spread of the multipath channel. The received signal will be the convolution of the channel and the transmitted signal. We assume that the fading process remains static during each OFDM block (one time-slot) and that it varies from one block to another. The fading processes associated with different transmit-receive antenna pairs are considered to be uncorrelated. We also assume a perfect synchronization that implies the MT receive signal components (from the MT antennae) arriving with negligible differences in their path delays. After removing the cyclic prefix and FFT at the receiver side, the demodulated received signal can be expressed as

527

for codes G2 , H3 (and H4 ), and G3 (and G4 ), respectively [23]). This will be a reasonable assumption when the OFDM block size, N, is large enough. One may conclude that the adaptive modulation in STBC and SFBC–OFDM systems will be a subband-by-subband basis adaptation, instead of a subcarrier-by-subcarrier basis adaptation.

3.2. Adaptive SFBC–OFDM with subcarrier-by-subcarrier basis antenna selection In this section, we determine adaptive bit allocation for maximizing the spectral efficiency of SFBC–OFDM employing antenna selection (selecting MT out of KT transmit antennae). We assume that M-QAM is employed for each subcarrier and [n, k] bits/symbol are assigned for the kth subcarrier in the nth block, and M = 2[n,k] . The negligible degradation due to the cyclic prefix in OFDM is not being considered. Consider an SFBC system employing MT transmit antennae and one receive antenna. Similar to the expression in [36] for the received signal in STBC, the received signal for SFBC–OFDM after SFBC decoding can be written as s˜ [n, k] = C

MT 

|Hi [n, k]|2 s[n, k] + [n, k],

(12)

i=1

r(n) =

MT 

Hi (n)Si (n) + W(n),

(11)

i=1

where r(n) = (r[n, 0], . . . , r[n, N − 1])T . W(n) = (W [n, 0], . . . , W [n, N − 1])T denotes the AWGN, and Hi (n) represents a diagonal matrix whose elements (Hi [n, k], i = 1, 2, . . . , MT , k = 0, 1, . . . , N − 1) are the gains of the selected subchannels. Knowing the channel information at the receiver, the maximum likelihood (ML) decoding can be used for SFBC decoding of the received signal. At the end, the elements of block Nt −1 T ˜ S(n)=({˜ s [n, m]}m=0 ) are demodulated to extract the data. The encoding and decoding of the SFBC–OFDM system with some examples have been illustrated in [35]. Note that by employing adaptive modulation, different modulation modes may be assigned to N/q sub-blocks inside the block Si (n), while the same modulation mode will be assigned to the corresponding q signals inside each subblock si [n, k  ]. This is because in SFBC (similar to STBC) -for the purpose of SFBC decoding-we assume that the channel gains of q adjacent subchannels are approximately equal (q is the period of STBC and it is equal to 2, 4 and 8 1 For

example, in the well-known STBC code G2 (Alamouti’s code) [22], where q = 2, MT = 2, sub-blocks s1 [n, k  ] and s2 [n, k  ] can be simply constructed as s1 [n, k  ] = (s[n, 2k  ] − s ∗ [n, 2k  + 1]) and s2 [n, k  ] = (s[n, 2k  + 1] s ∗ [n, 2k  ]), for k  = 0, 1, . . . , N2 − 1. Therefore, the output of SFBC encoder will be  T S1 (n)= s[n, 0] − s ∗ [n, 1] · · · s[n, N − 2] − s ∗ [n, N − 1] and   T S2 (n) = s[n, 1] s ∗ [n, 0] · · · s[n, N − 1] s ∗ [n, N − 2] .

where Hi [n, k] is the selected subchannel associated with ith transmit antenna, [n, k] is a complex Gaussian noise with MT  2 0 distribution N(0, CN i=1 |Hi [n, k]| ) per dimension and 2 C is a constant that depends on the STBC coding matrix; for example, C = 1 for codes G2 , H3 , and H4 , in [23] and C = 2 for G3 and G4 in [23]. The total energy of the symbol transmitted through the MT antennae can be normalized to MT and, therefore, similar to the expression in [36], we can express the instantaneous SNR per symbol at the receiver as =

MT s  |Hi [n, k]|2 , MT Rc

(13)

i=1

where s = Es /N0 is the average SNR per receive antenna. Similar to the conventional OFDM (2), we can express the instantaneous BER of each subcarrier of M-QAMSFBC–OFDM over a frequency-selective fading channel as   2 1 BER[n, k] = 1− √ [n, k] 2[n,k] ⎛

⎞ MT  1.5s i=1 |Hi [n, k]|2 ⎠. · erfc ⎝ (14) MT Rc (2[n,k] − 1) This can be approximated as    T  2 1.6s M |H [n, k]| i=1 i BER[n, k] = 0.2 exp − . MT Rc (2[n,k] − 1)

(15)

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M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

By inverting (15), the suitable modulation scheme and the corresponding number of bits can be calculated from ⎛ ⎞ MT  ⎜ 1.6s i=1 |Hi [n, k]|2 ⎟  ⎟  [n, k] = log2 ⎜ (16) ⎝1 + ⎠. 0.2 MT Rc ln BERt The average modulation throughput (bits/s/Hz) will be given by R =EH  [n,k] {[n, k]}. Taking the SFBC code rate into account, the effective average spectral efficiency will be equal to Reff = Rc R in terms of bits/s/Hz.

3.3. Non-adaptive SFBC–OFDM (without antenna selection) In the case of non-adaptive SFBC–OFDM and without antenna selection, and assuming MT available transmit antennae, the average BER can be written as BER = EH1 [n,k],...,HMT [n,k] {BER[n, k]}.

(17)

Since |Hi [n, k]|, i = (1, . . . , MT ) is i.i.d. Rayleighdistributed, |Hi [n, k]|2 has a chi-square probability distribution with two degrees of freedom. Consequently, i = s |Hi [n, k]|2 is also chi-square-distributed with the probability density function as follows:   1 i p(i ) = exp − , i 0 (18) ¯ i ¯ i and since  BER =



∞

Similarly, we can express the average SNR gain obtained from antenna selection as  M T  1  2 ¯ = (24) G E |Hi [n, k]| . AS MT i=1

Finally, for the comparison of these two gains we define    0.2 2 MT − 1 MT ¯ G BER  t K = AM =  (25) . ¯ MT  0.2 G 2 AS E ln |H [n, k]| i=1 i BERt

BER[n, k] 0

· p(1 ) . . . p(M )d1 . . . dM , T

T

substituting (15) and (18) into (19 ) we obtain  −MT 1.6s BER = 0.2 1 + . MT Rc (2 − 1)

(19)

4. Simulation results (20)

We can invert (20) to express  as a function of s and the target BER ⎛ ⎞ ⎜ ⎜  = log2 ⎜ ⎜1 + ⎝

Therefore, the average SNR gain obtained from adaptive modulation will be  0.2 MT −1 BERt ¯ = E{G } =  MT .  G (23) AM AM 0.2 ln BERt

If K 1, we can conclude that adaptive modulation provides a higher SNR gain than does antenna selection.

∞

... 0

with adaptive modulation). We also assume that there are MT available transmit antennae. Therefore, [n, k] will be the same as (16), but Hi [n, k]=Hi [n, k]. Now, comparing the resulting [n, k], with  from the non-adaptive SFBC–OFDM case (21), we can express the SNR gain obtained from adaptive modulation as  0.2 MT −1 MT BERt   GAM = |Hi [n, k]|2 . (22) 0.2 i=1 ln BERt

MT Rc

1.6  s 0.2 M T

BERt

⎟ ⎟ ⎟ ⎟. ⎠ −1

(21)

In this case, all subcarriers use the same modulation mode.

3.4. Adaptive modulation vs. antenna selection Here, we compare the gains obtained from adaptive modulation (AM) and antenna selection (AS). In order to find the SNR gain obtained from adaptive modulation, we consider an A-SFBC–OFDM without antenna selection (only

The performance of the proposed system is evaluated by computer simulation over a frequency-selective fading channel. The considered fading channel is a multipath fading channel with coherence bandwidth smaller than the total bandwidth of the multicarrier system and thus seen as frequency-selective fading. The fading process is assumed to be stationary and slowly varying compared to the symbol duration of the multicarrier signal, such that it is approximately constant during one-OFDM block length. The fading process impulse response at the antenna j (j =1, 2, . . . , KT ) can be expressed as hj (t) = L−1 l=0 l,j (t)(t − l (t)) [26], where the tap weight l,j (t) is a complex Gaussian random process with zero mean and variance 1/L (equal power) and l (t) is the time delay of the lth path and L is the total number of resolvable paths (L = 4). With this model we have assumed that the path delays, l (t), are multiples of the symbol duration Ts . The OFDM system includes N = 512 subcarriers and a cyclic prefix which is longer than the channel delay. It is assumed that the channel state information is available at the receiver and transmitter.

M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

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14 AOFDM Target-BER = 10–3 AOFDM Target-BER = 10–4 AOFDM Target-BER = 10–5 AOFDM Target-BER = 10–6 OFDM Target-BER = 10–3 OFDM Target-BER = 10–4 OFDM Target-BER = 10–5 OFDM Target-BER = 10–6

Average Spectral Efficiency (Bits/sec/Hz)

12

10

8

6

4

2

0 0

5

10

15

20

25 30 SNR (dB)

35

40

45

50

Fig. 5. Average spectral efficiency of non-adaptive OFDM and AOFDM.

The average spectral efficiencies of AOFDM and nonadaptive OFDM were expressed in (5) and (9). They are compared in Fig. 5 for different target BERs. It can be seen that adaptive OFDM can greatly improve the performance of OFDM. For example, at a target BER of 10−3 and a spectral efficiency of 6 bits/s/Hz about 14 dB gain can be obtained, and at a target BER of 10−5 and a spectral efficiency of 2 bits/s/Hz about 32 dB gain can be obtained. Using the expressions in (16) and (21), we can compare the throughput of A-SFBC–OFDM and non-adaptive SFBC–OFDM. Fig. 6 compares the average throughput of the SFBC–OFDM with that of A-SFBC–OFDM, both employing code G4 (MT = 4, Rc = 0.5). The results indicate that A-SFBC–OFDM is better than SFBC–OFDM in terms of throughput for the given target BER. For example, at a throughput of 6 bits/s/Hz, gains of 7.65, 9.18, and 10.65 dB can be obtained when KT is 4, 6, and 10, respectively. Furthermore, it can be seen that the SNR gain (7.65 dB) obtained from adaptive modulation only (no selection) is much higher than that obtained from antenna selection (1.53 dB, when selecting 4 out of 6 antennae). Fig. 7 compares the throughput of the SFBC–OFDM with that of A-SFBC–OFDM, both employing code H4 (MT = 4, Rc = 0.75). As shown, A-SFBC–OFDM is better than SFBC–OFDM in terms of throughput for the given target BER. For example, at a throughput of 6 bits/s/Hz, gains of 7.66, 9.19, and 10.42 dB can be obtained when KT is 4, 6, and 10, respectively. Previous examples indicated the continuous-rate adaptation, where the number of bits/symbol is not restricted

to integer values. In discrete-rate adaptation, numbers of bits/symbol are integer values. Herein, we consider a discrete-rate version of A-SFBC–ODFDM, where discreterate adaptive modulation responds to the instantaneous SNR (13) by varying the constellation size. We divide the SNR range into a number of regions, each representing a constellation size. As stated earlier, the set of switching thresholds can be obtained from the required instantaneous SNR to achieve the target BER over an AWGN channel. For example, as shown in Fig. 3 for a target BER of 10−4 , the thresholds are 8.4, 11.4, 18.2, and 24.3 dB for BSPK (1 bit), 4-QAM (2 bits), 16-QAM (4 bits), and 64-QAM (6 bits), respectively. For SNR<8.4 dB, zero bit will be allocated (no Transmission). Fig. 8 shows the average throughput curves of continuous-rate (C–R) and discreterate (D–R) A-SFBC–OFDM (2Tx–1Rx) (with and without antenna selection), using the code G2 (MT = 2, Rc = 1) for the target BER of 10−4 . It can be observed that in terms of throughput, A-SFBC–OFDM with no antenna selection is better than the non-adaptive case, and employing antenna selection in A-SFBC–OFDM provides further improvement. Also, it can be seen that the achievable throughput of discrete-rate A-SFBC–OFDM comes within about 3 dB of the continuous-rate A-SFBC–OFDM, whereas non-adaptive SFBC–OFDM suffers a larger throughput penalty. On the other hand, the throughput curve corresponding to discreterate A-SFBC–OFDM (employing antenna selection) undulates dramatically and exhibits a staircase-like shape. This is because employing antenna selection increases the

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M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533 12 4 out of 10 Tx, A-SFBC-OFDM-G4 (4Tx-1Rx) Target-BER = 10–6 4 out of 6 Tx, A-SFBC-OFDM-G4 (4Tx-1Rx) Target-BER = 10–6 4 out of 4 Tx, A-SFBC-OFDM-G4 (4Tx-1Rx) Target-BER = 10–6 Non-Adaptive SFBC-OFDM-G4 (4Tx-1Rx) Target-BER = 10–6

Throughput (Bits/Sec/Hz)

10

8

6

4

2

0 0

5

10

15

20

25

30

35

40

SNR (dB)

Fig. 6. Average throughput of non-adaptive SFBC–OFDM (4Tx–1Rx) and A-SFBC–OFDM (4Tx–1Rx) using code G4 .

12 4 out of 10 Tx, A-SFBC-OFDM-H4 (4Tx-1Rx) Target-BER = 10–6 4 out of 6 Tx, A-SFBC-OFDM-H4 (4Tx-1Rx) Target-BER = 10–6 4 out of 4 Tx, A-SFBC-OFDM-H4 (4Tx-1Rx) Target-BER = 10–6 Non-Adaptive SFBC-OFDM-H4 (4Tx1-Rx) Target-BER = 10–6

Throughput (Bits/Sec/Hz)

10

8

6

4

2

0 0

5

10

15

20

25

30

35

40

SNR (dB)

Fig. 7. Average throughput of non-adaptive SFBC–OFDM (4Tx–1Rx) and A-SFBC–OFDM (4Tx–1Rx) using code H4 .

M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

531

10 CR CR DR DR CR

9

Throughput (Bits/Sec/Hz)

8 7

ASFBCOFDM (2 out of 8 Tx Selection) ASFBCOFDM (No Selection) ASFBCOFDM (2 out of 8 Tx Selection) ASFBCOFDM (No Selection) Non Adaptive SFBCOFDM

Continuous Rate Adaptive

6 5 Discrete Rate Adaptive

4 3 2 Continuous Rate Non Adaptive

1 0 0

5

10

15

20

25

30

35

40

SNR (dB)

Mode Selection Probability

Fig. 8. Average throughput of continuous-rate (C–R) and discrete-rate (D–R) A-SFBC–OFDM (2Tx–1Rx) using code G2 and with 5 switching modes (64-QAM, 16-QAM, QPSK, BPSK, no Transmission (no TX)) of D–R case for a target BER of 10−4 .

1 0.8 No TX, 0 bit BPSK, 1 bit QPSK, 2 bits 16-QAM, 4 bits 64-QAM, 6 bits

0.6 0.4 0.2 0

Mode Selection Probability

0

5

10

15

20 SNR (dB)

25

30

35

40

1 0.8 No TX, 0 bit BPSK, 1 bit QPSK, 2 bits 16-QAM, 4 bits 64-QAM, 6 bits

0.6 0.4 0.2 0 0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 9. Mode selection probability of D-R A-SFBC–OFDM (2Tx–1Rx) with 5 switching modes (64–QAM, 16–QAM, QPSK, BPSK, no Transmission (no TX)) for a target BER of 10−4 , (a) without antenna selection, (b) with selection of 2 out of 8 Tx antennae.

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M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

30 AM, No Selection AM, AS, 2 out of 4 Tx AM, AS, 2 out of 8 Tx 25

GAM /GAS

20

15

10

5

0 100

101 BER t

102

/ 10–5

Fig. 10. The ratio of average SNR obtained from adaptive modulation over that of antenna selection in SFBC–OFDM (2Tx–1Rx).

antenna diversity gain and makes the effective subchannel, H [n, k], a less severe fading channel. Therefore, the mode selection becomes more “discriminating” in comparison with the no antenna selection case. Note that in the extreme case of an AWGN-like channel, the throughput curve will be similar to the curve in Fig. 3. For further explanation of this phenomenon, we show the corresponding mode selection probabilities in Fig. 9, with the same parameter settings for discrete-rate A-SFBC–OFDM as explained in Fig. 8. By comparing Fig. 9(a) with (b), we can see that antenna selection increases the probability of the most appropriate modulation mode at a certain channel SNR and causes the corresponding curve to exhibit a narrower shape. This is clearly demonstrated in the Fig. 9(b) by the increased peaks of each modulation mode, more specifically around SNRs of 12 and 18 dB for QPSK and 16-QAM modes, respectively, where the staircase-like behavior happens in Fig. 8. This explains why the throughput curve for the antenna selection case increases in a staircase-like manner. This problem may be mitigated by introducing power-efficient 8-QAM and 32-QAM modes [16] because our scheme does not use 3 and 5 BPS modulation modes in the set of constituent modulation modes employed. Finally, Fig. 10 shows the comparison of SNR gain obtained from adaptive modulation and from antenna selection in the SFBC–OFDM system, for the range of target BER ¯ AM /G ¯ AS of 10−5 to 10−3 . It can be seen that the ratio of G is greater than one. Therefore, we can conclude that, in this

example, adaptive modulation provides a higher SNR gain than antenna selection does for the SFBC–OFDM systems.

5. Conclusion In this article, a new scheme of adaptive modulation for SFBC–OFDM aided antenna selection is presented. The spectral efficiency advantage of the proposed system is examined. Both continuous-rate and discrete-rate adaptive modulation schemes have been considered. It is shown that antenna selection with adaptive modulation can greatly improve the performance of conventional SFBC–OFDM systems. This improvement is provided on the assumption of perfect channel estimation at the receiver and perfect feedback (noise-free, and delayless) to the transmitter. From a practical point of view, the effects of channel estimation errors and feedback delay on the overall performance of the presented system is currently under investigation.

References [1] Hayes JF. Adaptive feedback communications. IEEE Trans Commun Technol 1968;16:29–34. [2] Goldsmith A, Varaiya P. Capacity of fading channels with channel side information. IEEE Trans Inf Theory 1997;43:1986–92. [3] Alouini M-S, Goldsmith A. Capacity of rayleigh fading channels under different adaptive transmission and

M. Torabi / Int. J. Electron. Commun. (AEÜ) 62 (2008) 521 – 533

[4] [5]

[6]

[7] [8]

[9] [10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18]

[19]

[20]

[21]

diversity-combining techniques. IEEE Trans Veh Technol 1999;48:1165–81. Goldsmith A, Chu S. Variable-rate variable-power MQAM for fading channels. IEEE Trans Commun 1997;45:1218–30. Torrance J, Hanzo L. Optimisation of switching levels for adaptive modulation in slow rayleigh fading channel. Electronic Letters 1996;32:1167–9. Choi BJ, Hanzo L. Optimum mode-switching assisted adaptive modulation. In: Proceedings of IEEE global telecommunication Conference (GLOBECOM’01), vol. 6, 2001, p. 3316–20. Hanzo L, Web W, Keller T. Single and multicarrier quadrature amplitude modulation. New York, USA: Wiley; 2000. Keller T, Hanzo L. Adaptive muticarrier modulation: a convenient frame work for time–frequency processing wireless communications. Proc IEEE 2000;88:611–40. Kalet I. The multitone channel. IEEE Trans Commun 1989;37:119–24. Chow PS, Cioffi JM, Bingham JA. A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channel. IEEE Trans Commun 1995;43:773–5. Czylwik A. Adaptive OFDM for wideband radio channels. In: Proceedings of IEEE global telecommunication conference (GLOBECOM’96), 1996. p. 713–8. Souryal MR, Pickholtz RL. Adaptive modulation with imperfect channel information in OFDM. In: Proceedings of IEEE international conference on communications, (ICC’01), vol. 6, 2001. p. 1861–5. Chung ST, Goldsmith A. Adaptive multicarrier modulation for wireless systems. In: Proceedings of the 34th Asilomar conference on signals, systems and computers, vol. 2, 2000. p. 1603–7. Ye S, Blum RS, Cimini LJ. Adaptive modulation for variablerate OFDM systems. In: Proceedings of IEEE on vehicular technology conference (VTC’02 Spring), vol. 2, 2002. p. 767–1. Chung T, Goldsmith A. Degrees of freedom in adaptive modulation: a unified view. IEEE Trans Commun 2001;49: 1561–71. Hanzo L, Wong CH, Yee MS. Adaptive wireless transceivers. New York, USA: Wiley; 2002. Fan P, Li J, Cao Z. Transmit diversity scheme with power control for wireless communications. IEICE Trans Commun 2001;E84-B:1720–6. Catreux S, Erceg V, Gesbert D, Heath RW. Adaptive modulation and MIMO coding for broadband wireless data networks. IEEE Commun Mag 2002;40:108–14. Liew TH, Hanzo L. Space–time block coded adaptive modulation aided OFDM. In: Proceedings of IEEE global telecommunication conference (GLOBECOM’01) vol. 1, 2001. p. 136–40. Torabi M, Soleymani M. Adaptive bit allocation for space–time block coded OFDM system. In: Proceedings of IEEE international conference on acoustics, speech, and signal processing, (ICASSP’03), vol. 4, 2003. p. 409–12. Torabi M, Soleymani M. Adaptive OFDM with space–time coding and antenna selection for broadband wireless communications. In: Proceedings of IEEE international symposium on information theory, (ISIT-2003), 2003, p. 35.

533

[22] Alamouti SM. A simple transmit diversity technique for wireless communications. IEEE J Select Areas Commun 1998;16:1451–8. [23] Tarokh V, Jafarkhani H, Calderbank AR. Space–time block codes from orthogonal designs. IEEE Trans Inf Theory 1999;45:1456–67. [24] Tarokh V, Jafarkhani H, Calderbank AR. Space–time block coding for wireless communications: performance results. IEEE J Select Areas Commun 1999;17:451–60. [25] Forney JGD, Gallager RG, Lang GR, Longstaff FM, Qureshi SU. Efficient modulation for band-limited channels. IEEE J Select Areas Commun 1984;SAC-2:632–46. [26] Proakis JG. Digital communications. New York, USA: McGraw-Hill; 2000. [27] Gitlin RD, Hayes J, Weinstein SB. Data communications principles. New York, USA: Plenum Press; 1992. [28] Foschini GJ, Salz J. Digital communications over fading radio channels. Bell System Tech J 1983;62:429–56. [29] Win M, Winters J. Virtual branch analysis of symbol error probability for hybrid selection/maximal-ratio combining in rayleigh fading. IEEE Trans Commun 2001;49:1926–34. [30] Gore D, Paulraj A. MIMO antenna subset selection with space–time coding. IEEE Trans Signal Process 2002;50: 2580–8. [31] Blum RS, Winters JH. On optimum MIMO with antenna selection. IEEE Commun Lett 2002;6:322–4. [32] Zeng XN, Ghrayeb A. Performance bounds for space–time block codes with receive antenna selection. IEEE Trans Inf Theory 2004;50:2130–7. [33] Hui S, Katayama M, Yamazato T, Okada H. An adaptive antenna selection scheme for transmit diversity in OFDM systems. In: Proceedings of IEEE vehicular technology conference (VTC’01 Fall), vol. 4, 2001. p. 2168—72. [34] Bahceci I, Duman T, Altunbasak Y. Performance of MIMO antenna selection for space-time coded OFDM systems. In: Proceedings of IEEE wireless communication and networking conference WCNC’04 2004;2:987–92. [35] Torabi M. Multicarrier systems with antenna diversity for wireless communications. Dissertation, Electrical and Computer Engineering Department, Concordia University, Montreal, Canada, 2004. [36] Femenias G. BER performance of linear STBC from orthogonal designs over MIMO correlated nakagami-m fading channel. IEEE Trans Veh Technol 2004;53:307–17. Mohammad Torabi received the Ph.D. degree in electrical engineering from Concordia University, Montreal, QC, Canada, in 2004. He was a Post-Doctoral Fellow in Electrical and Computer Engineering Department at Concordia University. Currently he is with the Department of Electrical Engineering, École Polytechnique de Montréal, Montreal, QC, Canada as a Post-Doctoral Fellow. His research interests include digital signal processing, communication theory and wireless communications with emphasis on OFDM, adaptive modulation, resource allocation, MIMO systems and space time coding.