Multiple Antenna Systems

Multiple Antenna Systems
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Dr.MohitBansal,Canada,Teacher
Published Date:25-10-2017
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Multiple Antenna Systems Multiple antenna systems became popular roughly a decade ago as a result of the funda- mental work of Alamouti (1998), Foschini (1996), Foschini and Gans (1998), Kuhn ¨ and Kammeyer (2004), Seshadri and Winters (1994), Seshadri et al. (1997), Tarokh et al. (1998), Telatar (1995), Wittneben (1991), Wolniansky et al. (1998), and many others. The reason for the great interest is that multiple antennas offer an efficient way to increase the spec- tral efficiency of mobile radio systems by exploiting the resource space. After their large potential has been widely recognized, they found their way into several standards. As an example, very simple structures can be found in Release 99 of UMTS systems (Holma and Toskala 2004). More sophisticated methods are under discussion for further evolutions (3GPP 2005a; Hanzo et al. 2002b). This chapter gives a brief overview of different multiple-input multiple-output (MIMO) strategies for point-to-point communications without claiming to be comprehensive. Mul- tiuser scenarios are briefly discussed in Section 2.4. After the introduction, Section 6.2 addresses spatial diversity concepts. Starting with simple receive diversity, different possi- bilities to obtain a diversity gain with multiple transmit antennas are discussed. While these techniques improve the link reliability, multilayer transmission presented in Section 6.3 multiply the data rate without increasing the signal bandwidth. Linear dispersion (LD) codes introduced in Section 6.4 represent a comprehensive description of space–time cod- ing (STC) and multilayer transmission and allow optimal trade-offs between diversity and multiplexing gains. Finally, the high potential of multiple antenna systems is illustrated by looking at the channel capacity in Section 6.5. 6.1 Introduction There exist a multitude of reasons for using multiple antenna systems. This section gives a brief overview of different strategies without claiming to be comprehensive. Principally, two different categories can be distinguished. The first objective is to improve the link reliability, that is, the ergodic error probability or the outage probability are reduced. This can be accomplished by enhancing the instantaneous signal-to-noise ratio (SNR) (beamforming)276 MULTIPLE ANTENNA SYSTEMS or by decreasing the variations of the SNR (diversity). If multiple access or cochannel interference in cellular networks disturbs the transmission, interferers that are separable in space can be suppressed with multiple antennas, resulting in an improved signal to interference plus noise ratio (SINR). Section 6.3 is restricted to diversity techniques for reducing SNR variations. The second objective discussed in Section 6.3 is to multiply the data rate by transmitting several data streams simultaneously over different antennas. This approach is denoted as space division multiple access (SDMA) and can certainly be combined with other multiple access schemes. Since bandwidth became a very valuable and expensive resource, using the space for increasing data rates without expanding the bandwidth is very attractive. Moreover, we will see in Section 6.5 dealing with channel capacity aspects that the potential capacity gain of multiple antenna systems is much larger than the gain obtained by simply increasing the transmit power. A technique termed multi-stratum codes offers a combination of diversity and multiplexing gains (Bohnke ¨ et al. 2004a,b,c; Wachsmann 2001). The manner in which multiple antennas should be used depends on the properties of the channel, especially on the rank r of H or its covariance matrix  .As anexample, HH we know from Section 1.5 that correlation among the subchannels reduces the diversity gain. In the case of a strong line-of-sight component (Rice fading), diversity is also not an appropriate means because fading is not a severe problem. If we can exploit other sources of diversity, for example, frequency diversity with the Rake receiver or time diversity due to coding over time-varying channels, we are probably already close to the additive white Gaussian Noise (AWGN) performance and little can be gained by a further increase of the diversity degree. In each of these cases, multiple antennas should be used in a different way. If we look for spatial multiplexing, we know from Section 2.3 that we need a channel whose rank is larger than one. Otherwise, we cannot reliably transmit parallel data streams. Hence, for highly correlated channels with a rank r= 1, beamforming that exploits only the strongest eigenmode of a channel would be an appropriate choice instead of multilayer transmission. Therefore, the manner in which multiple antennas are used has to be properly adapted to the general propagation conditions. In order to simplify notation, this chapter is restricted to frequency-nonselective chan- nels. Hence, the impulse response h , κ between transmit antenna µ and receive antenna ν,µ ν reduces to scalar coefficients h  and the channel matrix in (1.33) of Section 1.2.4 ν,µ becomes H= H, 0. Figure 6.1 illustrates the resulting structure of the communication system. The received signal can be described by y= H· x+ n. (6.1) h  1,1 n  1 h  2,1 x  y  1 1 n  2 x  y  2 2 h  2,N T n  N R h  x  N ,N y  N R T N R T Figure 6.1 Structure of MIMO channelMULTIPLE ANTENNA SYSTEMS 277 Some extensions for frequency-selective environments are discussed in Al-Dhahir (2001), Al-Dhahir and Sayed (2000), Wubben ¨ (2006), Wubben ¨ and Kammeyer (2003), Wubben ¨ et al. (2003b). 6.2 Spatial Diversity Concepts This section addresses the application of multiple antennas at the receiver and/or the trans- mitter for the purpose of increasing the diversity degree. As already mentioned, only frequency-nonselective channels are considered for notational as well as conceptual simplic- ity. Another reason is that spatial diversity concepts achieve the highest gains for channels that do not provide diversity in other dimensions such as frequency or time. Moreover, only Rayleigh fading channels without a line-of-sight component are considered. Since we know that correlations among the contributing channels reduce the diversity gain (see Section 1.5) we further assume that the channels are totally uncorrelated. For Rice and correlated fading channels, refer to the theoretical results presented in Section 1.5. Uncorrelated channels can be achieved by an appropriate antenna spacing depending on the spatial channel characteristics, for example, the angle spread. Assuming a uniform linear array with equidistantly arranged antennas and an isotropic scattering environment where signals impinge from all directions with the same probability, a small distance d= λ/2 between neighboring elements may be sufficient. The parameter λ denotes the wavelength and is related to the carrier frequency f by λ= c/f where c describes the speed of light. 0 0 On the contrary, d λ/2 must hold in scenarios with small angle spread and d can take values up to 10λ. This obviously requires a device large enough to host several antennas with appropriate distances. This section is divided into three parts: First, receive diversity is shortly explained. Next, orthogonal space–time block codes (STBCs) are addressed. Nonorthogonal block codes are not considered here and the interested reader is referred to Bossert et al. (2000, 2002), Gabidulin et al. (2000), Lusina et al. (2001, 2003, 2002). In the last subsection, space–time trellis codes providing an additional coding gain are introduced. An overview of space–time coding can also be found in Liew and Hanzo (2002). 6.2.1 Receive Diversity The simplest method to achieve spatial diversity is to use multiple antennas at the receiver. The structure of the system is depicted in Figure 6.2. It can be mathematically described with y= h· x+ n (6.2) T where h= h , ..., h  comprises all contributing channel coefficients. Since 1 N R there is no interference, a simple matched filter performing maximum ratio combining represents the optimum receiver and we obtain H h r= · y= x+˜ n (6.3) 2 h H 2 where n˜= h · n/h denotes the noise at the matched filter output.278 MULTIPLE ANTENNA SYSTEMS n  1 h  1 y  1 n  2 y  h  2 2 x n  N R h  N R y  N R Figure 6.2 Structure of receive diversity system Comparing (6.3) with the theoretical result from Section 1.5, we recognize that the full diversity degree D= N is achieved as long as the channel coefficients remain uncorre- R lated. The single-input multiple-output (SIMO) channel is transformed by matched filtering into an equivalent single-input single-output system (SISO) channel with smaller variations of the SNR. The only difference between (6.3) and (1.104) affects the total received power √ √ that has not been normalized to E /T (missing factor D= N ). The reason for this is s s R that the application of multiple receive antennas yields not only a diversity gain but also an array gain because the N -fold power is collected, leading to a gain of 10 log (N ) dB. R R 10 This gain is independent of diversity considerations and is also available for totally cor- related channels. Since Section 1.5 illustrates only the diversity effect, this array gain was suppressed by normalizing the received power. Hence, we always have to look carefully at the definition of the SNR when multiple antennas are applied. Figure 6.3 illustrates this difference by showing the results known from Section 1.5. In Figure a, the total transmitted energy per information bit is fixed at E . Hence, it is b b) SNR per receive antenna a) SNR per bit 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 0 5 10 15 20 −10 −5 0 5 10 15 20 E /N in dB→ E /N in dB→ b 0 s 0 Figure 6.3 Performance of receive diversity for BPSK and uncorrelated Rayleigh fading channels; bold dashed line denotes AWGN channel (‘◦’: N = 1, ‘×’: N = 2, ‘’: N = R R R 4, ‘∇’: N = 8, ‘’: N = 16) R R P → s P → sMULTIPLE ANTENNA SYSTEMS 279 independent of N and is equally distributed onto the diversity paths so that only the N th R R part of E can be exploited at each receive antenna. In this scenario, the gain obtained b solely by diversity can be observed. On the contrary, Figure b depicts the error rate versus the average E /N at each receive antenna. Therefore, the total transmit power increases s 0 linearly with N and the entire SNR after maximum ratio combining becomes N times R R larger, indicating the additional array gain. Comparing the difference between adjacent curves in both the plots, we recognize a difference of 3 dB that exactly represents the gain obtained by doubling the number of receive antennas. We can conclude that receive diversity is an efficient and simple possibility to increase the link reliability. However, its applicability becomes immediately limited if the size of the receiving terminal is very small. Cell phones for mobile radio communications have become smaller and smaller in recent years so that it is a difficult task to place several antennas on such small devices. Even if we succeed, it is questionable whether the spacing would be large enough to guarantee uncorrelated channels. Although different polarizations represent a further dimension to obtain diversity, the decoupling is generally imperfect, leading to cross talk. In this situation, the question arises whether diversity can also be exploited with multiple antennas at the transmitter. 6.2.2 Performance Analysis of Space–Time Codes In this subsection, the general concept of space–time transmit diversity is addressed, that is, using multiple antennas at the transmitter. A straightforward implementation where a signal x is transmitted simultaneously over several antennas will not provide the desired diversity gain. Looking at the received signal N T  1 y= √ · x· h + n (6.4) ν N T ν=1 we see that an incoherent superposition is obtained, resulting in a new Rayleigh-distributed 1 channel. Hence, the equivalent SISO channel still has SNR variations as large as the orig- inal single-input single-output system and no diversity has been gained. To overcome this dilemma, appropriate coding is required at the transmitter. This coding is performed in the dimensions space and time leading to the name space–time codes. First, this subsection dis- cusses the potential of STCs and derives some guidelines concerning the code construction. In the next two subsections, specific codes, namely, orthogonal space–time block codes (oSTBCs) and space–time trellis codess (STTCs) are introduced. The general structure of the considered system is depicted in Figure 6.4. The data bits   T di are fed into the space–time encoder that outputs L vectors xk= x k··· x k 1 N T of length N . They are transmitted over a MIMO channel according to (6.1). The channel T coefficients h k= h are assumed to be constant during one encoded frame so that µ,ν µ,ν the received signal becomes yk= H· xk+ nk. (6.5) 1 Note that the total transmit power has been normalized according to the agreement on page 289 so that it is independent of the number of antennas.280 MULTIPLE ANTENNA SYSTEMS − Figure 6.4 Structure of transmit diversity system with N receive antennas R Combining all L vectors xk, yk, and nk within one coded frame as column vectors into the matrices X, Y,and N, respectively, results in Y= H· X+ N, (6.6) where the N × L matrix T   x 0 x 1 ··· x L− 1 1 1 1   x 0 x 1 ··· x L− 1   2 2 2   X= x0x1··· xL− 1 =   (6.7) . . . . . .   . . . x 0 x 1 ··· x L− 1 N N N T T T denotes the entire data frame encoded in space and time. The code comprising all possible code matrices is termed X. The matrices N and Y have the dimensions N × L. R Next, we derive some general results concerning the achievable diversity and coding gains that can be used for the code design. An optimum maximum likelihood decision and a perfectly known channel matrix H are assumed at the receiver. We start with the ˜ pairwise error probability between two competing codewords X and X already known from Section 1.3. Contrary to Section 1.3, we now receive a mixture of all transmit signals at each receive antenna. Therefore, we have to look at the squared Frobenius (see Appendix C on 2 ˜ page 336) norm of the noiseless received signalsHX− HX of both codewords instead F 2 ˜ ofX− X . The conditional pairwise error probability of (1.49) then becomes F   C D5 5 2 , - D5 ˜5 HX− HX 1   E F ˜ Pr X→ X H = · erfc . (6.8)   2 2 4σ N √ √ ˜ ˜ We now normalize the space–time codewords to B= X/ E /T and B= X/ E /T in s s s s the same way as was done in Section 1.3. This changes the squared Euclidean distance to 5 5 5 5 E 2 2 s 5 5 5 5 ˜ ˜ H· (X− X) = H· (B− B) · (6.9) F F T s 2 and (6.8) becomes with σ = N /T for complex-valued signals 0 s N " , - 5 5 1 E 2 s ˜ 5 ˜ 5 Pr X→ X H = · erfc H(B− B) · . (6.10) F 2 4N 0MULTIPLE ANTENNA SYSTEMS 281 √ −x The complementary error function can be upper bounded by erfc( x) e . Denoting the µth row of H with h leads to an upper bound µ + , - 5 5 1 E 2 s ˜ 5 ˜ 5 Pr B→ B H ≤ · exp − H(B− B) · F 2 4N 0   N R  5 5 1 E 2 s  5 ˜ 5  ≤ · exp − h (B− B) · µ 2 4N 0 µ=1 + N R 7  1 E s H H ˜ ˜ ≤ · exp − h (B− B)(B− B) h · . (6.11) µ µ 2 4N 0 µ=1 H ˜ ˜ Obviously, the matrix A= (B− B)(B− B) is Hermitian and its rank r equals that of ˜ B− B. Moreover, it is positive semidefinite and its r nonzero eigenvalues λ obtained ν H by an eigenvalue decomposition A= UU are real and positive. The pairwise error probability can now be expressed as + N R , - 7  1 E s H H ˜ Pr B→ B H ≤ · exp − h UU h · µ µ 2 4N 0 µ=1 + N R 7 1 E s H ≤ · exp −β β · . (6.12) µ µ 2 4N 0 µ=1 The new row vectors β = h U= β ··· β still consist of complex rotationally µ,1 µ,N µ T µ invariant Gaussian distributed random variables β because U is unitary (Naguib et al. µ,ν 1997). Hence, the squared magnitudes of their elements are chi-squared distributed with two degrees of freedom. In order to obtain a pairwise error probability that is independent of the instantaneous channel matrix H, we have to calculate the expectation of (6.12) with respect to H. This results in , -     ˜ ˜ Pr B→ B = E Pr B→ B H H +F N r R 7 7 1 E s 2 ≤ · E exp −λ ·β · β ν µ,ν 2 4N 0 µ=1 ν=1 + N  R r ∞ 7 7 1 E s −ξ ≤ · e · exp −ξ· λ dξ ν 2 4N 0 0 µ=1 ν=1 3 4 N R r 7 1 1 ≤ · (6.13) E s 2 1+ λ · ν ν=1 4N 0 where r denotes the rank of A, that is, the number of nonzero eigenvalues. A further upper bound that is tight for large SNRs is obtained by dropping the +1 in the denominator.282 MULTIPLE ANTENNA SYSTEMS Rewriting (6.13) finally leads to the expression   −rN R 1/r r 7   1 E s ˜   Pr B→ B · · λ . (6.14) ν 2 4N 0 ν=1 From (6.14), the following conclusions can be drawn. Owing to the similarity with (1.112) where the reciprocal of the SNR is taken to the power of D, the exponent rN is called R the diversity gain . Hence, in order to achieve the maximum possible diversity degree, the ˜ minimum rank r among all pairwise differences B− B should be maximized, leading to the diversity gain  ˜ g = N · min rank B− B . (6.15) d R ˜ (B,B) On the other hand, the coding gain leading to a horizontal shift of the error rate curves can be described by 1/r r 7 g = min λ . (6.16) c ν ˜ (B,B) ν=1 If the code design ensures full-rank differences with r= rankA= N , the product of the T eigenvalues equals the determinant det(A) 1/N T N T / 0 7 1/N T ˜ g = min λ = min det(B− B) . (6.17) c ν ˜ ˜ (B,B) (B,B) ν=1 We obtain the code design criteria according to (Tarokh et al. 1998): • rank criterion: In order to obtain the maximum diversity gain, the first design goal is ˜ to maximize the minimum rank r of all matrices X− X. The diversity degree equals rN ; its maximum is N N . R T R • determinant criterion: For a diversity gain of rN , the coding gain is maximized if R  r 1/r the minimum of ( λ ) is maximized over all codeword pairs. ν ν=1 A code optimization according to these criteria cannot be performed analytically but has to be carried out as a computer-based code search. The next two subsections introduce examples for space–time coding schemes. First, orthogonal STBCs are presented. Since their codewords are obtained by orthogonal matrix design, the determinant is constant and no coding gain is obtained. However, full diversity gains are achievable and the receiver structures are very simple. Second, space–time trellis codes are briefly described, providing additional coding gains at the expense of much higher decoding complexity. 6.2.3 Orthogonal Space–Time Block Codes Figure 6.5 shows the principle structure of a space–time block coding system for N = 1 R receive antenna. The subsequent derivation includes more generally the application of an arbitrary number of receive antennas. As a variation from the general concept of space–time coding depicted in Figure 6.4, the signal mapper and space–time encoder are separated. First, the data bits are mapped onto symbols a that are elements of a finite signalMULTIPLE ANTENNA SYSTEMS 283 − Figure 6.5 System structure for space–time block codes with N = 1 receive antenna R constellation according to the linear modulation schemes presented in Section 1.4. Next, the space–time block encoder collects a block of K successive symbols a and maps them   T onto a sequence of L consecutive vectors xk= x k··· x k ,0≤kL. Hence, 1 N T the generated symbols a are encoded in two dimensions, namely, in space and time explaining the name space–time coding. The code rate amounts to K R = . (6.18) c L The system can certainly be improved by an outer forward error correction (FEC) coding scheme. In the following part, we make the widely used assumption that the channel remains constant during one coding block. Therefore, we can drop the time indices of the channel coefficients (h k → h ) in subsequent derivations. µ µ Alamouti’s Scheme In order to illustrate how oSTBCs work, a simple example introduced by Alamouti (1998) is used. Originally, it employs N = 2 transmit antennas and N = 1 receive antenna. T R However, it can be easily extended to more receive antennas. To be precise, we have to consider blocks of K= 2 consecutive symbols, say a = a2and a = a2+ 1. 1 2 These two symbols are now encoded in the following way. At time instant 2k= 2, sym- √ √ bol x 2k= a / 2 is transmitted at the first antenna and x 2k= a / 2 at the second 1 1 2 2 √ ∗ antenna. At the next time instant 2k+ 1, the symbols are flipped and x 2k+ 1=−a / 2 1 2 √ ∗ as well as x 2k+ 1= a / 2 hold. The whole codeword arranged in space and time can 2 1 be described using vector notations + ∗   1 a −a 1 2 X = x2k x2k+ 1 = √ · (6.19) 2 ∗ a a 2 2 1 √ where the factor 1/ 2 ensures that the total average transmit power per symbol equals E /T . The entire set of codewords is denoted by X . The columns comprise the sym- s s 2 bols transmitted at a certain time instant, while the rows represent the symbols transmitted over a certain antenna. Since K= 2 symbols a and a are transmitted during L= 2 1 2 time instants, the rate of this code is R = K/L= 1. It is important to mention that c the columns in X are orthogonal and so Alamouti’s scheme does not provide a cod- 2 ing gain.284 MULTIPLE ANTENNA SYSTEMS A different implementation was chosen in the UMTS standard (3GPP 1999) without changing the achievable diversity gain. Here, the code matrix has the form +   1 a a 1 2 X = x2k x2k+ 1 = √ · . (6.20) 2 ∗ ∗ −a a 2 2 1 The advantage of this implementation is that the original symbols a and a are transmitted 1 2 over the same antenna. Therefore, the first antenna is used in the same way as without space–time coding. Switching from N =1to N = 2 just requires the activation of the T T second antenna without influencing the data stream x . Nevertheless, we will restrict our 1 analysis on the first notation of (6.19). The corresponding two received symbols can be expressed by 1 y2k= √ · (h a + h a )+ n2k (6.21a) 1 1 2 2 2 1 ∗ ∗ y2k+ 1= √ · (h (−a )+ h a )+ n2k+ 1. (6.21b) 1 2 2 1 2 Using vector notations, we can combine the two received symbols and the two noise samples     T T into vectors y= y2k y2k+ 1 and n= n2k n2k+ 1 , respectively. This yields the compact description + + + + 1 y a a h n 1 1 2 1 1 y= = √ · · + = X · h+ n. (6.22) 2 ∗ ∗ y −a a h n 2 2 2 2 2 1 Rewriting (6.22) by taking the conjugate complex of the second line, we obtain + + + + 1 1 y h h a n 1 1 2 1 1 y˜= = √ · · + = √ · HX · a+ n˜. (6.23) ∗ ∗ ∗ ∗ 2 y h −h a n 2 2 2 2 1 2 2 With this slight modification, we have transformed the multiple-input single-output (MISO) channel h into an equivalent MIMO channel HX . The matrix describing this equiva- 2 lent channel has orthogonal columns. In this case, we already know from Chapter 4 that the matched filter represents the optimum detector according to the maximum likelihood principle. The matched filter output becomes + 2 2 1 h +h 0 1 2 H H r˜= H X · y˜= √ · · a+ H X · n˜. (6.24) 2 2 2 2 0 h +h 1 2 2 Looking at the diagonal elements that equal the squared norm of the contributing channel coefficients, we observe that the Alamouti scheme provides the full diversity degree D= N = 2 that can be achieved with two transmit antennas. Moreover, no interference between T H a and a disturbs the transmission because H X HX is a diagonal matrix. Owing to 1 2 2 2 this reason and the fact that the noise remains white when multiplied by a matrix consisting of orthogonal columns, the ML decision with respect to the vector a can be split into element-wise decisions . . 2 2 2 . . aˆ = argmin r˜ − (h +h )a˜ . (6.25) µ µ 1 2 a˜MULTIPLE ANTENNA SYSTEMS 285 Although (6.24) looks similar to the result of simple receive diversity, there exists a major difference. Indeed, the diversity gain is exactly the same for receive and transmit √ diversity concepts. However, the factor 1/ 2 in (6.24) leads to an SNR loss of 3 dB. The reason is that the receiver was assumed to have perfect channel knowledge so that beamforming with an antenna gain of 10 log (N )≈ 3 dB is possible. On the contrary, R 10 we have no channel knowledge at the transmitter so that space–time transmit diversity techniques do not achieve any antenna gain. As all space–time coding schemes, the Alamouti scheme can be easily combined with multiple receive antennas. According to (6.23), we obtain a vector y˜ = H X a+ n˜ (6.26) µ µ 2 µ containing two successive symbols at each receive antenna 1≤ µ≤ N . They are now R included in the vector   T T T y˜= y˜ ··· y˜ . 1 N R Consequently, the equivalent channel matrix HX also has to be extended. Following the 2 notation in (6.23) it becomes   h h 1,1 1,2   ∗ ∗   h −h H X 1 2 1,2 1,1       . . . . . . HX = =   . (6.27) 2   . . .     H X h h N 2 N ,1 N ,2 R R R ∗ ∗ h −h N ,2 N ,1 R R The receiver now consists of a bank of matched filters, one for each receive antenna. Their outputs are simply summed, yielding N R   1 H 2 2 H r˜= H X · y˜= √ h +h · a+ H X · n˜. (6.28) 2 µ,1 µ,2 2 2 µ=1 As long as all channels remain uncorrelated, a maximum diversity degree of D= 2N can R be achieved. Extension to More than Two Transmit Antennas Using some basic results from matrix theory, one can show that Alamouti’s scheme is the only orthogonal space–time code with rate 1. For more than two transmit antennas, several orthogonal codes have been found with lower rates, so that spectral efficiency is lost. The code matrix X generally consists of N rows and L columns and contains the symbols N T T ∗ ∗ a , ..., a as well as the conjugate complex counterparts a , ..., a . The construction 1 K 1 K of X has to be performed such that X has orthogonal rows, that is, N N T T H X X = P· I (6.29) N N T N T T holds, where P is a constant depending on the symbol powers that will be discussed on page 289. In the following part, all codeword matrices are presented without normalization.286 MULTIPLE ANTENNA SYSTEMS In Tarokh et al. (1999a), it is shown that there exist half-rate codes for an arbitrary number of transmit antennas. The code matrices for N =3and N = 4 are presented as T T examples. For N = 3, we obtain T   ∗ ∗ ∗ ∗ a −a −a −a a −a −a −a 1 2 3 4 1 2 3 4 ∗ ∗ ∗ ∗   X = a a a −a a a a −a (6.30) 3 2 1 4 3 2 1 4 3 ∗ ∗ ∗ ∗ a −a a a a −a a a 3 4 1 2 3 4 1 2 providing a diversity degree of D= N = 3. Obviously, X consists of L= 8 columns and T 3 K= 4 different symbols a , ..., a are encoded, leading to the rate R = K/L= 1/2. 1 4 c Each symbol a occurs six times with full energy in X. From (6.30), we can write the µ received vector as    h h h 000 0 0 a 1 2 3 1    h −h 0 −h 00 0 0 a 2 1 3 2       h 0 −h h 00 0 0 a 3 1 2 3       0 h −h −h 00 0 0 a 3 2 1 4    y= + n. (6.31) ∗    00 0 0 h h h 0 a 1 2 3 1    ∗    00 0 0 h −h 0 −h a 2 1 3 2    ∗    00 0 0 h 0 −h h a 3 1 2 3 ∗ 00 0 0 0 h −h −h a 3 2 1 4 We observe in (6.31) that the last four symbols in y only depend on the conjugate com- plex transmit symbols. Hence, conjugating the last four rows similar to the procedure for Alamouti’s scheme in (6.23) results in       y h h h 0 n 1 1 2 3 1       y h −h 0 −h n 2 2 1 3 2              y h 0 −h h a n 3 3 1 2 1 3              y 0 h −h −h a n 4 3 2 1 2 4        y˜= HX a+ n˜ ⇒ = + . (6.32) 3 ∗ ∗ ∗ ∗ ∗        y h h h 0 a n 3 5 1 2 3 5       ∗ ∗ ∗ ∗ ∗       y h −h 0 −h a n 4 6 2 1 3 6       ∗ ∗ ∗ ∗ ∗       y h 0 −h h n 7 3 1 2 7 ∗ ∗ ∗ ∗ ∗ y 0 h −h −h n 8 3 2 1 8 T Obviously, (6.32) uses only the original symbols a= a ··· a and not their conjugate 1 4 complex versions. Moreover, the columns in HX are orthogonal so that 3 N T   H 2 2 2 2 H X · HX = 2· h · I = 2· h +h +h · I (6.33) 3 3 µ 4 1 2 3 4 µ=1 holds. Therefore, the optimum receiver is again a matched filter that multiplies the modified H received vector y˜ with H X . In the case of multiamplitude modulation, an appropriate 3 scaling prior to the hard decision is necessary. For N = 4, a diversity gain of D= N = 4 is achieved with the code matrix T T   ∗ ∗ ∗ ∗ a −a −a −a a −a −a −a 1 2 3 4 1 2 3 4 ∗ ∗ ∗ ∗   a a a −a a a a −a 2 1 4 3 2 1 4 3   X = . (6.34) 4 ∗ ∗ ∗ ∗   a −a a a a −a a a 3 4 1 2 3 4 1 2 ∗ ∗ ∗ ∗ a a −a a a a −a a 4 3 2 1 4 3 2 1MULTIPLE ANTENNA SYSTEMS 287 Equivalent to the case of N = 3, we obtain a received vector y according to T    h h h h 00 0 0 a 1 2 3 4 1    h −h h −h 00 0 0 a 2 1 4 3 2       h −h −h h 00 0 0 a 3 4 1 2 3       h h −h −h 00 0 0 a 4 3 2 1 4    y= + n. (6.35) ∗    00 0 0 h h h h a 1 2 3 4   1 ∗    00 0 0 h −h h −h a 2 1 4 3   2 ∗    00 0 0 h −h −h h a 3 4 1 2 3 ∗ 00 0 0 h h −h −h a 4 3 2 1 4 Complex conjugation of the last four elements in y leads to y˜= HX · a+ n˜ with 4   h h h h 1 2 3 4   h −h h −h 2 1 4 3     h −h −h h 3 4 1 2     h h −h −h 4 3 2 1   HX = . (6.36) 4 ∗ ∗ ∗ ∗   h h h h 1 2 3 4   ∗ ∗ ∗ ∗   h −h h −h 2 1 4 3   ∗ ∗ ∗ ∗   h −h −h h 3 4 1 2 ∗ ∗ ∗ ∗ h h −h −h 4 3 2 1 Again, the columns of HX are mutually orthogonal and estimates aˆ are obtained by 4 H multiplying y˜ with H X and appropriate scaling. 4 Looking at higher spectral efficiencies, only two codes with N =3and N =4have T T been found for R 1/2 (Tarokh et al. 1999a,b). In order to distinguish them from the codes c presented so far, we use the notations T and T .For N = 3, the orthogonal space–time 3 4 T codeword is √ √   ∗ ∗ ∗ 2a −2a 2a 2a 1 2 3 3 √ √ ∗ ∗ ∗   T = 2a 2a 2a − 2a . (6.37) 3 2 1 3 3 √ √ ∗ ∗ ∗ ∗ 2a 2a −a − a + a − a a − a + a + a 3 3 1 2 1 2 1 2 1 2 Since it comprises four time instants for transmitting three symbols, the code rate amounts to R = 3/4. Using (6.37), the received vector can be written as c     a 1 h 3 √ h h 00 0 1 2   2 a 2    h 3 √ 00 h −h 0   2 1 a 3 2    y= 2 + n. (6.38) h h h h h +h ∗  3 3 3 3 1 2  √ √ √ √ √ a − 0 − −  1   2 2 2 2 2 ∗   h h h h h −h a 3 3 3 3 1 2 √ √ √ √ √ 2 0 − ∗ 2 2 2 2 2 a 3 Unfortunately, the channel matrix in (6.38) does not have the block diagonal structure so that a separation into rows associated only with the original symbols a , ..., a and 1 3 those associated with their complex conjugate versions is not possible. Hence, a direct construction of an equivalent matrix HT containing the complex channel coefficients is 3 not possible. However, we can separate real and imaginary parts of all components and stack them into vectors and matrices similar to the approach applied to linear multiuser288 MULTIPLE ANTENNA SYSTEMS detectors for real-valued modulation schemes discussed in Sections 5.2.1, 5.2.2, and 5.4.2.   Denoting the real part of a complex symbol y with y and the imaginary part with y ,we define the real-valued vectors   T r     y = y ··· y y ··· y (6.39a) 1 L 1 L   T r     n ··· n n ··· n n = (6.39b) L L 1 1   T r     a = a ··· a a ··· a . (6.39c) 1 K 1 K r r r r The received vector can now be expressed by y = H T a + n with 3     h h     3 3 √ √ h h −h −h − 1 2 1 2 2 2     h h     3   3 √ √ h −h h −h −   2 1 2 1 2 2       h +h h +h     1 2 1 2 √ √ −h 0 0 −h   3 3 2 2       h −h h −h   1 2  1 2  √ √ 0 h −h 0  3 3  r 2 2 H T =     . (6.40) 3 h h      3 3  √ √ h h h h  1 2 1 2  2 2     h h   3   3   √ √ h −h −h h  2 1 2 1  2 2       h +h h +h   1 2 1 2   √ √ −h 0 0 h − 3 3   2 2     h −h h −h  1 2  1 2 √ √ 0 h h 0 − 3 3 2 2 Owing to the separation of real and imaginary parts, we have again obtained a matrix with orthogonal columns N T    T r 2 2 2 2 H T · HT = 2 h · I = 2· h +h +h · I . 3 3 µ 3 1 2 3 3 µ=1  T r r After multiplying y with H T , real and imaginary parts of each symbol experience 3 a diversity gain of N . For multiamplitude modulation, they have to be normalized and T combined into a complex symbol again to allow the demodulation. Finally, a space–time coding scheme with N = 4 transmit antennas shall be presented. T The space–time codeword is √ √   ∗ ∗ ∗ 2a −2a 2a 2a 1 2 3 3 √ √ ∗ ∗ ∗   2a 2a 2a − 2a 2  1 3 3  √ √ T = . (6.41) 4 ∗ ∗ ∗ ∗   2a 2a −a − a + a − a a − a + a + a 3 3 1 2 1 2 1 2 1 2 √ √ ∗ ∗ ∗ ∗ 2a − 2a −a − a − a − a −(a + a + a + a ) 3 3 1 2 1 2 1 2 1 2 Again, three symbols are transmitted within a block covering four time instants, leading to R = 3/4. The received vector can be described using (6.41) yielding c     a h +h 1 3 4 √ h h 00 0 1 2   2 a 2    h −h 3 4 √ 00 h −h 0   2 1 a 3 2    y= 2 + n. (6.42) −h +h h −h h +h h +h h +h ∗  3 4 3 2 3 4 3 4 1 2  √ a 0 − − 1    2 2 2 2 2 ∗   h −h h −h h +h h +h h −h a 3 4 3 4 3 4 3 4 1 2 2 √ 0 − 2 2 2 2 ∗ 2 a 3MULTIPLE ANTENNA SYSTEMS 289 The channel matrix for the real-valued received vector can now be expressed as      h +h h +h 4   3   3 4 √ √ h h −h −h − 1 2 1 2 2 2       h −h −h +h       3 4 3 4 √ √ h −h h −h   2 1 2 1 2 2        h +h h +h    1 2   1 2 √ √ −h −h −h −h  3 4 4 3 2 2         h −h h −h     1 2 1 2   √ √ −h h −h h 4 3 3 4   r 2 2 H T = . (6.43)   4     h +h h +h    3 4   3 4  √ √ h h h h  1 2 1 2  2 2        h −h h −h      3 4 3 4 √ √  h −h −h h  2 1 2 1 2 2         h +h h +h   1 2   1 2  √ √  −h −h h h − 3 4 4 3 2 2       h −h h −h     1 2 1 2 √ √ −h h h −h − 4 3 3 4 2 2 Owing to the separation of real and imaginary parts, we have again obtained a matrix with orthogonal columns. Certainly, the real-valued description can also be applied to Alamouti’s scheme and to the codesX andX . Therefore, it is more general and can exploit more degrees of freedom 3 4 because it is not restricted to use complex symbols and their conjugate versions. Linear STBCs constructed with real-valued notations are called linear dispersion codes (Hassibi and Hochwald 2002) and are addressed in Section 6.5. As already explained for Alamouti’s scheme, each of the discussed STBCs can be combined with several receive antennas. In this case, we obtain several equivalent channel matrices which are stacked into a large matrix according to (6.27). The receiver consists of a bank of N matched filters and simply sums their outputs. This leads to an overall R diversity degree of D= N · N . T R Although oSTBCs do not provide a coding gain, they have the great advantage that decoding simply requires some linear combinations of the received symbols. Moreover, they provide the full diversity degree achievable with a certain number of transmit and receive antennas. Normalizing the Transmit Power We now have to consider the transmit power of the presented STBCs in more detail. Certainly, there exist several possibilities for normalizing the transmit power. From the channel coding perspective, we know to distinguish E and E . In the context of space–time s b coding, we have the possibility of fixing the average SNR per channel use, that is, per time instant. In this case, the constant P in (6.29) grows linearly with the length L of a space–time codeword and we obtain   E s H tr X X = L· . (6.44) N T N T T s Since the trace in (6.44) also depends on the number of transmit antennas, all codeword √ matrices have to be multiplied with the factor 1/ N . T290 MULTIPLE ANTENNA SYSTEMS In order to draw a fair comparison among the discussed STC approaches, we can also fix the average power spent per data symbol to E /T , leading to s s   E s H tr X X = K· . (6.45) N T N T T s Starting with Alamouti’s scheme, each of the two symbols a and a (including their 1 2 complex conjugate versions) is transmitted twice during one block. This leads to a scaling √ √ factor of 1/ K= 1/ 2 as already used on page 283. For the codes X and X , each 3 4 √ symbol is transmitted six and eight times, respectively. Hence, we obtain the factors 1/ 6 √ and 1/ 8. In relation to T and T , the scaling factors before the codeword matrices 3 4 amount to 1/2. With this normalization, the error rate is depicted against E /N . s 0 Finally, a comparison of schemes with different spectral efficiencies is generally drawn with respect to E /N instead of E /N . Normalizing to the number of receive antennas so b 0 s 0 that no array gain is measured, we obtain the following relationship between the average energy E per information bit and the symbol energy E b s m· R m· K c E = · E = · E , (6.46) s b b N L· N R R where m denotes the number of bits per symbol. Alternatively, the SNR at each receive antenna can also be used so that the array gain of the receiver becomes obvious. However, this must be explicitly mentioned. Simulation Results We now look at the error rate performance of the space–time block coding schemes explained so far. First, Figure 6.6a depicts the error rates of Alamouti’s scheme with b) different modulations a) BPSK 0 0 10 10 N = 1 QPSK, N = 1 R R N = 2 R QPSK, N = 4 R −1 −1 10 10 N = 3 8-PSK, N = 1 R R N = 4 8-PSK, N = 4 R R −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 E /N in dB→ E /N in dB→ b 0 b 0 Figure 6.6 Bit error rate of Alamouti’s scheme for different modulation types and number of receive antennas, (solid bold line: AWGN channel, solid dashed line: Rayleigh fading channel without diversity) BER→ BER→MULTIPLE ANTENNA SYSTEMS 291 b) error rate versus E /N a) error rate versus E /N b 0 s 0 0 0 10 10 X X 2 2 X X 3 3 −1 −1 10 10 X X 4 4 T T 3 3 T T −2 −2 4 4 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 0 5 10 15 20 0 5 10 15 20 E /N in dB→ E /N in dB→ s 0 b 0 Figure 6.7 Bit error rate for different orthogonal STBCs, BPSK, and N = 1 receive R antenna different number of receive antennas. Since X provides a diversity degree of D= 2, 2 additional receive antennas multiply this degree, leading to D= 4, D=6and D=8for N = 2, N = 3, and N = 4, respectively. A comparison between theoretical results from R R R Section 1.5 (lines) and simulation results (symbols) illustrates that both coincide perfectly. Hence, as long as the channel is ideally known to the receiver, optimum diversity per- formance is achieved. Figure 6.6b shows the performance of X for different modulation 2 schemes. Both quaternary phase shift keying (QPSK) and 8-PSK profit by an increased diversity degree. Next, we compare space–time coding schemes for binary phase shift keying (BPSK) and a single receive antenna. From Figure 6.7a, it becomes obvious that X and T have 3 3 identical diversity degrees in addition to X and T . The results are identical with those 4 4 obtained from Section 1.5. However, the codes have different rates R ,leadingtodifferent c spectral efficiencies. Therefore, we have to depict the error rates against E /N instead of b 0 E /N . Figure 6.7b shows the corresponding relations. The slopes of all curves are still s 0 the same as shown in Figure 6.7a but those of X , X , T ,and T are shifted horizontally 3 4 3 4 by 10 log (R ). The half-rate codes X and X perform worse especially at small SNRs c 3 4 10 compared to T and T . Despite its higher diversity degree, X outperforms Alamouti’s 3 4 3 scheme only for SNRs above 15 dB. Similar intersections exist for X and T . 4 3 A fair comparison between different space–time coding schemes can be guaranteed if it is drawn for identical spectral efficiencies. This can be achieved by choosing an appropriate modulation scheme for each STC. Table 6.1 summarizes some constellations considered here. For η= 2 bits/s/Hz, Alamouti’s scheme employs a QPSK while X and 3 X have to use a 16-QAM or 16-PSK because of their lower code rate of R = 1/2. For 4 c η= 3 bits/s/Hz, we use the 8-PSK for X and 16-QAM for T and T . 2 3 4 The results for η= 1 bit/s/Hz are depicted in Figure 6.8a. Since BPSK and QPSK show the same bit error rate (BER) performance against E /N , X and X do not suf- b 0 3 4 fer from a higher sensitivity of the modulation scheme and can fully exploit the larger BER→ BER→292 MULTIPLE ANTENNA SYSTEMS Table 6.1 Combinations of space–time codes and modulation schemes for different overall spectral efficiencies η X X X T T 2 3 4 3 4 1 bit/s/Hz BPSK QPSK QPSK – – 2 bits/s/Hz QPSK 16-QAM 16-QAM – – 3 bits/s/Hz 8-PSK – – 16-QAM 16-QAM b) η= 2 bits/s/Hz a) η= 1 bit/s/Hz 0 0 10 10 X , BPSK X , QPSK 2 2 X , QPSK X , 16-QAM 3 3 −1 −1 10 10 X , QPSK X , 16-QAM 4 4 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 0 5 10 15 20 0 5 10 15 20 E /N in dB→ E /N in dB→ b 0 b 0 Figure 6.8 Bit error rate for different orthogonal STCs, N = 1 receive antenna and dif- R ferent spectral efficiencies (solid bold line: AWGN, bold dashed line: flat Rayleigh fading; in Figure b) both for 16-QAM) diversity degree. In Figure 6.8b, we observe different results for η= 2 bit/s/Hz. QPSK is much more robust than 16-QAM against the influence of noise. Hence, the higher diversity degree becomes obvious only for high SNRs. At low SNRs, Alamouti’s scheme with QPSK still performs best. Finally, Figure 6.9 illustrates the results obtained for a spectral efficiency of η= 3 bit/s/Hz. Because of the relative high code rate of R = 0.75, we have to just switch c between 8-PSK and 16-QAM. However, 16-QAM performs nearly as good as 8-PSK because it exploits the signal space more efficiently (cf. Section 1.4). Therefore, the loss obtained by changing from 8-PSK to 16-QAM is rather low and the diversity gain dominates the bit error rate for T and T . 3 4 The following conclusion can be drawn in relation to the trade-off between diversity degree and modulation type for a fixed spectral efficiency η. In the high SNR regime, diversity is most important and overcompensates the larger sensitivity of high-order modu- lation schemes. At low SNRs, robust modulation schemes such as QPSK should be preferred because the diversity gain is smaller than the loss associated with a change of the modulation scheme. BER→ BER→MULTIPLE ANTENNA SYSTEMS 293 0 10 X , 8-PSK 2 T , 16-QAM 3 −1 T , 16-QAM 10 4 −2 10 −3 10 −4 10 −5 10 0 5 10 15 20 25 30 E /N in dB→ b 0 Figure 6.9 Bit error rate for different orthogonal STCs and N = 1 receive antenna, spectral R efficiency η= 3 bit/s/Hz 6.2.4 Space–Time Trellis Codes Contrary to the previously presented oSTBCs, STTCs can also provide a coding gain. First, optimization criteria and some handmade codes have been presented in Seshadri et al. (1997), Tarokh et al. (1997, 1998). Results of a systematic computer-based code search can be found in Baro ¨ et al. (2000a,b) and some implementation aspects in Naguib et al. (1997, 1998). Figure 6.10 shows the general structure of an encoder with N = 2 transmit antennas. T Obviously, STTCs are related to convolutional codes explained in Section 3.3. At each time   T instant , a vector d= d ··· d  is fed into the linear shift register consisting 1 K of L blocks each comprising K bits. The old content is shifted by K positions to the right. c Hence, the total length of the register is L K bits and L represents the constraint length c c as for convolutional codes. The variable Q= L − 1 denotes the memory of the register. c The major difference compared to binary convolutional codes is the way in which the register content   d    .  T . q= = q ··· q  q ··· q  (6.47)   1 K K+1 L K . c     d− Q input vector d state is combined to form the outputs b and b . Assuming an M-ary linear modulation 1 2 scheme according to Section 1.4, the generator coefficients g are generally nonbinary i,j with g ∈0, 1,··· M− 1. They can be included in the generator matrix i,j   g g ··· g 1,1 1,2 1,L K c   g g ··· g 2,1 2,2 2,L K c   G= (6.48)  . .  . . . .   . ··· . . g g ··· g N ,1 N ,2 N ,L K T T T c BER→294 MULTIPLE ANTENNA SYSTEMS b  x  1 1 + M g g g g 1,1 1,QK+1 1,K 1,L K c d q  q  q  q  1 K QK+1 L K c g g g g 2,QK+1 2,1 2,K 2,L K c b  x  2 2 + M Figure 6.10 General structure of space–time trellis encoder for N = 2 transmit antennas T (Q= L − 1) c   with which the output vector b= b ··· b  can be described by 1 N T  b= G· q mod M. (6.49) TheN integersb ∈0,··· M− 1 are then mapped ontoM-ary phase shift keying (PSK) T µ or quadrature amplitude modulation (QAM) symbols by N independent signal mappers. T In Tarokh et al. (1998), it is shown that the maximum K is restricted by the modulation scheme if maximum diversity degree of N N should be achieved. Hence, K= log (M) T R 2 holds for M-ary modulation schemes. The number of states naturally depends on the mem- ory of the register. However, it may happen that the left-most and the right-most bit tuples dand d− Q are not fully connected to the generators. Assuming that the last τ ele- ments of a are not connected to the generators, only QK− τ memory elements are used QK−τ and the number of states reduces to 2 . In this case, the generator matrix is not fully loaded (Blum 2000). Similar to convolutional codes, STTCs can also be graphically described with a trel- lis diagram. An example with four states and N = 2 transmit antennas is depicted in T 2 Figure 6.11 where K= 2and L = 2 hold, resulting in 2 = 4 states. At each time instant, c two input bits d and d  are encoded in a register with memory Q= 1, resulting in 1 2 four branches leaving each state. On the left-hand side, the binary representation of each state, that is, the register content q q , is depicted. On the right-hand side, the output 3 4 symbols x and x  belonging to different branches are listed, wherein the first symbol 1 2 pair belongs to the uppermost branch leaving a state and the last belongs to the lowest branch. Generally, natural mapping (see Section 1.4) is applied as can be seen from the signal space of QPSK. Decoding Space–Time Trellis Codes Owing to the equivalence between convolutional codes and STTCs, we can use the Viterbi algorithm for decoding. However, there exists a major difference. In the case of binary

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