MULTIPLE ANTENNA SYSTEMS 279 independent of N R and is equally distributed onto the diversity paths so that only the N R th part of E b can be exploited at each receive antenna. In this scenario, the gain obtained solely by diversity can be observed. On the contrary, Figure b depicts the error rate versus the average E s /N 0 at each receive antenna. Therefore, the total transmit power increases linearly with N R and the entire SNR after maximum ratio combining becomes N R times 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 y[] = 1 √ N T · x[] · N T  ν=1 h ν + n[] (6.4) we see that an incoherent superposition is obtained, resulting in a new Rayleigh-distributed channel. 1 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 d[i] are fed into the space–time encoder that outputs L vectors x[k] =  x 1 [k] ···x N T [k]  T of length N T . They are transmitted over a MIMO channel according to (6.1). The channel coefficients h µ,ν [k] = h µ,ν are assumed to be constant during one encoded frame so that the received signal becomes y[k] = H · x[k] + n[k]. (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 R receive antennas Combining all L vectors x[k], y[k], and n[k] within one coded frame as column vectors into the matrices X, Y,andN, respectively, results in Y = H · X + N, (6.6) where the N T × L matrix X =  x[0] x[1] ···x[L − 1]  =      x 1 [0] x 1 [1] ··· x 1 [L − 1] x 2 [0] x 2 [1] ··· x 2 [L − 1] . . . . . . . . . x N T [0] x N T [1] ··· x N T [L − 1]      (6.7) 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 R × L. 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 page 336) norm of the noiseless received signals HX − H ˜ X 2 F of both codewords instead of X − ˜ X 2 F . The conditional pairwise error probability of (1.49) then becomes Pr  X → ˜ X | H  = 1 2 · erfc          HX − H ˜ X   2 F 4σ 2 N    . (6.8) We now normalize the space–time codewords to B = X/ √ E s /T s and ˜ B = ˜ X/ √ E s /T s in the same way as was done in Section 1.3. This changes the squared Euclidean distance to   H · (X − ˜ X)   2 F =   H · (B − ˜ B)   2 F · E s T s (6.9) and (6.8) becomes with σ 2 N = N 0 /T s for complex-valued signals Pr  X → ˜ X | H  = 1 2 · erfc     H(B − ˜ B)   2 F · E s 4N 0  . (6.10) MULTIPLE ANTENNA SYSTEMS 281 The complementary error function can be upper bounded by erfc( √ x) < e −x . Denoting the µth row of H with h µ leads to an upper bound Pr  B → ˜ B | H  ≤ 1 2 · exp  −   H(B − ˜ B)   2 F · E s 4N 0  ≤ 1 2 · exp   − N R  µ=1   h µ (B − ˜ B)   2 · E s 4N 0   ≤ 1 2 · N R  µ=1 exp  −  h µ (B − ˜ B)(B − ˜ B) H h H µ  · E s 4N 0  . (6.11) Obviously, the matrix A = (B − ˜ B)(B − ˜ B) H is Hermitian and its rank r equals that of B − ˜ B. Moreover, it is positive semidefinite and its r nonzero eigenvalues λ ν obtained by an eigenvalue decomposition A = UU H are real and positive. The pairwise error probability can now be expressed as Pr  B → ˜ B | H  ≤ 1 2 · N R  µ=1 exp  −  h µ UU H h H µ  · E s 4N 0  ≤ 1 2 · N R  µ=1 exp  −β µ β H µ · E s 4N 0  . (6.12) The new row vectors β µ = h µ U = [β µ,1 ···β µ,N T ] still consist of complex rotationally invariant Gaussian distributed random variables β µ,ν because U is unitary (Naguib et al. 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 H  Pr  B → ˜ B | H   ≤ 1 2 · N R  µ=1 r  ν=1 E β  exp  −λ ν ·|β µ,ν | 2 · E s 4N 0  ≤ 1 2 · N R  µ=1 r  ν=1  ∞ 0 e −ξ · exp  −ξ · λ ν E s 4N 0  dξ ≤ 1 2 ·  r  ν=1 1 1 + λ ν · E s 4N 0  N R (6.13) 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 Pr  B → ˜ B  < 1 2 ·   E s 4N 0 ·  r  ν=1 λ ν  1/r   −rN R . (6.14) 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 R is called 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 d = N R · min (B, ˜ B) rank  B − ˜ B  . (6.15) On the other hand, the coding gain leading to a horizontal shift of the error rate curves can be described by g c = min (B, ˜ B)  r  ν=1 λ ν  1/r . (6.16) If the code design ensures full-rank differences with r = rank{A}=N T , the product of the eigenvalues equals the determinant det(A ) g c = min (B, ˜ B)  N T  ν=1 λ ν  1/N T = min (B, ˜ B)  det(B − ˜ B)  1/N T . (6.17) 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 R ; its maximum is N T N R . • determinant criterion: For a diversity gain of rN R , the coding gain is maximized if the minimum of (  r ν=1 λ ν ) 1/r is maximized over all codeword pairs. 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 R = 1 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 signal MULTIPLE ANTENNA SYSTEMS 283 − Figure 6.5 System structure for space–time block codes with N R = 1 receive antenna 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 onto a sequence of L consecutive vectors x[k] =  x 1 [k] ···x N T [k]  T ,0≤ k<L. Hence, 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 R c = K L . (6.18) 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 µ [k] → h µ ) in subsequent derivations. Alamouti’s Scheme In order to illustrate how oSTBCs work, a simple example introduced by Alamouti (1998) is used. Originally, it employs N T = 2 transmit antennas and N R = 1 receive antenna. 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 1 = a[2]anda 2 = a[2 + 1]. These two symbols are now encoded in the following way. At time instant 2k = 2, sym- bol x 1 [2k] = a 1 / √ 2 is transmitted at the first antenna and x 2 [2k] = a 2 / √ 2 at the second antenna. At the next time instant 2k + 1, the symbols are flipped and x 1 [2k +1] =−a ∗ 2 / √ 2 as well as x 2 [2k +1] = a ∗ 1 / √ 2 hold. The whole codeword arranged in space and time can be described using vector notations X 2 =  x[2k] x[2k +1]  = 1 √ 2 ·  a 1 −a ∗ 2 a 2 a ∗ 1  (6.19) where the factor 1/ √ 2 ensures that the total average transmit power per symbol equals E s /T s . The entire set of codewords is denoted by X 2 . The columns comprise the sym- bols transmitted at a certain time instant, while the rows represent the symbols transmitted over a certain antenna. Since K = 2 symbols a 1 and a 2 are transmitted during L = 2 time instants, the rate of this code is R c = K/L = 1. It is important to mention that the columns in X 2 are orthogonal and so Alamouti’s scheme does not provide a cod- 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 X 2 =  x[2k] x[2k + 1]  = 1 √ 2 ·  a 1 a 2 −a ∗ 2 a ∗ 1  . (6.20) The advantage of this implementation is that the original symbols a 1 and a 2 are transmitted over the same antenna. Therefore, the first antenna is used in the same way as without space–time coding. Switching from N T = 1toN T = 2 just requires the activation of the second antenna without influencing the data stream x 1 []. Nevertheless, we will restrict our analysis on the first notation of (6.19). The corresponding two received symbols can be expressed by y[2k] = 1 √ 2 · (h 1 a 1 + h 2 a 2 ) + n[2k] (6.21a) y[2k +1] = 1 √ 2 · (h 1 (−a ∗ 2 ) + h 2 a ∗ 1 ) + n[2k +1]. (6.21b) Using vector notations, we can combine the two received symbols and the two noise samples into vectors y =  y[2k] y[2k + 1]  T and n =  n[2k] n[2k +1]  T , respectively. This yields the compact description y =  y 1 y 2  = 1 √ 2 ·  a 1 a 2 −a ∗ 2 a ∗ 1  ·  h 1 h 2  +  n 1 n 2  = X 2 · h + n. (6.22) Rewriting (6.22) by taking the conjugate complex of the second line, we obtain ˜ y =  y 1 y ∗ 2  = 1 √ 2 ·  h 1 h 2 h ∗ 2 −h ∗ 1  ·  a 1 a 2  +  n 1 n ∗ 2  = 1 √ 2 · H[X 2 ] ·a + ˜ n. (6.23) With this slight modification, we have transformed the multiple-input single-output (MISO) channel h into an equivalent MIMO channel H[X 2 ]. The matrix describing this equiva- 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 ˜ r = H H [X 2 ] · ˜ y = 1 √ 2 ·  |h 1 | 2 +|h 2 | 2 0 0 |h 1 | 2 +|h 2 | 2  · a + H H [X 2 ] · ˜ n. (6.24) 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 T = 2 that can be achieved with two transmit antennas. Moreover, no interference between a 1 and a 2 disturbs the transmission because H H [X 2 ]H[X 2 ] is a diagonal matrix. Owing to 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 ˆa µ = argmin ˜a   ˜r µ − (|h 1 | 2 +|h 2 | 2 ) ˜a   2 . (6.25) 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 10 (N R ) ≈ 3 dB is possible. On the contrary, 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 µ = H µ [X 2 ]a + ˜ n µ (6.26) containing two successive symbols at each receive antenna 1 ≤ µ ≤ N R . They are now included in the vector ˜ y =  ˜ y T 1 ··· ˜ y T N R  T . Consequently, the equivalent channel matrix H[X 2 ] also has to be extended. Following the notation in (6.23) it becomes H[X 2 ] =    H 1 [X 2 ] . . . H N R [X 2 ]    =        h 1,1 h 1,2 h ∗ 1,2 −h ∗ 1,1 . . . . . . h N R ,1 h N R ,2 h ∗ N R ,2 −h ∗ N R ,1        . (6.27) The receiver now consists of a bank of matched filters, one for each receive antenna. Their outputs are simply summed, yielding ˜ r = H H [X 2 ] · ˜ y = 1 √ 2 N R  µ=1  |h µ,1 | 2 +|h µ,2 | 2  · a + H H [X 2 ] · ˜ n. (6.28) As long as all channels remain uncorrelated, a maximum diversity degree of D = 2N R can 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 N T generally consists of N T rows and L columns and contains the symbols a 1 , , a K as well as the conjugate complex counterparts a ∗ 1 , , a ∗ K . The construction of X N T has to be performed such that X N T has orthogonal rows, that is, X N T X H N T = P · I N T (6.29) 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 T = 3andN T = 4 are presented as examples. For N T = 3, we obtain X 3 =   a 1 −a 2 −a 3 −a 4 a ∗ 1 −a ∗ 2 −a ∗ 3 −a ∗ 4 a 2 a 1 a 4 −a 3 a ∗ 2 a ∗ 1 a ∗ 4 −a ∗ 3 a 3 −a 4 a 1 a 2 a ∗ 3 −a ∗ 4 a ∗ 1 a ∗ 2   (6.30) providing a diversity degree of D = N T = 3. Obviously, X 3 consists of L = 8 columns and K = 4 different symbols a 1 , , a 4 are encoded, leading to the rate R c = K/L = 1/2. Each symbol a µ occurs six times with full energy in X. From (6.30), we can write the received vector as y =             h 1 h 2 h 3 000 0 0 h 2 −h 1 0 −h 3 00 0 0 h 3 0 −h 1 h 2 00 0 0 0 h 3 −h 2 −h 1 00 0 0 00 0 0h 1 h 2 h 3 0 00 0 0h 2 −h 1 0 −h 3 00 0 0h 3 0 −h 1 h 2 00 0 00h 3 −h 2 −h 1                         a 1 a 2 a 3 a 4 a ∗ 1 a ∗ 2 a ∗ 3 a ∗ 4             + n. (6.31) 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[X 3 ]a + ˜ n ⇒             y 1 y 2 y 3 y 4 y ∗ 5 y ∗ 6 y ∗ 7 y ∗ 8             =             h 1 h 2 h 3 0 h 2 −h 1 0 −h 3 h 3 0 −h 1 h 2 0 h 3 −h 2 −h 1 h ∗ 1 h ∗ 2 h ∗ 3 0 h ∗ 2 −h ∗ 1 0 −h ∗ 3 h ∗ 3 0 −h ∗ 1 h ∗ 2 0 h ∗ 3 −h ∗ 2 −h ∗ 1                 a 1 a 2 a 3 a 4     +             n 1 n 2 n 3 n 4 n ∗ 5 n ∗ 6 n ∗ 7 n ∗ 8             . (6.32) Obviously, (6.32) uses only the original symbols a = [a 1 ···a 4 ] T and not their conjugate complex versions. Moreover, the columns in H[X 3 ] are orthogonal so that H H [X 3 ] · H[X 3 ] = 2 · N T  µ=1 |h µ | 2 · I 4 = 2 ·  |h 1 | 2 +|h 2 | 2 +|h 3 | 2  · I 4 (6.33) holds. Therefore, the optimum receiver is again a matched filter that multiplies the modified received vector ˜ y with H H [X 3 ]. In the case of multiamplitude modulation, an appropriate scaling prior to the hard decision is necessary. For N T = 4, a diversity gain of D = N T = 4 is achieved with the code matrix X 4 =     a 1 −a 2 −a 3 −a 4 a ∗ 1 −a ∗ 2 −a ∗ 3 −a ∗ 4 a 2 a 1 a 4 −a 3 a ∗ 2 a ∗ 1 a ∗ 4 −a ∗ 3 a 3 −a 4 a 1 a 2 a ∗ 3 −a ∗ 4 a ∗ 1 a ∗ 2 a 4 a 3 −a 2 a 1 a ∗ 4 a ∗ 3 −a ∗ 2 a ∗ 1     . (6.34) MULTIPLE ANTENNA SYSTEMS 287 Equivalent to the case of N T = 3, we obtain a received vector y according to y =             h 1 h 2 h 3 h 4 00 0 0 h 2 −h 1 h 4 −h 3 00 0 0 h 3 −h 4 −h 1 h 2 00 0 0 h 4 h 3 −h 2 −h 1 00 0 0 00 0 0h 1 h 2 h 3 h 4 00 0 0h 2 −h 1 h 4 −h 3 00 0 0h 3 −h 4 −h 1 h 2 00 0 0h 4 h 3 −h 2 −h 1                         a 1 a 2 a 3 a 4 a ∗ 1 a ∗ 2 a ∗ 3 a ∗ 4             + n. (6.35) Complex conjugation of the last four elements in y leads to ˜ y = H[X 4 ] ·a + ˜ n with H[X 4 ] =             h 1 h 2 h 3 h 4 h 2 −h 1 h 4 −h 3 h 3 −h 4 −h 1 h 2 h 4 h 3 −h 2 −h 1 h ∗ 1 h ∗ 2 h ∗ 3 h ∗ 4 h ∗ 2 −h ∗ 1 h ∗ 4 −h ∗ 3 h ∗ 3 −h ∗ 4 −h ∗ 1 h ∗ 2 h ∗ 4 h ∗ 3 −h ∗ 2 −h ∗ 1             . (6.36) Again, the columns of H[X 4 ] are mutually orthogonal and estimates ˆ a are obtained by multiplying ˜ y with H H [X 4 ] and appropriate scaling. Looking at higher spectral efficiencies, only two codes with N T = 3andN T = 4have been found for R c > 1/2 (Tarokh et al. 1999a,b). In order to distinguish them from the codes presented so far, we use the notations T 3 and T 4 .ForN T = 3, the orthogonal space–time codeword is T 3 =   2a 1 −2a ∗ 2 √ 2a ∗ 3 √ 2a ∗ 3 2a 2 2a ∗ 1 √ 2a ∗ 3 − √ 2a ∗ 3 √ 2a 3 √ 2a 3 −a 1 − a ∗ 1 + a 2 − a ∗ 2 a 1 − a ∗ 1 + a 2 + a ∗ 2   . (6.37) Since it comprises four time instants for transmitting three symbols, the code rate amounts to R c = 3/4. Using (6.37), the received vector can be written as y = 2       h 1 h 2 h 3 √ 2 00 0 00 h 3 √ 2 h 2 −h 1 0 − h 3 √ 2 h 3 √ 2 0 − h 3 √ 2 − h 3 √ 2 h 1 +h 2 √ 2 h 3 √ 2 h 3 √ 2 0 − h 3 √ 2 h 3 √ 2 h 1 −h 2 √ 2               a 1 a 2 a 3 a ∗ 1 a ∗ 2 a ∗ 3         + n. (6.38) 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 1 , , a 3 and those associated with their complex conjugate versions is not possible. Hence, a direct construction of an equivalent matrix H[T 3 ] containing the complex channel coefficients is 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 multiuser 288 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 y r =  y  1 ··· y  L y  1 ··· y  L  T (6.39a) n r =  n  1 ··· n  L n  1 ··· n  L  T (6.39b) a r =  a  1 ··· a  K a  1 ··· a  K  T . (6.39c) The received vector can now be expressed by y r = H r [T 3 ]a r + n r with H r [T 3 ] =                    h  1 h  2 h  3 √ 2 −h  1 −h  2 − h  3 √ 2 h  2 −h  1 h  3 √ 2 h  2 −h  1 − h  3 √ 2 −h  3 0 h  1 +h  2 √ 2 0 −h  3 h  1 +h  2 √ 2 0 h  3 h  1 −h  2 √ 2 −h  3 0 h  1 −h  2 √ 2 h  1 h  2 h  3 √ 2 h  1 h  2 h  3 √ 2 h  2 −h  1 h  3 √ 2 −h  2 h  1 h  3 √ 2 −h  3 0 h  1 +h  2 √ 2 0 h  3 − h  1 +h  2 √ 2 0 h  3 h  1 −h  2 √ 2 h  3 0 − h  1 −h  2 √ 2                    . (6.40) Owing to the separation of real and imaginary parts, we have again obtained a matrix with orthogonal columns  H r [T 3 ]  T · H[T 3 ] = 2 N T  µ=1 |h µ | 2 · I 3 = 2 ·  |h 1 | 2 +|h 2 | 2 +|h 3 | 2  · I 3 . After multiplying y r with  H r [T 3 ]  T , real and imaginary parts of each symbol experience a diversity gain of N T . For multiamplitude modulation, they have to be normalized and combined into a complex symbol again to allow the demodulation. Finally, a space–time coding scheme with N T = 4 transmit antennas shall be presented. The space–time codeword is T 4 =     2a 1 −2a ∗ 2 √ 2a ∗ 3 √ 2a ∗ 3 2a 2 2a ∗ 1 √ 2a ∗ 3 − √ 2a ∗ 3 √ 2a 3 √ 2a 3 −a 1 − a ∗ 1 + a 2 − a ∗ 2 a 1 − a ∗ 1 + a 2 + a ∗ 2 √ 2a 3 − √ 2a 3 −a 1 − a ∗ 1 − a 2 − a ∗ 2 −(a 1 + a ∗ 1 + a 2 + a ∗ 2 )     . (6.41) Again, three symbols are transmitted within a block covering four time instants, leading to R c = 3/4. The received vector can be described using (6.41) yielding y = 2       h 1 h 2 h 3 +h 4 √ 2 000 00 h 3 −h 4 √ 2 h 2 −h 1 0 −h 3 +h 4 2 h 3 −h 2 2 0 − h 3 +h 4 2 − h 3 +h 4 2 h 1 +h 2 √ 2 h 3 −h 4 2 h 3 −h 4 2 0 − h 3 +h 4 2 h 3 +h 4 2 h 1 −h 2 √ 2               a 1 a 2 a 3 a ∗ 1 a ∗ 2 a ∗ 3         + n. (6.42) [...]... provide a coding gain First, optimization criteria and some handmade codes have been presented in Seshadri et al ( 199 7), Tarokh et al ( 199 7, 199 8) Results of a systematic computer-based code search can be found in B¨ ro et al (2000a,b) and some implementation aspects in Naguib et al ( 199 7, a 199 8) Figure 6.10 shows the general structure of an encoder with NT = 2 transmit antennas Obviously, STTCs are related... signal processing at the receiver This leads to the well-known Bell Labs Layered Space–Time (BLAST) architecture (Foschini 199 6; Foschini and Gans 199 8; Foschini et al 199 9) depicted in Figure 6.22 We see that the parallel data streams are transmitted with identical power levels over different antennas without further preprocessing Neglecting the interleaver, MULTIPLE ANTENNA SYSTEMS 307 − Figure 6.22... and finding the optimum power distribution is still an unsolved problem In uncoded systems, appropriate bit and power-loading strategies have been proposed by Hughes-Hartogs ( 198 9), Fischer and Huber ( 199 6), Krongold et al ( 199 9, 2000), and Mutti and Dahlhaus (2004) 306 MULTIPLE ANTENNA SYSTEMS Beamforming Classical beamforming exploits only a single eigenmode of the channel and is beneficial if the... of the symbol transmitted over antenna ν for the transition between states s and s Consequently, z(s →s) comprises all NT hypotheses The remaining parts of the Viterbi algorithm are identical to that of convolutional codes Examples for Space–Time Trellis Codes In the following part some codes, derived by Wittneben, Tarokh, Yan, and Bro (Tarokh et al 199 8; Wittneben 199 1, 199 3; Yan and Blum 2000), are... and the number of transmit antennas NT All codes achieve the maximum diversity gain NT NR so that only the coding gain has to be considered  296 MULTIPLE ANTENNA SYSTEMS Delay diversity by Wittneben The delay diversity scheme proposed by Wittneben ( 199 1, 199 3) represents an exception because it provides no coding gain However, it can be interpreted as the simplest STTC and is illustrated in Figure... coding gain gc with only two transmit antennas Table 6.2 lists the codes T(4, Z, 2) and Y(4, Z, 2) by Tarokh et al ( 199 8), Yan and Blum (2000) for Z states and QPSK, Table 6.3 the codes B(4, Z, 2) by Bro (B¨ ro et al a Table 6.2 List of space–time trellis codes taken from Tarokh et al ( 199 8), Yan and Blum (2000) for NT = 2, QPSK, η = 2 bits/s/Hz and diversity degree D = 2 Z 4 8 16 32 T(4, Z, 2) 1 0 2... Multilayer transmission with perfect channel knowledge at the transmitter and receiver either time-selective channels or many antennas at the transmitter or receiver do coding gains become visible Naturally, STBCs as well as STTCs can be combined with classical error correction coding schemes (Bauch 199 9; Bauch et al 2000) This leads to concatenated schemes that can be processed iteratively according to... accomplished by means of sphere detection (Agrell et al 2002; Fincke and Pohst 198 5; Schnoor and Euchner 199 4) with lower computational costs MULTIPLE ANTENNA SYSTEMS BER → 10 10 10 10 10 a) ZF detection 0 −1 linear QLD SQLD SQLD+PSA −1 10 −2 −3 −4 linear QLD SQLD SQLD+PSA −2 10 −3 10 −4 10 −5 0 b) MMSE detection 0 10 BER → 10 3 09 −5 5 10 15 Eb /N0 in dB → 20 10 0 5 10 15 Eb /N0 in dB → 20 Figure 6.24 Bit... and overcompensates the larger sensitivity of high-order modulation 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 MULTIPLE ANTENNA SYSTEMS 293 0 10 X2 , 8-PSK T3 , 16-QAM T4 , 16-QAM −1 BER → 10 −2 10 −3 10 −4 10 −5 10 0 5 10 15 20 Eb /N0 in dB → 25 30 Figure 6 .9 Bit... QPSK system with NT = 4 and NR = 6 antennas (bold line: maximum likelihood detection) The loss compared to the algorithm with optimum post sorting (SQLD+PSA) known as V-BLAST detection (Foschini et al 199 9) and described in Section 5.4.2 is neglectable This performance is obtained with only a fraction of the computational costs of the V-BLAST The MMSE criterion outperforms the ZF approach by 0.6–0.7 dB . al. ( 199 7), Tarokh et al. ( 199 7, 199 8). Results of a systematic computer-based code search can be found in B ¨ aro et al. (2000a,b) and some implementation aspects in Naguib et al. ( 199 7, 199 8). Figure. following part some codes, derived by Wittneben, Tarokh, Yan, and Bro (Tarokh et al. 199 8; Wittneben 199 1, 199 3; Yan and Blum 2000), are presented. This list does not claim to be comprehensive the coding gain has to be considered. 296 MULTIPLE ANTENNA SYSTEMS Delay diversity by Wittneben The delay diversity scheme proposed by Wittneben ( 199 1, 199 3) represents an exception because it