10.3.4 Alamouti space–time code

The coding scheme proposed in [111] exploits two transmit antennas, operates with no extra bandwidth and offers maximal possible diversity gain for two antennas nd ¼ nT ¼ 2. Let b0 and b1 be two successive data symbols standing for even and odd time positions, respectively, and belonging to some fixed modulation alphabet (PSK, QAM etc.). Codewords of the Alamouti space–time code are 2 2 arrays of the form:

securing separability of superimposed signals of different subchannels in the receiver. Actually, however, there is no need to fulfil the separation of subchannels as a special procedure, since the optimal (ML) detection of data symbols b0 and b1 automatically includes it, as well as maximal ratio combining. For a clear reason we assume that a single code symbol transmitted currently by one antenna utilizes on average half of the total average symbol energy Es. Let Y_ ¼ (Y_0, Y_1) be an observation vector whose components Y_t, t ¼ 0, 1 are samples of the complex envelope at the symbol matched filter output for even and odd positions, respectively, normalized for convenience by the

where n is a two-dimensional vector of independent complex Gaussian noise samples

^

with zero means and equal variances. Then the ML rule (see Chapter 2) gives out b0 and

^

b1 as estimations of data symbols b0 and b1 if they minimize the Euclidean (squared) distance between the observation Y_ and the useful component H_ 1u1 þ H_ 2u2:

Distributivity and symmetry ((u, v) ¼ (v, u) ) axioms of the inner product along with orthogonality of u1, u2 allow getting:

where d2 is a shortened designation for the squared distance in question, or after substituting u1, u2 from (10.37):

as well as the norm of the observation vector do not depend on variables b0, b1, with respect to which d2 has to be minimized. Therefore, in the equation above we are

It is evident now that minimizing d2 in b0, b1 breaks into a separate minimization of two

functions of one variable d 2(b0) ¼ jz0 b0j2þH2jb0j2 and d2(b1) ¼ jz1 b1j2þH2jb1j2, where H2 ¼ H_ 1 2þ H_ 2 2 1, with respect to b0 and b1. Thus, the estimates of the data

symbols b0, b1 are found as:

where zl is defined by (10.39) and minimization is done over all values of bl within the

given data symbol alphabet. In the particular case of PSK data modulation jblj2¼ 1 and the only component of the first term of (10.40) dependent on bl is Re(bl zl), which turns the decision rule into the ordinary form of PSK demodulation (see Section 7.1.2), but based on the modified matched filter statistics zl:

useful components of the decision statistics zl, l ¼ 0, 1 can be evaluated by averaging

Y_ 0 and Y_ 1 in (10.39) with respect to an additive noise. Denoting this operation by Enf g, we obtain (see (10.38)) EnfY_ 0g ¼ H_ 1b0 þ H_ 2b1, EnfY_ 1g ¼ H_ 1b1 þ H_ 2b0, so that:

Enfz0g ¼ ðA21 þ A22Þb0; Enfz1g ¼ ðA21 þ A22Þb1

where, as before, Ai ¼ H_ i , i ¼ 1, 2. It may be seen now that the useful component of zl is formed as though no separation problem existed and each symbol were transmitted over two independent diversity branches, further maximal-ratio combined (see Section 3.6.1).

Now take into consideration the randomness of Ai, i ¼ 1, 2, bl, l ¼ 0, 1, and average q2zl

with respect to all random factors to come to the mean SNR q2zl for each of the statistics zl, l ¼ 0, 1. Under a natural normalization of fading coefficients and modulation

the Alamouti scheme, as well as the time-switched code, preserves the same average symbol SNR as the no-diversity scheme, providing diversity gain nd ¼ nT ¼ 2. We stress again that the advantage of the Alamouti code against the time-switched one is the absence of pauses in emission, entailing better peak-factor and higher spectral efficiency.

The Alamouti code is a full-rate one, meaning that two independent modulation symbols are transmitted over two-symbol duration. In general, a space–time code of length n, which allows transmitting k independent data modulation symbols, has rate R ¼ k/n. Full-rate codes are preferable, since they involve no extra bandwidth compared to a single-antenna transmission. The existence of more full-rate codes securing maximal diversity gain nd ¼ nT strongly depends on the modulation alphabet. For real modulation alphabets (e.g. BPSK) full-rate codes exist for several values of the number nT of transmit antennas, while for complex modulation symbols (QPSK, QAM etc.) the Alamouti code is unique2 [112]. At the same time, several interesting constructions of space–time block codes become available for both complex and real alphabets if the full-rate restriction is removed [110,112] (see also Problems 10.16 and 10.17). We refer the reader wishing to gain more information on this issue and become familiar with other aspects of space–time coding to works [110–113] and the papers cited in [110].

10.3.5 Transmit diversity in spread spectrum applications

Seemingly the spread spectrum concept offers a very direct and easy way of providing transmit diversity. Indeed, the main bottleneck of the transmit diversity is separation of signals emitted simultaneously by different transmit antennas at the receiver side. One may get round this stumbling block by spreading the signals of different antennas using different (orthogonal) spreading codes. This at first sight obvious tool of the transmit diversity is, however, far from universal. As a matter of fact, orthogonal sequences in CDMA systems are a deficit resource, since their number determines the potential number of users. Thus, in a saturated (K ¼ N ) or, more so, oversaturated (K > N ) CDMA downlink there are no spare orthogonal sequences for arranging transmit diversity, which makes the space–time bandwidth saving codes equally valuable in CDMA applications, too.

The way of incorporating the Alamouti code into the DS CDMA downlink is straightforward and does not require an extra signature resource. Let S_k(t) be the complex envelope of the kth user signature (treated as a signal of the same duration Tp as the data symbol) and bk, 0, bk, 1 be even and odd data modulation symbols sent to the kth user. Then it is enough only to use in the array (10.37) the DS spread symbols bk, 0S_k(t) and bk, 1S_k(t) in place of b0, b1, respectively, to arrange the transmit diversity on the basis of a fixed signature S_k(t).

2 We do not consider as different trivial modifications of (10.37) preserving row orthogonality and row norms, like a common conjugation or/and multiplication of rows by 1 as well as by any fixed complex number of magnitude one.