10.3 Transmit diversity and space–time coding in CDMA systems

an appropriate fashion. To provide a chance of such separation, the data transmission through nT parallel transmit antennas should be arranged carefully, and ways of doing this constitute a subject of the problem called space–time coding. The name reflects the fact that the group of transmitted data bits is mapped one-to-one onto the two-dimensional nT n codeword [uit]. The i, t entry uit of the array is a code symbol transmitted by the ith antenna at the tth time moment, n being the code length. Note that in some cases the receive diversity may appear infeasible, e.g. in the mobile radio downlink, where the small dimensions of a handset does not give enough space for several receive antennas. In these scenarios the transmit diversity and, hence, an adequate space–time coding become especially valuable. In what follows we assume that only a single receive antenna is used to concentrate only on investigating the efficiency of the transmit diversity. This allows us to simplify designations of the channel fading coefficients, retaining only a single subscript pointing at the transmit antenna: H_ i1 ¼ H_ i.

10.3.2 Efficiency of transmit diversity

A lot of research has been undertaken to evaluate the Shannon capacity, i.e. the potential rate of error-free data transmission, of MIMO channels, and the profitable role of antenna multiplicity has been proved for the basic fading models [108–110]. There is no wonder in the benefits of receive diversity, since extra receive antennas utilize signal energies from extra space points which would be lost irrevocably with a single antenna. With the maximal ratio combining of nd identical receive diversity branches, average power SNR grows nd times (see Section 3.6.1), and although this factor is not central in the improvement of error probability and channel capacity, it still makes this improvement readily predictable. Unlike this, the nature of gaining capacity or reducing error probability through transmit diversity is not that obvious, considering the division of the limited power resource between multiple transmit antennas. In fact, no gain in average SNR takes place under the maximal ratio combining of identical diversity branches with fixed overall power, unless the transmitter knows the channel state and is able to coordinate transmitting through different branches so that subchannel signals are summed coherently in the receive antenna. Indeed, let channel state information be unavailable to the transmitter and the total power P be equally divided between nd identical diversity branches (antennas, frequency channels etc.). Then the average power SNR per branch is q2/nd , where q2 is the average power SNR, which would exist at the receiver with no diversity. Clearly, the maximal ratio combining would only increase average power SNR per branch nd times, making it equal to the one with no diversity.

Accordingly, there are two contradictory trends in the transmit (as well as frequency etc.) diversity. On the one hand, increasing the number of branches, given the total power, provides a greater number of independent subchannels, which, supporting each other, secure higher probability that at least some of them are not poor. On the other hand, conditions (branch SNR) in each of the diversity subchannels become poorer with growth of nd . An accurate theoretical analysis shows that the first of these factors overweighs the second. We put aside mathematical derivations concerning the channel capacity, which may be found in the literature (e.g. [108–110]), but Problems 10.11 and 10.12 contain the plainest examples illustrating the issue. As for the positive effect of

328 Spread Spectrum and CDMA

transmit diversity on the error probability, it becomes obvious from the following

To come to the unconditional bit error probability Pe we have to average (10.34) in all subchannel amplitudes Ai, i ¼ 1, 2, . . . , nd using their joint PDF W(A1, A2, . . . , And ). Due to the independence of branches this PDF is just a product of nd one-dimensional PDFs of all amplitudes, and:

To make the integration variables of (10.35) separable, let us approximate the complementary error function Q(x) by its upper bound (see Problem 10.13) Q(x) (1/2) exp ( x2/2), x 0, coming to:

For the channel with Rayleigh fluctuations PDF W(Ai) obeys the law (3.12), which leads to:

Figure 10.9 demonstrates the behaviour of the bit error probability depending on the total average SNR q for 1, 2, 3, 4, 5 and 6 diversity branches. It is of great importance that two diversity branches provide a significant energy gain, error probability preassigned. For example, if a tolerable bit error probability is no greater than 10 4, two diversity branches cut down the necessary transmitted energy by more than 15 dB. At the same time, with further addition of diversity branches the energy gain grows at a dropping rate, and, say, transition from 5 to 6 branches promises a saving of only around 1 dB of emitted energy. This explains why in many practical systems (e.g. mobile radio downlinks) two transmit antennas are chosen as a good balance between the diversity gain and equipment complexity.

Note that when the number of branches tends to infinity the right-hand side of (10.36) turns into (1/2) exp ( q2/2) (see Problem 10.14), i.e. an upper bound (dashed line in Figure 10.9) of the bit error probability for the case of a non-fading Gaussian channel. In other words, by increasing the number of transmit diversity branches one can in the limit (at least theoretically) completely eliminate the harmful effect of multipath propagation.

q (dB)

Figure 10.9 Bit error probability versus overall SNR in the transmit diversity scheme for 1, 2, 3, 4, 5 and 6 diversity branches

10.3.3 Time-switched space–time code

There may be different approaches to designing space–time codes depending on the fading model. Fast fading (see Section 3.5) implies such rapid fluctuations of the multipath pattern that values of the fading coefficient of the same subchannel at two adjacent symbol intervals are independent. In what follows we are dealing with the opposite case of slow fading, assuming that subchannel fading coefficients remain constant over all codeword duration.

To accentuate the non-trivial character of the problem of designing codes securing separability of the signals of different transmit antennas at the receiving side, let us start with the simplest example.

Example 10.3.1. Let the fixed power resource P be divided equally between nT ¼ 2 transmit antennas sending simultaneously the same data symbol with no measures allowing separation of subchannel signals at the receiver (repetition space–time code). Let a single receive antenna be used and the fading coefficients H_ 1, H_ 2 of two subchannels be independent Gaussian complex numbers with zero means and equal variances. This is exactly the case of Rayleigh

A2 ¼ 1, too. Therefore, the resultant channel is absolutely identical to each of the subchannels, and using two antennas in this case cannot give any benefit as compared to a single antenna. In other words, the repetition code is a degenerated one, providing no real diversity.

As the example shows, the number of effective diversity branches may appear smaller than the number of transmit antennas. One of the most important parameters of any space–time code is the diversity gain, i.e. the number of really contributing diversity branches secured by the coding scheme (see Problem 10.15).

The obvious way to provide separation of signals of different transmit antennas in the receiver is to emit the same data symbol by nT antennas by turns, i.e. with no time overlapping. In other words, one and only one of the transmit antennas emits the signal at each time, employing the total power resource. Then the situation is identical to that of multiuser TDMA communications, i.e. the signal of each antenna is identified by its time position and is orthogonal to others due to non-overlapping in the time domain. Thereby, the receiver observes all the subchannel signals not in the mixture but following successively in time with no mutual interference. Then, knowing the current fading coefficients H_ i, the receiver is entirely certain of which of these signals are more or less reliable, and able to combine them in any appropriate way, e.g. maximize SNR by using maximal ratio weighting. This simplest coding scheme corresponds to the time-switched space–time code.

Example 10.3.2. Consider again the case of two antennas (nT ¼ 2) repeating transmission of the same current data symbol, this time operating in an intermittent manner: when the first emits energy, the second is inactive, and vice versa. The receiver observes the signals, one after another, passing through the subchannels with fading coefficients H_ 1 and H_ 2, both distorted by additive noise of power 2. To realize the maximal ratio combining, these observations are summed with weights H_ i , i ¼ 1, 2, respectively, resulting in power SNR (see (3.15)) qr2 ¼ (A21 þ A22)qs2/2, where qs2 is an overall ‘non-fading’ power SNR per one transmitted data symbol and halving arises due to splitting the fixed symbol energy between two antennas.

For identical subchannels with magnitudes normalized to one (A2i ¼ 1) average power SNR is, of course, again the same as in the case of transmitting the whole symbol energy through a single subchannel: qr2 ¼ 2qs2/2 ¼ qs2. However, if the combiner is treated as the output of the resultant channel, the latter is no longer Rayleigh and has error probability smaller than a Rayleigh one under the same average SNR (see Figure 10.9). Thus, two diversity branches really exist and provide a predicted gain, which is achieved in return for two times smaller transmission rate per bandwidth unit. Indeed, the transmission rate in bit/s fixed, each antenna now transmits every data symbol over duration Tp /2, i.e. occupying a doubled bandwidth. One more instructive comparison is with the two-branch receive diversity system (nT ¼ 1, nR ¼ 2). It is easy to see that if a single transmit antenna uses the full interval Tp to transmit a current

symbol, the receiver maximal ratio combiner provides average power SNR qr2 ¼ 2qs2, i.e. 3 dB higher versus that in the transmit diversity scheme, energies per symbol equal. The roots of the energy loss of the transmit diversity against the receive diversity have been repeatedly pointed out: splitting total fixed energy between antennas in the transmit diversity scheme.

Time-switched space–time codes are very simple but they realize maximal diversity gain nT in exchange for widening bandwidth (rate fixed) and discontinuity of transmission, i.e. increasing signal peak-factor. This strongly motivates the search for space–time codes allowing separation of signals from different antennas despite their overlapping in time. The simplest, but very important, example of such codes is introduced in the next subsection.