i.e. 50 times higher. This drastic energy loss due to the fading gets greater when higher transmission reliability is necessary and becomes close to 25 dB (300 times) for Pe ¼ 10 4.

The physical explanation of a quite detrimental fading effect is rather straightforward. Sporadic sharp drops in signal intensity due to the multipath interference are rather likely in the Rayleigh channel. Elementary integration of PDF (3.12) shows, for instance, that the probability of Ar falling below the level 0.4 (SNR decreases by 8 dB) is about 0.15. Then, as is seen from Figure 3.16 (dashed line), in those sessions where such drops happen, the error probability can not be lower than 0.1 if reference SNR is 10 dB. Since the share of such sessions is 0.15, their contribution to the total (average) error probability will be no less than 0:1 0:15 ¼ 0:015, i.e. 15 times bigger than the value corresponding to the reference SNR. This effect cannot in any way be compensated by possible favourable sessions with high SNR, since their contribution to the total error probability is never negative.

The consequence of multipath propagation may potentially be even more dramatic when the fading is frequency selective. This term, as opposed to the attribute ‘flat’, defines the situation where the channel transfer function is not uniform within the signal bandwidth. This happens if the delay spread covers several transmitted bits so that at the channel output the previous bits overlap with the current one. To counter this ISI, special filters (equalizers) are used, which rectify the channel transfer function nonuniformity. On the other hand, frequency selectivity when used properly is a good resource for countering fading by arranging the multipath diversity discussed in Section 3.7.

3.6 Diversity

The general idea of combating destructive multipath effects consists in diversity, which means arranging several independent transmission channels or branches. Thanks to this, despite every individual branch remaining liable to Rayleigh (or other) fading, the probability that the interference patterns in all of them are simultaneously poor is defined by the multiplication rule and thus diminishes radically. Take the figures of the example at the end of the previous section and suppose that two identical independent branches are somehow organized. Then the probability of the same fall of signal power in both of them at once is 0:152 ¼ 2:25 10 2, i.e. perceptibly smaller compared to the probability of poor conditions in an individual branch. With a larger number of branches this diversity gain becomes more and more substantial. Branches operate in parallel, as though they secured each other, mitigating fading impairment.

In other words, we know that the poor performance resulting from multipath fading is entirely due to the deep drops of SNR occurring from time to time. Hence, the final objective of diversity techniques is to process jointly signals of the branches in a manner that makes ‘better’ (higher SNR) branches more influential on the overall performance in comparison to the worse ones. Such joint processing is called combining.

3.6.1 Combining modes

Various strategies for combining the results of processing signals arriving via different branches may be used at the receiver. Suppose that there are nd diversity branches

altogether and let Ai, i and i be current signal amplitude, signal phase and noise standard deviation in the ith branch, where i ¼ 1, 2, . . . , nd . What, then, is the best linear processing producing the maximum possible resultant SNR? Any linear form of the

magnitude of the deterministic component of this sum and the variance of its noise component. The latter is simply the sum of branch noise variances weighted by jwij2, since branches are independent. Therefore:

i¼1

i¼1

where qi ¼ Ai/ i is voltage SNR of the ith diversity branch. When optimal weights:

Ai

wi ¼ 2i expð j iÞ

are taken, inequality (3.15) becomes an equality, i.e. maximal possible resultant SNR is achieved. Such weights, as is readily seen, realize joined matched filtering of the responses of the diversity branches. Technically it is possible only when accurate values of all signal amplitudes and phases are known. Then signals can be summed coherently with an appropriate amplitude weighting. This combining technique is often referred to in the literature as the maximal ratio technique [5,18].

To assess the efficiency of combining, denote maximal SNR over all the diversity

i¼1 qi nd qmax no combining scheme can provide gain greater than nd , and the latter is achievable only in the maximal ratio scheme under the additional stipulation that all the diversity branches have the same SNR.

In practice, some other combining schemes find application, too, because maximal ratio processing is rather demanding as to the extra arrangements necessary (to measure SNR and phase in a diversity branch some special pilot signal may appear necessary etc.). Alternative combining modes are equal-weight combining and selection of a maximum SNR branch. The first approaches the maximal ratio mode in effectiveness if all the diversity branches have nearly equal SNR. The gain of the second is close to that of the optimal scheme if one of the diversity branches dominates over the rest in value of SNR.

Certainly, these strategies can be combined with each other, e.g. several branches with the best SNR values are selected and then their outputs are summed with equal weights.

Consider now traditional ways of organizing independent diversity branches.

3.6.2 Arranging diversity branches

The traditional ways to set up independent diversity branches may be categorized as follows:

. Space diversity

. Frequency diversity

. Time diversity

. Polarization diversity

. Multipath diversity.

Space diversity implies creating several independent propagation paths at the expense of involving multiple antennas, which explains the other popular name for this technique: antenna diversity. Duplicating antennas may be used at the receiving side as well as at the transmitting side. Being spaced from each other by a distance of 7–10 wavelengths or more, they provide practical independence of parallel interference patterns at the receiver input. When used at the receiver (Figure 3.17) (receive diversity) antenna diversity is most effective, since additional antennas utilize signal energy which otherwise would not be captured at all. In this case diversity signals are separated automatically since different antennas receive them. Being matched-filtered individually, they may be further combined as described above.

Transmitting antenna diversity (transmit diversity) is not that straightforward. First, as is seen from Figure 3.18, a limited total transmitter energy resource should be divided between several transmitting antennas. Second, the receiver antenna receives the mixture of signals emitted by all transmitting antennas. Therefore, some measures should be taken to provide an opportunity for separation and individual processing of those signals by the receiver before combining. These factors make this sort of diversity a sophisticated optimization problem, solving which is the subject of a special branch of communication theory called space–time coding (see Section 10.3).

Reflector

Figure 3.17 Receive antenna diversity

Reflector

Figure 3.18 Transmit antenna diversity

Certainly, when the technological constraints allow it, the combination of transmit and receive antenna diversity may be used to gain maximal benefits.

The idea of frequency diversity is based on the concept of the channel coherence bandwidth. This notion determines the frequency range within which fading is considered as flat, i.e. distortion of signal frequency components is strongly dependent. On the other hand, the harmonics with frequency space beyond the coherence bandwidth may be treated as independently distorted by the channel. As was already underlined in the previous section, the frequency range of the flat fading depends inversely on the delay spread, so the wider the range of dispersing signals in time, the shorter the coherence bandwidth. Evidently, transmitting the same signal simultaneously at nd carriers whose frequencies are offset by coherence bandwidth or more creates nd diversity branches. We may say that frequency diversity puts frequency selectivity of fading to good use. Figure 3.19 gives an elementary clarification of the idea. The waves of two wavelengths w1 and w2 propagating along the same couple of paths have identical geometrical propagation differences . However, the phase differences between the signals of the two paths are individual for each wavelength and equal to 2 / w1 and 2 / w2, respectively. When one of these phase differences leads to attenuation of the resultant signal the other may appear less destructive. With many parallel propagation paths present statistical interpretation comes into force, and frequency difference exceeding the channel coherence bandwidth provides the independence of diversity branches in this scheme. An appropriate choice of frequencies in this diversity scheme provides separation of branches at the receiver with the help of bandpass filtering.

Reflector

φ1 = 2πδ/λw1

φ2 = 2πδ/λw2

λw1

λw2

Figure 3.19 Frequency diversity