The importance of mobile communication systems nowadays needs not to be em-phasized. Worldwide millions of people rely daily on their mobile phone. While for the user a mobile phone looks very similar to an old-fashioned wired telephone, the engineering technique behind it is very much different. The reason for this is that in a wireless communication system several physical effects occur that change the behavior of the channel completely compared with wired communication:
• The signal may find many different paths from the sender to the receiver via various different reflections (buildings, trees, etc.). Therefore the receiver re-ceives multiple copies of the same signal, however, since each path has different length and different attenuation, the various copies of the signal will arrive at different times and with different strength.
• Since the transmitter and/or the receiver might be in motion while transmit-ting, a physical phenomenon called Doppler effect occurs: the frequency of the transmitted signal is shifted depending on the relative movement between receiver and transmitter.
• Since receiver and transmitter are moving and because the environment is changing permanently (e.g., movements by wind, passing cars, people, etc.), the different signal paths are constantly changing, too.
The first two effects lead to a channel that not only adds noise to the transmitted signal (as this is the case for the traditional wired communication channel), but also changes the amplitude of the signal (so called fading) and in extreme cases intro-duces inter-symbol interference. Both effects can be combatted using appropriate transmissions schemes and coding.
The fact of the time variant nature of the channel is more difficult to deal with.
Nowadays, usually a wireless communication system uses training sequences that are regularly transmitted between real data in order to measure the channel state, and then this knowledge is used to detect the data. This approach has the advantage that the system design can be split into two parts: one part dealing with estimating the channel and one part doing the detection under the assumption that the channel state is perfectly known.
The big disadvantage of the separate estimation and detection is that it is rather inefficient because bandwidth is lost for the transmission of the training sequences.
Particularly, if the channel is fast changing, the estimates will quickly become poor and the amount of needed training data will be exuberantly large.
A more promising approach is to design a system that uses the received data carrying the information at the same time for estimating the channel state. Such a joint estimation and detection approach will be particularly important for future systems where the required data rates are considerably larger than the rates provided by present systems (like, e.g., GSM).
A further advantage of such joint estimation and detection systems is that they allow fair and realistic approximations to the physically feasible data rates. To elab-orate more on this point, we need to briefly review some basic facts from information theory: in his famous landmark paper “A Mathematical Theory of Communication”
[1] Claude E. Shannon proved that for every communication channel there exists a maximal rate—denoted capacity—above which one cannot transmit information re-liably, i.e., the probability of making decoding errors tends to one. On the other hand for every rate below the capacity it is theoretically possible to design a system such that the error probability is as small as one wishes. Of course, depending on the aimed probability of error, the system design will be rather complex and one will encounter possibly very long delays between the start of the transmission until the signal can be decoded. Particularly the latter is a large obstacle in real systems because most communication systems cannot afford large delays. Nevertheless, the capacity shows the ultimate limit of communication rate of the available channel and is therefore fundamental for the understanding of the channel and also for the judgment of implemented systems regarding their efficiency.
As mentioned above, in the situation of wireless communication channels the channel capacity is limited due to two main sources of transmission errors. Firstly, the receiver introduces thermal noise that can be well modeled by an additive random noise process. Secondly, because the signals are electromagnetic waves transmitted through air, the received signals suffer from random fluctuations in the magnitude and phase. This effect, known as fading, can be described by a multiplicative random noise process.
While the additive noise can be well approximated by an independent and identi-cally distributed (IID) complex Gaussian process for almost all channels of interest, the detailed properties of the multiplicative noise depends on many parameters, system-internal and -external, and should therefore be kept as general as possible.
Unfortunately, the analysis of the channel capacity in such generality is very difficult so that commonly the model is simplified in certain aspects.
One possible simplification is to assume that the receiver perfectly knows the fading realizations. This assumption is based on the idea that the transmitter will firstly transmit some known training symbols from which the receiver learns the current state of the multiplicative noise process. The capacity is then computed without taking into account the estimation scheme. It is common to call this the coherent capacity of fading channels. Such an approach will definitely lead to an overly optimistic capacity value because
• even with a large amount of training data the channel knowledge will never be perfect, but only an estimate; and because
• the data rate that is wasted for the training symbols is completely ignored.
In this project we will not make this simplification, but stick with noncoherent detection where the receiver has no additional knowledge about the channel state.
Note that the receiver is free to do anything in its power to gain knowledge about the fading based on the received signals.
Marzetta and Hochwald [2] simplify the noncoherent channel model by assuming that during blocks consisting of several symbol periods the fading remains constant, while the fading coefficients corresponding to different blocks are assumed to be independent. This model is generally known as block fading model. Note that it is pessimistic to assume that the blocks are independent of each other because memory provides additional information about the current fading level which in general will increase capacity. However, it is more problematic to conjecture that the fading coefficients are perfectly constant during one block. This means that for high enough signal-to-noise ratios (SNR) and for long enough blocks the receiver can get an (almost) perfect estimate of the fading value within a block and use this knowledge to decode the received signal similarly to coherent detection. For larger SNR this seems to be overly optimistic. Indeed, as shown in [2] for single-input–
single-output (SISO) Gaussian block fading and in [3] for multiple-input–multiple-output (MIMO) Gaussian block fading, the capacity of the block fading channel grows logarithmically in the SNR at high SNR, i.e., the capacity has the same growth rate as the coherent capacity (and, as a matter of fact, as the capacity of an additive noise channel without fading, too).
In [4] Liang and Veeravalli generalize the SISO Gaussian block fading model by allowing some temporal correlation between the different fading coefficients within one block. They show that the rank of the block correlation matrix is crucial when determining the high-SNR channel capacity: if we have a rank-deficient correlation matrix, the effect of perfect predictability comes into play again similar to the situ-ation of Marzetta and Hochwald [2]. This then again leads to a logarithmic growth of capacity. For a full-rank correlation matrix this is not true anymore. In this case the channel model reduces to a special case of the more general model described next.
The most general models only restrict the random noise processes to be sta-tionary and ergodic, with additional variations in the exact fading law, the number of antennas, and the memory [5]–[13]. In [5] the authors investigate a memory-less SISO Rayleigh fading channel and derive some bounds. In [6] it is shown that the capacity-achieving input distribution for the memoryless SISO Rayleigh fading channel is discrete. In [7]–[9] the channel model is then generalized to MIMO and to general non-Gaussian fading distributions (possibly with memory) where the fading process is assumed to be regular, i.e., its differential entropy rate is finite. The com-plementary situation of nonregular fading processes has been studied in [10]–[13].
It turns out that the capacity at high SNR is very sensitive to the exact assump-tions of the channel model, in particular to the regularity assumption. If we assume a regular fading process, then the capacity grows only double-logarithmically in the SNR at high SNR [7, Theorem 4.2], [9, Theorem 6.10]. This means that at high power such a channel becomes extremely power-inefficient in the sense that whenever the capacity shall be increased by only one bit, the SNR needs to be squared or, on a dB-scale, the SNR needs to be doubled! So the high-SNR behavior is dramatically different from the optimistic models mentioned above.
For nonregular Gaussian fading the high-SNR behavior of capacity depends on the specific power spectral density and can be anything between the logarithmic and the double-logarithmic growth [11].
However, it is interesting to observe that for low SNR the difference between the different models is relatively small. Indeed, the capacity of regular fading channels
−100 0 10 20 30 40 50 60 1
2 3 4 5 6 7 8 9 10
Output-SNR ρ [dB]
C[natsperchanneluse]
|d| = 0
|d| = 1
|d| = 2
|d| = 4
|d| = 8
|d| = 16
|d| = 32
H= |d|2+ 1 H∼ NC(0, 1) H∼ NC(d, 1)
Figure 1: An upper bound on the capacity of a Rician fading channel as a function of the output-SNR ρ = (1 + |d|2)snr for different values of the specular component d. The dashed line corresponds to the situation of a Rayleigh fading channel with a zero line-of-sight component d = 0. The dotted line depicts the capacity of an additive Gaussian noise channel (without fading) of equal output-SNR ρ, namely log(1 + ρ).
usually shows a very distinct turn at a certain SNR level where the growth rate changes from logarithmic to double-logarithmic. As an example Figure 1 shows the capacity of a noncoherent Rician fading channel with various values of the line-of-sight component. One clearly sees that the capacity curve, while growing logarith-mically at lower SNR, suddenly has a sharp bend at a certain threshold where its growth becomes very slow. Moreover, one sees that this threshold depends strongly on the channel law, i.e., on the line-of-sight component.
We conclude that at lower SNR the exact choice of the channel model has only a small impact on the capacity analysis, i.e., the described simplifications (even the assumption of coherent detection) are useful in that regime. However, at high SNR many simplifications seem to lose their validity. Based on this observation we imme-diately ask ourselves whether we can say something about the separation between these two regimes. Particularly, in the situation of a regular fading model, we would like to know more about the threshold between the efficient low- to medium-SNR regime where the capacity grows logarithmically in the SNR and the highly ineffi-cient high-SNR regime with a double-logarithmic growth. The dependence of this threshold on some system parameters like the number of antennas, the memory in the channel, or the availability of feedback might give valuable insight in good design criteria of wireless and mobile communication systems.