The importance of mobile communication system nowadays needs not to be empha-sized. Worldwide millions of people rely daily on their mobile phone. While for the user a mobile phone looks very similar to a 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 effect called Doppler effect occurs: the frequency of the trans-mitted signal is shifted depending on the relative movement between receiver and transmitter.
• Since receiver and transmitter are moving and because the environment is permanently changing (e.g., movements by wind, passing cars, people, etc.), the different signal paths are constantly changing.
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.
So far the capacity analysis of above mentioned wireless communication channels were based on the assumption that the receiver has perfect knowledge of the channel state due to the training sequences. The capacity was then computed without taking into account the estimation scheme. Such an approach will definitely lead to an overly optimistic capacity, because
• even with 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 sequences is completely ignored.
The new approach of joint estimation and detection now allows to incorporate the estimation into the capacity analysis. As a matter of fact, we don’t even need to make some assumption about how a particular estimation scheme might work, but can directly try to derive the ultimate data rate that the theoretically best system could achieve. The capacity of such a system is also known as the non-coherent capacity of fading channels.
Unfortunately, the evaluation of the non-coherent channel capacity involves an optimization that is very difficult—if not infeasible—to evaluate analytically or nu-merically.1 Therefore, the question arises how one could get knowledge about the ultimate limit of reliable communication over fading channels without having to solve this infeasible expression.
A promising and interesting approach to this goal is the study of good upper and lower bounds to channel capacity. However, one needs to be aware that finding upper bounds to an expression that itself is a maximization might be rather challenging, too.
1As a matter of fact, this optimization is infeasible for most channels of interest.
In [2] and extracts thereof published before [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], large progress has been made in tackling this problem: a technique has been proposed for the derivation of upper bounds on channel capacity.2 It is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relative entropy) over distributions on the channel output alphabet. Every choice of an output distribution leads to an upper bound on mutual information.
The chosen output distribution need not correspond to some distribution on the channel input. With a judicious choice of output distributions one can often derive tight upper bounds on channel capacity.
Furthermore, in [2] a technique has been proposed for the analysis of the asymp-totic capacity of general cost-constrained channels. The technique is based on the observation that—under fairly mild conditions on the channel—every input distri-bution that achieves a mutual information with the same growth-rate in the cost constraint as the channel capacity must escape to infinity; i.e., under such a distri-bution for some finite cost, the probability of the set of input symbols of lesser cost tends to zero as the cost constraint tends to infinity. For more details about this concept see Section 3.1.1.
Both techniques have been proven very successful: they have been successfully applied to various channel models:
• the free-space optical intensity channel [2], [6], [8];
• an optical intensity channel with input-dependent noise [2];
• the Poisson channel [2], [6], [8];
• multiple-antenna flat fading channels with memory where the fading process is assumed to be regular (i.e., of finite entropy rate3) and where the realization of the fading process is unknown at the transmitter and unknown (or only partially known) at the receiver [2], [4], [7];
• multiple-antenna flat fading channels with memory where the fading process may be irregular (i.e., of possibly infinite entropy rate) and where the realiza-tion of the fading process is unknown (or only partially known) at the receiver [14], [15], [16], [17], [18];
• fading channels with feedback [19], [2], [5];
• non-coherent fading networks [20], [21];
• a phase noise channel [22], [23].
The bounds that have been derived in these contributions are often very tight. For various cases the asymptotic capacity in the limit when the available power (signal-to-noise ratio SNR) tends to infinity has been derived precisely. This is for example the case for the regular single-input multiple-output (SIMO) fading channel with memory and for the regular memoryless multiple-input single-output (MISO) fading channel. In other cases the capacity pre-log (i.e., the ratio of channel capacity to the logarithm of the SNR in the limit when the SNR tends to infinity) could be quantified.
2The technique works for general channels, not fading channels only.
3I.e., a process is called regular when the actual fading realization cannot be predicted even if the infinite past of the process is known.
Some of these results have been very unexpected. E.g., it has been shown in [2]
that regular fading processes have a capacity that grows only double-logarithmically in the SNR at high SNR. This means that at high power these channels become ex-tremely power-inefficient in the sense that for every additional bit capacity the SNR needs to be squared or, respectively, on a dB-scale the SNR needs to be doubled!
This behavior is independent of the particular law of the fading process, the law of the noise process, or the number of antennas at the transmitter or receiver. More-over, the capacity-growth at high SNR is double-logarithmic irrespective whether there is memory in the fading process or not, and it even remains this slow when introducing noiseless feedback [19]! This is in stark contrast to the situation of ad-ditive noise channels and even to the so far known capacity results when assuming prefect knowledge of the channel state at the receiver: there the capacity grows log-arithmically in the power and the mentioned factors (like, e.g., number of antennas, memory, or feedback) have a strong (positive) impact on the capacity. For addi-tive white Gaussian noise (AWGN) channels, e.g., the number of receiver antennas multiplies the capacity and is therefore very beneficial!
−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 1.1: An upper bound on the capacity of a Rician fading channel for different values of the specular component d. The dotted line depicts the capacity of a Gaussian channel of equal output-SNR, namely log(1 + ρ).
Therefore the question arises whether in the case of non-coherent fading channels multiple antennas or feedback is useful at all. It turns out that although the asymp-totic growth rate of capacity is unchanged by these parameters, they still do have a large influence on the systems: the threshold above which the capacity growth changes from logarithmic to double-logarithmic is highly dependent on them! As an example Figure 1.1 shows the capacity of non-coherent Rayleigh fading channels with various numbers of receive antennas.