Shannon Theory 2.0

Telecommunication technology has been developed significantly using various equipments including dialing telephones.

Shannon theory has been greatly useful in the research and develop of wireless communication systems. Shannon theory is one of the effective analysis tools to find the maximum achievable capacity of a given communication channel and offers a hint to find a signal processing method to achieve the optimum performance bound. By the help of Shannon theory, we have been able to find effective solutions for modern wireless communication systems, leading to near the optimal performance.

Shannon theory has been crucial to develop simplified wireless communication systems which are forced to have a number of strict assumptions such as hard limitation on inter-user interference. For the simplified communication systems, it has been revealed that the throughput performance of the system can be unbelievably close to the theoretical bound provided by the Shannon theory. However, communication researchers recently found that it is necessary to evolve the original Shannon theory to analyze emerging complex wireless communication systems. Intermediate examples of complex wireless communication systems are the MIMO and MU-MIMO systems and the regular example is the cooperative relay system, where to find the optimal performance, the former systems require the reasonable update of the original Shannon theory and the later system requires fundamental evolution of the original Shannon theory.

J. Andrews and et al. presented that the following points should be revisited or revised to apply the Shannon theory to cooperative relay wireless systems: unbound delay and reliability, spatial and timescale decomposition and overhead consideration.

Optimization for Simple Communication Systems

Main article: Effective Spectral Efficiency for Enhanced Shannon Theory

For simple wireless communication systems including the single antenna and peer-to-peer (single user) system, Shannon theory has been applied successfully to analysis the performance of the simple system. The basic link capacity formula, given by

$C_{SU} = log2 (1+ \mathrm{SNR} )$

where SNR is the signal-to-noise ratio. The term SINR can be replaced to the term SNR. The theoretical capacity of the simple communication system assuming orthogonal transmission is close to the net throughput after implementation.

The principle Shannon capacity formula of $C_{SU}$ ignores the performance loss by overhead signaling. Even if we replace SNR by SINR in the formula, it is fundamentally limited to consider the performance of more realistic cases.

Let's consider limited feedback beamforming systems including multuser ZF-BF. The capacity of the feedback system ($C_{SU}$) can be improved by increasing the amount of feedback signaling as the SINR is increased. However, as we know the overall performance can be, on the contrary, decreased because the normalized data transmission time is decreased such as $T_{DL}/T =( T - T_{FB} ) / T$.

We define the overall performance as the effective spectral efficiency

$R_{eff} = (1-T_{FB}/T) * R_{ZF}$

where $R_{ZF}$ is the spectral efficiency of limited feedback ZFBF. Hence, the optimization of the feedback time is necessary to achieve the maximum effective spectral efficiency.

Spatial and timescale decomposition

For space-time decomposition, I'll explain detail later on except to my understanding, it represents the performance variation according to locations of multiple nodes in a same wireless network. If two node is highly close each other, information transmision from one mobile to another requires almost no wireless resource. Hence, the consideration of location in terms of space and time is important in mobile ad-hoc networks.

Overhead consideration: pilot, feedback

Main article: Overhead consideration: pilot, feedback

We consider the effective spectral efficiency of overhead signaling systems. Assume that $R$ is the achievable throughput of the perfect information system and $\Delta R$ is the difference of the achievable throughput performance between the perfect information system and the limited information system. For overhead signalling, we use time resource of $T_\mathrm{OS}$ when the total time resource is $T$. Then, the effective throughput performance of the limited information system is defined as

$\mathcal{R}_\mathrm{eff} = (1 - \tau_\mathrm{OS}) R_\mathrm{OS}(\tau_\mathrm{OS})$

where $\tau_\mathrm{OS} = T_\mathrm{OS}/T$ and $R_\mathrm{OS}(\tau_\mathrm{OS}) = R - \Delta R_\mathrm{OS}(\tau_\mathrm{OS})$. The equation above shows that if the time resource for overhead signalling $\tau_\mathrm{OS}$ goes to large, the real transmission time of $T_\mathrm{DL} = 1 - \tau_\mathrm{OS}$ becomes smaller and the performance difference of $\Delta R_\mathrm{OS}(\tau_\mathrm{OS})$ becomes smaller. Therefore, it is obvious that there is an optimal point for $\tau_\mathrm{OS}$ to achieve the maximum of $\mathcal{R}_\mathrm{eff}$

Important Examples

We have now several examples which show the possibility to find better solutions than conventional approaches. Candidate examples are Alamouti space-time diversity coding, MIMO, interference alignment and XOR network coding. I will explain why those example schemes can outperform conventional approaches.

Alamouti space-time diversity coding

Many researchers thought that two different antenna should transmit their signal independently using a duplexing method such as time division or frequency division duplexing.