In this paper we review the most peculiar and interesting information-theoretic and communications features of fading channels. We first describe the statistical models of fading channels which are frequently used in the analysis and design of communication systems. Next, we focus on the information theory of fading channels, by emphasizing capacity as the most important performance measure. Both single-user and multiuser transmission are examined. Further, we describe how the structure of fading channels impacts code design, and finally overview equalization of fading multipath channels.

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Fading channels: information-theoretic and communications aspects

In this paper, we introduce a mathematical framework for the characterization of network interference in wireless systems. We consider a network in which the interferers are scattered according to a spatial Poisson process and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We start by determining the statistical distribution of the aggregate network interference. We then investigate four applications of the proposed model: 1) interference in cognitive radio networks; 2) interference in wireless packet networks; 3) spectrum of the aggregate radio-frequency emission of wireless networks; and 4) coexistence between ultrawideband and narrowband systems. Our framework accounts for all the essential physical parameters that affect network interference, such as the wireless propagation effects, the transmission technology, the spatial density of interferers, and the transmitted power of the interferers.

A Mathematical Theory of Network Interference and Its Applications

The emerging massive/large-scale multiple-input multiple-output (LS-MIMO) systems that rely on very large antenna arrays have become a hot topic of wireless communications. Compared to multi-antenna aided systems being built at the time of this writing, such as the long-term evolution (LTE) based fourth generation (4G) mobile communication system which allows for up to eight antenna elements at the base station (BS), the LS-MIMO system entails an unprecedented number of antennas, say 100 or more, at the BS. The huge leap in the number of BS antennas opens the door to a new research field in communication theory, propagation and electronics, where random matrix theory begins to play a dominant role. Interestingly, LS-MIMOs also constitute a perfect example of one of the key philosophical principles of the Hegelian Dialectics, namely, that “quantitative change leads to qualitative change.” In this treatise, we provide a recital on the historic heritages and novel challenges facing LS-MIMOs from a detection perspective. First, we highlight the fundamentals of MIMO detection, including the nature of co-channel interference (CCI), the generality of the MIMO detection problem, the received signal models of both linear memoryless MIMO channels and dispersive MIMO channels exhibiting memory, as well as the complex-valued versus real-valued MIMO system models. Then, an extensive review of the representative MIMO detection methods conceived during the past 50 years (1965–2015) is presented, and relevant insights as well as lessons are inferred for the sake of designing complexity-scalable MIMO detection algorithms that are potentially applicable to LS-MIMO systems. Furthermore, we divide the LS-MIMO systems into two types, and elaborate on the distinct detection strategies suitable for each of them. The type-I LS-MIMO corresponds to the case where the number of active users is much smaller than the number of BS antennas, which is currently the mainstream definition of LS-MIMO. The type-II LS-MIMO corresponds to the case where the number of active users is comparable to the number of BS antennas. Finally, we discuss the applicability of existing MIMO detection algorithms in LS-MIMO systems, and review some of the recent advances in LS-MIMO detection.

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Fifty Years of MIMO Detection: The Road to Large-Scale MIMOs

It has been well established that the appropriate criterion for optimum trellis-coded modulation design on the additive white Gaussian noise channel is maximization of the free Euclidean distance. It is shown that when the trellis-coded modulation is used on a Rician fading channel with interleaving/deinterleaving, the design of the code of optimum performance is guided by other factors, in particular, the length of the shortest error-event path, and the product of branch distances (possibly normalized by the Euclidean distance of the path) along the path. Although maximum free distance (d/sub free/) is still an important consideration, it plays a less significant role the more severe the fading is on the channel. These considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over Rician fading channels. If no interleaving/deinterleaving is used, then once again the design of the trellis code is guided by maximizing d/sub free/. >

The design of trellis coded MPSK for fading channels: performance criteria

Using the notion of the "spacing" between ordered exponential random variables, a performance analysis of the generalized selection combining (GSC) diversity scheme over Rayleigh fading channels is presented and compared with that of the conventional maximal-ratio combining and selection combining schemes. Starting with the moment generating function (MGF) of the GSC output signal-to-noise ratio (SNR), we derive closed-form expressions for the average combined SNR, outage probability, and average error probability of a wide variety of modulation schemes operating over independently, identically distributed (i.i.d.) diversity paths. Because of their simple form, these expressions readily allow numerical evaluation for cases of practical interest. The results are also extended to the case of non-i.i.d. diversity paths.

An MGF-based performance analysis of generalized selection combining over Rayleigh fading channels

The performance of various coherent 8-ary and 16-ary modulations in additive white Gaussian noise (AWGN) and slowly fading channels are analyzed. New expressions for the exact symbol error rates (SER) in fading with diversity combining are derived for any two-dimensional signaling format having polygonal decision boundaries. Maximal ratio combining, equal gain combining, and selection combining are considered. The SER formulas obtained make it possible for the first time to optimize parameters of various constellations precisely and to determine,which constellation has the lowest probability of error. For example, a star constellation such as that specified in the CCITT V.29 standard can be improved by adjusting the amplitude ratios of the points in the constellation to save about 0.63 dB power in AWGN without sacrificing the phase error tolerance, while maintaining the same error rate. The sensitivity of each constellation to phase error is presented and comparisons are made. Six 8-ary signal sets and 11 16-ary signal sets are examined using the new symbol error probability formulas to determine best signal sets for fading channels.

Signaling constellations for fading channels

The performance of 8-PSK and 8-DPSK trellis codes is presented for a class of fast fading, land mobile satellite communication channels. The fading model is Rician but, in addition, the line-of-sight path is subjected to a fast lognormal attenuation that represents tree shadowing. The fading parameters used in this study represent the degree of shadowing and are based on measured data. The primary application considered is for digital speech transmission and thus, bit error probabilities in the order of 10/sup -3/ are emphasized. Sensitivity of the bit error probability to amplitude fading, amplitude and phase fading, and decoding delay is presented. Performance is determined via digital computer simulation. Optimal four- and eight-state codes are determined and optimality is found to be dependent on the presence of lognormal shadowing. >

PSK and DPSK trellis codes for fast fading, shadowed mobile satellite communication channels

A new exact method for computing the average symbol error probability of two-dimensional M-ary signaling in slow fading is presented. The method is generally applicable to polygonal decision regions. The exact average symbol error rate of coherent 16-star-QAM is obtained and compared to that of 16-rectangular-QAM. New results for optimum ring ratios of 16-star-QAM in additive white Gaussian noise (AWGN) and in slow fading are also given.

Error probabilities of two-dimensional M-ary signaling in fading

This letter proposes cyclic prefixed single carrier transmission with frequency domain equalization (SC-FDE) as an alternative physical layer solution for UWB communications. The performance of SC-FDE over IEEE 802.15.3a UWB channel models is analyzed, simulated and compared with that of impulse based single carrier UWB (SC-UWB) and multicarrier UWB employing orthogonal frequency division multiplexing (OFDM-UWB). The impact of channel coding on the performance of OFDM and SC-FDE in UWB is also studied. Our results demonstrate performance advantage of the SC-FDE scheme, especially when implementation issues such as low complexity and low power consumption for UWB are taken into consideration. The performance of SC-FDE with diversity combining using oversampling is also investigated

Cyclic prefixed single carrier transmission in ultra-wideband communications

This paper presents a technique for optimizing the baseband pulse shapes in digital angle modulated signals to minimize the fraction of out-of-band power for a given channel bandwidth. As examples of practical interest, the optimization is carried out for channel bandwidths up to three times the bit rate and for the range of modulation indices usually encountered in digital transmission. Results for MSK-type signals appear as a special case.

P.H. Wittke

Papers

Signaling constellations for fading channels

PSK and DPSK trellis codes for fast fading, shadowed mobile satellite communication channels

Error probabilities of two-dimensional M-ary signaling in fading

Cyclic prefixed single carrier transmission in ultra-wideband communications

Optimum Pulse Shaping in Digital Angle Modulation