Mitchell D. Trott

Efficient use of side information in multiple-antenna data transmission over fading channels

We derive performance limits for two closely related communication scenarios involving a wireless system with multiple-element transmitter antenna arrays: a point-to-point system with partial side information at the transmitter, and a broadcast system with multiple receivers. In both cases, ideal beamforming is impossible, leading to an inherently lower achievable performance as the quality of the side information degrades or as the number of receivers increases. Expected signal-to-noise ratio (SNR) and mutual information are both considered as performance measures. In the point-to-point case, we determine when the transmission strategy should use some form of beamforming and when it should not. We also show that, when properly chosen, even a small amount of side information can be quite valuable. For the broadcast scenario with an SNR criterion, we find the efficient frontier of operating points and show that even when the number of receivers is larger than the number of antenna array elements, significant performance improvements can be obtained by tailoring the transmission strategy to the realized channel.

The capacity of discrete-time memoryless Rayleigh-fading channels

We consider transmission over a discrete-time Rayleigh fading channel, in which successive symbols face independent fading, and where neither the transmitter nor the receiver has channel state information. Subject to an average power constraint, we study the capacity-achieving distribution of this channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. We numerically compute the capacity and the corresponding optimal distribution as a function of the signal-to-noise ratio (SNR). The behavior of the channel at low SNR is studied and finally a comparison is drawn with the ideal additive white Gaussian noise channel.

The dynamics of group codes: state spaces, trellis diagrams, and canonical encoders

A group code C over a group G is a set of sequences of group elements that itself forms a group under a component-wise group operation. A group code has a well-defined state space Sigma /sub k/ at each time k. Each code sequence passes through a well-defined state sequence. The set of all state sequences is also a group code, the state code of C. The state code defines an essentially unique minimal realization of C. The trellis diagram of C is defined by the state code of C and by labels associated with each state transition. The set of all label sequences forms a group code, the label code of C, which is isomorphic to the state code of C. If C is complete and strongly controllable, then a minimal encoder in controller canonical (feedbackfree) form may be constructed from certain sets of shortest possible code sequences, called granules. The size of the state space Sigma /sub k/ is equal to the size of the state space of this canonical encoder, which is given by a decomposition of the input groups of C at each time k. If C is time-invariant and nu -controllable, then mod Sigma /sub k/ mod = Pi /sub 1 >

Sphere-bound-achieving coset codes and multilevel coset codes

A simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array L and its error probability on an additive white Gaussian noise (AWGN) channel. It is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding, including certain binary lattices. These codes have structure of the kind that has been found to be useful in practice. Capacity curves and design guidance for practical codes are given. Exponential error bounds for coset codes are developed, generalizing Poltyrevs (1994) bounds for lattices. These results are based on the channel coding theorems of information theory, rather than the Minkowski-Hlawka theorem of lattice theory.

Performance limits of coded diversity methods for transmitter antenna arrays

Several aspects of the design and optimization of coded multiple-antenna transmission diversity methods for slowly time-varying channels are explored from an information-theoretic perspective. Both optimized vector-coded systems, which can achieve the maximum possible performance, and suboptimal scalar-coded systems, which reduce complexity by exploiting suitably designed linear precoding, are investigated. The achievable rates and associated outage characteristics of these spatial diversity schemes are evaluated and compared, both for the case when temporal diversity is being jointly exploited and for the case when it is not. Complexity and implementation issues more generally are also discussed.

Efficient signal processing techniques for exploiting transmit antenna diversity on fading channels

A class of powerful and computationally efficient strategies for exploiting transmit antenna diversity on fading channels is developed. These strategies, which require simple linear processing at the transmitter and receiver, have attractive asymptotic characteristics. In particular, given a sufficient number of transmit antennas, these techniques effectively transform a nonselective Rayleigh fading channel into a nonfading, simple white marginally Gaussian noise channel with no intersymbol interference. These strategies, which we refer to as linear antenna precoding, can be efficiently combined with trellis coding and other popular error-correcting codes for bandwidth-constrained Gaussian channels. Linear antenna precoding requires no additional power or bandwidth and is attractive in terms of robustness and delay considerations. The resulting schemes have powerful and convenient interpretations in terms of transforming nonselective fading channels into frequency- and time-selective ones.

Minimality and observability of group systems

Abstract Group systems are a generalization of Willems-type linear systems that are useful in error control coding. It is shown that the basic ideas of Willemss treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems. The interplay between systems (behaviors) and trellises (evolution laws) is discussed with respect to completeness, minimality, controllability, and observability. It is pointed out that, for trellises of group systems and Willems-type linear systems, minimality is essentially the same as observability. The development is universal-algebraic in nature and holds unconditionally for linear systems over the real numbers.

Rotational invariance of trellis codes. I. Encoders and precoders

We present a theoretical framework for rotational invariance of trellis codes. The distinction between codes and encoders plays a pivotal role. Necessary and sufficient conditions for rotational invariance are derived under general assumptions, and a construction is presented that obtains a rotationally invariant encoder for almost any rotationally invariant code, independent of the codes algebraic structure. Encoders that use a differential precoder are considered as a separate case, where a system-theoretic characterization of precoding is used to find two alternative and slightly less general encoder constructions.

The dynamics of group codes: Dual abelian group codes and systems

Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian (LCA) groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent. If C and C/sup /spl perp// are dual codes, then the state spaces of C act as the character groups of the state spaces of C/sup /spl perp//. The controllability properties of C are the observability properties of C/sup /spl perp//. In particular, C is (strongly) controllable if and only if C/sup /spl perp// is (strongly) observable, and the controller memory of C is the observer memory of C/sup /spl perp//. The controller granules of C act as the character groups of the observer granules of C/sup /spl perp//. Examples of minimal observer-form encoder and syndrome-former constructions are given. Finally, every observer granule of C is an end-around controller granule of C.

Rotational invariance of trellis codes. II. Group codes and decoders

For pt.I see ibid., vol.42, no.3, p.751-65 (1996). In Part I, general results on rotationally invariant codes and encoders were derived assuming no algebraic structure. In Part II, trellis codes based on group systems are considered as a special case for which code and encoder constructions are particularly simple. Rotational invariance is expressed as an algebraic constraint on a group code, and algebraic constructions are found for both absorbed precoder encoders and for encoders with separate differential precoders. Finally, the various encoder forms used to achieve rotational invariance are compared based on their performance on an AWGN channel.