Abstract Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

Multiple communication standards including 3GPP LTE and IEEE 802.11, employ codebooks of antenna precoders. For these standards, feedback from the receiver is used to optimize the antenna precoder selection performed at the base station. This optimization process implicitly assumes that the antenna precoders which have equal Euclidean norm are mapped to antenna patterns with equal transmit power. However, for coupled antennas, it can be shown that that if the precoders are implemented as a vector voltage source in a Thevenin source model or as a vector current source in a Norton source model, the equal power condition is not satisfied. The effect of transmission lines between the source and the antenna array are included in the analysis, as this affects the power delivered to the array, as does the use of matching networks. To illustrate, an example is considered using two half-wavelength dipoles for which the variation in transmit power can be in excess of 3 dB even when the Euclidean norm of the precoders is held constant. Given this problem, two methods are proposed which can be used to map equal norm antenna precoders to equal power antenna patterns. The first of these two methods requires signaling of scale factors to the receiver, while the second method can be implemented in a manner that is completely transparent to the receiver.

Orthogonal frequency division multiplexing (OFDM) has become a promising physical layer modulation technology for beyond 3G or 4G wireless communications due to effective inter-symbol interference mitigation for high speed data transmission. However, the timing of the OFDM symbol, i.e., the placement of the DFT collection window in a multi-path time dispersive channel remains an important and challenging issue in OFDM receiver design. An erroneous timing decision creates inter-symbol interference (ISI), inter-carrier interference (ICI), channel attenuation, and channel estimation error, which leads to a penalty on the collected OFDM symbol signal to noise ratio (SNR) resulting in an irreducible error floor. In this paper we quantify such effects and derive an optimal OFDM symbol timing solution in the sense of maximizing the signal to interference ratio (SIR) of the collected OFDM symbol. A practical timing algorithm, referred to as the equilibrium algorithm, is then developed to approximate the optimal timing decision. Compared with existing schemes in the literature, the proposed approach does not rely on explicit detection of individual channel paths or the delay spread boundary and therefore greatly reduces timing complexity. The equilibrium algorithm performs nearly as well as the optimal solution over a variety of channel delay spreads, is simple to implement, and is robust to channel estimation errors.

In this paper, we describe a CRC-based optimal blind transport format determination algorithm with application to the UMTS uplink. The derivation, based on MAP detection, is presented and simulation results under both AWGN and fading channel conditions are given.

In digital communication systems, variable-length Walsh codes, also called orthogonal variable spreading factor codes, are used to support simultaneous variable-rate data transmission. This paper describes algorithms for dynamic variable- length Walsh code assignment. Optimal assignment criterion is derived and an efficient data structure for maintaining variable- length Walsh codes is introduced. The benefits of code reallocation are also studied. Performance is evaluated via simulation.

The channel code bit log likelihood ratio (LLR) for soft decision decoding is derived for quadrature amplitude modulated signals (QAM). The effect of imperfect channel knowledge on soft decision decoding performance is studied. Our results indicate this effect increases with channel estimation error and/or QAM modulation level. A metric based on generalized log likelihood ratio (GLLR) is derived for soft decision channel decoding with imperfect channel knowledge. Numerical results show that the GLLR-based metric outperforms the conventional minimum distance decoding metric that does not take into account channel estimation error.

Multiuser detection has been shown to be an attractive alternative to conventional matched filter detection of CDMA signals. Most of the detection schemes studied however assume perfect channel knowledge. We study two multiuser detection schemes which operate over diversity channels and have access to only noisy channel estimates. The first is an optimum approach and the second is based on decorrelation detection. These detectors must combat not only multiuser interference but also channel estimation noise. Performance measures which separately gauge a multiuser detectors susceptibility to both of these impairments are introduced and evaluated for the proposed detectors. The decorrelation approach, while inferior in terms of rejecting multiuser interference, has the advantage of reduced sensitivity to channel estimation noise. This is illustrated in a simple two user, two diversity branch example.