Locally Stationary Functional Time Series

The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cram\'er representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary

Instantaneous Power Spectrum

The estimation of time varying spectra is a complicated one. The use of classical techniques coupled with the local stationarity assumption is met with only moderate success. Of the many time frequency distribution functions used in the signal analysis, none present fully satisfactory spectra. The performance of the spectrogram, Instantaneous Power Spectra (IPS) the Wigner-Ville Distribution (WD) and various aspects of the Rihaczek distribution (RD) for a variety of signal nonstationarities are compared. WD has the most narrow main-lobes but suffers from spectral cross-terms. IPS, the real part of the RD consistently shows a broadened main-lobe without cross-terms. The squared magnitude of the RD places sharp peaks along the crest of the main-lobe and is otherwise very similar to IPS. The imaginary part of the RD shows a sensitivity to discontinuous frequency changes i.e., frequency shift keying.http://archive.org/details/instantaneouspow1094537553Lieutenant, Unuted States NavyApproved for public release; distribution is unlimited

An Adaptive Block-Based Eigenvector Equalization for Time-Varying Multipath Fading Channels

In this paper we present an adaptive Block-Based EigenVector Algorithm (BBEVA) for blind equalization of time-varying multipath fading channels. In addition we assess the performance of the new algorithm for different configurations and compare the results with the least mean squares (LMS) algorithm. The new algorithm is evaluated in terms of intersymbol interference (ISI) suppression, mean squared error (MSE) and by examining the signal constellation at the output of the equalizer. Simulation results show that the BBEVA performs better than the non-blind LMS algorithm

Hybrid solutions to instantaneous MIMO blind separation and decoding: narrowband, QAM and square cases

Future wireless communication systems are desired to support high data rates and high quality transmission when considering the growing multimedia applications. Increasing the channel throughput leads to the multiple input and multiple output and blind equalization techniques in recent years. Thereby blind MIMO equalization has attracted a great interest.Both system performance and computational complexities play important roles in real time communications. Reducing the computational load and providing accurate performances are the main challenges in present systems. In this thesis, a hybrid method which can provide an affordable complexity with good performance for Blind Equalization in large constellation MIMO systems is proposed first. Saving computational cost happens both in the signal sep- aration part and in signal detection part. First, based on Quadrature amplitude modulation signal characteristics, an efficient and simple nonlinear function for the Independent Compo- nent Analysis is introduced. Second, using the idea of the sphere decoding, we choose the soft information of channels in a sphere, and overcome the so- called curse of dimensionality of the Expectation Maximization (EM) algorithm and enhance the final results simultaneously. Mathematically, we demonstrate in the digital communication cases, the EM algorithm shows Newton -like convergence.Despite the widespread use of forward -error coding (FEC), most multiple input multiple output (MIMO) blind channel estimation techniques ignore its presence, and instead make the sim- plifying assumption that the transmitted symbols are uncoded. However, FEC induces code structure in the transmitted sequence that can be exploited to improve blind MIMO channel estimates. In final part of this work, we exploit the iterative channel estimation and decoding performance for blind MIMO equalization. Experiments show the improvements achievable by exploiting the existence of coding structures and that it can access the performance of a BCJR equalizer with perfect channel information in a reasonable SNR range. All results are confirmed experimentally for the example of blind equalization in block fading MIMO systems