Distortion-Rate Function of Sub-Nyquist Sampled Gaussian Sources

The amount of information lost in sub-Nyquist sampling of a continuous-time Gaussian stationary process is quantified. We consider a combined source coding and sub-Nyquist reconstruction problem in which the input to the encoder is a noisy sub-Nyquist sampled version of the analog source. We first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate-limited version of its samples. This expression is a function of the sampling frequency and the average number of bits describing each sample. It is given as the sum of two terms: Minimum mean square error in estimating the source from its noisy but otherwise fully observed sub-Nyquist samples, and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source. We extend this result to multi-branch uniform sampling, where the samples are available through a set of parallel channels with a uniform sampler and a pre-sampling filter in each branch. Further optimization to reduce distortion is then performed over the pre-sampling filters, and an optimal set of pre-sampling filters associated with the statistics of the input signal and the sampling frequency is found. This results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering. These results thus unify the Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion theory for Gaussian sources.Comment: Accepted for publication at the IEEE transactions on information theor

An Analysis of Mutually Dispersive Brown Symbols for Non-Linear Ambiguity Suppression

This thesis significantly advances research towards the implementation of optimal Non-linear Ambiguity Suppression (NLS) waveforms by analyzing the Brown theorem. The Brown theorem is reintroduced with the use of simplified linear algebraic notation. A methodology for Brown symbol design and digitization is provided, and the concept of dispersive gain is introduced. Numerical methods are utilized to design, synthesize, and analyze Brown symbol performance. The theoretical performance in compression and dispersion of Brown symbols is demonstrated and is shown to exhibit significant improvement compared to discrete codes. As a result of this research a process is derived for the design of optimal mutually dispersive symbols for any sized family. In other words, the limitations imposed by conjugate LFM are overcome using NLS waveforms that provide an effective-fold increase in radar unambiguous range. This research effort has taken a theorem from its infancy, validated it analytically, simplified it algebraically, tested it for realizability, and now provides a means for the synthesis and digitization of pulse coded waveforms that generate an N-fold increase in radar effective unambiguous range. Peripherally, this effort has motivated many avenues of future research

Wavelets and Subband Coding

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book

Orthogonal transmultiplexers : extensions to digital subscriber line (DSL) communications

An orthogonal transmultiplexer which unifies multirate filter bank theory and communications theory is investigated in this dissertation. Various extensions of the orthogonal transmultiplexer techniques have been made for digital subscriber line communication applications. It is shown that the theoretical performance bounds of single carrier modulation based transceivers and multicarrier modulation based transceivers are the same under the same operational conditions. Single carrier based transceiver systems such as Quadrature Amplitude Modulation (QAM) and Carrierless Amplitude and Phase (CAP) modulation scheme, multicarrier based transceiver systems such as Orthogonal Frequency Division Multiplexing (OFDM) or Discrete Multi Tone (DMT) and Discrete Subband (Wavelet) Multicarrier based transceiver (DSBMT) techniques are considered in this investigation. The performance of DMT and DSBMT based transceiver systems for a narrow band interference and their robustness are also investigated. It is shown that the performance of a DMT based transceiver system is quite sensitive to the location and strength of a single tone (narrow band) interference. The performance sensitivity is highlighted in this work. It is shown that an adaptive interference exciser can alleviate the sensitivity problem of a DMT based system. The improved spectral properties of DSBMT technique reduces the performance sensitivity for variations of a narrow band interference. It is shown that DSBMT technique outperforms DMT and has a more robust performance than the latter. The superior performance robustness is shown in this work. Optimal orthogonal basis design using cosine modulated multirate filter bank is discussed. An adaptive linear combiner at the output of analysis filter bank is implemented to eliminate the intersymbol and interchannel interferences. It is shown that DSBMT is the most suitable technique for a narrow band interference environment. A blind channel identification and optimal MMSE based equalizer employing a nonmaximally decimated filter bank precoder / postequalizer structure is proposed. The performance of blind channel identification scheme is shown not to be sensitive to the characteristics of unknown channel. The performance of the proposed optimal MMSE based equalizer is shown to be superior to the zero-forcing equalizer

Optimal channel equalization for filterbank transceivers in presence of white noise

Filterbank transceivers are widely employed in data communication networks to cope with inter-symbol-interference (ISI) through the use of redundancies. This dissertation studies the design of the optimal channel equalizer for both time-invariant and time-varying channels, and wide-sense stationary (WSS) and possible non-stationary white noise processes. Channel equalization is investigated via the filterbank transceivers approach. All perfect reconstruction (PR) or zero-forcing (ZF) receiver filterbanks are parameterized in an affine form, which eliminate completely the ISI. The optimal channel equalizer is designed through minimization of the mean-squared-error (MSE) between the detected signals and the transmitted signals. Our main results show that the optimal channel equalizer has the form of state estimators, and is a modified Kalman filter. The results in this dissertation are applicable to discrete wavelet multitone (DWMT) systems, multirate transmultiplexers, orthogonal frequency division multiplexing (OFDM), and direct-sequence/spread-spectrum (DS/SS) based code division multiple access (CDMA) networks. Design algorithms for the optimal channel equalizers are developed for different channel models, and white noise processes, and simulation examples are worked out to illustrate the proposed design algorithms

Feature Extraction for image super-resolution using finite rate of innovation principles

To understand a real-world scene from several multiview pictures, it is necessary to find the disparities existing between each pair of images so that they are correctly related to one another. This process, called image registration, requires the extraction of some specific information about the scene. This is achieved by taking features out of the acquired images. Thus, the quality of the registration depends largely on the accuracy of the extracted features. Feature extraction can be formulated as a sampling problem for which perfect re- construction of the desired features is wanted. The recent sampling theory for signals with finite rate of innovation (FRI) and the B-spline theory offer an appropriate new frame- work for the extraction of features in real images. This thesis first focuses on extending the sampling theory for FRI signals to a multichannel case and then presents exact sampling results for two different types of image features used for registration: moments and edges. In the first part, it is shown that the geometric moments of an observed scene can be retrieved exactly from sampled images and used as global features for registration. The second part describes how edges can also be retrieved perfectly from sampled images for registration purposes. The proposed feature extraction schemes therefore allow in theory the exact registration of images. Indeed, various simulations show that the proposed extraction/registration methods overcome traditional ones, especially at low-resolution. These characteristics make such feature extraction techniques very appropriate for applications like image super-resolution for which a very precise registration is needed. The quality of the super-resolved images obtained using the proposed feature extraction meth- ods is improved by comparison with other approaches. Finally, the notion of polyphase components is used to adapt the image acquisition model to the characteristics of real digital cameras in order to run super-resolution experiments on real images

Wavelet-based multi-carrier code division multiple access systems

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