Fractional biorthogonal partners in channel equalization and signal interpolation

The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers, hence, the name fractional biorthogonal partners. The conditions for the existence of stable and of finite impulse response (FIR) fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely, the fractionally spaced equalization in digital communications. The goal is to construct zero-forcing equalizers (ZFEs) that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the all-FIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners

Biorthogonal partners and applications

Two digital filters H(z) and F(z) are said to be biorthogonal partners of each other if their cascade H(z)F(z) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. We first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above-mentioned applications

Blind channel identification based on second-order statistics: a frequency-domain approach

In this communication, necessary and sufficient conditions are presented for the unique blind identification of possibly nonminimum phase channels driven by cyclostationary processes. Using a frequency domain formulation, it is first shown that a channel can be identified by the second-order statistics of the observation if and only if the channel transfer function does not have special uniformly spaced zeros. This condition leads to several necessary and sufficient conditions on the observation spectra and the channel impulse response. Based on the frequency-domain formulation, a new identification algorithm is proposed

Generic Feasibility of Perfect Reconstruction with Short FIR Filters in Multi-channel Systems

We study the feasibility of short finite impulse response (FIR) synthesis for perfect reconstruction (PR) in generic FIR filter banks. Among all PR synthesis banks, we focus on the one with the minimum filter length. For filter banks with oversampling factors of at least two, we provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR almost surely. The prescribed length is as short or shorter than the analysis filters and has an approximate inverse relationship with the oversampling factor. Our results are in form of necessary and sufficient statements that hold generically, hence only fail for elaborately-designed nongeneric examples. We provide extensive numerical verification of the theoretical results and demonstrate that the gap between the derived filter length prescriptions and the true minimum is small. The results have potential applications in synthesis FB design problems, where the analysis bank is given, and for analysis of fundamental limitations in blind signals reconstruction from data collected by unknown subsampled multi-channel systems.Comment: Manuscript submitted to IEEE Transactions on Signal Processin

All-adaptive blind matched filtering for the equalization and identification of multipath channels: a practical approach

Blind matched filter receiver is advantageous over the state-of-the-art blind schemes due the simplicity in its implementation. To estimate the multipath communication channels, it uses neither any matrix decomposition methods nor statistics of the received data higher than the second order ones. On the other hand, the realization of the conventional blind matched filter receiver requires the noise variance to be estimated and the equalizer parameters to be calculated in state-space with relatively costly matrix operations. In this paper, a novel architecture is proposed to simplify a potential hardware implementation of the blind matched filter receiver. Our novel approach transforms the blind matched filter receiver into an all-adaptive format which replaces all the matrix operations. Furthermore, the novel design does not need for any extra step to estimate the noise variance. In this paper we also report on a comparative channel equalization and channel identification scenario, looking into the performances of the conventional and our novel all-adaptive blind matched filter receiver through simulations

Differentiable Artificial Reverberation

Artificial reverberation (AR) models play a central role in various audio applications. Therefore, estimating the AR model parameters (ARPs) of a target reverberation is a crucial task. Although a few recent deep-learning-based approaches have shown promising performance, their non-end-to-end training scheme prevents them from fully exploiting the potential of deep neural networks. This motivates to introduce differentiable artificial reverberation (DAR) models which allows loss gradients to be back-propagated end-to-end. However, implementing the AR models with their difference equations "as is" in the deep-learning framework severely bottlenecks the training speed when executed with a parallel processor like GPU due to their infinite impulse response (IIR) components. We tackle this problem by replacing the IIR filters with finite impulse response (FIR) approximations with the frequency-sampling method (FSM). Using the FSM, we implement three DAR models -- differentiable Filtered Velvet Noise (FVN), Advanced Filtered Velvet Noise (AFVN), and Feedback Delay Network (FDN). For each AR model, we train its ARP estimation networks for analysis-synthesis (RIR-to-ARP) and blind estimation (reverberant-speech-to-ARP) task in an end-to-end manner with its DAR model counterpart. Experiment results show that the proposed method achieves consistent performance improvement over the non-end-to-end approaches in both objective metrics and subjective listening test results.Comment: Manuscript submitted to TASL

Subband adaptive filtering for acoustic echo control using allpass polyphase IIR filterbanks

Published versio

Identification of linear periodically time-varying (LPTV) systems

A linear periodically time-varying (LPTV) system is a linear time-varying system with the coefficients changing periodically, which is widely used in control, communications, signal processing, and even circuit modeling. This thesis concentrates on identification of LPTV systems. To this end, the representations of LPTV systems are thoroughly reviewed. Identification methods are developed accordingly. The usefulness of the proposed identification methods is verified by the simulation results. A periodic input signal is applied to a finite impulse response (FIR)-LPTV system and measure the noise-contaminated output. Using such periodic inputs, we show that we can formulate the problem of identification of LPTV systems in the frequency domain. With the help of the discrete Fourier transform (DFT), the identification method reduces to finding the least-squares (LS) solution of a set of linear equations. A sufficient condition for the identifiability of LPTV systems is given, which can be used to find appropriate inputs for the purpose of identification. In the frequency domain, we show that the input and the output can be related by using the discrete Fourier transform (DFT) and a least-squares method can be used to identify the alias components. A lower bound on the mean square error (MSE) of the estimated alias components is given for FIR-LPTV systems. The optimal training signal achieving this lower MSE bound is designed subsequently. The algorithm is extended to the identification of infinite impulse response (IIR)-LPTV systems as well. Simulation results show the accuracy of the estimation and the efficiency of the optimal training signal design

A system identification approach to non-invasive central cardiovascular monitoring

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2008.Includes bibliographical references (leaves 180-187).This thesis presents a new system identification approach to non-invasive central cardiovascular monitoring problem. For this objective, this thesis will develop and analyze blind system identification and input signal reconstruction algorithms for a class of 2-channel IIR and Wiener systems. In particular, this thesis will present blind identifiability conditions for a class of 2-channel IIR and Wiener wave propagation systems and develop the associated blind identification algorithms. It will be shown that the blind identifiability conditions can be achieved in many real-world applications by appropriate selection of channel lengths, sensor locations, and sampling frequency which are the specifications that the system design can exploit for blind identifiability In addition, this thesis will develop a novel input signal reconstruction algorithm that is applicable to general class of multi-channel IIR and Wiener systems. Furthermore, this thesis will rigorously analyze and evaluate three analytic measures for determining the system order and other key parameters of the black-box dynamics as well as for quantifying the quality of the identified gray-box dynamics, without any direct use of unknown input signal: persistent excitation, model identifiability and asymptotic variance. The blind identification and input signal reconstruction algorithms will first be applied to 2-sensor central cardiovascular monitoring problem using two distinct peripheral blood pressure measurements, where the cardiovascular wave propagation dynamics is blindly identified and the aortic blood pressure and flow signals are reconstructed by exploiting black-box and physics-based gray-box model structures of the cardiovascular system.(cont.) The validity of the 2-sensor central cardiovascular monitoring methodology will be illustrated by experimental data from swine subjects and simulation data from a full-scale human cardiovascular simulator across diverse physiologic conditions. The 2-sensor central cardiovascular monitoring methodology will then be extended to address noninvasive, 1-sensor cardiovascular monitoring problem, where the specific challenges involved are 1) identifying the cardiovascular wave propagation dynamics and reconstructing the aortic blood pressure signal by exploiting the measurement from a single peripheral sensor, and 2) identifying the scale for calibrating the blood pressure signal. In order to address these challenges, this thesis will propose a heuristics-based system order estimation algorithm and a model-based blood pressure calibration algorithm, which will be combined with the blind identification of the cardiovascular wave propagation dynamics to realize the non-invasive 1-sensor central cardiovascular monitoring. The non-invasive 1-sensor central cardiovascular monitoring methodology will be illustrated by experimental data from swine subjects, simulation data from a full-scale human cardiovascular simulator, and experimental data from human subjects across diverse physiologic conditions.by Jin-Oh Hahn.Ph.D