Chair: Dimitrios Hatzinakos, University of Toronto (CANADA)
J.C. Ralston, Queensland University of Technology (AUSTRALIA)
A. M. Zoubir, Queensland University of Technology (AUSTRALIA)
We consider the identification of systems which are both time-varying and nonlinear. This class of systems is more likely to be encountered in practice, but is often avoided due to the difficulties that arise in modelling and estimation. We attempt to address this problem by considering a new time-varying nonlinear model, the time-varying Hammerstein model, which effectively characterises time-variation and nonlinearity in a simple manner. Using this model we formulate a procedure to find least-squares estimates of the coefficients. The model is general and can be used when little is known about the time-variation of the system. In addition, we do not require that the input is stationary or Gaussian. Finally, an application to automobile knock modelling is presented, where a time-varying nonlinear model is seen to more accurately characterise the system than a time-varying linear one.
Olivier Michel, ENS/Lyon (FRANCE)
Alfred Hero, University of Michigan (USA)
We develop a non-parametric method of nonlinear prediction based on adaptive partitioning of the phase space associated with the process. The partitioning method is implemented with a recursive tree-structured vector quantization algorithm which successively refines the partition by binary splitting where the splitting threshold is determined by a penalized maximum entropy criterion. A complexity penalty is derived and applied to protect against high statistical variability of the predictor structure. We establish an important relation between our tree-structured model for the process and generalized non-linear thresholded AR model (ART). We illustrate our method for two cases where classical linear prediction is ineffective: a chaotic double-scroll signal measured at the output of a Chua-type electronic circuit, and a simulated second order ART model.
S. Lawrence Marple Jr., Acuson Corporation (USA)
This paper presents a new fast computational algorithm for the solution of the least squares normal equations of the two-dimensional (2-D) covariance method of linear prediction. The fast algorithm exploits the near-to-doubly-Toeplitz structure of the normal equations when expressed in matrix form. This algorithm is useful for generating high resolution imagery from coherent imaging system in-phase/quadrature (I/Q) data, such as synthetic aperture radar (SAR).
M. Kemal Sonmez, University of Maryland (USA)
John S. Baras, University of Maryland (USA)
We derive estimators for the multiple lag process, a generalization of the lag process, via spectral representations of stationary processes by complex random spectral measures. We present estimators of transfer functions for the multiple lag model with a given vector of lags and derive a multiple-lag (quadratic) coherence which can be maximized to choose the best vector of lags in the minimum mean squared error sense from a given set of lag vectors. We also demonstrate the estimation scheme by a simulation example and point out possible applications for the multiple-lag model in speech processing.
Unto K.Laine, Helsinki University of Technology (FINLAND)
The conventional theory of linear prediction (LP) is renewed and extended to form a more flexible algorithm called generalized linear prediction (GLP). There are three new levels of generalization available: On the first level (I) the predictor FIR is replaced with a generalized FIR constructed out of allpass sections having complex coefficients. On the second level (II) the allpass filters have distributed coefficients, i.e. they are unequal, and on the third and the most general level (III) the filter sections may have different characteristics. The theory of GLP is presented and the algorithm is tested with speech signals. The results show that GLP works as desired: nonuniform frequency resolution can be achieved and the resolution is controlled by the choice of the allpass parameters. On level I, the angle of the pole-zero-pair of the allpass sections defines the highest resolution area while the radius of the pole controls the degree of the resolution improvement. The GLP prediction error decreases rapidly with the order of the predictor. Its normalized RMS value falls off exponentially (!) and its spectral flatness improves efficiently. On the average the results are much better than those of conventional LP. Levels II and III are only briefly discussed.
Miki Haseyama, Hokkaido University (JAPAN)
Yoshihiro Aketa, Hokkaido University (JAPAN)
Hideo Kitajima, Hokkaido University (JAPAN)
This paper proposes a method for realization of an ARMAX lattice filter. ARMAX (Autoregressive Moving Average model with Exogenous Variable) model identification is significant because the ARMAX model is a standard tool in the control field, and it can be performed by the proposed algorithm. One of the recursive least-square methods for the ARMAX model identification is the ELS (Extended Least Squares). Applied to the ARMAX model identification, the ELS uses O($N^2$) multiplications, where N = AR order + MA order + X order. When the proposed realization method of the ARMAX lattice filter is used, o(M) multiplications are needed for the ARMAX model identification, where M = max{AR order, MA order, X order}.
Jean-Yves Tourneret, ENSEEIHT/ GAPSE (FRANCE)
Bernard Lacaze, ENSEEIHT/ GAPSE (FRANCE)
Being linked by non-linear relations, AR parameters and reflection coefficients cannot both be gaussian. However, a statistical study can show that these two sets of parameters are gaussian asymptotically. The aim of this paper is to show that the convergence rate of reflection coefficient distribution to the gaussian one depends on the position of AR model poles in the unit circle. An analysis of the reflection coefficient Taylor expansion around AR parameters is proposed to determine this convergence rate.
Mats Cedervall, Uppsala University
Petre Stoica, Chalmers University of Technology (SWEDEN)
This paper considers the estimation of the parameters of a linear discrete-time system from noisy input and output measurements. The conditions imposed on the system are quite general. The proposed method makes use of an instrumental variable (IV)-vector whose cross-covariance with the system's regression vector is pre- and post-multiplied by some prechosen weights. The singular vectors of this matrix possess complete information on the system parameters. A weighted subspace fitting (WSF) method is then applied to these singular vectors to consistently estimate the parameters of the system. The proposed method is non-iterative, easy to implement and has a small computational burden. The asymptotic distribution of its estimation errors is derived and the result is used to motivate the choice of the weighting matrix in the WSF step and also to predict the estimation accuracy. A numerical example is included to illustrate the performance.
Jin-Jen Hsue, University of Michigan (USA)
Andrew E. Yagle, University of Michigan (USA)
We provide a comparison between one-sided linear prediction (OSP) and two-sided linear prediction (TSP) with respect to prediction error, relationships to AR modeling and to two-sided AR modeling, and the application to time series interpolation, linear-phase filter design, and spectral estimation. New contributions of this paper include: (1) proof that TSP produces smaller, non-white residuals than OSP, extending previous results; (2) specification of the frequency-domain error criterion minimized by TSP, and comparison with the analogous OSP criterion; (3) demonstration that TSP and two-sided AR modeling are different problems, unlike OSP; (4) interpretation of performance of TSP interference-rejection filters.
A. Ferrari, CNRS/UNSA (FRANCE)
R. Lorion, CNRS/UNSA (FRANCE)
G. Alengrin, CNRS/UNSA (FRANCE)
This paper treats the problem of ambiguity resolution using non-uniform sampling. This problem occurs for Doppler estimation in coherent pulsed Doppler radar. In this paper, we study the case where the duration between two samples is a linear function of time: quadratic sampling. Assuming that the continuous signal is stationary, the sampled signal will be non-stationary. The autocorrelation of this signal is derived and the Wigner distribution of the sampled signal related to the spectrum of the continuous signal. As a consequence, a time frequency relief of the signal will verify symmetries. These constraints, assuming an AR evolutive model for the sampled signal and band limitation for the continuous signal, allow the derivation of a particular time varying model for the samples. An associated estimation algorithm, leading to the unfolded spectrum is then proposed.