102 research outputs found
Robust Padé approximation via SVD
Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity
Universal analytic properties of noise. Introducing the J-Matrix formalism
We propose a new method in the spectral analysis of noisy time-series data
for damped oscillators. From the Jacobi three terms recursive relation for the
denominators of the Pad\'e Approximations built on the well-known Z-transform
of an infinite time-series, we build an Hilbert space operator, a J-Operator,
where each bound state (inside the unit circle in the complex plane) is simply
associated to one damped oscillator while the continuous spectrum of the
J-Operator, which lies on the unit circle itself, is shown to represent the
noise. Signal and noise are thus clearly separated in the complex plane. For a
finite time series of length 2N, the J-operator is replaced by a finite order
J-Matrix J_N, having N eigenvalues which are time reversal covariant. Different
classes of input noise, such as blank (white and uniform), Gaussian and pink,
are discussed in detail, the J-Matrix formalism allowing us to efficiently
calculate hundreds of poles of the Z-transform. Evidence of a universal
behaviour in the final statistical distribution of the associated poles and
zeros of the Z-transform is shown. In particular the poles and zeros tend, when
the length of the time series goes to infinity, to a uniform angular
distribution on the unit circle. Therefore at finite order, the roots of unity
in the complex plane appear to be noise attractors. We show that the
Z-transform presents the exceptional feature of allowing lossless undersampling
and how to make use of this property. A few basic examples are given to suggest
the power of the proposed method.Comment: 14 pages, 8 figure
Discrete structure of the brain rhythms
Neuronal activity in the brain generates synchronous oscillations of the
Local Field Potential (LFP). The traditional analyses of the LFPs are based on
decomposing the signal into simpler components, such as sinusoidal harmonics.
However, a common drawback of such methods is that the decomposition primitives
are usually presumed from the onset, which may bias our understanding of the
signal's structure. Here, we introduce an alternative approach that allows an
impartial, high resolution, hands-off decomposition of the brain waves into a
small number of discrete, frequency-modulated oscillatory processes, which we
call oscillons. In particular, we demonstrate that mouse hippocampal LFP
contain a single oscillon that occupies the -frequency band and a
couple of -oscillons that correspond, respectively, to slow and fast
-waves. Since the oscillons were identified empirically, they may
represent the actual, physical structure of synchronous oscillations in
neuronal ensembles, whereas Fourier-defined "brain waves" are nothing but
poorly resolved oscillons.Comment: 17 pages, 9 figure
Matrix methods for Pad\'e approximation: numerical calculation of poles, zeros and residues
A representation of the Pad\'e approximation of the -transform of a signal
as a resolvent of a tridiagonal matrix is given. Several formulas for the
poles, zeros and residues of the Pad\'e approximation in terms of the matrix
are proposed. Their numerical stability is tested and compared. Methods for
computing forward and backward errors are presented
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