384 research outputs found
A note on "Relaxation Oscillators with Exact Limit Cycles"
In this note we give a family of planar polynomial differential systems with
a prescribed hyperbolic limit cycle. This family constitutes a corrected and
wider version of an example given in the work of M.A. Abdelkader entitled
``Relaxation Oscillators with Exact Limit Cycles'', which appeared in J. Math.
Anal. Appl. 218 (1998), 308--312. The result given in this note may be used to
construct models of Li\'enard differential equations exhibiting a desired limit
cycle.Comment: 8 pages, no figure
The role of algebraic solutions in planar polynomial differential systems
We study a planar polynomial differential system, given by \dot{x}=P(x,y),
\dot{y}=Q(x,y). We consider a function I(x,y)=\exp \{h_2(x) A_1(x,y) \diagup
A_0(x,y) \} h_1(x) \prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}, where g_i(x) are
algebraic functions, A_1(x,y)=\prod_{k=1}^r (y-a_k(x)), A_0(x,y)=\prod_{j=1}^s
(y-\tilde{g}_j(x)) with a_k(x) and \tilde{g}_j(x) algebraic functions, A_0 and
A_1 do not share any common factor, h_2(x) is a rational function, h(x) and
h_1(x) are functions with a rational logarithmic derivative and \alpha_i are
complex numbers. We show that if I(x,y) is a first integral or an integrating
factor, then I(x,y) is a Darboux function. In order to prove this result, we
show that if g(x) is such that there exists an irreducible polynomial f(x,y)
with f(x,g(x)) \equiv 0, then f(x,y)=0 is an invariant algebraic curve of the
system. In relation with this fact, we give some characteristics related to
particular solutions and functions of the form I(x,y) such as the structure of
their cofactor. Moreover, we consider a function of the form \Phi(x,y):= \exp
\{h_2(x) A_1(x,y) / A_0 (x,y) \}. We show that if the derivative of \Phi(x,y)
with respect to the flow is well defined over A_0(x,y)=0 then \Phi(x,y) gives
rise to an exponential factor.Comment: 28 pages, no figure
Conditional maximum likelihood timing recovery
The conditional maximum likelihood (CML) principle, well known in the context of sensor array processing, is applied to the problem of timing recovery. A new self-noise free CML-based timing error detector is derived. Additionally, a new (conditional) Cramer-Rao bound (CRB) for timing estimation is obtained, which is more accurate than the extensively used modified CRB (MCRB).Peer ReviewedPostprint (published version
Integrability of planar polynomial differential systems through linear differential equations
In this work, we consider rational ordinary differential equations dy/dx =
Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real
coefficients. We give a method to construct equations of this type for which a
first integral can be expressed from two independent solutions of a
second-order homogeneous linear differential equation. This first integral is,
in general, given by a non Liouvillian function. We show that all the known
families of quadratic systems with an irreducible invariant algebraic curve of
arbitrarily high degree and without a rational first integral can be
constructed by using this method. We also present a new example of this kind of
families. We give an analogous method for constructing rational equations but
by means of a linear differential equation of first order.Comment: 24 pages, no figure
On the cyclicity of weight-homogeneous centers
Let W be a weight-homogeneous planar polynomial differential system with a
center. We find an upper bound of the number of limit cycles which bifurcate
from the period annulus of W under a generic polynomial perturbation. We apply
this result to a particular family of planar polynomial systems having a
nilpotent center without meromorphic first integral.Comment: 13 pages, no figure
Non-data-aided frequency offset and symbol timing estimation for binary CPM: performance bounds
The use of (spectrally efficient) CPM modulations may lead to a serious performance degradation of the classical non-data-aided (NDA) frequency and timing estimators due to the presence of self noise. The actual performance of these estimators is usually much worse than that predicted by the classical modified Cramer-Rao bound. We apply some well known results in the field of signal processing to these two important problems of synchronization. In particular we propose and explain the meaning of the unconditional CRB in the synchronization task. Simulation results for MSK and GMSK, along with the performance of some classical and previously proposed synchronizers, show that the proposed bound (along with the MCRB) is useful for a better prediction of the ultimate performance of the NDA estimators.Peer ReviewedPostprint (published version
Polynomial and rational first integrals for planar homogeneous polynomial differential systems
In this paper we find necessary and suffcient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems
Robust beamforming for interference rejection in mobile communications
The problem of robust beamformer design in the presence of moving sources is considered. A new technique based on a generalization of the constrained minimum variance beamformer is proposed. The method explicitly takes into account changes in the scenario due to the movement of the desired and interfering sources, requiring only estimation of the desired DOA. Computer simulations show that the resulting performance constitutes a compromise between interference and noise rejection, computational complexity, and sensitivity to source movement.Peer ReviewedPostprint (published version
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