6,573 research outputs found
Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function
A common problem in cosmology is to integrate the product of two or more
spherical Bessel functions (sBFs) with different configuration-space arguments
against the power spectrum or its square, weighted by powers of wavenumber.
Naively computing them scales as with the number of
configuration space arguments and the grid size, and they cannot be
done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the
sBFs as sums of products of sine and cosine and then using the product to sum
identities, these integrals can then be performed using 1-D FFTs with scaling. This "rotation" method has the potential to
accelerate significantly a number of calculations in cosmology, such as
perturbation theory predictions of loop integrals, higher order correlation
functions, and analytic templates for correlation function covariance matrices.
We implement this approach numerically both in a free-standing,
publicly-available \textsc{Python} code and within the larger,
publicly-available package \texttt{mcfit}. The rotation method evaluated with
direct integrations already offers a factor of 6-10 speed-up over the
naive approach in our test cases. Using FFTs, which the rotation method
enables, then further improves this to a speed-up of
over the naive approach. The rotation method should be useful in light of
upcoming large datasets such as DESI or LSST. In analysing these datasets
recomputation of these integrals a substantial number of times, for instance to
update perturbation theory predictions or covariance matrices as the input
linear power spectrum is changed, will be one piece in a Monte Carlo Markov
Chain cosmological parameter search: thus the overall savings from our method
should be significant
Analytic solutions to a Marcum Q-function-based integral and application in energy detection of unknown signals over multipath fading channels
This work presents analytic solutions for a useful integral in wireless
communications, which involves the Marcum function in combination with an
exponential function and arbitrary power terms. The derived expressions have a
rather simple algebraic representation which renders them convenient both
analytically and computationally. Furthermore, they can be useful in wireless
communications and particularly in the context of cognitive radio
communications and radar systems, where this integral is often encountered. To
this end, we derive novel expressions for the probability of detection in
energy detection based spectrum sensing over fading channels.
These expressions are given in closed-form and are subsequently employed in
analyzing the effects of generalised multipath fading conditions in cognitive
radio systems. As expected, it is shown that the detector is highly dependent
upon the severity of fading conditions as even slight variation of the fading
parameters affect the corresponding performance significantly.Comment: Latest/Preprint Versio
Local observers on linear Lie groups with linear estimation error dynamics
This paper proposes local exponential observers for systems on linear Lie
groups. We study two different classes of systems. In the first class, the full
state of the system evolves on a linear Lie group and is available for
measurement. In the second class, only part of the system's state evolves on a
linear Lie group and this portion of the state is available for measurement. In
each case, we propose two different observer designs. We show that, depending
on the observer chosen, local exponential stability of one of the two
observation error dynamics, left- or right-invariant error dynamics, is
obtained. For the first class of systems these results are developed by showing
that the estimation error dynamics are differentially equivalent to a stable
linear differential equation on a vector space. For the second class of system,
the estimation error dynamics are almost linear. We illustrate these observer
designs on an attitude estimation problem
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
- …