1) The document discusses using the residue theorem to evaluate a complex contour integral to calculate the Laplace transform of the output of an ideal sampler. This provides a closed-form solution that is less painful than the infinite series form.
2) An ideal sampler can be modeled as a carrier signal modulated by the input signal. The output of the sampler is then sent to a zero-order hold.
3) By choosing an appropriate contour, the complex contour integral can be evaluated using the residue theorem. This gives the Laplace transform of the sampler output in terms of the residues of the integrand's poles.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Application of Residue Inversion Formula for Laplace Transform to Initial Val...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Application of Residue Inversion Formula for Laplace Transform to Initial Val...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
I am Grey Nolan. Currently associated with matlabassignmentexperts.com as an assignment helper. After completing my master's from the University of British Columbia, I was in search for an opportunity that expands my area of knowledge hence I decided to help students with their Signals and Systems assignments. I have written several assignments till date to help students overcome numerous difficulties they face in Signals and Systems Assignments.
My talk in the MCQMC Conference 2016, Stanford University. The talk is about Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks.
La introducción de la incertidumbre en modelos epidemiológicos es un área de incipiente actividad en la actualidad. En la mayor parte de los enfoques adoptados se asume un comportamiento gaussiano en la formulación de dichos modelos a través de la perturbación de los parámetros vía el proceso de Wiener o movimiento browiniano u otro proceso discretizado equivalente.
En esta conferencia se expone un método alternativo de introducir la incertidumbre en modelos de tipo epidemiológico que permite considerar patrones no necesariamente normales o gaussianos. Con el enfoque adoptado se determinará en contextos epidemiológicos que tienen un gran número de aplicaciones, la primera función de densidad de probabilidad del proceso estocástico solución. Esto permite la determinación exacta de la respuesta media y su variabilidad, así como la construcción de predicciones probabilísticas con intervalos de confianza sin necesidad de recurrir a aproximaciones asintóticas, a veces de difícil legitimación. El enfoque adoptado también permite determinar la distribución probabilística de parámetros que tienen gran importancia para los epidemiólogos, incluyendo la distribución del tiempo hasta que un cierto número de infectados permanecen en la población, lo cual, por ejemplo, permite tener información probabilística para declarar el estado de epidemia o pandemia de una determinada enfermedad. Finalmente, se expondrá algunos de los retos computacionales inmediatos a los que se enfrenta la técnica expuesta.
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Seminar Talk: Multilevel Hybrid Split Step Implicit Tau-Leap for Stochastic R...Chiheb Ben Hammouda
In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics are dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. These stochastic models constitute the theory of Stochastic Reaction Networks (SRNs). In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (explicit-TL) method, can be very slow. In this talk, we propose a novel implicit scheme, split-step implicit tau-leap (SSI-TL), to improve numerical stability and provide efficient simulation algorithms for those systems. Furthermore, to estimate statistical quantities related to SRNs, we propose a novel hybrid Multilevel Monte Carlo (MLMC) estimator in the spirit of the work by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012). This estimator uses the SSI-TL scheme at levels where the explicit-TL method is not applicable due to numerical stability issues, and then, starting from a certain interface level, it switches to the explicit scheme. We present numerical examples that illustrate the achieved gains of our proposed approach in this context.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
Joint blind calibration and time-delay estimation for multiband rangingTarik Kazaz
In this presentation, we focus on the problem of blind joint calibration of multiband transceivers and time-delay (TD) estimation of multipath channels. We show that this problem can be formulated as a particular case of covariance matching. Although this problem is severely ill-posed, prior information about radio-frequency chain distortions and multipath channel sparsity is used for regularization. This approach leads to a biconvex optimization problem, which is formulated as a rank-constrained linear system and solved by a simple group Lasso algorithm.
% This method is general and can be also applied for calibration of sensors arrays and in direction of arrival estimation.
Numerical experiments show that the proposed algorithm provides better calibration and higher resolution for TD estimation than current state-of-the-art methods.
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
Very often one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (\pdf) and/or by the corresponding probability characteristic functions (\pcf), or by a function representation. Here the interest is mainly to compute characterisations like the entropy, or
relations between two distributions, like their Kullback-Leibler divergence, or more general measures such as $f$-divergences,
among others. These are all computed from the \pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. It is an even stronger numerical challenge to then actually compute said characterisations in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format.
Low rank tensor approximation of probability density and characteristic funct...
residue
1. • page 1
To: Dr. Jim Gillilan
From: Rob Arnold
Re: An application of the “residue theorem”
I asked you once in Applied Complex Variables class, “What do you do with a complex contour integral?”
Well, I found out when I got to Digital Control Systems class that some nasty Laplace transform stunts are
better handled by evaluating a complex contour integral, usually resulting in much less pain. An example
follows:
Suppose f(z) is analytic inside and on a simple closed contour C except for finitely many isolated singular
points z1, z2, . . .zn interior to C.
Then by the residue theorem,
f z dz j
C
( ) =∫ 2π (sum of residues of f at points z1, z2, . . .zn)
Consider a continuous function, x(t). The action of sampling x(t) at discrete instants of time (t = kT, k =
0,1,2,3 . . .) can (ideally) be expressed as the multiplication of x(t) with a train of “impulse functions” (the
Dirac delta function, given byδ( ) ,t t= ≠0 0 δ( )t dt =
−∞
∞
∫ 1 ). If we symbolize the impulse train by
δ δT
n
t t nT( ) ( )= −
=
∞
∑0
, then an ideal sampling of x(t) can be represented mathematically by
x t x t tT*( ) ( ) ( )= δ . This can be interpreted as a carrier signal, δT t( ) , modulated by a signal x(t).
Functions with discontinuities at t=kT can cause an ambiguity in the above definition. Therefore, we define
the output signal of an ideal sampler as follows:
X s x nT nTs
n
*( ) ( )exp( )= −
=
∞
∑0
, where x(t) is the signal at the input of the ideal sampler. Typically,
the output of the sampling stage is sent to a zero-order hold, a circuit which maintains the sampler’s output
value for the duration of the sample interval. The sample-and-hold output can be expressed as:
x t x u t u t T x T u t T u t T x T u t T u t T( ) ( )[ ( ) ( )] ( )[ ( ) ( )] ( )[ ( ) ( )]= − − + − − − + − − − +0 2 2 2 3 K
which has Laplace transform
X s x
s
Ts
s
x T
Ts
s
Ts
s
x T
Ts
s
Ts
s
( ) ( )
exp( )
( )
exp( ) exp( )
( )
exp( ) exp( )
= −
−
+
−
−
−
+
−
−
−
+0
1 2
2
2 3
K
[ ]=
− −
+ − + − + − +
1
0 2 2 3 3
exp( )
( ) ( )exp( ) ( )exp( ) ( )exp( )
Ts
s
x x T Ts x T Ts x T Ts K
= −
− −
=
∞
∑x nT nTs
Ts
sn
( )exp( )
exp( )
0
1
The first factor depends on the input signal and the sample period T. The second factor is independent of
the input signal. Note that the first term corresponds to X*(s). The second factor then expresses the effect
of the zero-order hold.
How can we evaluate X*(s)? One method involves taking the Laplace transform by evaluating the
convolution integral:
X s
j
X s dTc j
c j
*( ) ( ) ( )= −
− ∞
+ ∞
∫
1
2π
λ λ λ∆ ,where ∆T s( ) is the Laplace Transform of δT t( )
2. • page 2
∆T s Ts Ts Ts
Ts
( ) exp( ) exp( ) exp( )
exp( )
= + − + − + − + =
− −
1 2 3
1
1
K
∆T s( ) has poles at values of s satisfying exp( )− =Ts 1, or s j
n
T
jn n=
= = ± ±
2
0 1 20
π
ω , , , ,K
Choose a contour γ such that all the poles of X(λ) lie within γ, and all the poles of ∆T ( )λ lie to the
exterior of γ. This can be done by choosing the real constant c appropriately.
Pole locations for the integrand given above
The λ-plane is translated horizontally from the s-plane by real
constant c.
Then, for the contour shown above, X s
j
X s dT*( ) ( ) ( )= −∫
1
2π
λ λ λ
γ
∆
which can be evaluated by the “residue theorem.” (We picked a contour to ensure that the integrand is
analytic on and around contour γ.) This gives
X s residues of X
T spoles of X
*( ) _ _ ( )
exp( ( ))_ _ ( )
=
− − −
∑ λ
λλ
1
1
which can be evaluated
somewhat less painfully than the infinite series form of X*(s). In practice, you can use the above result
without resorting to the actual setup of any convolution integral or consideration of pole locations and
contours in the λ-plane.
Q: “What do you do with a complex contour integral?”
A: “Calculate a closed form for the Laplace transform of the output of an “ideal” sampler, a device
which models the analog-to-digital converter, increasingly used in control systems applications as well as in
communications, digital audio (i.e. your CD player or digital VCR), and data acquisition. It turns out that
the residue theorem is an extremely useful result, producing results in these problems not readily obtainable
by other means.”
References:
Charles L. Phillips & H. Troy Nagle, Digital Control System Analysis and Design. Englewood Cliffs, New
Jersey: Prentice Hall, 1995.
B. P. Lathi, Linear Systems and Signals. Carmichael, California: Berkeley-Cambridge Press, 1992.
Math 407 Class Notes.