This document introduces the Z-transform and its properties for analyzing discrete signals and systems. It begins by defining the Z-transform as an extension of the discrete Fourier transform with a convergence factor, allowing it to represent a wider class of sequences. Properties discussed include the unilateral and bilateral Z-transforms, time shifting, and the region of convergence. Several examples of direct Z-transforms are provided, such as for exponential, unit step, and impulse sequences.
These notes include some results from frame theory. These results are useful for sparse representations and compressive sensing theory. Errors and omissions are all mine.
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
This document provides an overview of character theory for finite groups and its analogy to representation theory for the infinite compact group S1. It discusses several key concepts:
1) Character theory characterizes representations of a finite group G by their characters, which are class functions that map G to complex numbers. Characters determine representations up to isomorphism.
2) The irreducible representations of S1 are 1-dimensional and in bijection with integers n, where each representation maps z to zn.
3) By analogy to Fourier analysis, the characters of S1 form an orthonormal basis for L2(S1) and decompose representations into irreducibles in the same way as for finite groups.
This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
The document discusses analyzing functions using calculus concepts like derivatives. It introduces analyzing functions to determine if they are increasing, decreasing, or constant on intervals based on the sign of the derivative. The sign of the derivative indicates whether the graph of the function has positive, negative, or zero slope at points, relating to whether the function is increasing, decreasing, or constant. It also introduces the concept of concavity, where the derivative indicates whether the curvature of the graph is upward (concave up) or downward (concave down) based on whether tangent lines have increasing or decreasing slopes. Examples are provided to demonstrate these concepts.
Double Robustness: Theory and Applications with Missing DataLu Mao
When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
These notes include some results from frame theory. These results are useful for sparse representations and compressive sensing theory. Errors and omissions are all mine.
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
This document provides an overview of character theory for finite groups and its analogy to representation theory for the infinite compact group S1. It discusses several key concepts:
1) Character theory characterizes representations of a finite group G by their characters, which are class functions that map G to complex numbers. Characters determine representations up to isomorphism.
2) The irreducible representations of S1 are 1-dimensional and in bijection with integers n, where each representation maps z to zn.
3) By analogy to Fourier analysis, the characters of S1 form an orthonormal basis for L2(S1) and decompose representations into irreducibles in the same way as for finite groups.
This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
The document discusses analyzing functions using calculus concepts like derivatives. It introduces analyzing functions to determine if they are increasing, decreasing, or constant on intervals based on the sign of the derivative. The sign of the derivative indicates whether the graph of the function has positive, negative, or zero slope at points, relating to whether the function is increasing, decreasing, or constant. It also introduces the concept of concavity, where the derivative indicates whether the curvature of the graph is upward (concave up) or downward (concave down) based on whether tangent lines have increasing or decreasing slopes. Examples are provided to demonstrate these concepts.
Double Robustness: Theory and Applications with Missing DataLu Mao
When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
This document discusses limits and continuity in calculus. It begins by explaining how limits were used to define instantaneous rates of change in velocity and acceleration, which were fundamental to the development of calculus. The chapter then aims to develop the concept of the limit intuitively before providing precise mathematical definitions. Limits are introduced as the value a function approaches as the input gets arbitrarily close to a given value, without actually reaching it. Several examples are provided to illustrate how to determine limits through sampling inputs and making conjectures.
The document introduces the concept of generalized quasi-nonexpansive (GQN) maps. Some key results are:
1) GQN maps generalize quasi-nonexpansive maps but the fixed point set may not always be closed or convex.
2) If a subset satisfies certain conditions, it is a GQN-retract of the space.
3) Under these conditions, the class of GQN-retracts is closed under intersection and the common fixed point set of an increasing sequence of GQN maps is a GQN-retract.
Topic: Fourier Series ( Periodic Function to change of interval)Abhishek Choksi
The document discusses Fourier series and their properties. Fourier series can be used to represent periodic functions as an infinite sum of sines and cosines. The key points are:
- Fourier series can represent functions over any interval length by transforming the variable.
- Examples show how to calculate the Fourier coefficients for specific functions over given intervals.
- The Fourier series representation allows periodic functions to be broken down into their constituent trigonometric components.
This document summarizes a research paper on obtaining local estimates for minimizers of convex integral functionals involving the gradient.
The paper considers functionals whose lagrangians depend only on the gradient and satisfy a generalized symmetry assumption. It adapts a method to obtain local estimates for minimizers near points that are not local extrema. These estimates show that minimizers are bounded from below (above) by a linear function associated with the Minkowski functional. The estimates are then used to deduce versions of the strong maximum principle in the variational setting.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document summarizes an investigation into extending the Strong Maximum Principle for integral functionals involving Minkowski gauges. It begins by introducing the integral functional and definitions related to Minkowski gauges. It then discusses prior work establishing the Strong Maximum Principle under certain conditions, and extending the class of comparison functions used. The document aims to further generalize the Strong Maximum Principle by considering inf- and sup-convolutions of functions with the Minkowski gauge, which allows the principle to unify previous properties into a single extremal extension principle. It provides background, definitions, and auxiliary results to support the generalization proposed in Section 3.
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
Chapter 12 vectors and the geometry of space mergedEasyStudy3
This document discusses vectors and geometry in 3D space. It covers topics like 3D coordinate systems, vectors, dot and cross products, equations of lines and planes, cylinders and quadric surfaces. There are also tables listing examples of quadric surface graphs. The document provides information on representing and analyzing geometric objects in 3D space using vectors and coordinate systems.
Numerical integration;Gaussian integration one point, two point and three poi...vaibhav tailor
The document discusses numerical integration using Gaussian quadrature. It describes one-point, two-point, and three-point Gaussian quadrature rules. For each rule, it provides the formula used to approximate a definite integral of a function over an interval by calculating a weighted sum of the function values at specified points. Examples are included to demonstrate applying the one-point, two-point, and three-point rules to evaluate definite integrals.
The document discusses issues with pinning and facetting in lattice Boltzmann simulations of multiphase flows. It presents a lattice Boltzmann model for propagating sharp interfaces using a phase field approach. Sharpening the phase field interface causes it to become pinned to the lattice or develop facets. Introducing randomness via a random projection method or random threshold prevents pinning and delays facetting, allowing the interface to propagate at the correct speed even for very sharp boundaries.
This document discusses implicit differentiation and exponential growth and decay models. It contains:
1) An example of using implicit differentiation to find the derivative of a circle equation and the equation of the tangent line.
2) An explanation of how exponential growth and decay models take the form of y' = ky, leading to solutions of y = Ce^kt where k is the constant relative growth or decay rate.
3) An example modeling world population growth from 1950-2020 using an exponential growth model that estimates the 1993 population and predicts the 2020 population.
The lattice Boltzmann equation: background and boundary conditionsTim Reis
The document summarizes the lattice Boltzmann equation (LBE) method for computational fluid dynamics. It begins by discussing how the LBE is derived from kinetic theory and discrete kinetic theory, rather than directly from fluid equations. Taking moments of the discrete Boltzmann equation recovers the compressible Navier-Stokes equations through Chapman-Enskog expansion. The document then derives an analytic solution for the LBE governing Poiseuille flow between stationary walls under the influence of gravity.
The document summarizes key concepts in vector calculus and linear algebra including:
- The gradient of a scalar field describes the direction of steepest ascent/descent and is defined as the vector of partial derivatives.
- Curl describes infinitesimal rotation of a 3D vector field and is defined as the cross product of the del operator and the vector field.
- Divergence measures the magnitude of a vector field's source or sink and is defined as the del operator dotted with the vector field.
- Solenoidal fields have zero divergence and irrotational fields have zero curl. The curl of a gradient is always zero and the divergence of a curl is always zero.
IRJET- Common Fixed Point Results in Menger SpacesIRJET Journal
This document presents a common fixed point theorem for five self-maps on a complete Menger space. The theorem proves that if the maps satisfy certain conditions, including being continuous, having a compatibility property, and satisfying a contraction condition, then the maps have a common fixed point. The conditions and proof involve the use of probabilistic distances, triangular norms, Cauchy sequences, and limits in Menger spaces. The theorem generalizes prior results on common fixed points and provides a way to establish the existence of solutions to equations involving multiple operators.
The double integral of a function f(x,y) over a bounded region R in the xy-plane is defined as the limit of Riemann sums that approximate the total value of f over R. This double integral is denoted by the integral of f(x,y) over R and its value is independent of the subdivision used in the Riemann sums. Properties and methods for evaluating double integrals are discussed, along with applications such as finding the area, volume, mass, and moments of inertia. Changes of variables in double integrals using the Jacobian are also covered.
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
This document presents a stochastic sharpening approach for improving the pinning and facetting of sharp phase boundaries in lattice Boltzmann simulations. It summarizes that multiphase lattice Boltzmann models are prone to unphysical pinning and facetting of interfaces. By replacing the deterministic sharpening threshold with a random variable, the approach delays the onset of these issues and better predicts propagation speeds, even with very sharp boundaries. The random projection method preserves the shape of a propagating circular patch, unlike the standard LeVeque model which results in facetting and non-circular shapes at high sharpness ratios.
This document discusses key concepts in vector calculus including:
1) The gradient of a scalar, which is a vector representing the directional derivative/rate of change.
2) Divergence of a vector, which measures the outward flux density at a point.
3) Divergence theorem, relating the outward flux through a closed surface to the volume integral of the divergence.
4) Curl of a vector, which measures the maximum circulation and tendency for rotation.
Formulas are provided for calculating these quantities in Cartesian, cylindrical, and spherical coordinate systems. Examples are worked through applying the concepts and formulas.
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
1. The doubling map on the circle has 2n-1 periodic points of period n. Its periodic points are dense.
2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
System monitoringu i optymalizacji łańcucha zaopatrzenia zakładów przetwórstw...BetterSolutions
Prezentacja systemu zarządzania transportem oraz wyników projektu badawczo-rozwojowego FLOTA++, w którym stworzono algorytmy marszrutyzacji pojazdów przeznaczone dla optymalizacji planów transportowych.
Дні Польського Кіно в Україні — власний проект Польського Інституту у Києві. Тра-
диційно у програмі представлено шість найкращих польських фільмів, створених
впродовж останніх двох років.
This document discusses limits and continuity in calculus. It begins by explaining how limits were used to define instantaneous rates of change in velocity and acceleration, which were fundamental to the development of calculus. The chapter then aims to develop the concept of the limit intuitively before providing precise mathematical definitions. Limits are introduced as the value a function approaches as the input gets arbitrarily close to a given value, without actually reaching it. Several examples are provided to illustrate how to determine limits through sampling inputs and making conjectures.
The document introduces the concept of generalized quasi-nonexpansive (GQN) maps. Some key results are:
1) GQN maps generalize quasi-nonexpansive maps but the fixed point set may not always be closed or convex.
2) If a subset satisfies certain conditions, it is a GQN-retract of the space.
3) Under these conditions, the class of GQN-retracts is closed under intersection and the common fixed point set of an increasing sequence of GQN maps is a GQN-retract.
Topic: Fourier Series ( Periodic Function to change of interval)Abhishek Choksi
The document discusses Fourier series and their properties. Fourier series can be used to represent periodic functions as an infinite sum of sines and cosines. The key points are:
- Fourier series can represent functions over any interval length by transforming the variable.
- Examples show how to calculate the Fourier coefficients for specific functions over given intervals.
- The Fourier series representation allows periodic functions to be broken down into their constituent trigonometric components.
This document summarizes a research paper on obtaining local estimates for minimizers of convex integral functionals involving the gradient.
The paper considers functionals whose lagrangians depend only on the gradient and satisfy a generalized symmetry assumption. It adapts a method to obtain local estimates for minimizers near points that are not local extrema. These estimates show that minimizers are bounded from below (above) by a linear function associated with the Minkowski functional. The estimates are then used to deduce versions of the strong maximum principle in the variational setting.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document summarizes an investigation into extending the Strong Maximum Principle for integral functionals involving Minkowski gauges. It begins by introducing the integral functional and definitions related to Minkowski gauges. It then discusses prior work establishing the Strong Maximum Principle under certain conditions, and extending the class of comparison functions used. The document aims to further generalize the Strong Maximum Principle by considering inf- and sup-convolutions of functions with the Minkowski gauge, which allows the principle to unify previous properties into a single extremal extension principle. It provides background, definitions, and auxiliary results to support the generalization proposed in Section 3.
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
Chapter 12 vectors and the geometry of space mergedEasyStudy3
This document discusses vectors and geometry in 3D space. It covers topics like 3D coordinate systems, vectors, dot and cross products, equations of lines and planes, cylinders and quadric surfaces. There are also tables listing examples of quadric surface graphs. The document provides information on representing and analyzing geometric objects in 3D space using vectors and coordinate systems.
Numerical integration;Gaussian integration one point, two point and three poi...vaibhav tailor
The document discusses numerical integration using Gaussian quadrature. It describes one-point, two-point, and three-point Gaussian quadrature rules. For each rule, it provides the formula used to approximate a definite integral of a function over an interval by calculating a weighted sum of the function values at specified points. Examples are included to demonstrate applying the one-point, two-point, and three-point rules to evaluate definite integrals.
The document discusses issues with pinning and facetting in lattice Boltzmann simulations of multiphase flows. It presents a lattice Boltzmann model for propagating sharp interfaces using a phase field approach. Sharpening the phase field interface causes it to become pinned to the lattice or develop facets. Introducing randomness via a random projection method or random threshold prevents pinning and delays facetting, allowing the interface to propagate at the correct speed even for very sharp boundaries.
This document discusses implicit differentiation and exponential growth and decay models. It contains:
1) An example of using implicit differentiation to find the derivative of a circle equation and the equation of the tangent line.
2) An explanation of how exponential growth and decay models take the form of y' = ky, leading to solutions of y = Ce^kt where k is the constant relative growth or decay rate.
3) An example modeling world population growth from 1950-2020 using an exponential growth model that estimates the 1993 population and predicts the 2020 population.
The lattice Boltzmann equation: background and boundary conditionsTim Reis
The document summarizes the lattice Boltzmann equation (LBE) method for computational fluid dynamics. It begins by discussing how the LBE is derived from kinetic theory and discrete kinetic theory, rather than directly from fluid equations. Taking moments of the discrete Boltzmann equation recovers the compressible Navier-Stokes equations through Chapman-Enskog expansion. The document then derives an analytic solution for the LBE governing Poiseuille flow between stationary walls under the influence of gravity.
The document summarizes key concepts in vector calculus and linear algebra including:
- The gradient of a scalar field describes the direction of steepest ascent/descent and is defined as the vector of partial derivatives.
- Curl describes infinitesimal rotation of a 3D vector field and is defined as the cross product of the del operator and the vector field.
- Divergence measures the magnitude of a vector field's source or sink and is defined as the del operator dotted with the vector field.
- Solenoidal fields have zero divergence and irrotational fields have zero curl. The curl of a gradient is always zero and the divergence of a curl is always zero.
IRJET- Common Fixed Point Results in Menger SpacesIRJET Journal
This document presents a common fixed point theorem for five self-maps on a complete Menger space. The theorem proves that if the maps satisfy certain conditions, including being continuous, having a compatibility property, and satisfying a contraction condition, then the maps have a common fixed point. The conditions and proof involve the use of probabilistic distances, triangular norms, Cauchy sequences, and limits in Menger spaces. The theorem generalizes prior results on common fixed points and provides a way to establish the existence of solutions to equations involving multiple operators.
The double integral of a function f(x,y) over a bounded region R in the xy-plane is defined as the limit of Riemann sums that approximate the total value of f over R. This double integral is denoted by the integral of f(x,y) over R and its value is independent of the subdivision used in the Riemann sums. Properties and methods for evaluating double integrals are discussed, along with applications such as finding the area, volume, mass, and moments of inertia. Changes of variables in double integrals using the Jacobian are also covered.
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
This document presents a stochastic sharpening approach for improving the pinning and facetting of sharp phase boundaries in lattice Boltzmann simulations. It summarizes that multiphase lattice Boltzmann models are prone to unphysical pinning and facetting of interfaces. By replacing the deterministic sharpening threshold with a random variable, the approach delays the onset of these issues and better predicts propagation speeds, even with very sharp boundaries. The random projection method preserves the shape of a propagating circular patch, unlike the standard LeVeque model which results in facetting and non-circular shapes at high sharpness ratios.
This document discusses key concepts in vector calculus including:
1) The gradient of a scalar, which is a vector representing the directional derivative/rate of change.
2) Divergence of a vector, which measures the outward flux density at a point.
3) Divergence theorem, relating the outward flux through a closed surface to the volume integral of the divergence.
4) Curl of a vector, which measures the maximum circulation and tendency for rotation.
Formulas are provided for calculating these quantities in Cartesian, cylindrical, and spherical coordinate systems. Examples are worked through applying the concepts and formulas.
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
1. The doubling map on the circle has 2n-1 periodic points of period n. Its periodic points are dense.
2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
System monitoringu i optymalizacji łańcucha zaopatrzenia zakładów przetwórstw...BetterSolutions
Prezentacja systemu zarządzania transportem oraz wyników projektu badawczo-rozwojowego FLOTA++, w którym stworzono algorytmy marszrutyzacji pojazdów przeznaczone dla optymalizacji planów transportowych.
Дні Польського Кіно в Україні — власний проект Польського Інституту у Києві. Тра-
диційно у програмі представлено шість найкращих польських фільмів, створених
впродовж останніх двох років.
This document lists and defines different types of geometric shapes including rectangles, circles, hexagons, triangles, trapezoids, cylinders, cubes, parallelograms, stars and hearts. It also mentions that lines and shapes are features of faces and provides the titles of two artworks.
Suponiendo que nuestro barco se hunde cerca de una isla desierta, descubrimos cómo apareció el dinero y sus funciones. La aparición de los bancos y la necesidad del banco central (con sus funciones). ¿Por qué nació la bolsa? ¿Para qué sirve?
Por último cómo la política monetaria permite el control de la inflación.
Este documento describe la modulación AM de banda lateral única (SSB), que elimina la portadora y una de las bandas laterales de la señal AM para mejorar la eficiencia espectral y la relación señal-ruido. SSB reduce el ancho de banda requerido a la mitad y aumenta la potencia transmitida en las bandas laterales, mejorando la relación señal-ruido en más de 7 dB en comparación con AM convencional. SSB transmite solo una de las bandas laterales que contienen la información de audio, eliminando
Shirley J. Willis has over 30 years of experience in special education as a teacher, administrator, and consultant in Maine and Arizona. She holds Master's and Bachelor's degrees in education from the University of Southern Maine and the University of Maine at Presque Isle. Her resume lists her extensive certifications and employment history working with students from kindergarten through high school, as well as her involvement in professional organizations and additional experiences in curriculum development, workshops, and adjunct teaching.
Knowledge management and information systemnihad341
this file would help you in writing your assignment on knowledge management and information system. I did this for a student of UK. He got a very satisfactory marks from it. Then i thought that why not help others. The course is a complex one. So, this would be my pleasure if someone really found this useful.
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This document summarizes a seminar report on discrete time systems and the Z-transform. It defines discrete time systems and different types of systems including causal/noncausal, linear/nonlinear, time-invariant/variant, static/dynamic. It then explains the Z-transform, its properties including region of convergence and time shifting. Some common Z-transform pairs are provided along with methods for the inverse Z-transform. Advantages of the Z-transform for analysis of discrete systems and signals are mentioned.
La clasificación de Kennedy es una clasificación topográfica que provee información rápida sobre el diagnóstico, tratamiento y pronóstico de pacientes parcialmente desdentados. Divide los arcos dentarios en clases basadas en la ubicación y número de dientes faltantes, ayudando a la comunicación entre profesionales dentales, pacientes y laboratorios. No solo se usa para prótesis removibles sino en todo tipo de rehabilitación protésica.
A knowledge management system (KMS) is a system for applying and using knowledge management principles. These include data-driven objectives around business productivity, a competitive business model, business intelligence analysis and more.
This document provides an analytic-combinatoric proof of Pólya's recurrence theorem for simple random walks on lattices Zd. It introduces generating functions to represent combinatorial classes of random walks. The generating function for simple random walks yields a formula for the probability of being at a given position at a given time. Laplace's method is then used to estimate these probabilities asymptotically, showing the random walk is recurrent if d = 1, 2 and transient if d ≥ 3, proving Pólya's theorem.
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
Brian Covello: Research in Mathematical Group Representation Theory and SymmetryBrian Covello
Brian Covello's research review on group representation theory and symmetry. In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, a graphical method exists to determine their finite representations that associates with each representation a Young tableau (also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
Cauchy's integral theorem, Cauchy's integral formula, Cauchy's integral formula for derivatives, Taylor's Series, Maclaurin’s Series,Laurent's Series,Singularities and zeros, Cauchy's Residue theorem,Evaluation various types of complex integrals.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
This document provides exercises on Hawking radiation using scalar field theory in the Kruskal spacetime. It asks the student to find the radial equation for the scalar field and show that near the horizon, the field takes the form of ingoing and outgoing waves that are analytic in different coordinate systems. The document then derives the Klein-Gordon equation in Schwarzschild coordinates and uses separation of variables to obtain approximate solutions near the horizon. It shows that ingoing waves are regular on the future horizon but outgoing waves are not, and vice versa for the past horizon.
The document discusses the eigenvalues of finite CMV matrices, which are matrices that represent multiplication by z in an orthonormal basis of polynomials on the unit circle called the CMV basis. It explores how the eigenvalues of truncated CMV matrices behave asymptotically for different classes of measures, such as Lebesgue measure or measures with inserted mass points. In particular, it shows that for measures with a single inserted mass point, the eigenvalues exhibit "clock" behavior, clustering uniformly around the unit circle in the limit as the matrix size increases.
Chapter 6 frequency domain transformation.pptxEyob Adugnaw
This document discusses the z-transform, which is a generalization of the Fourier transform that can be used to analyze discrete-time signals and systems. It introduces the z-transform and its properties such as the region of convergence, poles and zeros. Examples are provided to illustrate how to determine the region of convergence from the sequence and identify stable and unstable systems based on whether the unit circle is included in the region of convergence. Theorems and properties of the z-transform such as linearity, time shifting, and differentiation are also covered.
- The z-transform is a mathematical tool that converts discrete-time sequences into complex functions, analogous to how the Laplace transform handles continuous-time signals.
- Key properties and sequences that are transformed include the unit impulse δn, unit step un, and geometric sequences an.
- The z-transform is computed by taking the z-transform definition, which is an infinite summation, and obtaining closed-form expressions using properties like linearity and geometric series sums.
- Common transforms include U(z) for the unit step, 1/1-az^-1 for geometric sequences an, and expressions involving z, sinh/cosh, and sin/cos for exponential and trigonometric sequences.
The document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. The z-transform plays a similar role for discrete-time systems as the Laplace transform does for continuous-time systems. Some key properties of the z-transform discussed include the region of convergence, properties and theorems like the shifting theorem and initial/final value theorems, and applications to feedback control systems.
The document discusses line integrals of complex functions along curves. It defines a line integral as the integral of a continuous function f(z) along a curve c from point A to B, which is divided into n parts. It also provides the definition of a line integral in terms of a parameterization z(t) of the curve from a to b. Finally, it lists five properties of line integrals: linearity, sense reversal, partitioning of paths, integral inequality, and the Mean Value inequality.
The document discusses permutations and the symmetric group S3. It defines what a permutation is and introduces the six permutations that make up S3: the identity E, a 120 degree rotation R, a 240 degree rotation R2, a vertical reflection V, and reflections RV and R2V. It explains that S3 forms a group under composition of permutations. It also introduces the alternating group A3, which is the subgroup of S3 made up of the even permutations E, R, and R2.
Frequency Analysis using Z Transform.pptxDrPVIngole
This document provides an overview of frequency analysis using the Z-transform. It defines the Z-transform, discusses its properties and relationship to the Fourier transform. Examples are provided to demonstrate calculating the Z-transform of different signals and determining the region of convergence. Key topics covered include the definition of the Z-transform, its region of convergence, properties, inverse Z-transform, and analyzing discrete time linear time-invariant systems using the Z-transform.
The document defines parameterized surfaces and discusses their surface area. A parameterized surface S is defined by a function f that maps points (s,t) in a parameter region T to points in R3. S has a smooth parameterization if the Jacobian of f is continuous and the normal vector is never zero. Surface area is approximated by dividing S into small parameter rectangles and taking the limit. The area of a parameterized surface S is given by the integral over the parameter region T of the cross product of the partial derivatives of the parameterization f with respect to s and t.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
I am Ahmed M. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from New York University, Abu Dhabi. I have been helping students with their assignments for the past 10 years. I solve assignments related to Signals and Systems.
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Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
I am Anastasia S. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Clemson University, USA. I have been helping students with their assignments for the past 6 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.
Some properties of m sequences over finite field fpIAEME Publication
1) The document discusses properties of M-sequences over finite fields Fp when p is an odd prime.
2) Some key properties discussed are: the set of cyclic permutations of a non-zero period is not closed under addition; the matrix of these permutations is symmetric about the second diagonal; and the sum of any two rows with one translated by half the period is the zero sequence.
3) The document also presents theorems about the characteristics and representations of M-sequences over finite fields, including their relation to irreducible polynomials and representation as matrices.
Similar to Signals and transforms in linear systems analysis (20)
The chemistry of the actinide and transactinide elements (set vol.1 6)Springer
Actinium is the first member of the actinide series of elements according to its electronic configuration. Actinium closely resembles lanthanum chemically. The three most important isotopes of actinium are 227Ac, 228Ac, and 225Ac. 227Ac is a naturally occurring isotope in the uranium-actinium decay series with a half-life of 21.772 years. 228Ac is in the thorium decay series with a half-life of 6.15 hours. 225Ac is produced from 233U with applications in medicine.
Transition metal catalyzed enantioselective allylic substitution in organic s...Springer
This document provides an overview of computational studies of palladium-mediated allylic substitution reactions. It discusses the history and development of quantum mechanical and molecular mechanical methods used to study the structures and reactivity of allyl palladium complexes. In particular, density functional theory methods like B3LYP have been widely used to study reaction mechanisms and factors controlling selectivity. Continuum solvation models have also been important for properly accounting for reactions in solvent.
1) Ranchers in Idaho observed lambs born with cyclopia (one eye) due to ewes grazing on corn lily plants. Cyclopamine was identified as the compound responsible and was later found to inhibit the Hedgehog signaling pathway.
2) Nakiterpiosin and nakiterpiosinone were isolated from cyanobacterial sponges and shown to inhibit cancer cell growth. Their unique C-nor-D-homosteroid skeleton presented synthetic challenges.
3) The authors developed a convergent synthesis of nakiterpiosin involving a carbonylative Stille coupling and a photo-Nazarov cyclization. Model studies led them to propose a revised structure for n
This document reviews solid-state NMR techniques that have been used to determine the molecular structures of amyloid fibrils. It discusses five categories of NMR techniques: 1) homonuclear dipolar recoupling and polarization transfer via J-coupling, 2) heteronuclear dipolar recoupling, 3) correlation spectroscopy, 4) recoupling of chemical shift anisotropy, and 5) tensor correlation methods. Specific techniques described include rotational resonance, dipolar dephasing, constant-time dipolar dephasing, REDOR, and fpRFDR-CT. These techniques have provided insights into the hydrogen-bond registry, spatial organization, and backbone torsion angles of amyloid fibrils.
This document discusses principles of ionization and ion dissociation in mass spectrometry. It covers topics like ionization energy, processes that occur during electron ionization like formation of molecular ions and fragment ions, and ionization by energetic electrons. It also discusses concepts like vertical transitions, where electronic transitions occur much faster than nuclear motions. The document provides background information on fundamental gas phase ion chemistry concepts in mass spectrometry.
Higher oxidation state organopalladium and platinumSpringer
This document discusses the role of higher oxidation state platinum species in platinum-mediated C-H bond activation and functionalization. It summarizes that the original Shilov system, which converts alkanes to alcohols and chloroalkanes under mild conditions, involves oxidation of an alkyl-platinum(II) intermediate to an alkyl-platinum(IV) species by platinum(IV). This "umpolung" of the C-Pt bond facilitates nucleophilic attack and product formation rather than simple protonolysis back to alkane. Subsequent work has validated this mechanism and also demonstrated that platinum(IV) can be replaced by other oxidants, as long as they rapidly oxidize the
Principles and applications of esr spectroscopySpringer
- Electron spin resonance (ESR) spectroscopy is used to study paramagnetic substances, particularly transition metal complexes and free radicals, by applying a magnetic field and measuring absorption of microwave radiation.
- ESR spectra provide information about electronic structure such as g-factors and hyperfine couplings by measuring resonance fields. Pulse techniques also allow measurement of dynamic properties like relaxation.
- Paramagnetic species have unpaired electrons that create a magnetic moment. ESR detects transition between spin energy levels induced by microwave absorption under an applied magnetic field.
This document discusses crystal structures of inorganic oxoacid salts from the perspective of periodic graph theory and cation arrays. It analyzes 569 crystal structures of simple salts with the formulas My(LO3)z and My(XO4)z, where M are metal cations, L are nonmetal triangular anions, and X are nonmetal tetrahedral anions. The document finds that in about three-fourths of the structures, the cation arrays are topologically equivalent to binary compounds like NaCl, NiAs, and FeB. It proposes representing these oxoacid salts as a quasi-binary model My[L/X]z, where the cation arrays determine the crystal structure topology while the oxygens play a
Field flow fractionation in biopolymer analysisSpringer
This document summarizes a study that uses flow field-flow fractionation (FlFFF) to measure initial protein fouling on ultrafiltration membranes. FlFFF is used to determine the amount of sample recovered from membranes and insights into how retention times relate to the distance of the sample layer from the membrane wall. It was observed that compositionally similar membranes from different companies exhibited different sample recoveries. Increasing amounts of bovine serum albumin were adsorbed when the average distance of the sample layer was less than 11 mm. This information can help establish guidelines for flow rates to minimize fouling during ultrafiltration processes.
1) The document discusses phonons, which are quantized lattice vibrations in crystals that carry thermal energy. It describes modeling crystal vibrations using a harmonic lattice approach.
2) Normal modes of the lattice vibrations can be described as a set of independent harmonic oscillators. Quantum mechanically, these normal modes are quantized as phonons with discrete energy levels.
3) Phonons can be thought of as quasiparticles that carry momentum and energy in the crystal lattice. Their propagation is described using a phonon field approach rather than independent normal modes.
This chapter discusses 3D electroelastic problems and applied electroelastic problems. For 3D problems, it presents the potential function method for solving problems involving a penny-shaped crack and elliptic inclusions. It derives the governing equations and introduces potential functions to obtain the general static and dynamic solutions. For applied problems, it discusses simple electroelastic problems, laminated piezoelectric plates using classical and higher-order theories, and piezoelectric composite shells. It also presents a unified first-order approximate theory for electro-magneto-elastic thin plates.
Tensor algebra and tensor analysis for engineersSpringer
This document discusses vector and tensor analysis in Euclidean space. It defines vector- and tensor-valued functions and their derivatives. It also discusses coordinate systems, tangent vectors, and coordinate transformations. The key points are:
1. Vector- and tensor-valued functions can be differentiated using limits, with the derivatives being the vector or tensor equivalent of the rate of change.
2. Coordinate systems map vectors to real numbers and define tangent vectors along coordinate lines.
3. Under a change of coordinates, components of vectors and tensors transform according to the Jacobian of the coordinate transformation to maintain geometric meaning.
This document provides a summary of carbon nanofibers:
1) Carbon nanofibers are sp2-based linear filaments with diameters of around 100 nm that differ from continuous carbon fibers which have diameters of several micrometers.
2) Carbon nanofibers can be produced via catalytic chemical vapor deposition or via electrospinning and thermal treatment of organic polymers.
3) Carbon nanofibers exhibit properties like high specific area, flexibility, and strength due to their nanoscale diameters, making them suitable for applications like energy storage electrodes, composite fillers, and bone scaffolds.
Shock wave compression of condensed matterSpringer
This document provides an introduction and overview of shock wave physics in condensed matter. It discusses the assumptions made in treating one-dimensional plane shock waves in fluids and solids. It briefly outlines the history of the field in the United States, noting that accurate measurements of phase transitions from shock experiments established shock physics as a discipline and allowed development of a pressure calibration scale for static high pressure work. It describes some of the practical applications of shock wave experiments for providing high-pressure thermodynamic data, understanding explosive detonations, calibrating pressure scales, and enabling studies of materials under extreme conditions.
Polarization bremsstrahlung on atoms, plasmas, nanostructures and solidsSpringer
This document discusses the quantum electrodynamics approach to describing bremsstrahlung, or braking radiation, of a fast charged particle colliding with an atom. It derives expressions for the amplitude of bremsstrahlung on a one-electron atom within the first Born approximation. The amplitude has static and polarization terms. The static term corresponds to radiation from the incident particle in the nuclear field, reproducing previous results. The polarization term accounts for radiation from the atomic electron and contains resonant denominators corresponding to intermediate atomic states. The full treatment allows various limits to be taken, such as removing the nucleus or atomic electron, reproducing known results from quantum electrodynamics.
Nanostructured materials for magnetoelectronicsSpringer
This document discusses experimental approaches to studying magnetization and spin dynamics in magnetic systems with high spatial and temporal resolution.
It describes using time-resolved X-ray photoemission electron microscopy (TR-XPEEM) to image the temporal evolution of magnetization in magnetic thin films with picosecond time resolution. Results are presented showing the changing domain structure in a Permalloy thin film following excitation with a magnetic field pulse. Different rotation mechanisms are observed depending on the initial orientation of the magnetization with respect to the applied field.
A novel pump-probe magneto-optical Kerr effect technique using higher harmonic generation is also discussed for addressing spin dynamics in magnetic systems with femtosecond time resolution and element selectivity.
This document discusses nanomaterials for biosensors and implantable biodevices. It describes how nanostructured thin films have enabled the development of more sensitive electrochemical biosensors by improving the detection of specific molecules. Two common techniques for creating nanostructured thin films are described - Langmuir-Blodgett films and layer-by-layer films. These techniques allow for the precise control of film thickness at the nanoscale and have been used to immobilize biomolecules like enzymes to create biosensors. Recent research is also exploring how these nanostructured films and biomolecules can be used to create implantable biosensors for real-time monitoring inside the body.
Modern theory of magnetism in metals and alloysSpringer
This document provides an introduction to magnetism in solids. It discusses how magnetic moments originate from electron spin and orbital angular momentum at the atomic level. In solids, electron localization determines whether magnetic properties are described by localized atomic moments or collective behavior of delocalized electrons. The key concepts of metals and insulators are introduced. The document then presents the basic Hamiltonian used to describe magnetism in solids, including terms for kinetic energy, electron-electron interactions, spin-orbit coupling, and the Zeeman effect. It also discusses how atomic orbitals can be used as a basis set to represent the Hamiltonian and describes the symmetry properties of s, p, and d orbitals in cubic crystals.
This chapter introduces and classifies various types of damage that can occur in structures. Damage can be caused by forces, deformations, aggressive environments, or temperatures. It can occur suddenly or over time. The chapter discusses different damage mechanisms including corrosion, excessive deformation, plastic instability, wear, and fracture. It also introduces concepts that will be covered in more detail later such as damage mechanics, fracture mechanics, and the influence of microstructure on damage and fracture. The chapter aims to provide an overview of damage types before exploring specific mechanisms and analyses in later chapters.
This document summarizes research on identifying spin-wave eigen-modes in a circular spin-valve nano-pillar using Magnetic Resonance Force Microscopy (MRFM). Key findings include:
1) Distinct spin-wave spectra are observed depending on whether the nano-pillar is excited by a uniform in-plane radio-frequency magnetic field or by a radio-frequency current perpendicular to the layers, indicating different excitation mechanisms.
2) Micromagnetic simulations show the azimuthal index φ is the discriminating parameter, with only φ=0 modes excited by the uniform field and only φ=+1 modes excited by the orthogonal current-induced Oersted field.
3) Three indices are used to label resonance
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Building Production Ready Search Pipelines with Spark and MilvusZilliz
Spark is the widely used ETL tool for processing, indexing and ingesting data to serving stack for search. Milvus is the production-ready open-source vector database. In this talk we will show how to use Spark to process unstructured data to extract vector representations, and push the vectors to Milvus vector database for search serving.
Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
Get an inside look at the latest Neo4j innovations that enable relationship-driven intelligence at scale. Learn more about the newest cloud integrations and product enhancements that make Neo4j an essential choice for developers building apps with interconnected data and generative AI.
Full-RAG: A modern architecture for hyper-personalizationZilliz
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
Infrastructure Challenges in Scaling RAG with Custom AI modelsZilliz
Building Retrieval-Augmented Generation (RAG) systems with open-source and custom AI models is a complex task. This talk explores the challenges in productionizing RAG systems, including retrieval performance, response synthesis, and evaluation. We’ll discuss how to leverage open-source models like text embeddings, language models, and custom fine-tuned models to enhance RAG performance. Additionally, we’ll cover how BentoML can help orchestrate and scale these AI components efficiently, ensuring seamless deployment and management of RAG systems in the cloud.
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?Speck&Tech
ABSTRACT: A prima vista, un mattoncino Lego e la backdoor XZ potrebbero avere in comune il fatto di essere entrambi blocchi di costruzione, o dipendenze di progetti creativi e software. La realtà è che un mattoncino Lego e il caso della backdoor XZ hanno molto di più di tutto ciò in comune.
Partecipate alla presentazione per immergervi in una storia di interoperabilità, standard e formati aperti, per poi discutere del ruolo importante che i contributori hanno in una comunità open source sostenibile.
BIO: Sostenitrice del software libero e dei formati standard e aperti. È stata un membro attivo dei progetti Fedora e openSUSE e ha co-fondato l'Associazione LibreItalia dove è stata coinvolta in diversi eventi, migrazioni e formazione relativi a LibreOffice. In precedenza ha lavorato a migrazioni e corsi di formazione su LibreOffice per diverse amministrazioni pubbliche e privati. Da gennaio 2020 lavora in SUSE come Software Release Engineer per Uyuni e SUSE Manager e quando non segue la sua passione per i computer e per Geeko coltiva la sua curiosità per l'astronomia (da cui deriva il suo nickname deneb_alpha).
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
2. 312 6 The Z-Transform and Discrete Signals
for function that may grow at infinity. This suggests that with the aid of a suit-
able convergence factor in (6.1) we should be able to obtain DFTs of divergent
infinite sequences. Thus suppose f[n] grows as ρn
min for n → ∞. By analogy
with the notation employed in connection with the Laplace transform theory
we shall denote this by
f[n] ∼ O(ρn
min
n∼∞
). (6.3)
To simplify matters let us first suppose that f[n] = 0 for n < 0. We shall refer to
such a sequence as a causal sequence. If we introduce the real positive number
ρ, the series
∞
n=0
f[n]ρ−n
e−inθ
will converge provided ρ > ρmin. Since the left side of (6.1) is a function of eiθ
the introduction of the convergence factor permits us to write the preceding as
follows:
F+
(ρeiθ
) =
∞
n=0
f[n](ρeiθ
)−n
, (6.4)
where we have added the superscript to draw attention to the causal nature
of the corresponding sequence. Of course, (6.4) is still an FS with coefficients
f[n]ρ−n
so that the usual inversion formula applies which gives
f[n]ρ−n
=
1
2π
π
−π
F+
(ρeiθ
)einθ
dθ.
Clearly no substantive change will result if we simply multiply both sides by ρn
and write
f[n] =
1
2π
π
−π
F+
(ρeiθ
)(ρeiθ
)n
dθ. (6.5)
As a notational convenience we now introduce the complex variable z = ρeiθ
in
(6.4) in which case the series assumes the form
F+
(z) =
∞
n=0
f[n]z−n
. (6.6)
In view of the convergence factor just introduced this series converges for all
complex z such that |z| = ρ > ρmin, i.e., outside a circle with radius ρmin.
Equation (6.6) will be recognized as the principal part of the Laurent se-
ries expansion of an analytic function (see Appendix) in the annular region
ρmin < |z| < ∞. The formula for the FS coefficients (6.5) can now be inter-
preted as a contour integral carried out in the counterclockwise direction along
a circular path of radius ρ. We see this directly by changing the variable of in-
tegration from θ to z = ρeiθ
. Computing dz = ρieiθ
dθ = izdθ and substituting
in (6.5) give
f[n] =
1
2πi
|z|=ρ
F+
(z)zn−1
dz, (6.7)
3. 6.1 The Z-Transform 313
where ρ > ρmin. The integration contour and the region of analyticity of F+
(z)
are shown in Fig. 6.1. Equations (6.6) and (6.7) represent, respectively, the
ℑm(z)
ℜe(z)
rmin
r
Figure 6.1: Region of analyticity and inversion contour for the unilateral Z-
transform F+
(z)
(single-sided or unilateral) Z-transform and its inverse. We shall denote it by
f [n]
Z
⇐⇒ F+
(z) . (6.8)
Just like the analogous inversion formula for the unilateral Laplace trans-
form, (6.8) yields zero for negative times, i.e., for n < 0. This is automatically
guaranteed by the analyticity of F(z) outside on the integration contour. We
can see this directly by applying the Cauchy residue theorem to a closed con-
tour formed by adding to the circular contour in Fig. 6.1 a circle at infinity.
The contribution from the integral along the latter vanishes as we see from the
following bounding argument:
1
2π
|z|=ρ
F+
(z)zn−1
dz ≤
1
2π
π
−π
F+
(ρeiθ
) ρn
dθ
≤ F+
(ρeiθ
) max
ρn
. (6.9)
Because F+
(z) is finite as ρ → ∞ the last term tends to zero for negative n.
When ρmin < 1 the integration path in Fig. 6.1 can be chosen to coincide
with the unit circle. In that case the inversion formula (6.7) may be replaced
by
f[n] =
1
2π
π
−π
F+
(eiθ
)einθ
dθ (6.10)
4. 314 6 The Z-Transform and Discrete Signals
and we may interpret F+
(eiθ
) either as the DFT of a causal sequence or as the
FT of a bandlimited analogue signal whose sample values on the negative time
axis are zero.
The restriction to causal sequences is not always convenient. To encompass
negative indexes we suppose, by analogy with (6.3), that
f[n] ∼ O(ρn
max
n∼−∞
). (6.11)
Let us suppose that this sequence is zero for nonnegative indexes. The transform
for such an “anti-causal” sequence may be defined by
F−
(z) =
−1
n=−∞
f[n]z−n
. (6.12)
With z = ρeiθ
we see that this series converges for 0 ≤ ρ < ρmax and may be
interpreted as an FS, i.e.,
F−
(ρeiθ
) =
−1
n=−∞
f[n]ρ−n
e−inθ
, (6.13)
wherein the coefficients are given by
f[n]ρ−n
=
1
2π
π
−π
F−
(ρeiθ
)einθ
dθ. (6.14)
Changing the integration variable to z transforms (6.14) into the contour integral
f[n] =
1
2πi
|z|=ρ
F−
(z)zn−1
dz, (6.15)
where the radius of the integration contour ρ < ρmax lies in the shaded region
of Fig. 6.2, corresponding to the region of analyticity of F−
(z). Note that the
transforms F+
(z) and F−
(z) are analytic functions in their respective regions of
analyticity. In general these regions would be disjoint. If, however, ρmax > ρmin,
the regions of analyticity of F+
(z) and F−
(z) overlap, the overlap being the
annular region
ρmin < |z| < ρmax (6.16)
shown in Fig. 6.3. In this case F(z),
F(z) = F+
(z) + F−
(z) (6.17)
is an analytic function within this annulus (6.16) and defines the bilateral Z-
transform of the sequence f[n] represented by the series
F(z) =
∞
n=−∞
f[n]z−n
. (6.18)
5. 6.1 The Z-Transform 315
ℑm(z)
ℜe(z)
r
rmax
Figure 6.2: Region of analyticity and inversion contour for F−
(z)
1
ℑm z
ℜe z
rmax
rmin
Figure 6.3: Typical annular region of analyticity of the doublesided Z-transform
F(z)
The annular analyticity region is also the region of convergence (ROC) of (6.18).
This convergence is guaranteed as long as the sequence f[n] exhibits the asymp-
totic behavior specified by (6.3) and (6.11). Equation (6.18) will be recognized
as the Laurent series (see Appendix) expansion of F(z) about z = 0. The inver-
sion formula is now given by the sum of (6.7) and (6.15), i.e.,
f[n] =
1
2πi
|z|=ρ
F(z)zn−1
dz (6.19)
6. 316 6 The Z-Transform and Discrete Signals
with ρ lying within the shaded annulus in Fig. 6.3. Of course due to analyticity
of the integrand we may use any closed path lying within the annular region of
analyticity. If the ROC includes the unit circle as, e.g., in Fig. 6.4, the inversion
formula (6.19) may be replaced by (6.2) in which case F(z) = F(eiθ
) reduces to
the FT.
1
rmax
rmin
ℑm z
ℜe z
Figure 6.4: Evaluation of the inversion integral of the bilateral Z-transform
along the unit circle
In summary (6.18) and (6.19) constitute, respectively, the direct and inverse
bilateral Z-transform (ZT) of a sequence. As is customary, we shall use in
the sequel the common symbol to denote both the bilateral and the unilateral
Z-transform. Thus our compact notation for both transforms will be
f [n]
Z
⇐⇒ F (z) . (6.20)
Other notations that will be employed in various contexts are : F(z) = Z{f [n]}
and f [n] = Z−1
{ F(z)}.
6.1.2 Direct ZT of Some Sequences
The principal tool in the evaluation of the direct ZT is the geometric series sum-
mation formula. For example, for the exponential sequence an
u[n], we obtain
F(z) =
∞
n=0
an
z−n
=
∞
n=0
(
a
z
)n
=
z
z − a
(6.21)
provided |z| > |a| = ρmin. This ZT has only one singularity, viz., a simple pole
at z = a. Using the notation (6.20) we write
an
u[n]
Z
⇐⇒
z
z − a
. (6.22)
7. 6.1 The Z-Transform 317
For the special case of the unit step we get
u[n]
Z
⇐⇒
z
z − 1
. (6.23)
Differentiating both sides of (6.21) k times with respect to a we get
∞
n=0
[n(n − 1) . . . (n − k + 1)] an−k
z−n
=
zk!
(z − a)
k+1
.
Dividing both sides by k! and using the binomial symbol results in the transform
pair
n
k
an−k Z
⇐⇒
z
(z − a)
k+1
(6.24)
a ZT with a k + 1-th order pole at z = a. Another canonical signal of interest
in discrete analysis is the unit impulse, denoted either by the Kronecker symbol
δmn or by its equivalent δ[n− m]. Formula (6.18) gives the corresponding ZT as
δ[n − m]
Z
⇐⇒ z−m
. (6.25)
For m > 0 this ZT is analytic everywhere except for a pole of order m at z = 0
which for m < 0 becomes a pole of the same order at z = ∞.
An example of a sequence that leads to a nonrational ZT is the sequence
f [n] = u[n − 1]/n
u[n − 1]/n
Z
⇐⇒ − ln
z − 1
z
, (6.26)
which may be obtained from a Taylor expansion of ln(1−w) for |w| < 1. This ZT
may be defined as analytic function for |z| > 1 by connecting the two logarithmic
singularities at z = 1 and z = 0 with the branch cut shown in Fig. 6.5.
As in case of the LT derivation of more complex transform pairs is facilitated
by the application of several fundamental properties of the ZT which we discuss
in the sequel.
6.1.3 Properties
Time Shift
Forward. We are given F(z) = Z{f[n]} and wish to compute Z{f[n+k]}
where k ≥ 0. Here we must distinguish between the unilateral and the bilateral
transforms. In the latter case we have
Z{f[n + k]} =
∞
n=−∞
f[n + k]z−n
= zk
∞
m=−∞
f[m]z−m
= zk
F(z). (6.27)
8. 318 6 The Z-Transform and Discrete Signals
1
BRANCH CUT
1+i00
ℑmz
ℜez
Figure 6.5: Branch cut that renders − ln [(z − 1)/z] analytic outside the unit
circle
On the other hand for the unilateral transform we obtain
Z{f[n + k]} =
∞
n=0
f[n + k]z−n
= zk
∞
m=k
f[m]z−m
. (6.28)
Since f[m] = 0 for m < 0 then whenever k ≤ 0 we may replace the lower limit of
the last sum by zero in which case we again obtain (6.27). A different situation
arises when k > 0 for then the last sum in (6.28) omits the first k − 1 values in
the sequence. Note that this is similar to the signal truncation for the unilateral
LT in (4.49) in 4.1. We can still express (6.28) in terms of F(z) by adding and
subtracting the series
k−1
m=0 f[m]z−m
. We then obtain
Z{f[n + k]} = zk
[F(z) −
k−1
m=0
f[m]z−m
]. (6.29a)
The last result is particularly useful in the solution of finite difference equations
with constant coefficients with specified initial conditions.
Backward Shift. For k ≥ 0 we also compute Z{f[n − k], referred to as a
backward shift. Instead of (6.28) we get
Z{f[n − k]} =
∞
n=0
f[n − k]z−n
= z−k
∞
m=−k
f[m]z−m
. (6.29b)
9. 6.1 The Z-Transform 319
Note that in this case the initial conditions are specified for negative indices.
When the sequence is causal1
these vanish and we get
Z{f[n − k]} = z−k
F (z) . (6.29c)
Time Convolution
By direct calculation we get
Z{
∞
k=−∞
f[n − k]g[k]} =
∞
k=−∞
Z{f[n − k]}g[k]
=
∞
k=−∞
F(z)z−k
g[k]
= F(z)G(z). (6.30)
Since
∞
k=−∞ u[n − k]f[k] ≡
n
k=−∞ u[n − k]f[k] ≡
n
k=−∞ f[k] we get as a
by-product the formula
n
k=−∞
f[k]
Z
⇐⇒
zF(z)
z − 1
, (6.31)
which is useful in evaluating the ZT of sequences defined by sums. For example,
applying to (6.26) results in
n
k=1
1
k
Z
⇐⇒ −
z ln z−1
z
z − 1
. (6.32)
Frequency Convolution
The ZT of the product of two sequences is
Z{f[n]g[n]} =
∞
n=−∞
f[n]g[n]z−n
=
∞
n=−∞
1
2πi
|z|=ρ
F(z )z n−1
dz g[n]z−n
=
1
2πi
|z|=ρ
F(z )G(z/z )z −1
dz . (6.33)
When the inversion contour is the unit circle the preceding becomes
Z{f[n}g[n]} =
1
2π
π
−π
F(θ )G(θ − θ )dθ . (6.34)
1This does not imply that the underlying system is not causal. The shift of initial condi-
tions to negative time is just a convenient way to handle forward differences.
10. 320 6 The Z-Transform and Discrete Signals
Initial Value Theorem
For a causal sequence we have
lim
|z|→∞
F(z) = f[0], (6.35)
which follows directly from (6.18) and analyticity of F(z) for |z| > ρmin.
Differentiation
Differentiating (6.18) we get the transform pair
Z{nf[n]} = −z
dF(z)
dz
. (6.36)
For example, starting with u[n] = f[n} and applying formula (6.36) twice we
get
n2
u[n]
Z
⇐⇒
z(z + 1)
(z − 1)
3 . (6.37)
6.2 Analytical Techniques in the Evaluation
of the Inverse ZT
From the preceding discussion we see that the ZT bears the same relationship to
the FS as the LT to the FT. Just like the LT inversion formula, the ZT inversion
formula (6.54) yields signals whose characteristics depend on the choice of the
integration contour. Whereas in case of the LT this nonuniqueness is due to the
possibility of the existence of several strips of analyticity within each of which
we may place a linear integration path, in case of the ZT we may have several
annular regions of analyticity wherein we may locate the circular integration
contour. We illustrate this with several examples.
Example 1 Let us return to the simple case of a ZT whose only singularity
is a simple pole at z = a, i.e.,
F(z) =
z
z − a
. (6.38)
There are two annular regions of analyticity (i) |a| < ρ < ∞ (ρmin = |a| and
ρmax = ∞) and (ii) 0 ≤ ρ < |a| (ρmin = 0 and ρmax = |a|). In accordance with
(6.19) we have to compute
f[n] =
1
2πi
|z|=ρ
zn
z − a
dz. (6.39)
Consider first case (i). For n < 0 the integrand decays as |z| → ∞. In fact
(6.39) taken over a circle of infinite radius vanishes. To see this let z = R eiθ
so
that dz = R eiθ
idθ and the integral may be bounded as follows
11. 6.2 Analytical Techniques in the Evaluation of the Inverse ZT 321
|z|=R
zn
z − a
dz ≤
π
−π
R
|R eiθ − a|
Rn
dθ. (6.40)
Since n < 0 the right side of the preceding expression approaches zero as R → ∞.
(It is not hard to see that the limit will also be zero for any F(z) that approaches
a constant as |z| → ∞ a result we shall rely on repeatedly in the sequel.)
Because there are no intervening singularities between the circle |z| = ρ and ∞
this integral vanishes. For n ≥ 0 the integrand in (6.39) is analytic except at
z = a and a residue evaluation gives an
. In summary, for case (i) we have the
causal sequence
f[n] =
0 ; n < 0,
an
; n ≥ 0.
(6.41)
so that (6.38) together with the choice of contour represents the unilateral trans-
form. For case (ii) the pole at z = a lies between the inversion contour and ∞.
Because the integral over the infinite circular contour vanishes for n < 0 equa-
tion (6.39) gives the negative residue at the pole, i.e., the integration around
the pole is, in effect, being carried out in the clockwise direction. This may be
visualized in terms of the composite closed contour shown in Fig. 6.6 comprised
of the circle with radius ρ, the circle at infinity, and two linear segments along
which the individual contributions mutually cancel. As we proceed along the
composite contour we see that the pole is being enclosed in the clockwise direc-
tion. For n ≥ 0 the integrand is analytic within the integration contour and we
get zero. Thus, in summary, for case (ii) we obtain the anti-causal sequence
R
x a
ℑm z
ℜe zr
Figure 6.6: Residue evaluation using an auxiliary contour
12. 322 6 The Z-Transform and Discrete Signals
f[n] =
−an
; n < 0,
0 ; n ≥ 0.
(6.42)
From the preceding example we see that if |a| > 1, i.e., the pole is outside the
unit circle, the unilateral ZT (case (i)) corresponds to a sequence that grows
with increasing n. Clearly for this sequence the FT does not exist. On the other
hand for case (ii) |a| > 1 results in a sequence that decays with large negative
n. Since in this case the inversion contour can be chosen to coincide with the
unit circle the FT exists and equals
F(ω) =
eiωΔt
eiωΔt − a
. (6.43)
For |a| < 1 one obtains the reverse situation: the unilateral ZT corresponds to
a decaying sequence whereas the anti-causal sequence in case (ii) grows. The
four possibilities are illustrated in Fig. 6.7.
-20 -15 -10 -5 0
case(I)
case(II)
5 10 15 20
0
1
2
3
4
5
abs(a)>1
abs(a)<1
-20 -15 -10 -5 0 5 10 15 20
0
1
2
3
4
5
6
abs(a)<1
abs(a)>1
Figure 6.7: Sequences corresponding to a ZT with a simple pole
Note, however, as long as |a| = 1 formula (6.43) remains valid and represents
the FT of either the causal or the anti-causal decaying sequence for |a| < 1 and
|a| > 1, respectively. What happens when |a| → 1? To answer this question
we cannot simply substitute the limiting form of a in (6.43) for (6.43) is not
valid for a on the unit circle. Rather we must approach the limit by evaluating
(6.19) along a contour just inside or just outside the unit circle. The answer,
not surprisingly, depends on which of the two options we choose. Suppose we
start with a contour just outside the unit circle. Since we are looking for the
FT we should like our integration contour to follow the path along the unit
13. 6.2 Analytical Techniques in the Evaluation of the Inverse ZT 323
circle as much as possible. Our only obstacle is the pole at z = eiθ0
. To remain
outside the circle we circumnavigate the pole with a small semicircle of radius
resulting in the composite integration path shown in Fig. 6.8. The contribution
to the integral along the circular contour is represented by a CPV integral.
For nonnegative n, the sum of the two contributions equals the residue at the
enclosed pole, and vanishes for negative n. This is, of course, identical to (6.41)
1
•
ℑm z
ℜe zq0
Figure 6.8: Integration path along unit circle in presence of a simple pole
with a = eiθ0
. Thus summing the two contributions along the closed contour
we have
1
2
einθ0
+
1
2π
CPV
π
−π
einθ
1 − e−i(θ−θ0)
dθ ≡ u [n] einθ0
= f [n] . (6.44)
Using the identity
1
1 − e−i(θ−θ0)
=
1
2
−
i
2
cot [(θ − θ0)/2]
we observe that (6.44) implies the transform pair
u [n] einθ0 F
⇐⇒ πδ (θ − θ0) +
1
2
−
i
2
cot [(θ − θ0)/2] . (6.45)
We leave it as an exercise to find the FT when the unit circle is approached
from the interior.
14. 324 6 The Z-Transform and Discrete Signals
Example 2 As another example consider the ZT with two simple poles
F(z) =
z2
(z − 1/2)(z − 2)
. (6.46)
In this case there are three annular regions of analyticity: (i) 2 < ρ1, (ii)
1/2 < ρ2 < 2, and (iii) 0 ≤ ρ3 < 1/2. We distinguish the three sequences by
the superscript k and evaluate
f(k)
[n] =
1
2πi
|z|=ρk
zn+1
(z − 1/2)(z − 2)
dz, k = 1, 2, 3.
As in the preceding example F(∞) is finite so that the contribution to the
inversion integral for n < 0 from a circle at ∞ is zero. The three sequences are
then found by a residue evaluation as follows.
case (i)
When n ≥ 0 we have
f(1)
[n] =
zn+1
(z − 1/2)
|z=2 +
zn+1
(z − 2)
z=1/2 =
4
3
2n
−
1
3
2−n
(6.47)
so that
f(1)
[n] =
0 ; n < 0,
4
3 2n
− 1
3 2−n
; n ≥ 0.
(6.48)
case (ii)
For n < 0 the negative of the residue at z = 2 contributes and for n ≥ 0 the
positive residue as in (6.47). Thus we obtain
f(2)
[n] =
−4
3 2n
; n < 0,
−1
3 2−n
; n ≥ 0.
(6.49)
case (iii)
No singularities are enclosed for n ≥ 0 so that f(3)
[n] = 0. For n < 0 the
negatives of the two residues in (6.48) contribute, so that the final result reads
f(3)
[n] =
−4
3 2n
+ 1
3 2−n
; n < 0,
0 ; n ≥ 0.
(6.50)
We note that only the sequence f(2)
[n] possesses an FT, which is
F(ω) =
e2iωΔt
(eiωΔt − 1/2)(eiωΔt − 2)
. (6.51)
This sequence is, however, not causal. Note that this is due to the presence of a
pole outside the unit circle. This same pole gives rise to the exponential growth
with n of the sequence in case (i). This sequence is causal and corresponds
to the single-sided ZT. Clearly to obtain a sequence which is both causal and
stable requires that all the poles of the ZT lie within the unit circle.
15. 6.2 Analytical Techniques in the Evaluation of the Inverse ZT 325
Example 3 In this next example
F(z) =
z
(z + 1/3) (z + 3)
2 (6.52)
we again have two poles one of which is a double pole. We have again three
annular regions of analyticity and we evaluate
f(k)
[n] =
1
2πi
|z|=ρk
zn
(z + 1/3) (z + 3)2 dz, k = 1, 2, 3 (6.53)
on each of three circular contours defined by 0 ≤ ρ1 < 1/3, 1/3 < ρ2 < 3, 3 < ρ3.
The residue evaluation gives
f(1)
[n] =
0 ; n ≥ 0,
− 9
64 (−1/3)
n
+ (9−8n)(−3)n
64 ; n < 0,
(6.54a)
f(2)
[n] =
⎧
⎨
⎩
(−1/3)n
(−1/3+3)2 = 9
64 (−1/3)n
; n ≥ 0,
− d
dz
zn
z+1/3 |z=−3 = (9−8n)(−3)n
64 ; n < 0,
(6.54b)
f(3)
[n] =
9
64 (−1/3)
n
− (9−8n)(−3)n
64 ; n ≥ 0,
0 ; n < 0.
(6.54c)
Of the three sequences only f(2)
[n] possesses an FT which is
F(ω) =
eiωΔt
(eiωΔt + 1/3)(eiωΔt + 3)
2 . (6.55)
Recall that the sample values of the corresponding analogue signal are f(2)
[n] /Δt
so that the reconstruction of this signal via the Shannon sampling theorem reads
f (t) =
1
Δt
−1
n=−∞
(9 − 8n) (−3)
n
64
sin [π (t/Δt − n)] / [π (t/Δt − n)]
+
1
Δt
∞
n=0
9
64
(−1/3)
n
sin [π (t/Δt − n)] / [π (t/Δt − n)] . (6.56)
Example 4 In the preceding examples the ZT was a proper rational function.
If this is not the case, a causal inverse does not exist. Consider, for example,
F(z) =
z4
z2 − 1/4
. (6.57)
In addition to the two simple poles at z = ±1/2 there is a second order pole at
infinity. Accordingly two analyticity regions are 1/2 < ρ1 < ∞ and 0 ≤ ρ2 <
1/2. The sequence corresponding to the first region is
f(1)
[n] =
1
2πi
|z|=ρ1
zn+3
z2 − 1/4
dz. (6.58)
16. 326 6 The Z-Transform and Discrete Signals
The integral over the circle at infinity vanishes provided n + 3 ≤ 0 so that
f(1)
[n] = 0 for n ≤ −3. For n > −3 the only singularities within |z| < ρ1 are
the two simple poles of F (z). Summing the two residues gives
f(1)
[n] =
1
8
1
2
n
+ −
1
2
n
u [n + 2] . (6.59)
For the second sequence
f(2)
[n] =
1
2πi
|z|=ρ2
zn+3
z2 − 1/4
dz (6.60)
the integration yields a null result for n + 3 ≥ 0. For n + 3 < 0 the integral over
the infinite circle again vanishes so that we can sum the residues of the poles
lying outside the integration contour resulting in
f(2)
[n] = −
1
8
1
2
n
+ −
1
2
n
u [−n − 4] . (6.61)
An alternative approach to dealing with an improper rational function is to
employ long division and reduce the given function to a sum of a polynomial
and a proper rational function. The inverse ZT of the polynomial is then just a
sum of Kronecker deltas while the inverse ZT of the proper rational function is
evaluated by residues. Using this approach in the present example we have
z4
z2 − 1/4
= z2
+
1
4
z2
z2 − 1/4
. (6.62)
The inverse of z2
is δ [n + 2] while the residue evaluation involving the second
term yields either a causal or anticausal sequence. In the former case we get for
the final result
f(1)
[n] = δ [n + 2] +
1
8
1
2
n
+ −
1
2
n
u [n] , (6.63)
which is easily seen as just an alternative way of writing (6.59).
Example 5 Let us find the FT of the causal sequence whose ZT is
F (z) =
z − 2
(z − 1/2)(z − eiθ0 )(z − e−iθ0 )
. (6.64)
This function has two simple poles on the unit circle and one interior pole.
Consequently the inversion contour for the FT along the unit circle will have
to be modified by two small semi-circles surrounding the poles (instead of one
as in Fig. 6.8). The integration along each of these semi-circles will contribute
17. 6.3 Finite Difference Equations and Their Use in IIR and FIR Filter Design 327
one-half the residue at the respective pole while the integral along the unit circle
must be defined as a CPV integral. As a result we obtain
f [n] =
1
2π
CPV
π
−π
F eiθ
einθ
dθ +
+
1
2
eiθ0(n−1)
eiθ0
− 2
(eiθ0 − 1/2)(eiθ0 − e−iθ0 )
+
1
2
e−iθ0(n−1)
e−iθ0
− 2
(e−iθ0 − 1/2)(e−iθ0 − eiθ0 )
. (6.65)
By absorbing the two residue contributions as multipliers of delta functions we
can write the complete FT as follows:
F (θ) = F eiθ
+ π
e−iθ0
eiθ0
− 2
(eiθ0 − 1/2)(eiθ0 − e−iθ0 )
δ (θ − θ0)
+π
eiθ0
e−iθ0
− 2
(e−iθ0 − 1/2)(e−iθ0 − eiθ0 )
δ (θ + θ0) . (6.66)
6.3 Finite Difference Equations and Their
Use in IIR and FIR Filter Design
The “method of finite differences” generally refers to the approximation of
derivatives in a differential equation using finite increments of the independent
variable. The approximate solution for the dependent variable is then found by
algebraic means. The finite difference approximation can be of the forward or
of the backward type. Thus if the finite difference is defined as
y (t + Δt) − y(t)
Δt
(6.67)
it is of the forward type. If it is defined by
y(t) − y (t − Δt)
Δt
(6.68)
it is referred to as a backward difference. Whereas (6.67) is more common
when dealing directly with numerical solutions of differential equations (6.68)
is generally preferred in digital signal processing mainly because y (t − Δt) has
direct physical interpretation of a step in time delay. To illustrate the connection
between a differential equation and the associated difference equation consider
the simple case of the first-order equation
dy(t)
dt
+ a0 y(t) = f (t) . (6.69)
With the forward difference approximation we get
y (t + Δt) − y(t)
Δt
+ a0 y(t) ≈ f (t) . (6.70)
18. 328 6 The Z-Transform and Discrete Signals
If we are interested in y (t) only at discrete time intervals we can set t = nΔt so
that y (t + Δt) = y [Δt (n + 1)] ≡ y [n + 1] , y(t) = y (nΔt) ≡ y [n] and Δtf (t) =
Δtf (nΔt) ≡ f [n]. Making these changes transforms (6.70) into the difference
equation
y [n + 1] + (a0Δt − 1) y [n] = f [n] . (6.71)
With Z{y [n]} = Y (z) , Z{f [n]} = F (z) we get
Y (z) =
F (z) + z y [0]
z + a0Δt − 1
=
F (z)
z + a0Δt − 1
+
z y [0]
z + a0Δt − 1
and upon inversion the solution of (6.71):
y [n] = Z−1
{
F (z)
z + a0Δt − 1
} + (a0Δt − 1)
n
y [0] . (6.72)
As far as the difference equation (6.71) is concerned (6.72) is its exact solution.
The actual solution to the differential equation (the analogue problem) has to
be gotten via a limiting process. For simplicity we do this when F (z) = 0 and
look for the limit of
y [n] = (1 − a0Δt)n
y [0] (6.73)
as Δt → 0. This is easily done by noting that discretization tells us that t may
be replaced by nΔt. If in addition we replace a0Δt by − δ, (6.73) becomes
y [n] = y [0] (1 + δ)−(1/δ)a0t
. (6.74)
Recalling the definition of e we get
lim
δ→0
y [n] = (1 + δ)−(1/δ)
→ e−1
so that (6.73) approaches y [0] e−a0t
which is the solution of (6.69). On the
other hand, the physical problem of interest may have been formulated ab initio
as a difference equation (6.71) where a finite increment Δt has a direct physical
significance. In that case the limiting form would constitute a wrong answer.
The application of the forward difference operation to an N-th order dif-
ferential equation with constant coefficients (3.145) in 3.3 leads, after some
unpleasant algebra, to the N-th order difference equation
N
k=0
ak y [n − k] = f [n] . (6.75)
Taking the Z-Transform yields
N
k=0
akzk
Y (z) = F (z) +
N
k=0
ak
k−1
=0
f [ ] zk−
, (6.76)
19. 6.3 Finite Difference Equations and Their Use in IIR and FIR Filter Design 329
where the excitation on the right includes, in addition to the forced excitation
F (z), the contribution from the initial conditions as given by (6.29a). Solving
for the transform of the output
Y (z) =
F (z)
N
k=0 akzk
+
N
k=0 ak
k−1
=0 y [ ] zk−
N
k=0 akzk
(6.77)
we identify the quantity
H+ (z) =
1
N
k=0 akzk
(6.78)
as the system transfer function.2
If all poles are within the unit circle then, in
accordance with the results in the preceding section, the application of the in-
version integral to (6.78) along the unit circle yields a causal and stable sequence
h [n]. Evaluation of (6.78) on the unit circle gives the FT
H (θ) =
1
N
k=0 akeikθ
= A (θ) eiψ(θ)
, (6.79)
where A (θ) is the amplitude and ψ (θ) the phase. The amplitude is an even and
the phase an odd function of frequency just like for continuous signals. An im-
portant class of transfer functions is characterized by the absence of zeros outside
the unit circle. They are called minimum phase-shift functions. For functions
of this type the phase can be determined from the amplitude (see 6.4.2).
Similar to the filter structures in Fig. 3.16 and 3.17 that were derived from
differential equations, difference equations lead to topologically similar repre-
sentations. They play an important role in the design of DSP algorithms. Here
we will switch from the representation based on forward differencing we used
to derive (6.78) to backward differencing which is more common in DSP appli-
cations. Figure 6.9 shows a feedback-type structure similar to that in Fig. 3.16.
The integrators have been replaced by unit delay elements (denoted by −1
within the upper circles). For each delay element an input y [n] gives an output
y [n − 1] consistent with backward differencing (6.68). Referring to the figure,
if we subtract the sum of the outputs from the difference operators from the
input f [n ] and multiply the result by 1/a0 we get
N
k=0
ak y [n − k] = f [n ] . (6.80)
Assuming y [n] = 0 for n < 0 we have for the Z-transform
Y (z)
N
k=0
ak z−k
= F (z) (6.81)
2The subscript (+) identifies that it is based on forward differencing.
20. 330 6 The Z-Transform and Discrete Signals
−aN
+
+
+
++
+
−a2−a1
y[n−N]y[n−2]y[n−1]
f
y[n]
1/a0
+
•••••
•• •••
−1 −1
[n]
Figure 6.9: Infinite impulse response filter (IIR)
and for the transfer function
H− (z) =
1
N
k=0 akz−k
. (6.82)
Since (6.82) is based on backward differencing it does not agree with (6.78).
We can see the relationship between the two by changing the summation index
from k to m = N − k with the result
H− (z) =
zN
N
m=0 aN−mzm
. (6.83)
The denominator has the form of (6.78) but the polynomial coefficients have
been interchanged so that the pole positions are different.
The transfer function (6.82) forms the basis for the design of infinite impulse
response (IIR) filters. The name derives from the property that with poles
within the unit circle (the usual case) the impulse response is of infinite duration.
This is also true for all pole analogue filters. In fact design procedures for digital
IIR filters are essentially the same as for analogue filters. A second class of
filters is finite impulse response (FIR) filters. A representative structure of an
FIR filter is shown in Fig. 6.10.
[n]f [n]f
−1 −1
b0
b0
[n]fb0 [n-1]fb1
b1 b2 bN
[n]y
+
Figure 6.10: Finite impulse response filter (FIR)
21. 6.4 Amplitude and Phase Relations Using the Discrete Hilbert Transform 331
Adding the tapped and in unit steps delayed input to the input that has
been delayed and multiplied by the last tap we get
b0f [n] + b1f [n] + b2f [n − 2] + . . . bN−1f [n − (N − 1)] + bN f [n − N] = y [n]
or, in compact, notation
y [n] =
N
k=0
bk f [n − k] . (6.84)
Here the impulse response is the finite length sequence
h [n] = bn ; n = 0, 1, 2 . . .N (6.85)
with the transfer function
HF IR (z) =
N
k=0
bk z−k
(6.86)
and an FT3
HF IR (θ) =
N
k=0
bk e−ikθ
. (6.87)
Since this is also an FS we can easily compute the coefficients for a prescribed
H (θ) and hence the filter parameters in Fig. 6.10. The practical problem here
is that finite length sequences cannot be contained within a finite bandwidth.
Generally the biggest offenders here are steep changes in the frequency spectrum
as would be the case e.g., for band-pass filters with steep skirts. These problems
can be in part alleviated by tapering the sequence (i.e., in the time domain).
However, generally for this and other reasons FIR filters require many taps.
A general filter structure in DSP applications combines IIR and FIR transfer
functions into the form
H(z) =
N
k=0 bk z−k
M
k=0 akz−k
. (6.88)
Such transfer functions can be realized either by using FIR and IIR filters in
tandem or combining them into a single structure similar to that in Fig. 3.17.
6.4 Amplitude and Phase Relations Using
the Discrete Hilbert Transform
6.4.1 Explicit Relationship Between Real and Imaginary
Parts of the FT of a Causal Sequence
We recall from 2.2.6 that the real and imaginary parts of the FT of a real causal
analogue signal are related by the Hilbert transform. In 2.4.2 this relationship
3Here and in the entire discussion of difference equations we have increased the sequence
length from that used with the DFT in Chap. 5 from N to N + 1. Consistency is easily
restored by setting the Nth term to zero.
22. 332 6 The Z-Transform and Discrete Signals
is reestablished and is shown to be a direct consequence of the analytic proper-
ties of the FT of causal signals; its extension to amplitude and phase of transfer
functions is discussed in 2.4.3. In the following we show that a similar set of rela-
tionships holds also for causal discrete signals, or, equivalently, for bandlimited
functions whose Nyquist samples are identically zero for negative indices.
As in the analogue case, in (2.168) in 2.2 we start directly from the definition
of a causal signal, in this case a real sequence f [n] which we decompose into its
even and odd parts as follows:
f [n] = fe [n] + fo [n] , (6.89)
where
fe [n] =
f [n] + f [−n]
2
, (6.90a)
fo [n] =
f [n] − f [−n]
2
. (6.90b)
With
f [n]
F
⇐⇒ F (θ) = R (θ) + iX (θ) , (6.91)
wherein R (θ) and X (θ) are real and imaginary parts of F (θ) it is not hard to
show that
fe [n]
F
⇐⇒ R (θ) , (6.92a)
fo [n]
F
⇐⇒ iX (θ) , (6.92b)
i.e., just as in the case of analogue signals, the odd and even parts are defined,
respectively, by the real and the imaginary parts of the FT of the sequence. In
order to carry out the transformations analogous to in (2.172) 2.2 we need the
FT of the discrete sign function sign [n] defined as +1 for all positive integers,
−1 for negative integers and zero for n = 0. We can find this transform from
(6.45) by first expressing the unit step as follows:
u [n] =
1
2
(sign [n] + δ [n] + 1) , (6.93)
where δ [n] = 1 for n = 0 and 0 otherwise. With the aid of (6.45) we then obtain
sign [n]
F
⇐⇒ −i cot (θ/2) . (6.94)
Suppose the sequence f [n] is causal. Then in view of (6.89) and (6.90) we have
f [n] = 2fe [n] u [n] (6.95)
and also
fo [n] = sign [n] fe [n] (6.96a)
fe [n] = f [0] δ [n] + sign [n] fo [n] . (6.96b)
23. 6.4 Amplitude and Phase Relations Using the Discrete Hilbert Transform 333
Taking account of (6.92) and (6.94) and applying the frequency convolution
theorem to (6.96) yield
X (θ) = −
1
2π
CPV
π
−π
R θ cot θ − θ /2 dθ , (6.97a)
R (θ) = f [0] +
1
2π
CPV
π
−π
X θ cot θ − θ /2 dθ . (6.97b)
These relations are usually referred to as the discrete Hilbert transforms (DHT).
6.4.2 Relationship Between Amplitude and Phase
of a Transfer Function
We now suppose that A (θ) is the amplitude of the transfer function of a causal
digital filter with a real unit sample response. This means that there exists a
phase function ψ (θ) such that the FT A (θ) eiψ(θ)
= H (θ) has a causal inverse.
The problem before us is to find ψ (θ) given A (θ). As we shall demonstrate in
the sequel, a ψ (θ) that results in a causal transfer function can always be found
provided ln A (θ) can be expanded in a convergent Fourier series in (−π, π). As
in the corresponding analogue case the solution is not unique for we can always
multiply the resulting transfer function by an all pass factor of the form eiψ0(θ)
which introduces an additional time delay (and hence does not affect causality)
but leaves the amplitude response unaltered.
To find ψ (θ) we proceed as follows. Assuming that the FS expansion
ln A (θ) =
∞
n=−∞
w [n] e−inθ
(6.98)
exists we form the function
Q (θ) = ln H(θ) = ln A (θ) + iψ (θ) . (6.99)
Since the unit sample response is real, A (θ) is an even function of θ so that we
may regard ln A(θ) as the real part of the FT of a causal sequence q [n] with
ψ (θ) the corresponding imaginary part. In that case ψ (θ) can be expressed
in terms of ln A (θ) by the DHT (6.97a) so that the solution to our problem
appears in the form
ψ (θ) = −
1
2π
CPV
π
−π
ln A θ cot θ − θ /2 dθ . (6.100)
The evaluation of this integral is numerically much less efficient than the fol-
lowing alternative approach. It is based on the observation that for the causal
sequence q [n] the FT of its even part is just the given log amplitude ln A (θ),
i.e.,
w [n] = qe [n] =
q [n] + q [−n]
2
F
⇐⇒ ln A (θ) . (6.101)
24. 334 6 The Z-Transform and Discrete Signals
But in view of (6.95) we can write
q [n] = 2qe [n] u [n]
F
⇐⇒ ln A (θ) + iψ (θ) . (6.102)
Thus to find ψ (θ) from A (θ) we first take the inverse FT of ln A (θ), truncate
the sequence to positive n, multiply the result by 2, and compute the FT of the
new sequence. The desired phase function is then given by the imaginary part
of this FT. Expressed in symbols the procedure reads:
ψ (θ) = Im F 2F−1
{ln A (θ)}u [n] . (6.103)
The complete transfer function is then
H (θ) = A (θ) ei Im F{2F−1
{2 ln A(θ)}u[n]}. (6.104)
To prove that this H (θ) has a causal inverse we note that, by construction,
ln H eiθ
=
∞
n=0
q [n] e−inθ
(6.105)
so that
∞
n=0 q [n] z−n
is analytic in |z| > 1. But then ln H (z) must also be
analytic outside the unit circle. In particular, this means that H (z) cannot
have any poles or zeros outside the unit circle. Since the exponential is an
analytic function
H (z) = e
∞
n=0 q[n]z−n
(6.106)
is necessarily analytic in |z| > 1 and hence has a causal inverse.
Analytic functions devoid of zeros outside the unit circle are referred to as
minimum phase-shift functions, a terminology shared with Laplace transforms
of analogue signals that are analytic in the right half of the s plane and free of
zeros there. Despite this common terminology minimum phase-shift functions
of sampled causal signals are endowed with an important feature not shared
by transfer functions of causal analogue signals, viz., the absence of zeros as
well as analyticity outside the unit circle also ensures the analyticity of 1/H (z)
outside the unit circle. Hence the reciprocal of minimum phase-shift function
also possesses a causal inverse, a feature of great importance in the design of
feedback control systems.
6.4.3 Application to Design of FIR Filters
The preceding procedure can be used to synthesize a minimum phase FIR filter
having a prescribed amplitude response. With A [m], m = 0, 1, . . .N − 1 the
prescribed amplitude function we compute the transfer function via (6.104)
H [m] = A [m] ei Im F{2F−1
{2 ln A(k)}u[n]}. (6.107)
The unit sample response h [n] is then given by
h [n] =
1
N
N−1
m=0
H [m] ei 2πnm
N ; n = 0, 1, . . . N − 1. (6.108)
25. Problems 335
Equations (6.108) and (6.109) are exact but unfortunately the required number
of filter coefficients (taps) equals the number of samples, which would normally
be impractically large. It turns out, however, that for many practically useful
filter functions the h [n] decay rapidly with increasing n so that the number of
taps can be chosen much less than N. Thus instead of the exact form of H [m]
we shall use the first M values of h [n] in (6.108) and define
H(M)
[m] =
M−1
n=0
h [n] ei 2πnm
N ; m = 0, 1.2 . . .N − 1. (6.109)
The degree of permissible truncation depends on the specified performance level
(e.g., stopband attenuation, relative bandwidth) and is strongly dependent on
the functional form of the chosen amplitude function. The recommended proce-
dure is to start with (6.108) and progressively reduce the number of taps until
the observed amplitude spectrum begins to show significant deviations from the
prescribe amplitude response.
Problems
1. The Z-transform
F (z) =
z
(z2 + 1/2) (z + 16)
can represent several sequences.
(a) Find all the sequences.
(b) One of the sequences represents the coefficients of the Fourier series
expansion of the Fourier transform of a band-limited function. Iden-
tify the sequence and find the Fourier transform of the corresponding
bandlimited function.
2. The Fourier transform of a bandlimited function f(t) is given by
F (ω) =
1
5/4−cos(πω/Ω) ; |ω| ≤ Ω
0; |ω| > Ω
(a) The function is sampled at intervals Δt = π/Ω. Find the Z-transform
of the sequence f [n] = f(n Δt)Δt.
(b) Find the n-th sampled value f(n Δt) of f(t).
(c) Suppose the function is sampled at intervals Δt = 2π/Ω and its
Fourier transform ˆF (ω) is reconstructed using the formula.
ˆF (ω) = Δt
∞
n=−∞
f(nΔt )e−iωnΔt
Compute ˆF (ω) and sketch ˆF (ω) in the interval |ω| ≤ Ω.
26. 336 6 The Z-Transform and Discrete Signals
3. Using Z-transforms solve the following difference equation:
y[n + 2] + 2y [n + 1] +
1
4
y [n] = (1/2)
n
, n ≥ 0, y [0] = 1, y [1] = −1.
4. The Fourier transform F (ω) of a bandlimited signal f (t) is given by
F (ω) =
sin4 πω
Ω ; |ω| ≤ Ω,
0 ; |ω| > Ω.
(a) The signal is sampled at intervals Δt = π/Ω. Find the samples.
(b) Find the Z-transform of the sampled sequence f[n] = f(nΔt)Δt.
(c) Find f (t).
(d) Suppose the signal is sampled at intervals Δt = 2π/Ω and its Fourier
transform is approximated by
ˆF (ω) = Δt
∞
n=−∞
f (nΔt ) e−iωnΔt
.
Compute and sketch ˆF (ω) within the band |ω| ≤ Ω.
5. The sequence y [n] satisfies the following difference equation:
y[n + 2] +
1
6
y [n + 1] −
1
6
y [n] = u [n] ,
where
u [n] =
1 ; n ≥ 0,
0 ; n < 0.
(a) Assuming y [0] = 0 and y [1] = 0 find the causal solution of the
difference equation using Z transforms.
(b) Suppose the y(nΔt) = y [n] /Δt represent samples of a bandlimited
signal with a bandwidth of 10 Hz. Find the Fourier transform of the
signal.
6. Using Z-transforms solve the following difference equation:
y[n + 2] + y [n + 1] +
1
4
y [n] = (1/3)
n
, n ≥ 0, y [0] = 1, y [1] = 2.