The document discusses linear transformations and their properties. It defines key concepts such as the kernel, range, rank, and nullity of a linear transformation. The kernel is the set of all vectors that map to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. The rank is the dimension of the range, and the nullity is the dimension of the kernel. A linear transformation is one-to-one if different vectors always map to different outputs, and onto if its range is equal to the codomain. An isomorphism is a linear transformation that is both one-to-one and onto, and maps spaces to spaces of the same dimension.
This document discusses linear transformations between vector spaces. It begins by defining a linear transformation as a function between vector spaces that satisfies the properties of vector addition and scalar multiplication. It then provides examples of standard linear transformations like the matrix transformation and zero transformation. The document also covers properties of linear transformations such as how they are determined by the images of basis vectors. Finally, it provides applications of linear operators like reflection, rotation, and shear transformations.
This document provides an introduction to linear transformations. It defines a linear transformation as a function that maps one vector space to another while preserving vector addition and scalar multiplication. Key concepts discussed include the domain, co-domain, range, and pre-image of a linear transformation. Examples are given to demonstrate linear transformations and functions that are not linear transformations. The relationship between linear transformations and matrices is also explained.
Quiz 2 will cover sections 1.4, 1.5, 1.7, and 1.8 on Wednesday January 27. Students with issues on quiz 1 should discuss with the instructor as soon as possible. The solution to quiz 1 will be posted on the website by Monday.
The document discusses linear transformations and provides examples of applying linear transformations to vectors. It defines key concepts such as the domain, co-domain, and range of a transformation. Examples are provided of interesting linear transformations including rotation and reflection transformations. Solutions to examples involving finding the image of vectors under given linear transformations are shown.
This document discusses isomorphism, one-to-one and onto linear transformations, and provides examples. It defines isomorphism as when two vector spaces are said to be isomorphic if there exists a linear transformation between them that is both one-to-one and onto. It provides theorems relating the dimension of isomorphic spaces and characterizations of one-to-one and onto linear transformations in terms of the kernel, rank, and nullity. Examples are given to determine whether linear transformations are one-to-one and/or onto based on their rank and nullity. Finally, several vector spaces are stated to be isomorphic to each other.
The document discusses linear transformations and their properties. It defines key concepts such as the kernel, range, rank, and nullity of a linear transformation. The kernel is the set of all vectors that map to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. The rank is the dimension of the range, and the nullity is the dimension of the kernel. A linear transformation is one-to-one if different vectors always map to different outputs, and onto if its range is equal to the codomain. An isomorphism is a linear transformation that is both one-to-one and onto, and maps spaces to spaces of the same dimension.
This document discusses linear transformations between vector spaces. It begins by defining a linear transformation as a function between vector spaces that satisfies the properties of vector addition and scalar multiplication. It then provides examples of standard linear transformations like the matrix transformation and zero transformation. The document also covers properties of linear transformations such as how they are determined by the images of basis vectors. Finally, it provides applications of linear operators like reflection, rotation, and shear transformations.
This document provides an introduction to linear transformations. It defines a linear transformation as a function that maps one vector space to another while preserving vector addition and scalar multiplication. Key concepts discussed include the domain, co-domain, range, and pre-image of a linear transformation. Examples are given to demonstrate linear transformations and functions that are not linear transformations. The relationship between linear transformations and matrices is also explained.
Quiz 2 will cover sections 1.4, 1.5, 1.7, and 1.8 on Wednesday January 27. Students with issues on quiz 1 should discuss with the instructor as soon as possible. The solution to quiz 1 will be posted on the website by Monday.
The document discusses linear transformations and provides examples of applying linear transformations to vectors. It defines key concepts such as the domain, co-domain, and range of a transformation. Examples are provided of interesting linear transformations including rotation and reflection transformations. Solutions to examples involving finding the image of vectors under given linear transformations are shown.
This document discusses isomorphism, one-to-one and onto linear transformations, and provides examples. It defines isomorphism as when two vector spaces are said to be isomorphic if there exists a linear transformation between them that is both one-to-one and onto. It provides theorems relating the dimension of isomorphic spaces and characterizations of one-to-one and onto linear transformations in terms of the kernel, rank, and nullity. Examples are given to determine whether linear transformations are one-to-one and/or onto based on their rank and nullity. Finally, several vector spaces are stated to be isomorphic to each other.
The document discusses linear transformations between vector spaces. It defines key concepts such as the domain, codomain, range, and images/preimages of vectors under a linear transformation. A linear transformation preserves vector addition and scalar multiplication. It provides examples of linear transformations, such as rotations and projections in planes and spaces. It also discusses when a transformation defined by a matrix represents a linear transformation.
This document discusses linear transformations and their properties. It defines a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. The kernel of a linear transformation is the set of vectors mapped to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. Matrices can represent linear transformations, with the matrix equation representing the transformation of vectors. Examples are provided to illustrate key concepts such as kernels, ranges, and matrix representations of linear transformations.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
1) The document discusses various linear transformations including reflection, rotation, and shear applied to geometric shapes like triangles and rectangles.
2) Reflection is illustrated for a triangle across the x-axis and y=-x line. Rotation is shown counterclockwise for different triangles around the origin.
3) Clockwise rotation and shear transformations in the x-direction are also demonstrated through examples using matrices applied to triangle and rectangle vertices. Plots of the transformed shapes are provided.
The document defines and provides examples of linear transformations. It then presents a question asking to find the matrix of a linear transformation between two vector spaces. The solution shows that:
1) The vector spaces have standard bases of {[1,0,0],[0,1,0],[0,0,1]} and {[1,0],[0,1]}.
2) The matrix of the linear transformation is A = [[3,4,9],[5,3,2]].
3) For a given linear transformation T defined by a matrix A, the transformation can be expressed in terms of coordinates as T(x)=A*x.
This document discusses linear transformations and their matrix representations. It defines a linear transformation as a function between vector spaces that respects the underlying linear structure. The matrix of a linear transformation uniquely represents the transformation and maps vectors from the domain to the range by matrix multiplication. Several examples are provided of finding the matrix of linear transformations between Rn and Rm spaces based on their actions on the standard basis vectors.
Linear Combination, Span And Linearly Independent, Dependent SetDhaval Shukla
In this presentation, the topic of Linear Combination, Span and Linearly Independent and Linearly Dependent Sets have been discussed. The sums for each topic have been given to understand the concept clearly for viewers.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
Fourier Transform ,LAPLACE TRANSFORM,ROC and its Properties Dr.SHANTHI K.G
1. The document discusses Fourier transforms, Laplace transforms, and their applications. Fourier transforms represent non-periodic signals as a function of frequency by decomposing them into simpler constituent parts. Laplace transforms transform time domain signals to the complex s-domain.
2. Key aspects covered include the Fourier analysis and synthesis equations, properties of the Fourier transform such as its magnitude and phase spectra. Conditions for the existence of Fourier transforms are explained. The region of convergence where the Laplace transform converges is also defined.
3. Examples are provided to demonstrate the calculation of Fourier and Laplace transforms of simple signals and determining their regions of convergence. Poles and zeros of transforms are also explained.
The document summarizes key concepts about the Laplace transform. It defines the Laplace transform, discusses properties like linearity and time shifting. It provides examples of taking the Laplace transform of unit step functions. It also covers computing the inverse Laplace transform using partial fraction expansion and handling cases with repeated or complex poles.
The document discusses unit step functions and their use in describing abrupt changes in function values that occur at specific times, such as a voltage being switched on or off in an electrical circuit. It defines the unit step function u(t) as having a value of 0 for negative t and 1 for positive t. Shifted and rectangular pulse functions are also described. Examples are provided of writing functions in terms of unit step functions and sketching the corresponding waveforms.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
Fourier-transform analysis of a unilateral fin line and its derivativesYong Heui Cho
This document presents a Fourier-transform analysis of a unilateral n-line and its derivatives. The key points are:
1) A unilateral n-line is transformed into an equivalent problem of multiple suspended substrate microstrip lines using the image theorem and Fourier transform.
2) New rigorous dispersion relations are derived in the form of rapidly convergent series solutions, providing an accurate yet efficient method for numerical computation.
3) The dispersion relations for derivatives of the unilateral n-line including suspended substrate microstrip lines, microstrip lines, slot lines and coplanar waveguides are also presented.
1. The document defines the Fourier series as an expansion of a function in a series of sines and cosines.
2. Fourier series can be used to represent even functions as a cosine series and odd functions as a sine series.
3. Examples are provided of calculating the Fourier coefficients for different functions, including finding the Fourier series of the function f(x)=x on the interval [0,π].
The Laplace transform generalizes the Fourier transform by allowing the parameter s to be any complex number rather than purely imaginary. The Laplace transform of a signal x(t) is defined as the integral of x(t) multiplied by e^-st from negative infinity to positive infinity. The region of convergence specifies the values of s for which the Laplace transform converges. Within the region of convergence, the Laplace transform provides information about both the growth and frequency content of a signal, unlike the Fourier transform.
Classification of signals
Deterministic and Random signals
Continuous time and discrete time signal
Even (symmetric) and Odd (Anti-symmetric) signal
Periodic and Aperiodic signal
Energy and Power signal
Causal and Non-causal signal
1. The Laplace transform converts differential equations describing systems from the time domain to the frequency domain by replacing functions of time with functions of a complex variable dependent on frequency.
2. The inverse Laplace transform converts the solution back from the frequency domain to the time domain to obtain the solution in terms of the time variable.
3. Partial fraction expansion is often used to break solutions into simpler terms that can be inverted using Laplace transform tables to find the solution in the time domain.
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
The Laplace transform allows solving differential equations using algebra by transforming differential operators into algebraic operations. It transforms a function of time (f(t)) into a function of a complex variable (F(s)), allowing differential equations describing systems to be solved for variables of interest. Key properties include linearity, time and frequency shifting, and relationships between derivatives, integrals, and the Laplace transform that enable solving differential equations by taking the transform, performing algebra, and applying the inverse transform.
This presentation contributes towards understanding the periodic function of a Laplace Transform. A sum has been included to relate the method for this topic and a video also so that the learning can be easy.
This document discusses linear transformations. It begins by defining a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. It provides examples of linear transformations and functions that are not linear transformations. It then discusses properties of linear transformations including the zero and identity transformations. It introduces the concept that a linear transformation can be represented by a matrix and that matrix multiplication defines a linear transformation. It concludes by stating that a linear transformation defined by a matrix satisfies the properties of being linear.
This document discusses the Fourier transform method for analyzing linear systems. It begins by introducing Fourier series as a way to represent periodic functions as an infinite series of sinusoids. It then discusses how the Fourier transform can be used to represent non-periodic functions. The key steps of the Fourier transform method are outlined, including determining the Fourier coefficients, representing signals in the frequency domain, and taking the inverse transform. Properties of Fourier series and examples of periodic and non-periodic signals are also briefly covered. The document provides an overview of the Fourier transform method for analyzing input/output signals of linear networks in both the time and frequency domains.
The document discusses linear transformations between vector spaces. It defines key concepts such as the domain, codomain, range, and images/preimages of vectors under a linear transformation. A linear transformation preserves vector addition and scalar multiplication. It provides examples of linear transformations, such as rotations and projections in planes and spaces. It also discusses when a transformation defined by a matrix represents a linear transformation.
This document discusses linear transformations and their properties. It defines a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. The kernel of a linear transformation is the set of vectors mapped to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. Matrices can represent linear transformations, with the matrix equation representing the transformation of vectors. Examples are provided to illustrate key concepts such as kernels, ranges, and matrix representations of linear transformations.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
1) The document discusses various linear transformations including reflection, rotation, and shear applied to geometric shapes like triangles and rectangles.
2) Reflection is illustrated for a triangle across the x-axis and y=-x line. Rotation is shown counterclockwise for different triangles around the origin.
3) Clockwise rotation and shear transformations in the x-direction are also demonstrated through examples using matrices applied to triangle and rectangle vertices. Plots of the transformed shapes are provided.
The document defines and provides examples of linear transformations. It then presents a question asking to find the matrix of a linear transformation between two vector spaces. The solution shows that:
1) The vector spaces have standard bases of {[1,0,0],[0,1,0],[0,0,1]} and {[1,0],[0,1]}.
2) The matrix of the linear transformation is A = [[3,4,9],[5,3,2]].
3) For a given linear transformation T defined by a matrix A, the transformation can be expressed in terms of coordinates as T(x)=A*x.
This document discusses linear transformations and their matrix representations. It defines a linear transformation as a function between vector spaces that respects the underlying linear structure. The matrix of a linear transformation uniquely represents the transformation and maps vectors from the domain to the range by matrix multiplication. Several examples are provided of finding the matrix of linear transformations between Rn and Rm spaces based on their actions on the standard basis vectors.
Linear Combination, Span And Linearly Independent, Dependent SetDhaval Shukla
In this presentation, the topic of Linear Combination, Span and Linearly Independent and Linearly Dependent Sets have been discussed. The sums for each topic have been given to understand the concept clearly for viewers.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
Fourier Transform ,LAPLACE TRANSFORM,ROC and its Properties Dr.SHANTHI K.G
1. The document discusses Fourier transforms, Laplace transforms, and their applications. Fourier transforms represent non-periodic signals as a function of frequency by decomposing them into simpler constituent parts. Laplace transforms transform time domain signals to the complex s-domain.
2. Key aspects covered include the Fourier analysis and synthesis equations, properties of the Fourier transform such as its magnitude and phase spectra. Conditions for the existence of Fourier transforms are explained. The region of convergence where the Laplace transform converges is also defined.
3. Examples are provided to demonstrate the calculation of Fourier and Laplace transforms of simple signals and determining their regions of convergence. Poles and zeros of transforms are also explained.
The document summarizes key concepts about the Laplace transform. It defines the Laplace transform, discusses properties like linearity and time shifting. It provides examples of taking the Laplace transform of unit step functions. It also covers computing the inverse Laplace transform using partial fraction expansion and handling cases with repeated or complex poles.
The document discusses unit step functions and their use in describing abrupt changes in function values that occur at specific times, such as a voltage being switched on or off in an electrical circuit. It defines the unit step function u(t) as having a value of 0 for negative t and 1 for positive t. Shifted and rectangular pulse functions are also described. Examples are provided of writing functions in terms of unit step functions and sketching the corresponding waveforms.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
Fourier-transform analysis of a unilateral fin line and its derivativesYong Heui Cho
This document presents a Fourier-transform analysis of a unilateral n-line and its derivatives. The key points are:
1) A unilateral n-line is transformed into an equivalent problem of multiple suspended substrate microstrip lines using the image theorem and Fourier transform.
2) New rigorous dispersion relations are derived in the form of rapidly convergent series solutions, providing an accurate yet efficient method for numerical computation.
3) The dispersion relations for derivatives of the unilateral n-line including suspended substrate microstrip lines, microstrip lines, slot lines and coplanar waveguides are also presented.
1. The document defines the Fourier series as an expansion of a function in a series of sines and cosines.
2. Fourier series can be used to represent even functions as a cosine series and odd functions as a sine series.
3. Examples are provided of calculating the Fourier coefficients for different functions, including finding the Fourier series of the function f(x)=x on the interval [0,π].
The Laplace transform generalizes the Fourier transform by allowing the parameter s to be any complex number rather than purely imaginary. The Laplace transform of a signal x(t) is defined as the integral of x(t) multiplied by e^-st from negative infinity to positive infinity. The region of convergence specifies the values of s for which the Laplace transform converges. Within the region of convergence, the Laplace transform provides information about both the growth and frequency content of a signal, unlike the Fourier transform.
Classification of signals
Deterministic and Random signals
Continuous time and discrete time signal
Even (symmetric) and Odd (Anti-symmetric) signal
Periodic and Aperiodic signal
Energy and Power signal
Causal and Non-causal signal
1. The Laplace transform converts differential equations describing systems from the time domain to the frequency domain by replacing functions of time with functions of a complex variable dependent on frequency.
2. The inverse Laplace transform converts the solution back from the frequency domain to the time domain to obtain the solution in terms of the time variable.
3. Partial fraction expansion is often used to break solutions into simpler terms that can be inverted using Laplace transform tables to find the solution in the time domain.
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
The Laplace transform allows solving differential equations using algebra by transforming differential operators into algebraic operations. It transforms a function of time (f(t)) into a function of a complex variable (F(s)), allowing differential equations describing systems to be solved for variables of interest. Key properties include linearity, time and frequency shifting, and relationships between derivatives, integrals, and the Laplace transform that enable solving differential equations by taking the transform, performing algebra, and applying the inverse transform.
This presentation contributes towards understanding the periodic function of a Laplace Transform. A sum has been included to relate the method for this topic and a video also so that the learning can be easy.
This document discusses linear transformations. It begins by defining a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. It provides examples of linear transformations and functions that are not linear transformations. It then discusses properties of linear transformations including the zero and identity transformations. It introduces the concept that a linear transformation can be represented by a matrix and that matrix multiplication defines a linear transformation. It concludes by stating that a linear transformation defined by a matrix satisfies the properties of being linear.
This document discusses the Fourier transform method for analyzing linear systems. It begins by introducing Fourier series as a way to represent periodic functions as an infinite series of sinusoids. It then discusses how the Fourier transform can be used to represent non-periodic functions. The key steps of the Fourier transform method are outlined, including determining the Fourier coefficients, representing signals in the frequency domain, and taking the inverse transform. Properties of Fourier series and examples of periodic and non-periodic signals are also briefly covered. The document provides an overview of the Fourier transform method for analyzing input/output signals of linear networks in both the time and frequency domains.
The document discusses linear transformations between vector spaces. It defines key concepts such as the domain, codomain, image, and preimage of a transformation. A linear transformation preserves vector addition and scalar multiplication. Any function defined by a matrix T(v) = Av is a linear transformation from Rn to Rm. The matrix representing a linear transformation describes how it rotates vectors in the plane.
An introduction to discrete wavelet transformsLily Rose
This document provides an overview of wavelet transforms and their applications. It introduces continuous and discrete wavelet transforms, including multiresolution analysis and the fast wavelet transform. It discusses how wavelet transforms can be used for image compression, edge detection, and digital watermarking due to properties like decomposing images into different frequency subbands. The fast wavelet transform allows efficient computation of wavelet coefficients by exploiting relationships between scales.
The document discusses a stochastic volatility model that incorporates jumps in volatility and the possibility of default. It describes the dynamics of the model and how it can be used to price volatility and credit derivatives. Analytical and numerical methods are presented for solving the pricing problem. As an example application, the model is fit to data on General Motors to analyze the implications.
LECT 1 Part 2 - The Transmission Line Theory.pptxahmedmohamedn92
This document discusses transmission line theory and the telegrapher's equations. It introduces key concepts such as series inductance, shunt conductance, and shunt capacitance per unit length. It derives the telegrapher's equations and shows how they can be used to model voltage and current on a transmission line as a function of position and time. It also introduces the use of phasors to represent sinusoidal signals on lines and discusses characteristic impedance and wave propagation effects.
I am John G. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from Glasgow University. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems assignments.
The Fourier transform relates a signal in the time domain, x(t), to its frequency domain representation, X(jw). It represents the frequency content of the signal. The Fourier transform is a linear operation, and time shifts in the time domain result in phase shifts in the frequency domain. Differentiation in the time domain corresponds to multiplication by jw in the frequency domain. Convolution becomes simple multiplication in the frequency domain. These properties allow differential equations and systems with convolution to be solved using algebraic operations by working in the frequency domain.
2. Power Computations and Analysis Techniques_verstud.pdfLIEWHUIFANGUNIMAP
This document provides an overview of power computations and analysis techniques for power electronics. It covers topics such as sign convention, periodic waveforms, power and energy, capacitors, inductors, effective values including RMS, apparent power and power factor, sinusoidal and nonsinusoidal AC circuits. The key points covered are:
- The ability to evaluate the performance of various AC-DC and DC-DC converters is discussed.
- Concepts of instantaneous power, average power, energy storage in capacitors and inductors are explained.
- Effective voltage and current values using RMS are described for different waveforms including sinusoidal, pulse, triangular and general Fourier series.
- Apparent power, power factor and power computations
The Laplace transform converts a function of time (f(t)) into a function of complex frequency (F(s)). The Laplace transform is defined as the integral of the function multiplied by an exponential term. The document provides several examples of Laplace transform pairs, including the transforms of basic functions like step functions, impulses, and exponentials. It also outlines some important properties like linearity, time shifting, and differentiation/integration in the transform domain. Understanding common transform pairs and properties allows one to use the Laplace transform to solve differential equations.
The Laplace transform is defined as the integral of a function f(t) multiplied by e^-st from 0 to infinity, where s is a complex variable. Some basic Laplace transform properties and pairs are listed, including transforms of unit step functions, exponential functions, trigonometric functions, and their combinations. The document provides an introduction to basic Laplace transforms and lists some common transform pairs and properties.
1) Transmission lines carry signals between two points by propagating waves along two parallel conductors. Common types include coaxial cable and printed circuit board traces.
2) Transmission lines are characterized by their per-unit-length inductance, capacitance, resistance, and conductance. The behavior of signals on the line is described by telegrapher's equations.
3) Waves on transmission lines travel at the phase velocity, defined as the ratio of frequency to phase constant. The characteristic impedance is determined by the line's inductance and capacitance.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document discusses linear filters and their properties. It can be summarized as:
1. Linear filters relate an input time series {xt} to an output time series {yt} through a linear transformation and convolution.
2. The impulse response function of a linear filter describes how the filter responds to a unit impulse input and characterizes the filter.
3. Stability of a linear filter requires that the sum of the absolute values of the coefficients in the filter's transfer function converges.
4. Discrete dynamic models relate the difference of the output time series to the difference of the input time series through differencing operators, providing a way to model physical systems with differential equations using discrete time series data.
The document provides notes on signals and systems from an EECE 301 course. It includes:
- An overview of continuous-time (C-T) and discrete-time (D-T) signal and system models.
- Details on chapters covering differentials/differences, convolution, Fourier analysis (both C-T and D-T), Laplace transforms, and Z-transforms.
- Examples of calculating the Fourier transform of specific signals like a decaying exponential and rectangular pulse. These illustrate properties of the Fourier transform.
This document discusses Fourier cosine and sine integrals. It provides the definitions and formulas for the Fourier cosine transform, Fourier sine transform, and their inverses. It also discusses improper integrals of type 1 and 2, including definitions and convergence conditions. Examples are provided to illustrate the concepts. The physical interpretation of the Fourier integrals is that higher integration limits include more higher frequency sinusoidal components in the approximation of the real function.
This document discusses linear algebra concepts for digital filter design. It begins with definitions of filters and digital filters, then covers topics like FIR and IIR filter design. Key points include:
- A filter removes unwanted things from an item of interest passed through a structure. A digital filter removes unwanted frequency components from a discretized signal passed through a structure with delay, multiplier, and summer elements.
- FIR filter design involves choosing filter coefficients to satisfy desired frequency response specifications. IIR filter design similarly involves solving systems of equations to determine coefficients for structures using feedback.
- Matlab code for FIR filter design is provided in slide 17 for appreciation. A presentation on slides 3-4 and 17 is assigned.
The document discusses quasi-resonant converters and the half-wave zero-current-switching quasi-resonant switch cell. The switch cell uses a small resonant inductor and capacitor to achieve zero-current switching of the transistor. It operates in four subintervals per switching period: 1) transistor on, 2) resonant ringing, 3) capacitor discharging, 4) diode on. Mathematical analysis determines the waveforms and durations of each subinterval. Averaging the switch cell currents and voltages gives the conversion ratio, allowing the cell to be analyzed and incorporated into converter circuits.
This lecture discusses Fourier series and Fourier transforms. Fourier series represent periodic signals as a sum of sinusoids, while Fourier transforms represent both periodic and non-periodic signals as a function of frequency. Examples of calculating the Fourier series and Fourier transform of common signals like sinusoids, step functions, and exponentials are provided. Exercises are suggested to practice calculating Fourier transforms and using them to analyze the frequency content of signals.
- Convolution is used to represent the output of a continuous-time linear time-invariant (CT LTI) system when the input is an arbitrary signal. The output is defined as the convolution of the input signal with the system's impulse response.
- The impulse response completely characterizes a CT LTI system. The output can be computed by taking the convolution integral of the input signal with the impulse response.
- Properties of the convolution integral include associativity, commutativity, and relationships to differentiation, integration, shifting and scaling of signals.
- A CT LTI system can also be represented using the unit step response, which is the convolution of the impulse response with the unit step function.
Similar to Linear transformation vcla (160920107003) (20)
• Register Transfer Language
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Templates are a feature of the C++ programming language that allow functions and classes to operate with generic types. This allows a function or class to work on many different data types without being rewritten for each one.
This document describes various memory management techniques used in computer systems, including swapping, contiguous allocation, paging, segmentation, and the memory architecture of the Intel Pentium CPU. It discusses how paging uses a page table to map logical addresses to physical frames through an address translation process. Segmentation divides memory into variable-length segments and uses segment tables. The Pentium supports both pure segmentation and a hybrid of segmentation and paging to translate logical addresses to physical memory locations.
Reliable data transfer CN - prashant odhavani- 160920107003Prashant odhavani
transport layer services
multiplexing/demultiplexing
connectionless transport: UDP
principles of reliable data transfer
connection-oriented transport: TCP
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connection management
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To solve differential equation problem:
If f is “smooth enough”, then
a solution will exist and be unique
we will be able to approximate it accurately with a wide variety of method
Two ways of expressing “smooth enough”
Lipschitz continuity
Smooth and uniformly monotone decreasing
Five basic operations in relational algebra: Selection, Projection, Cartesian product, Union, and Set Difference.
These perform most of the data retrieval operations needed.
Also have Join, Intersection, and Division operations, which can be expressed in terms of 5 basic operations
Non linear data structure ds (2190702) - 169020107003Prashant odhavani
Non linear data structure is part of dynamic Non linear data structure are
1. Non linear data structure - Tree
2.Non linear data structure - Graph
this ppt in explain Non linear data structure - Tree in how to perform different types operation e.g. inorder, preorder and posorder .
the Non linear data structure - Tree use fro network connection of local area network(LAN), wide area network(WAN)
Digital Electronics is subject of Coputer, I.T., Electrical and Electronics Branch.
Number system is most important topic. Number system in various types of conversation e.g. binary, octal, decimal, hexadecimal.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
3. Linear Transformation
3
Zero transformation : VT vv ,0)(WVT :
Identity transformation : VVT : VT vvv ,)(
VWVT vu,,:Properties of linear transformations :
00 )((1)T )()((2) vv TT
)()()((3) vuvu TTT 1 1 2 2
1 1 2 2
1 1 2 2
(4) If
Then ( ) ( )
( ) ( ) ( )
n n
n n
n n
c v c v c v
T T c v c v c v
c T v c T v c T v
v
v
L
L
L
4. The Kernel and Range of a Linear Transformation
4
Kernel of a linear transformation T
Let be a linear transformation then the set of all vectors
v in V that satisfy is called the kernel of T and is denoted by ker(T)
},0)(|{)ker( VTT vvv
WVT :
5. The kernel is a subspace of V
The kernel of a linear transformation is a subspace of the domain V.
then.ofkernelin thevectorsbeandLet Tvu
000)()()( vuvu TTT
00)()( ccTcT uu )ker(Tc u
)ker(T vu
.ofsubspaceais)ker(Thus, VT
Note:
The kernel of T is sometimes called the nullspace of T.
WVT :
5
6. 6
Ex : Finding a basis for the kernel
82000
10201
01312
11021
andRiniswhere,)(bydefinedbe:Let 545
A
ATRRT xxx
Find a basis for ker(T) as a subspace of R5.
8. Range of a linear transformation T
)(bydenotedisandTofrangethecalledisVin
vectorofimagesarein W thatwvectorsallofsetThen the
L.T.abe:Let
Trange
WVT
}|)({)( VTTrange vv
8
9. .:Tnnsformatiolinear traaofrangeThe WWV foecapsbusasi
The range of T is a subspace of W
TTT ofrangein thevectorbe)(and)(Let vu
)()()()( TrangeTTT vuvu
)()()( TrangecTcT uu
),( VVV vuvu
)( VcV uu
.subspaceis)(Therefore, WTrange
9
11. 11
Ex: Finding a basis for the range of a linear transformation
82000
10201
01312
11021
andiswhere,)(bydefinedbe:Let 545
A
RATRRT xxx
Find a basis for the range of T.
13. Rank and Nullity of Linear Transformation
13
Rank of a linear transformation T:V→W:
TTrank ofrangetheofdimensionthe)(
Nullity of a linear transformation T:V→W:
TTnullity ofkerneltheofdimensionthe)(
Note:
)()(
)()(
then,)(bygivenL.T.thebe:Let
AnullityTnullity
ArankTrank
ATRRT mn
xx
14. Sum of rank and nullity
AmatrixnmT anbydrepresenteisLet
)ofdomaindim()ofkerneldim()ofrangedim(
)()(
TTT
nTnullityTrank
rArank )(Assume
Let T:V→W be a L.T. form an n dimation 1 Vector space
V into a Vector space W . Then
14
20. 20One-to-one and onto linear transformation
onto.isitifonlyandifone-to-oneisThen.dimension
ofbothandspaceorwith vectL.T.abe:Let
Tn
WVWVT
0))(dim(and}0{)(thenone,-to-oneisIf TKerTKerT
)dim())(dim())(dim( WnTKernTrange
onto.isly,Consequent T
0)ofrangedim())(dim( nnTnTKer
one.-to-oneisTherefore,T
nWTT )dim()ofrangedim(thenonto,isIf
21. 21
Inverse linear Transformation
ineveryfors.t.L.T.are:and:If 21
nnnnn
RRRTRRT v
))((and))(( 2112 vvvv TTTT
invertiblebetosaidisandofinversethecalledisThen 112 TTT
Note:
If the transformation T is invertible, then the
inverse is unique and denoted by T–1 .
22. 22Finding the inverse of a linear transformation
bydefinedis:L.T.The 33
RRT
)42,33,32(),,( 321321321321 xxxxxxxxxxxxT
142
133
132
formatrixstandardThe
A
T
321
321
321
42
33
32
xxx
xxx
xxx
100142
010133
001132
3IA
Show that T is invertible, and find its inverse.
22
Solution
24. 24
Finding the matrix of a linear transformation
.formatrixstandardtheFindaxis.-xtheontoinpointeach
projectingbygivenis:nnsformatiolinear traThe
2
22
TR
RRT
)0,(),( xyxT
00
01
)1,0()0,1()()( 21 TTeTeTA
Notes:
(1) The standard matrix for the zero transformation from Rn into Rm is the mn zero matrix.
(2) The standard matrix for the zero transformation from Rn into Rnis the nn identity matrix In
Solution
25. 25
Composition of T1:Rn→Rm with T2:Rm→Rp
n
RTTT vvv )),(()( 12
112 ofdomainofdomain, TTTTT
Composition of linear transformations
then,andmatricesstandardwith
L.T.be:and:Let
21
21
AA
RRTRRT pmmn
L.T.ais)),(()(bydefined,:ncompositioThe(1) 12 vv TTTRRT pn
12productmatrixby thegivenisformatrixstandardThe)2( AAATA
Note:
1221 TTTT