This document summarizes the key topics covered in Chapter 5 of the Signals & Systems course, which analyzes linear time-invariant (LTI) systems using time-domain and frequency-domain techniques. The chapter covers stability analysis using the impulse response and Routh-Hurwitz test, analyzing step responses for first-order and second-order systems, and the frequency response of LTI systems to sinusoidal and periodic inputs. Examples are provided to illustrate these time-domain and frequency-domain analysis methods.
This document provides an introduction to Hidden Markov Models (HMMs). It begins by explaining the key differences between Markov Models and HMMs, noting that in HMMs the states are hidden and can only be indirectly observed through observations. It then outlines the main elements of an HMM - the number of states, observations, state transition probabilities, observation probabilities, and initial state distribution. An example HMM is provided. Finally, it briefly introduces three common problems in HMMs - determining the most likely model given observations, determining the most likely state sequence, and determining the model parameters that are most likely to have generated the observations.
1) Markov models and hidden Markov models describe systems that transition between states based on probabilities, where the next state depends only on the current state.
2) Markov models assume each state corresponds to a directly observable event, while hidden Markov models allow states to be hidden and observations to depend probabilistically on the current state.
3) Transition and initial state probabilities can be described using a transition matrix in Markov models to calculate the probability of state sequences.
The document describes balanced homodyne detection and its use for linear optical sampling of light fields. Balanced homodyne detection measures the quadrature amplitudes of a signal field using a strong local oscillator field that acts as a phase reference. By varying the delay and phase of the local oscillator, different quadratures of the signal field can be sampled. This allows the direct measurement of mean quadrature amplitudes with sub-picosecond time resolution. It also enables the indirect measurement of photon number and photon number fluctuations in the signal by analyzing the statistics of the sampled quadratures over many repetitions. Linear optical sampling provides an alternative to nonlinear optical sampling for ultrafast optical sampling applications.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The document discusses applying counterexample guided abstraction refinement (CEGAR) to verifying properties of Petri nets. It summarizes using the Petri net state equation to represent reachable markings as solutions to a system of linear equations. It then describes using CEGAR to iteratively check solutions and refine the abstraction by adding increments when solutions are found to be infeasible. The approach is implemented in a tool called Sara which shows better performance than other tools on verification problems involving large Petri nets and parameterized systems.
This document provides formulas and properties for the Laplace transform and its inverse. It lists 10 formulas for common functions and their Laplace transforms, including 1, t, tn, sinat, cosat, sinhat, and coshat. It also lists 7 properties of the Laplace transform, such as how it is affected by scaling, derivatives, division by t, and multiplication by t. Finally, it lists 6 properties of the inverse Laplace transform, such as how it is affected by derivatives, division by s, and the convolution theorem.
This document discusses linear prediction analysis (LPC) for speech recognition. It begins by deriving the linear prediction equations and describing the autocorrelation method of LPC. It then interprets the LPC filter as a spectral whitener that flattens the spectrum of the prediction error. The document discusses alternative methods like covariance LPC and closed phase covariance LPC. It also describes alternative parameter sets that can represent the LPC filter, such as pole positions, reflection coefficients, and log area ratios.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document provides an introduction to Hidden Markov Models (HMMs). It begins by explaining the key differences between Markov Models and HMMs, noting that in HMMs the states are hidden and can only be indirectly observed through observations. It then outlines the main elements of an HMM - the number of states, observations, state transition probabilities, observation probabilities, and initial state distribution. An example HMM is provided. Finally, it briefly introduces three common problems in HMMs - determining the most likely model given observations, determining the most likely state sequence, and determining the model parameters that are most likely to have generated the observations.
1) Markov models and hidden Markov models describe systems that transition between states based on probabilities, where the next state depends only on the current state.
2) Markov models assume each state corresponds to a directly observable event, while hidden Markov models allow states to be hidden and observations to depend probabilistically on the current state.
3) Transition and initial state probabilities can be described using a transition matrix in Markov models to calculate the probability of state sequences.
The document describes balanced homodyne detection and its use for linear optical sampling of light fields. Balanced homodyne detection measures the quadrature amplitudes of a signal field using a strong local oscillator field that acts as a phase reference. By varying the delay and phase of the local oscillator, different quadratures of the signal field can be sampled. This allows the direct measurement of mean quadrature amplitudes with sub-picosecond time resolution. It also enables the indirect measurement of photon number and photon number fluctuations in the signal by analyzing the statistics of the sampled quadratures over many repetitions. Linear optical sampling provides an alternative to nonlinear optical sampling for ultrafast optical sampling applications.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The document discusses applying counterexample guided abstraction refinement (CEGAR) to verifying properties of Petri nets. It summarizes using the Petri net state equation to represent reachable markings as solutions to a system of linear equations. It then describes using CEGAR to iteratively check solutions and refine the abstraction by adding increments when solutions are found to be infeasible. The approach is implemented in a tool called Sara which shows better performance than other tools on verification problems involving large Petri nets and parameterized systems.
This document provides formulas and properties for the Laplace transform and its inverse. It lists 10 formulas for common functions and their Laplace transforms, including 1, t, tn, sinat, cosat, sinhat, and coshat. It also lists 7 properties of the Laplace transform, such as how it is affected by scaling, derivatives, division by t, and multiplication by t. Finally, it lists 6 properties of the inverse Laplace transform, such as how it is affected by derivatives, division by s, and the convolution theorem.
This document discusses linear prediction analysis (LPC) for speech recognition. It begins by deriving the linear prediction equations and describing the autocorrelation method of LPC. It then interprets the LPC filter as a spectral whitener that flattens the spectrum of the prediction error. The document discusses alternative methods like covariance LPC and closed phase covariance LPC. It also describes alternative parameter sets that can represent the LPC filter, such as pole positions, reflection coefficients, and log area ratios.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Po...IJERDJOURNAL
ABSTRACT: According to a Cauchy’s classical result all the zeros of a polynomial n j j j P z a z 0 ( ) of degree n lie in z 1 A , where n j j n a a A max0 1 . In this paper we find a ring-shaped region containing all or a specific number of zeros of P(z). Mathematics Subject Classification: 30C10, 30C15.
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].
This document discusses Hidden Markov Models (HMMs) and Markov chains. It begins with an introduction to Markov processes and how HMMs are used in various domains like natural language processing. It then describes the properties of a Markov chain, which has a set of states that the system transitions between randomly at discrete time steps based on transition probabilities. The Markov property is explained as the conditional independence of future states from past states given the present state. HMMs extend Markov chains by making the state sequence hidden and only allowing observation of the output states.
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 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 the Laplace transform, which transforms a signal from the time domain to the frequency domain. It defines the Laplace transform and inverse Laplace transform. Important properties include: linearity, shifting, scaling. Common Laplace transform pairs are presented in a table. Theorems allow taking derivatives and integrals of signals in the Laplace domain. Partial fraction expansion can be used to simplify rational functions.
This document provides an overview of signal fundamentals, including definitions, examples, and properties of signals. It discusses topics such as signal energy and power, signal transformations, periodic and exponential signals. Examples are provided to illustrate concepts such as determining if a signal has finite energy/power, applying signal transformations, decomposing signals into even and odd components, and plotting exponential signals. The document is from a university course on signal fundamentals and is intended to introduce basic signal processing concepts.
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
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
This document discusses methods for accelerating Google's PageRank algorithm. It provides background on how PageRank works and the problem of computing the PageRank vector. It then summarizes several methods studied, including the power method, solving a linear system, Langville and Meyer's algorithm, and the Iterative Aggregation/Disaggregation method. It finds that applying a row and column reordering to the power method matrix based on decreasing node degree almost always reduces the number of iterations needed for convergence. Future research directions are also outlined.
The document discusses quantization in analog-to-digital conversion. It describes the three processes of A/D conversion as sampling, quantization, and binary encoding. Quantization involves mapping amplitude values into a set of discrete values using a quantization interval or step size. The document discusses uniform quantization and how the range is divided into equal intervals. It also discusses non-uniform quantization which has smaller intervals near zero to better match real audio signals. Examples and MATLAB code demonstrations are provided to illustrate quantization of audio signals at different bit rates.
This document provides an overview of the continuous-time Fourier transform. It introduces the Fourier integral and defines the Fourier transform pair. It discusses properties of the Fourier transform including linearity, time scaling, time reversal, time shifting, frequency shifting, and properties for real functions. Examples are provided to illustrate these concepts and properties. The document also reviews the discrete Fourier transform and Fourier series to provide context and comparison to the continuous-time Fourier transform.
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.
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
DSP_FOEHU - MATLAB 02 - The Discrete-time Fourier AnalysisAmr E. Mohamed
This document discusses the discrete-time Fourier transform (DTFT) and its properties. It provides examples of calculating the DTFT of sequences and using it to analyze linear time-invariant (LTI) systems. The key points are:
1. The DTFT represents a discrete-time signal as a complex-valued continuous function of digital frequency. It has periodicity and symmetry properties useful for analysis.
2. LTI systems can be analyzed in the frequency domain using their frequency response, which is the DTFT of the system's impulse response.
3. The steady-state response of an LTI system to an input signal can be computed from the system's frequency response evaluated at the input signal's
fourier representation of signal and systemsSugeng Widodo
This document provides an overview of Fourier analysis concepts including:
- The Fourier transform decomposes a signal into its constituent frequencies.
- Properties of the Fourier transform like linearity, time/frequency shifting, and modulation are discussed.
- The Fourier transform of a time derivative or integral is related to the original Fourier transform.
- Convolution and correlation theorems explain how time domain operations translate to the frequency domain.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
The document discusses constrained impulsive motion and analytical methods for impulse response dynamics. It introduces the Hamiltonian and Lagrangian approaches for modeling constraints. Equations are presented for writing the general equation of constrained impulsive motion using Lagrangian multipliers. Examples are provided to illustrate constraints and solving for post-impact velocities and energies using impulsive constraint equations.
This document describes analysis of linear time-invariant (LTI) systems through impulse response and convolution. It defines LTI systems and explains that their behavior is characterized by impulse response. The input and output of an LTI system are related by convolution. Impulse response is the output of an LTI system when the input is a unit impulse. Given impulse response, the output can be found for any input using convolution. Difference equations are also used to represent LTI systems. The document provides examples of finding impulse responses and performing convolution in MATLAB. It discusses properties of convolution like associativity and commutativity. Tasks are given on finding impulse responses and convolving various signals.
This document discusses linear time-invariant (LTI) systems and convolution. It begins by defining LTI systems and convolution for both continuous and discrete time. Convolution is described as a way to construct the output of a system given its impulse response. Applications in digital signal processing and image processing are mentioned. Convolution filtering plays an important role in edge detection and related image processing algorithms. The mathematical definition of discrete time convolution is provided. An example problem calculating outputs for different inputs using convolution is given at the end.
A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Po...IJERDJOURNAL
ABSTRACT: According to a Cauchy’s classical result all the zeros of a polynomial n j j j P z a z 0 ( ) of degree n lie in z 1 A , where n j j n a a A max0 1 . In this paper we find a ring-shaped region containing all or a specific number of zeros of P(z). Mathematics Subject Classification: 30C10, 30C15.
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].
This document discusses Hidden Markov Models (HMMs) and Markov chains. It begins with an introduction to Markov processes and how HMMs are used in various domains like natural language processing. It then describes the properties of a Markov chain, which has a set of states that the system transitions between randomly at discrete time steps based on transition probabilities. The Markov property is explained as the conditional independence of future states from past states given the present state. HMMs extend Markov chains by making the state sequence hidden and only allowing observation of the output states.
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 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 the Laplace transform, which transforms a signal from the time domain to the frequency domain. It defines the Laplace transform and inverse Laplace transform. Important properties include: linearity, shifting, scaling. Common Laplace transform pairs are presented in a table. Theorems allow taking derivatives and integrals of signals in the Laplace domain. Partial fraction expansion can be used to simplify rational functions.
This document provides an overview of signal fundamentals, including definitions, examples, and properties of signals. It discusses topics such as signal energy and power, signal transformations, periodic and exponential signals. Examples are provided to illustrate concepts such as determining if a signal has finite energy/power, applying signal transformations, decomposing signals into even and odd components, and plotting exponential signals. The document is from a university course on signal fundamentals and is intended to introduce basic signal processing concepts.
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
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
This document discusses methods for accelerating Google's PageRank algorithm. It provides background on how PageRank works and the problem of computing the PageRank vector. It then summarizes several methods studied, including the power method, solving a linear system, Langville and Meyer's algorithm, and the Iterative Aggregation/Disaggregation method. It finds that applying a row and column reordering to the power method matrix based on decreasing node degree almost always reduces the number of iterations needed for convergence. Future research directions are also outlined.
The document discusses quantization in analog-to-digital conversion. It describes the three processes of A/D conversion as sampling, quantization, and binary encoding. Quantization involves mapping amplitude values into a set of discrete values using a quantization interval or step size. The document discusses uniform quantization and how the range is divided into equal intervals. It also discusses non-uniform quantization which has smaller intervals near zero to better match real audio signals. Examples and MATLAB code demonstrations are provided to illustrate quantization of audio signals at different bit rates.
This document provides an overview of the continuous-time Fourier transform. It introduces the Fourier integral and defines the Fourier transform pair. It discusses properties of the Fourier transform including linearity, time scaling, time reversal, time shifting, frequency shifting, and properties for real functions. Examples are provided to illustrate these concepts and properties. The document also reviews the discrete Fourier transform and Fourier series to provide context and comparison to the continuous-time Fourier transform.
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.
Introduction to Fourier transform and signal analysis宗翰 謝
The document discusses Fourier analysis techniques. It introduces continuous and discrete Fourier transforms, and covers properties like orthogonality, completeness of basis functions (e.g. cosines and sines), and Fourier series representations of periodic functions like step functions. It also defines the Fourier transform and its properties like linearity, translation, modulation, scaling, and conjugation. Concepts like Dirac delta functions and convolution theory are explained in relation to Fourier analysis.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
DSP_FOEHU - MATLAB 02 - The Discrete-time Fourier AnalysisAmr E. Mohamed
This document discusses the discrete-time Fourier transform (DTFT) and its properties. It provides examples of calculating the DTFT of sequences and using it to analyze linear time-invariant (LTI) systems. The key points are:
1. The DTFT represents a discrete-time signal as a complex-valued continuous function of digital frequency. It has periodicity and symmetry properties useful for analysis.
2. LTI systems can be analyzed in the frequency domain using their frequency response, which is the DTFT of the system's impulse response.
3. The steady-state response of an LTI system to an input signal can be computed from the system's frequency response evaluated at the input signal's
fourier representation of signal and systemsSugeng Widodo
This document provides an overview of Fourier analysis concepts including:
- The Fourier transform decomposes a signal into its constituent frequencies.
- Properties of the Fourier transform like linearity, time/frequency shifting, and modulation are discussed.
- The Fourier transform of a time derivative or integral is related to the original Fourier transform.
- Convolution and correlation theorems explain how time domain operations translate to the frequency domain.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
The document discusses constrained impulsive motion and analytical methods for impulse response dynamics. It introduces the Hamiltonian and Lagrangian approaches for modeling constraints. Equations are presented for writing the general equation of constrained impulsive motion using Lagrangian multipliers. Examples are provided to illustrate constraints and solving for post-impact velocities and energies using impulsive constraint equations.
This document describes analysis of linear time-invariant (LTI) systems through impulse response and convolution. It defines LTI systems and explains that their behavior is characterized by impulse response. The input and output of an LTI system are related by convolution. Impulse response is the output of an LTI system when the input is a unit impulse. Given impulse response, the output can be found for any input using convolution. Difference equations are also used to represent LTI systems. The document provides examples of finding impulse responses and performing convolution in MATLAB. It discusses properties of convolution like associativity and commutativity. Tasks are given on finding impulse responses and convolving various signals.
This document discusses linear time-invariant (LTI) systems and convolution. It begins by defining LTI systems and convolution for both continuous and discrete time. Convolution is described as a way to construct the output of a system given its impulse response. Applications in digital signal processing and image processing are mentioned. Convolution filtering plays an important role in edge detection and related image processing algorithms. The mathematical definition of discrete time convolution is provided. An example problem calculating outputs for different inputs using convolution is given at the end.
Z TRANSFORM PROPERTIES AND INVERSE Z TRANSFORMTowfeeq Umar
The document discusses various methods for computing the inverse z-transform including inspection, partial fraction expansion, and power series expansion. It provides examples to illustrate each method. The inverse z-transform finds the original time domain sequence from its z-transform. Key properties like linearity, time shifting, and convolution are also covered.
This document discusses linear time-invariant (LTI) systems and convolution. Convolution is a fundamental concept in signal processing that is used to determine the output of an LTI system given its impulse response and an input signal. The convolution of two signals is obtained by decomposing the input signal into scaled and shifted impulses, taking the scaled and shifted impulse response for each impulse, and summing them to find the overall output. Convolution amplifies or attenuates different frequency components of the input independently. It plays an important role in applications like image processing and edge detection. Examples are provided to demonstrate evaluating convolution of periodic sequences.
This document introduces concepts related to signals and systems. It defines a system as a mathematical model that relates an input signal to an output signal. Systems can be continuous-time or discrete-time, depending on whether the input and output signals are continuous or discrete. Other properties discussed include whether a system is linear or nonlinear, time-invariant or time-varying, causal or noncausal, stable, and whether it involves feedback. Examples are provided to illustrate different types of systems.
This lecture discusses linear time-invariant (LTI) systems and convolution. Any input signal can be represented as a sum of time-shifted impulse signals. The output of an LTI system is determined by its impulse response h[n] using convolution. Convolution involves multiplying and summing the input signal with time-shifted versions of the impulse response. This allows predicting a system's response to any input based only on its impulse response. Examples show calculating convolution by summing scaled signal segments and using the non-zero elements of h[n]. Exercises include reproducing an example convolution in MATLAB.
Dsp U Lec06 The Z Transform And Its Applicationtaha25
This document discusses the Z-transform and its application in digital signal processing. It covers topics such as:
1) Defining the Z-transform and how it can characterize linear time-invariant (LTI) systems.
2) Properties of LTI systems in the Z-domain, including causal and stable systems.
3) How the frequency response of a system can be obtained from its Z-transform.
4) Methods for finding the inverse Z-transform, including power series and partial fraction expansion.
5) Examples of using these techniques to analyze simple discrete systems.
This document provides an overview of discrete-time signals and systems in digital signal processing (DSP). It discusses key concepts such as:
1) Discrete-time signals which are represented by sequences of numbers and how common signals like impulses and steps are represented.
2) Discrete-time systems which take a discrete-time signal as input and produce an output signal through a mathematical algorithm, with the impulse response characterizing the system.
3) Important properties of linear time-invariant (LTI) systems including superposition, time-shifting of inputs and outputs, and representation using convolution sums or difference equations.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
This document discusses signals and systems. It defines signals as physical quantities that vary with respect to time, space, or another independent variable. Signals can be classified as discrete time or continuous time. It also defines unit impulse and unit step functions for discrete and continuous time. Periodic and aperiodic signals are discussed. The Fourier series and Fourier transform are introduced as ways to represent signals in the frequency domain. The Laplace transform, which generalizes the Fourier transform, is also mentioned. Key properties of linear time-invariant systems like superposition, time-invariance, and convolution are covered. Finally, sampling theory and the z-transform, which is analogous to the Laplace transform for discrete-time systems, are summarized at a high level
State equations model based on modulo 2 arithmetic and its applciation on rec...Anax Fotopoulos
1) The document discusses state equations that can model digital control systems based on modulo-2 arithmetic and their application to recursive convolutional coding.
2) State equations can be derived from the transfer function of a discrete-time controller and expressed using modulo-2 arithmetic.
3) A recursive convolutional encoder can be modeled by state equations in modulo-2 algebra, where the state at each time step is a function of the previous states and the input bit.
State Equations Model Based On Modulo 2 Arithmetic And Its Applciation On Rec...Anax_Fotopoulos
1) The document discusses state equations that can model digital control systems based on modulo-2 arithmetic and their application to recursive convolutional coding.
2) Key concepts covered include cyclic groups, rings, state equations derived from transfer functions expressed in z-transform, and direct hardware realizations of state equations.
3) An example of a recursive convolutional encoder is described based on a set of recursive signal equations, and its corresponding algebraic state equations in modulo-2 arithmetic are provided.
The document discusses the simplex method for solving linear programs in tabular form. It describes the key steps of the simplex method including initialization, optimality testing, and iterative steps to arrive at an optimal solution. The iterative step involves choosing an entering variable, determining a leaving variable using ratios, and performing Gaussian elimination to generate a new basis. The document also addresses tie-breaking rules and revising the simplex method to reduce storage requirements.
Generic parallelization strategies for data assimilationnilsvanvelzen
Presentation given at
The Ninth International Workshop on Adjoint Model Applications in Dynamic Meteorology, 10–14 October 2011, Cefalu, Sicily, Italy
Adjoint workshop 2011
The document discusses the design of IIR digital filters using different methods. It begins by describing the difference equation and transfer function of IIR filters. It then covers the Impulse Invariant Method and Bilinear Z-Transform (BZT) Method for designing IIR filters by transforming analog prototypes. Key steps include prewarping frequencies, designing analog filters, and applying the bilinear transform. Examples demonstrate applying these methods to design Butterworth filters.
This chapter discusses sampling and signal reconstruction from samples. It introduces the sampling theorem which states that a signal must be sampled at a rate at least twice its highest frequency component to avoid aliasing. It describes how to reconstruct the original signal from its samples using an interpolation formula. It also discusses the effects of undersampling and oversampling, and how aliasing can occur if the sampling rate is too low.
Reduced rcs uhf rfid gosper and hilbert tag antennasiaemedu
The document summarizes the design and analysis of conventional, Gosper, and Hilbert loop shape antennas with reduced radar cross section (RCS) for small object identification using radio frequency identification (RFID) technology. Fourier series analysis is used to calculate the input impedance of a conventional circular wire loop antenna. A T-matched chart method is introduced to match the antenna input impedance to the chip input impedance. Simulation results show the conventional loop antenna achieves an RCS reduction of 99.67% and 99.12% for the Gosper and Hilbert loop antennas, respectively, compared to a conventional design. Measurements of the fabricated antennas are also discussed.
Reduced rcs uhf rfid gosper and hilbert tag antennas no restrictioniaemedu
The document summarizes the design and analysis of conventional, Gosper, and Hilbert loop shape antennas with reduced radar cross section (RCS) for small object identification using radio frequency identification (RFID) technology. Fourier series analysis is used to calculate the input impedance of a conventional circular wire loop antenna. A T-matched chart method is introduced to match the antenna input impedance to the chip input impedance. Simulation results show the conventional loop antenna achieves an RCS reduction of 99.67% and 99.12% for the Gosper and Hilbert loop antennas, respectively, compared to a conventional design. Measurements of the fabricated antennas are also discussed.
This document provides an introduction to algorithm design and analysis. It discusses sorting as an example problem, comparing the insertion sort and merge sort algorithms. Insertion sort runs in O(n^2) time while merge sort runs in O(nlogn) time, making merge sort faster for large inputs. The document explains the recursive definitions and analyses of these algorithms' runtimes. It also introduces asymptotic notation and techniques for algorithm analysis such as recurrence relations and decision trees. Finally, it briefly discusses NP-complete problems.
This document contains homework problems related to physics concepts like crystal structure, band structure, diffraction, and ionic bonding energies. It includes 4 problems:
1) Identifying the element of a metallic sample based on x-ray diffraction data by matching Bragg angles to allowed crystal structures. The element is determined to be nickel.
2) Calculating the zero-point kinetic energy of particles confined in a linear potential well based on the de Broglie wavelength formula.
3) Analyzing the interatomic potential and compressibility of a 1D linear ionic crystal chain, expressing the work required to compress the chain in terms of material properties.
4) Computing the Madelung energies of barium oxide
This document contains homework problems related to physics concepts like crystal structure, band structure, diffraction, and ionic bonding energies. It includes 4 problems:
1) Identifying the element of a metallic sample based on x-ray diffraction data by matching Bragg angles to allowed crystal structures. The element is determined to be nickel.
2) Calculating the zero-point kinetic energy of particles confined in a linear potential well based on the de Broglie wavelength formula.
3) Analyzing the bonding, equilibrium distance, and compressibility of a linear ionic crystal chain based on the attractive and repulsive potential terms and Taylor expansion.
4) Computing the Madelung energies of divalent ionic
This document discusses statistical tools used in quality control laboratories and validation studies, including normal distributions, variance, ranges, coefficients of variation, F-tests, Student's t-tests, and paired t-tests. It provides the formulas and procedures for calculating and applying these statistical concepts to analyze laboratory data and test for significant differences between samples. Examples are given to demonstrate how to perform t-tests to compare averages from independent and paired samples with both known and unknown variances.
This document discusses correlative-level coding and its applications in baseband pulse transmission systems. Correlative-level coding introduces controlled intersymbol interference to increase signaling rate. It allows partial response signaling and maximum likelihood detection at the receiver. Specific techniques discussed include duobinary signaling and modified duobinary signaling. The document also covers tapped-delay line equalization using adaptive algorithms like least mean square to compensate for channel distortion. Decision feedback equalization and its implementation are summarized as well. Eye patterns are described as a tool to evaluate signal quality in such systems.
This document discusses algorithm design and analysis. It introduces sorting as an example problem and compares the insertion sort and merge sort algorithms. Insertion sort runs in O(n2) time in the worst case, while merge sort runs in O(nlogn) time. It provides pseudocode for insertion sort and merge sort and analyzes their time complexities. It also covers algorithm analysis techniques like recursion trees and asymptotic notation.
The document discusses several brute-force algorithms including bubble sort, selection sort, string matching, closest pair of points, convex hulls, traveling salesman problem, knapsack problem, and assignment problem. It analyzes the runtime of each algorithm, which are often quadratic or exponential time due to considering all possible combinations in a systematic way.
This document discusses algorithm analysis and asymptotic notation. It defines common asymptotic notations like O(N), Ω(N), and Θ(N) and provides examples of analyzing simple algorithms and determining their time complexities. The document also outlines general rules for analyzing algorithms with loops, nested loops, consecutive statements, and recursion to determine their asymptotic running times.
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
This document discusses the design of finite impulse response (FIR) filters. It begins by describing the basic FIR filter model and properties such as filter order and length. It then covers topics such as linear phase response, different filter types (low-pass, high-pass, etc.), deriving the ideal impulse response, and filter specification in terms of passband/stopband edges and ripple levels. The document concludes by outlining the common FIR design method of windowing the ideal impulse response, describing popular window functions, and providing a step-by-step example of designing a low-pass FIR filter using the Hamming window.
This document contains slides from a lecture on the z-transform and discrete-time linear time-invariant (LTI) systems. The slides cover properties of the z-transform including shift properties, convolution properties, and other properties related to scaling, time reversal, and initial values. They also discuss how to represent discrete-time LTI systems using difference equations and transfer functions in the z-domain. Methods for realizing LTI systems like direct form I and II structures are presented. The slides show examples of finding frequency responses of systems and mapping between the s-plane and z-plane representations of systems.
This weekly report discusses progress made on a project to develop a food spoilage detection platform. Modifications were made to address comments from a previous presentation. A diode was added to the charger circuit to protect the switching transformer when unplugged. Additional functions were also added, including an LED to indicate when batteries are charging by comparing the output voltage to a reference voltage. Both the modified charger circuit and switching transformer were installed inside the platform. While this integrated the components, a weakness is that the enclosed platform lacks ventilation, which could cause high temperatures. The next steps are to prepare materials for an upcoming presentation.
This weekly report summarizes the progress of an engineering student group working on a product design called P'Jub. The student tested a new charger circuit for their battery by decreasing the charging current to 300mA, which helped reduce the temperature of an LM317 component. The student designed a PCB board for the circuit, assembled it, and retested it. Strengths included cooling the LM317 and adjusting current flow. A weakness was unstable contact between the platform and body. The next week's task is to complete an English report.
This chapter discusses representing systems using transfer functions. It covers obtaining the transfer function by taking the Laplace transform of the input-output differential equation. Transfer functions allow representing systems in the frequency domain. Key concepts covered include poles and zeros of a system, frequency response functions, and practical passive filters using resistor-inductor-capacitor components. Transfer functions of interconnected systems are also addressed.
1. The weekly report summarizes work on improving a battery charger circuit. Issues with high temperatures in the previous LM317 circuit were addressed.
2. A new charger circuit was designed with current limiting using RV2 and automatic output voltage adjustment using BC337.
3. Next steps include testing the new charger circuit, designing an alarm for low 7806 output voltage, and designing a new PCB for the charger circuit.
The weekly report summarizes the work done on designing a power supply circuit. Key points:
- The student designed a charger circuit using an LM317 regulator to charge a 1500mAh 3.7V LI-PO battery within 60 minutes.
- Testing showed regulators 7806 and 7805 can generate stable 6V and 5V outputs with input voltages above 7.2V and 7V respectively.
- Issues identified were the LM317 running hot during charging and output voltage drops of 7806 below 6V at lower inputs.
- Next steps are to improve charging circuit and add an alarm for low 7806 output.
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Chapter5 system analysis
1. Signals & Systems
Chapter 5
Time-domain and
frequency-domain analysis
of LTI systems using TF
INC212 Signals and Systems : 2 / 2554
2. Overview
Stability and the impulse response
Routh-Hurwitz Stability Test
Analysis of the Step Response
Fourier analysis of CT systems
Response of LTI systems to sinusoidal inputs
Response of LTI systems to periodic inputs
Response of LTI systems to nonperiodic
(aperiodic) inputs
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
3. Stability and
the Impulse Response
bM s M + bM −1s M −1 + + b1s + b0 Numerator
H ( s) =
a N s N + a N −1s N −1 + + a1s + a0 Denominator
Assumption : N ≥ M
: H (s) dose not have any common poles and zeros
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
4. Stability and
the Impulse Response
H(s) has a real pole, p h(t) contains cept
H(s) has a complex pair poles, σ±jω
h(t) contains ceσtcos(ωt+θ)
H(s) has a repeated poles
h(t) contains ctiept or ctieσtcos(ωt+θ)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
5. Stability and
the Impulse Response
h(t) 0 as t ∞ Stability
cept p<0
ceσtcos(ωt+θ) σ<0
ctiept or ctieσtcos(ωt+θ) p < 0 or σ < 0
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
6. Stability and
the Impulse Response
h(t) 0 as t ∞ Stable
|h(t)| ≤ c for all t Marginally Stable
bounded
|h(t)| ∞ as t ∞ Unstable
unbounded
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
7. Stability and
the Impulse Response
OLHP ORHP
Stable Unstable
× × ×
× × ×
× × ×
Marginally Stable
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
8. Stability and
the Impulse Response
Example 8.1 Series RLC Circuit
1 LC R
H ( s) = 2 b<0 Re( p1 , p2 ) = − <0
2L
s + ( R L) s + 1 LC
R −
R
+ b <0
p1 , p2 = − ± b
2L b≥0 2L
2
R
2 b<
R 1 2L
b= −
2 L LC −
R
− b <0 b<
R
2L 2L
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
9. Routh-Hurwitz Stability Test
Table 8.1 Routh Array A( s ) = a N s N + a N −1s N −1 + + a1s + a0 , a N > 0
ai > 0 for i = 0,1,2, , N − 1
sN aN aN-2 aN-4 …
sN-1 aN-1 aN-3 aN-5 … a N −1a N − 2 − a N a N −3 a a
bN − 2 = = a N − 2 − N N −3
a N −1 a N −1
sN-2 bN-2 bN-4 bN-6 …
a N −1a N − 4 − a N a N −5 a a
bN − 4 = = a N − 4 − N N −5
sN-3 cN-3 cN-5 cN-7 … a N −1 a N −1
…
…
…
…
bN − 2 a N −3 − a N −1bN − 4 a b
c N −3 = = a N −3 − N −1 N − 4
s2 d2 d0 0 … bN − 2 bN − 2
s1 e1 0 0 … c N −5 =
bN − 2 a N −5 − a N −1bN −6 a b
= a N −5 − N −1 N −6
bN − 2 bN − 2
s0 f0 0 0 …
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
10. Routh-Hurwitz Stability Test
Table 8.1 Routh Array
If all elements > 0 sN aN aN-2 aN-4 …
Stable sN-1 aN-1 aN-3 aN-5 …
sN-2 bN-2 bN-4 bN-6 …
If 1 or more elements = 0
& no sign changes sN-3 cN-3 cN-5 cN-7 …
…
…
…
…
Marginally Stable
s2 d2 d0 0 …
If sign changes
s1 e1 0 0 …
Unstable s0 f0 0 0 …
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
11. Routh-Hurwitz Stability Test
Example 8.2 Second-Order Case
A( s ) = s 2 + a1s + a0 If a1 & a0 > 0
Stable
Routh Array in the N=2 Case
s2 1 a0 If a1 > 0 & a0 < 0
s1 a1 0 or a1 < 0 & a0 < 0
1 pole in ORHP
s0 a0 0
a1a0 − (1)(0) If a1 < 0 & a0 > 0
bN − 2 = = a0
a1 Unstable
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
12. Routh-Hurwitz Stability Test
Example 8.3 Third-Order Case
A( s ) = s + a2 s + a1s + a0
3 2
Routh Array in the N=3 Case
s3 1 a1 a0 a0
a1 − > 0 a1 >
s2 a2 a0 a2 a2
s1 a1-(a0/a2) 0 a0
a2 > 0, a1 > , a0 > 0
s0 a0 0 a2
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
13. Routh-Hurwitz Stability Test
Example 8.4 Higher-Order Case
A( s ) = 6 s 5 + 5s 4 + 4s 3 + 3s 2 + 2 s + 1
Example 8.4
s5 6 4 2
s4 5 3 1
s3 0.4 0.8 0
s2 -7 1 0
s1 6/7 0 0
s0 1 0 0
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
14. Routh-Hurwitz Stability Test
Example 8.5 Fourth-Order Case
A( s ) = s + s + 3s + 2 s + 2
4 3 2
Example 8.5
s4 1 3 2
s3 1 2 0
s2 1 2 0
s1 0 ≈ε 0 0
s0 2 0 0
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
15. Analysis of the Step Response
B(s) E (s) c
Y (s) = X ( s) Y ( s) = +
A( s ) A( s ) s
1 c = [ s Y ( s )]s =0 = H (0)
X (s) =
s E (s)
−1
y1 (t ) = L
Y (s) =
B(s) A( s )
A( s ) s y (t ) = y1 (t ) + H (0), t ≥ 0
Transient part Steady-state value (if stable)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
16. Analysis of the Step Response
First-Order Systems
k k
H (s) = y (t ) = − (1 − e pt ), t ≥ 0
s− p p
1 k pt
X ( s) = y1 (t ) = e , t ≥ 0
s p
−k p k p k
Y (s) = + H (0) = −
s s− p p
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
17. Analysis of the Step Response
First-Order Systems : k
y (t ) = − (1 − e pt ), t ≥ 0
p
Without bound
p = 3, 2, 1
Unstable
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
18. Analysis of the Step Response
First-Order Systems : k
y (t ) = − (1 − e pt ), t ≥ 0
p
Bound Stable
p = -5, -2, -1
k = -p H(0) = 1
Steady-state value = 1
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
19. Analysis of the Step Response
First-Order Systems :
(time constant, τ)
≈ 63% of H(0)
p = -5, -2, -1
τ =0.2 sec
τ = 0.5 sec
τ = 1 sec
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
20. Analysis of the Step Response
Determining the pole location from the step
response
y(t) ≈ 1.73 ;
t ≈ 0.1 s
y (0.1) = 1.73 = 2[1 − e p ( 0.1) ] t = 0.1 sec
p = −20
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
21. Analysis of the Step Response
Second-Order Systems
k p1 = −ζω n + ωn ζ 2 − 1
H ( s) = 2
s + 2ζω n s + ωn2
p2 = −ζω n − ωn ζ 2 − 1
ζ is called the damping ratio
ωn is called the natural frequency
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
22. Analysis of the Step Response
Second-Order Systems :
Case when both poles are real
k k
H ( s) = y (t ) = (k1e p1t + k 2 e p2t + 1), t ≥ 0
( s − p1 )( s − p2 ) p1 p2
k
k ytr (t ) = (k1e p1t + k 2 e p2t ), t ≥ 0
Y ( s) = p1 p2
( s − p1 )( s − p2 ) s
k k
H ( 0) = 2 =
ωn p1 p2
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
23. Analysis of the Step Response
Second-Order Systems :
k = 2, p1 = −1, p2 = −2 Case when both poles are real
2
H (s) =
( s + 1)( s + 2)
2
Y ( s) =
( s + 1)( s + 2) s
−2 1 1
Y ( s) = + +
s +1 s + 2 s
y (t ) = −2e −t + e − 2t + 1, t ≥ 0
ytr (t ) = −2e −t + e − 2t , t ≥ 0
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
24. Analysis of the Step Response
Second-Order Systems :
Case when poles are real and repeated
k
H (s) =
( s + ωn ) 2
k
Y (s) =
( s + ωn ) 2 s
y (t ) =
k
[1 − (1 + ωnt )e−ω t ], t ≥ 0
n
ωn2
k
ytr (t ) = − (1 + ωnt )e −ω t , t ≥ 0
n
ωn2
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
25. Analysis of the Step Response
Second-Order Systems :
Case when poles are real and repeated
k = 4, ωn = 2, p1 , p2 = −2
4
H (s) =
( s + 2) 2
4
Y ( s) =
( s + 2) 2 s
y (t ) = 1 − (1 + 2t )e − 2 t , t ≥ 0
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
26. Analysis of the Step Response
Second-Order Systems :
Location of poles in the complex plane
p1 = −ζω n + jωd
p2 = −ζω n − jωd
k
H (s) =
( s − p1 )( s − p2 )
k
H (s) =
( s + ζω n − jωd )( s + ζω n + jωd )
k
H (s) =
( s 2 + 2ζω n + ωn ) + ωd
2 2
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
27. Analysis of the Step Response
Second-Order Systems :
Case when poles are a complex pair
k k
H (s) = Y ( s) =
( s + ζω n ) 2 + ωd
2
( )
( s + ζω n ) 2 + ωd s
2
− ( k ωn ) s − 2kζ ωn k ωn
2 2
Y ( s) = +
( s + ζω n ) + ωd
2 2
s
− ( k ωn )( s + ζω n )
2
(kζ ωn ) k ωn2
= − +
( s + ζω n ) + ωd
2 2
( s + ζω n ) + ωd
2 2
s
k −ζω nt kζ −ζω nt k
y (t ) = − 2 e cos ωd t − e sin ωd t + 2 , t ≥ 0
ωn ωnωd ωn
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
28. Analysis of the Step Response
Second-Order Systems :
Case when poles are a complex pair
k −ζω nt kζ −ζω nt k
y (t ) = − e cos ωd t − e sin ωd t + 2 , t ≥ 0
ωn2
ωnωd ωn
C cos β + D sin β = C 2 + D 2 sin( β + θ )
tan −1 (C D), C ≥ 0
where θ =
π + tan (C D), C < 0
−1
k ωn −ζω nt
y (t ) = 2 1 − e sin(ωd t + φ ), t ≥ 0
ωn ωd
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
29. Analysis of the Step Response
Second-Order Systems :
Case when poles are a complex pair
17
H (s) =
s 2 + 2 s + 17
k = 17, ζ = 0.242, ωn = 17 , ωd = 4
p = −1 ± j 4
17 −t
y (t ) = 1 − e sin( 4t + 1.326), t ≥ 0
4
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
30. Analysis of the Step Response
Second-Order Systems :
Effect of Damping Ratio on the Step Response
k
H (s) =
( s + ζω n ) + ωd
2 2
ωn = 1, k = 1
for ζ = 0.1, 0.25, 0.7
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
31. Analysis of the Step Response
Second-Order Systems :
Effect of ωn on the Step Response
k
H (s) =
( s + ζω n ) 2 + ωd
2
ζ = 0.4, k = ωn 2
for ωn = 0.5, 1, 2 rad / s
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
32. Analysis of the Step Response
Second-Order Systems :
Comparison of cases
k
H ( s) = 2 0<ζ<1 underdamped
s + 2ζω n s + ωn2
ζ>1 overdamped
ζ=1 Critically damped
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
33. Analysis of the Step Response
Second-Order Systems :
Comparison of cases
k
H (s) = 2
s + 2ζω n s + ωn2
k = 4, ωn = 2
for ζ = 0.5, 1, 1.5
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
34. Analysis of the Step Response
Higher-Order Systems
bM s M + bM −1s M −1 + + b1s + b0
H ( s) = N N −1
s + a N −1s + + a1s + a0
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
35. Fourier Analysis of CT Systems
LTI systems
Time domain Frequency domain
x(t) h(t) y(t) X(ω) H(ω) Y(ω)
F
h(t) is Impulse Response H(ω) is Frequency Response function
∞
y (t ) = h(t ) ∗ x(t ) = ∫ h(λ ) x(t − λ )dλ Y (ω ) = H (ω ) X (ω )
−∞
Assume that the system is stable: Amplitude : Y (ω ) = H (ω ) ⋅ X (ω )
∞
∫−∞
h(t ) dt < ∞ Phase : ∠Y (ω ) = ∠H (ω ) + ∠X (ω )
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
36. Response of LTI System to
Sinusoidal Inputs X(ω) H(ω) Y(ω)
F
x(t ) = A cos(ω0t + θ ) X (ω ) = Aπ [e − jθ δ (ω + ω0 ) + e jθ δ (ω − ω0 )]
Y (ω ) = H (ω ) X (ω )
Y (ω ) = AH (ω )π [e − jθ δ (ω + ω0 ) + e jθ δ (ω − ω0 )]
= Aπ [e − jθ H (−ω0 )δ (ω + ω0 ) + e jθ H (ω0 )δ (ω − ω0 )]
= Aπ H (ω0 ) [e − j (θ + ∠H (ω0 ))δ (ω + ω0 ) + e j (θ + ∠H (ω0 ))δ (ω − ω0 )]
F -1
Response to
y (t ) = A H (ω0 ) cos(ω0t + θ + ∠H (ω0 ))
Sinusoidal Input
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
37. Response of LTI System to
Sinusoidal Inputs x(t) h(t) y(t)
x(t ) = A cos(ω0t + θ ) h(t) y (t ) = A H (ω0 ) cos(ω0t + θ + ∠H (ω0 ))
x(t ) = A1 cos(ω1t + θ1 ) + A2 cos(ω 2t + θ 2 ) h(t)
y (t ) = A1 H (ω1 ) cos(ω1t + θ1 + ∠H (ω1 )) + A2 H (ω2 ) cos(ω2t + θ 2 + ∠H (ω 2 ))
x(t ) = cos(100t ) + cos(3000t ) where ω1 = 100
h(t)
and ω2 = 3000
y (t ) = H (100) cos(100t + ∠H (100)) + H (3000) cos(3000t + ∠H (3000))
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
38. Response of LTI System to
Sinusoidal Inputs X(ω) H(ω) Y(ω)
Y (ω ) = H (ω ) X (ω )
Vout (ω ) = H (ω )Vin (ω )
Vout (ω )
dvout (t ) H (ω ) =
RC + vout (t ) = vin (t ) Vin (ω )
dt
jω RCVout (ω ) + Vout (ω ) = Vin (ω ) Vout (ω ) 1
=
( jω RC + 1)Vout (ω ) = Vin (ω ) Vin (ω ) ( jω RC + 1)
1 1
H (ω ) = ; H (ω ) = ; ∠ H (ω ) = − tan −1 ωRC
( jω RC + 1) (ω RC ) 2 + 1
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
39. Response of LTI System to
Sinusoidal Inputs
Low frequency
H(ω)
lim H (ω ) = 1
ω →0
High frequency
∠ H(ω)
lim H (ω ) = 0
ω →∞
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
40. Response of LTI System to
Sinusoidal Inputs x(t) h(t) y(t)
x(t ) = cos(100t ) + cos(3000t ) h(t)
y (t ) = H (100) cos(100t + ∠H (100)) + H (3000) cos(3000t + ∠H (3000))
1
H (ω ) =
(ω RC ) 2 + 1
x(t ) = cos(100t ) + cos(3000t ) ∠ H (ω ) = − tan −1 ωRC
y (t ) = H (100) cos(100t + ∠H (100)) + H (3000) cos(3000t + ∠H (3000))
ω = ω 1 = 100 ω = ω2 = 3000
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
41. Response of LTI System to
Sinusoidal Inputs
1
H (ω ) =
x(t ) = cos(100t ) + cos(3000t ) (ω RC ) 2 + 1
∠ H (ω ) = − tan −1 ωRC
y (t ) = H (100) cos(100t + ∠H (100)) + H (3000) cos(3000t + ∠H (3000))
RC = 0.001
H(ω)
x(t)
∠ H(ω)
y(t)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
42. Response of LTI System to
Sinusoidal Inputs
1
H (ω ) =
x(t ) = cos(100t ) + cos(3000t ) (ω RC ) 2 + 1
∠ H (ω ) = − tan −1 ωRC
y (t ) = H (100) cos(100t + ∠H (100)) + H (3000) cos(3000t + ∠H (3000))
RC = 0.01
H(ω)
x(t)
∠ H(ω)
y(t)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
43. Response of LTI System to
Periodic Inputs x(t) h(t) y(t)
∞
x (t ) = a0 + ∑ Ak cos(kω0t + θ k ), − ∞ < t < ∞ h(t)
k =1
∞
y (t ) = a0 H (0) + ∑ Ak H (kω0 ) cos(kω0t + θ k + ∠H ( kω0 )), − ∞ < t < ∞
k =1
1 x
A = A H (kω0 )
y x cky = Ak H (kω0 )
k k 2
θ ky = θ kx + ∠ H (kω0 ) ∠cky = θ kx + ∠ H (kω0 )
Akx , θ kx is the coefficients of the trigonometric FS for x(t)
Aky , θ ky is the coefficients of the trigonometric FS for y(t)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
44. Response of LTI System to
Periodic Inputs
Response to a rectangular pulse train
∞
x(t)
x(t ) = a0 + ∑ Ak cos(kω0t + θ k ), − ∞ < t < ∞
… …
k =1
∞
2.0 -0.5 0.5 2.0 t x(t ) = a0 + ∑ ak cos(kπt ), − ∞ < t < ∞
k =1
2
, k = 1,3,5,
0.5, k = 0 A = kπ
x
k
0,
k = 2,4,6,
ckx = 0, k = ±2,±4,±6,
1 π , k = 3,7,11,
kπ , k = ±1,±3,±5, θk =
x
0, all other k
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
45. Response of LTI System to
Periodic Inputs
Response to a rectangular pulse train
x(t)
… …
2.0 -0.5 0.5 2.0 t a0y = H (0)a0 = 0.5
x
ω = kω0 2 1
, k = 1,3,5,
2π Ak = kπ (kπRC ) + 1
y 2
ω0 = ,T = 2 0,
T k = 2,4,6,
ω = kπ π − tan −1 kπRC , k = 3,7,11,
θ ky =
− tan kπRC ,
−1
all other k
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
46. Response of LTI System to
Periodic Inputs
Response to a rectangular pulse train
• RC = 1
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
47. Response of LTI System to
Periodic Inputs
Response to a rectangular pulse train
• RC = 0.01
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
48. Response of LTI System to
Nonperiodic Inputs
Response to a rectangular pulse
X(ω) H(ω) Y(ω)
Y (ω ) = H (ω ) X (ω )
y (t ) = F - 1 { H (ω ) X (ω )}
1 ∞
y (t ) = ∫−∞ H (ω ) X (ω )e jωt dt
2π
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
49. Response of LTI System to
Nonperiodic Inputs
Response to a rectangular pulse
x (t ) y (t ) = F - 1 {Y (ω )}
1
= F - 1 { H (ω ) X (ω )}
t 1 ∞
y (t ) = ∫ H (ω ) X (ω )e jωt dt
-1/2 1/2 2π −∞
F F-1
ω Y (ω ) = H (ω ) X (ω )
X (ω ) = sinc
2π H(ω)
1
H (ω ) =
jωRC + 1
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
50. Response of LTI System to
Nonperiodic Inputs
Response to a rectangular pulse
• RC = 1
F-1
F
H(ω)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
51. Response of LTI System to
Nonperiodic Inputs
Response to a rectangular pulse
• RC = 0.1
F-1
F
H(ω)
INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF
52. INC212 Signals and Systems : 2 / 2554 Chapter 5 The analysis of LTI systems using TF