1. The document summarizes a lecture on discrete-time signals and systems.
2. It defines different types of signals, including discrete-time and discrete-valued signals which are relevant for digital filter theory.
3. It also classifies systems as static vs. dynamic, time-invariant vs. time-variable, linear vs. nonlinear, causal vs. non-causal, stable vs. unstable, and recursive vs. non-recursive.
4. It describes the time-domain representation of linear, time-invariant (LTI) systems using impulse response and convolution.
DSP_2018_FOEHU - Lec 03 - Discrete-Time Signals and SystemsAmr E. Mohamed
The document discusses discrete-time signals and systems. It defines discrete-time signals as sequences represented by x[n] and discusses important sequences like the unit sample, unit step, and periodic sequences. It then defines discrete-time systems as devices that take a discrete-time signal x(n) as input and produce another discrete-time signal y(n) as output. The document classifies systems as static vs. dynamic, time-invariant vs. time-varying, linear vs. nonlinear, and causal vs. noncausal. It provides examples to illustrate each classification.
The document discusses different types of discrete-time systems. It defines a discrete-time system as a device or algorithm that operates on a discrete-time input signal to produce an output signal. Discrete-time systems are then classified as static or dynamic, time-varying or time-invariant, linear or non-linear, causal or non-causal, and stable or unstable. Examples are provided for each type of system. Static systems have no memory, while dynamic systems can have finite or infinite memory. Time-invariant systems have input-output characteristics that do not change over time. Linear systems satisfy the principle of superposition, while non-linear systems do not. Causal systems only depend on past and present inputs.
1) Discrete-time signals are functions of independent integer variables where samples are only defined at discrete time instants. 2) A discrete-time system transforms an input signal into an output signal through operations like addition, multiplication, delay. 3) Linear, time-invariant (LTI) systems have properties of superposition and time-invariance, and their behavior can be characterized by the impulse response. The output of an LTI system is the convolution of the input and impulse response.
1. The document discusses continuous-time signals and systems. It defines signals and systems, and how they are classified based on properties like being continuous or discrete, and having one or more independent variables.
2. It describes various operations that can be performed on signals, including time shifting, time reversal, time compression/expansion, and amplitude scaling. These transformations change the signal while preserving the information content.
3. Systems are defined as entities that process input signals to produce output signals. Examples of signal processing systems include communication systems, control systems, and systems that interface between continuous and discrete domains.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document discusses three techniques for identifying linear systems: frequency chirp method, inverse filtering method, and coherence function method. The frequency chirp method uses a wideband excitation input like a frequency chirp and obtains the frequency response as the discrete Fourier transform of the system's output. The inverse filtering method uses pseudoinverse to find the impulse response. The coherence function method uses the coherence function and input-output cross-spectrum to estimate the system response directly and inversely. These three techniques were demonstrated on a 10th order Chebyshev filter. The inverse filtering method provided an identification closest to the actual system response compared to the other two techniques.
Classification of signals and systems as well as their properties are given in the PPT .Examples related to types of signals and systems are also given .
This document provides an overview of signals and systems classification. It discusses:
1) Signals can be continuous-time or discrete-time, periodic or non-periodic, deterministic or random, even or odd.
2) Systems can be causal or non-causal, linear or nonlinear, time-invariant or time-variant, stable or unstable.
3) Key system properties include memory/memoryless, and examples of discrete-time systems are presented.
This document summarizes key concepts in digital signal processing systems. It defines a system as a combination of elements that processes an input signal to produce an output signal. Systems are classified as continuous or discrete time, lumped or distributed parameter, static or dynamic, causal or non-causal, linear or non-linear, time variant or invariant, and stable or unstable. Convolution and the discrete Fourier transform (DFT) are also introduced as important tools in digital signal processing. The DFT transforms a signal from the time domain to the frequency domain.
DSP_2018_FOEHU - Lec 03 - Discrete-Time Signals and SystemsAmr E. Mohamed
The document discusses discrete-time signals and systems. It defines discrete-time signals as sequences represented by x[n] and discusses important sequences like the unit sample, unit step, and periodic sequences. It then defines discrete-time systems as devices that take a discrete-time signal x(n) as input and produce another discrete-time signal y(n) as output. The document classifies systems as static vs. dynamic, time-invariant vs. time-varying, linear vs. nonlinear, and causal vs. noncausal. It provides examples to illustrate each classification.
The document discusses different types of discrete-time systems. It defines a discrete-time system as a device or algorithm that operates on a discrete-time input signal to produce an output signal. Discrete-time systems are then classified as static or dynamic, time-varying or time-invariant, linear or non-linear, causal or non-causal, and stable or unstable. Examples are provided for each type of system. Static systems have no memory, while dynamic systems can have finite or infinite memory. Time-invariant systems have input-output characteristics that do not change over time. Linear systems satisfy the principle of superposition, while non-linear systems do not. Causal systems only depend on past and present inputs.
1) Discrete-time signals are functions of independent integer variables where samples are only defined at discrete time instants. 2) A discrete-time system transforms an input signal into an output signal through operations like addition, multiplication, delay. 3) Linear, time-invariant (LTI) systems have properties of superposition and time-invariance, and their behavior can be characterized by the impulse response. The output of an LTI system is the convolution of the input and impulse response.
1. The document discusses continuous-time signals and systems. It defines signals and systems, and how they are classified based on properties like being continuous or discrete, and having one or more independent variables.
2. It describes various operations that can be performed on signals, including time shifting, time reversal, time compression/expansion, and amplitude scaling. These transformations change the signal while preserving the information content.
3. Systems are defined as entities that process input signals to produce output signals. Examples of signal processing systems include communication systems, control systems, and systems that interface between continuous and discrete domains.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document discusses three techniques for identifying linear systems: frequency chirp method, inverse filtering method, and coherence function method. The frequency chirp method uses a wideband excitation input like a frequency chirp and obtains the frequency response as the discrete Fourier transform of the system's output. The inverse filtering method uses pseudoinverse to find the impulse response. The coherence function method uses the coherence function and input-output cross-spectrum to estimate the system response directly and inversely. These three techniques were demonstrated on a 10th order Chebyshev filter. The inverse filtering method provided an identification closest to the actual system response compared to the other two techniques.
Classification of signals and systems as well as their properties are given in the PPT .Examples related to types of signals and systems are also given .
This document provides an overview of signals and systems classification. It discusses:
1) Signals can be continuous-time or discrete-time, periodic or non-periodic, deterministic or random, even or odd.
2) Systems can be causal or non-causal, linear or nonlinear, time-invariant or time-variant, stable or unstable.
3) Key system properties include memory/memoryless, and examples of discrete-time systems are presented.
This document summarizes key concepts in digital signal processing systems. It defines a system as a combination of elements that processes an input signal to produce an output signal. Systems are classified as continuous or discrete time, lumped or distributed parameter, static or dynamic, causal or non-causal, linear or non-linear, time variant or invariant, and stable or unstable. Convolution and the discrete Fourier transform (DFT) are also introduced as important tools in digital signal processing. The DFT transforms a signal from the time domain to the frequency domain.
1. The document discusses different types of systems based on their properties, including static vs dynamic, time-variant vs time-invariant, linear vs non-linear, causal vs non-causal, and stable vs unstable.
2. A system is defined as a physical device or algorithm that performs operations on a discrete-time signal. Static systems have outputs that depend only on the present input, while dynamic systems have outputs that depend on present and past/future inputs.
3. Time-invariant systems have characteristics that do not change over time, while time-variant systems have characteristics that do change. Linear systems follow the superposition principle, while causal systems have outputs dependent only on present and past inputs.
The document summarizes several key properties of systems, including:
1) Memoryless systems have output that depends only on the present input, while causal systems have output that depends on present and past inputs.
2) Invertible systems have a one-to-one mapping between distinct inputs and outputs, allowing the input to be recovered from the output.
3) Time-invariant systems have parameters that do not vary over time, while time-varying systems have parameters dependent on time.
4) Impulse, unit step, and ramp signals are useful test signals that can reveal qualitative characteristics of a system's transient and steady-state response.
This document provides an introduction to discrete-time signals and linear time-invariant (LTI) systems. It defines discrete-time signals as sequences represented at discrete time instants. Basic discrete-time sequences including the unit sample, unit step, and periodic sequences are described. Discrete-time systems are defined as transformations that map an input sequence to an output sequence. Linear and time-invariant systems are introduced. For LTI systems, the impulse response is defined and convolution is used to represent the output as a summation of the input multiplied by delayed versions of the impulse response. Key properties of LTI systems including superposition, scaling, time-invariance, and the commutative property of convolution are covered.
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
The document summarizes key concepts about linear time-invariant (LTI) systems from Chapter 2. It discusses:
1) LTI systems can be modeled as the sum of their impulse responses weighted by the input signal. This is known as the convolution sum/integral for discrete/continuous-time systems.
2) Any signal can be represented as a linear combination of shifted unit impulses. The output of an LTI system is the convolution of the input signal with the system's impulse response.
3) The impulse response completely characterizes an LTI system. The output is found by taking the convolution integral or sum of the input signal with the impulse response.
This document provides an overview of digital signal processing systems. It defines a system as a physical device or algorithm that performs operations on a discrete time signal. A system has an input signal x(n), performs a process, and produces an output signal y(n). Systems can be classified based on their properties as static/dynamic, time-invariant/time-variant, linear/non-linear, causal/non-causal, and stable/unstable. Examples of each system type are provided. The key aspects covered are the definitions of each system property, how to determine if a given system has a particular property, and examples to illustrate the concepts.
The document summarizes key concepts about even and odd signals from a lecture on signals and systems:
1) Even signals have the property that x(-t)=x(t), while odd signals have the property that x(-t)=-x(t). Any signal can be expressed as the sum of an even and an odd component.
2) Discrete time linear and time-invariant (LTI) systems can be characterized by their impulse response h[n]. The output of an LTI system is the convolution of the input signal with the impulse response.
3) Convolution is a mathematical operation that mixes two signals together. It has applications in areas like edge detection. The output of an LTI system
2013.06.18 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on nonlinear analysis of time series as part of the Summer School on Modern Statisitical Analysis and Computational Methods hosted by the Social Sciences Compuing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
Fourier analysis of signals and systemsBabul Islam
This document discusses Fourier analysis of signals and linear time-invariant (LTI) systems. It defines LTI systems and explains that they are mathematically easy to analyze due to properties like superposition. Fourier analysis is used to represent signals in the frequency domain using techniques like the Fourier series for periodic signals and the Fourier transform for aperiodic signals. The frequency response of an LTI system is its output when the input is an impulse, and the output of any LTI system is the convolution of the input signal and impulse response.
1) A system transforms an input signal into an output signal through interconnections between components. Systems can be continuous-time or discrete-time depending on if the input and output signals are continuous or discrete.
2) Systems are linear if they satisfy superposition, meaning the response to a weighted sum of inputs equals the sum of responses. Nonlinear systems do not satisfy superposition.
3) A system is time-invariant if a time shift in the input results in the same time shift in the output. Otherwise, it is time-varying.
This document provides an overview of a lecture on time domain analysis of first order systems. The key topics covered include:
- Modeling systems using state-space representation and defining state, state vector, and state-space.
- Analyzing systems in the time domain by applying standard test inputs like impulse, step, ramp, and parabolic signals to determine the system response over time.
- The time response of first order systems consists of transient and steady-state responses. It is determined by applying standard inputs and using the system transfer function and properties like the time constant T and gain K.
1) Control theory deals with analyzing and designing closed-loop control systems to achieve desired output behaviors.
2) The document provides examples of modeling control systems using transfer functions and state-space representations. These include modeling an RF control system and passive and active filter circuits.
3) State-space representation involves expressing higher-order differential equations as a set of first-order equations and representing the system using matrix equations that can be analyzed and simulated on computers. This allows visualization and analysis of dynamic systems.
The document outlines the syllabus for a course on digital signal processing. It includes 5 units: 1) Introduction to signals and systems, 2) Discrete time system analysis using z-transforms, 3) Discrete Fourier transforms and computation including fast Fourier transforms, 4) Design of digital filters including FIR and IIR filters, and 5) Digital signal processors and their architecture. It allocates a total of 45 periods to cover these topics. Textbooks recommended for the course provide further information on digital signal processing principles, algorithms, and applications.
Controllability of Linear Dynamical SystemPurnima Pandit
The document discusses linear dynamical systems and controllability of linear systems. It defines dynamical systems as mathematical models describing the temporal evolution of a system. Linear dynamical systems are ones where the evaluation functions are linear. Controllability refers to the ability to steer a system from any initial state to any final state using input controls. The document provides the definition of controllability for linear time-variant systems using the controllability Gramian matrix. It also gives the formula for the minimum-norm control input that can steer the system between any two states. An example of checking controllability for a time-invariant linear system is presented.
The document provides an overview of discrete time signal processing concepts including:
1) Signals can be classified as continuous, discrete, deterministic, random, periodic, and non-periodic. Systems can be linear, time-invariant, causal, stable/unstable, and recursive/non-recursive.
2) Digital signal processing has advantages over analog such as precision, stability and easy implementation of operations. It also has drawbacks like needing ADCs/DACs and being limited by sampling frequency.
3) Discrete time signals are only defined at discrete time instances while continuous time signals are defined for all time. Both can be represented graphically, functionally, through tables or sequences.
This document provides an overview of chapter 2 on introduction to systems from a signals and systems textbook. It defines different types of systems, including continuous-time and discrete-time systems. It outlines the key topics that will be covered, including system properties like linearity and time-invariance. It defines linear time-invariant systems and discusses how discrete-time signals and systems can be represented using impulse functions and the convolution sum. Examples are provided to illustrate different system properties.
This document provides an overview and instructions for generating real-time digital audio signals using filtering on an ARM FM4 processor. It introduces the Cypress FM4 hardware and development tools used, including Keil uVision and LabVIEW. It then covers generating sine waves of different frequencies on the FM4 by programming an interrupt service routine with a lookup table approach. Debugging and visualizing the generated audio signals using an oscilloscope is demonstrated. Designing and implementing finite impulse response (FIR) filters for audio processing on the FM4 is also outlined.
This document contains 7 problems related to analyzing discrete-time systems. Problem 1 involves determining the input-output relationship of a system formed by connecting two other systems in series. Problem 2 analyzes properties like linearity and time-invariance for a given system. Problems 3-4 check these properties for additional systems and statements. Problem 5 calculates the step response and response to other inputs for a given system. Problem 6 determines the steady state periodic response of a system to a periodic input using MATLAB code. Problem 7 involves finding controller parameters to place closed loop poles and determining the steady state output for a feedback system.
The document discusses various properties of systems including memory, linearity, shift-invariance, stability, causality, and how these properties relate. It defines each property and provides examples. Memory refers to whether a system's output depends only on the current input or on past inputs as well. Linearity means a system's output to two combined inputs is the combination of the outputs to the individual inputs. Shift-invariance means shifting the input shifts the output by the same amount. Stability refers to bounded inputs producing bounded outputs. Causality means the output cannot depend on future input values. The document also presents theorems about how these properties interact, such as that zero input must produce zero output for linear or homogeneous systems.
This lecture covers signal and systems analysis, including:
1) Definitions of signals, systems, and their properties like time-invariance, linearity, stability, causality, and memory.
2) Classification of signals as continuous-time vs discrete-time, analog vs digital, deterministic vs random, periodic vs aperiodic.
3) Concepts of orthogonality, correlation, autocorrelation as they relate to signal comparison.
4) Review of the Fourier series and Fourier transform as tools to represent signals in the frequency domain.
Skybuffer SAM4U tool for SAP license adoptionTatiana Kojar
Manage and optimize your license adoption and consumption with SAM4U, an SAP free customer software asset management tool.
SAM4U, an SAP complimentary software asset management tool for customers, delivers a detailed and well-structured overview of license inventory and usage with a user-friendly interface. We offer a hosted, cost-effective, and performance-optimized SAM4U setup in the Skybuffer Cloud environment. You retain ownership of the system and data, while we manage the ABAP 7.58 infrastructure, ensuring fixed Total Cost of Ownership (TCO) and exceptional services through the SAP Fiori interface.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
1. The document discusses different types of systems based on their properties, including static vs dynamic, time-variant vs time-invariant, linear vs non-linear, causal vs non-causal, and stable vs unstable.
2. A system is defined as a physical device or algorithm that performs operations on a discrete-time signal. Static systems have outputs that depend only on the present input, while dynamic systems have outputs that depend on present and past/future inputs.
3. Time-invariant systems have characteristics that do not change over time, while time-variant systems have characteristics that do change. Linear systems follow the superposition principle, while causal systems have outputs dependent only on present and past inputs.
The document summarizes several key properties of systems, including:
1) Memoryless systems have output that depends only on the present input, while causal systems have output that depends on present and past inputs.
2) Invertible systems have a one-to-one mapping between distinct inputs and outputs, allowing the input to be recovered from the output.
3) Time-invariant systems have parameters that do not vary over time, while time-varying systems have parameters dependent on time.
4) Impulse, unit step, and ramp signals are useful test signals that can reveal qualitative characteristics of a system's transient and steady-state response.
This document provides an introduction to discrete-time signals and linear time-invariant (LTI) systems. It defines discrete-time signals as sequences represented at discrete time instants. Basic discrete-time sequences including the unit sample, unit step, and periodic sequences are described. Discrete-time systems are defined as transformations that map an input sequence to an output sequence. Linear and time-invariant systems are introduced. For LTI systems, the impulse response is defined and convolution is used to represent the output as a summation of the input multiplied by delayed versions of the impulse response. Key properties of LTI systems including superposition, scaling, time-invariance, and the commutative property of convolution are covered.
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
The document summarizes key concepts about linear time-invariant (LTI) systems from Chapter 2. It discusses:
1) LTI systems can be modeled as the sum of their impulse responses weighted by the input signal. This is known as the convolution sum/integral for discrete/continuous-time systems.
2) Any signal can be represented as a linear combination of shifted unit impulses. The output of an LTI system is the convolution of the input signal with the system's impulse response.
3) The impulse response completely characterizes an LTI system. The output is found by taking the convolution integral or sum of the input signal with the impulse response.
This document provides an overview of digital signal processing systems. It defines a system as a physical device or algorithm that performs operations on a discrete time signal. A system has an input signal x(n), performs a process, and produces an output signal y(n). Systems can be classified based on their properties as static/dynamic, time-invariant/time-variant, linear/non-linear, causal/non-causal, and stable/unstable. Examples of each system type are provided. The key aspects covered are the definitions of each system property, how to determine if a given system has a particular property, and examples to illustrate the concepts.
The document summarizes key concepts about even and odd signals from a lecture on signals and systems:
1) Even signals have the property that x(-t)=x(t), while odd signals have the property that x(-t)=-x(t). Any signal can be expressed as the sum of an even and an odd component.
2) Discrete time linear and time-invariant (LTI) systems can be characterized by their impulse response h[n]. The output of an LTI system is the convolution of the input signal with the impulse response.
3) Convolution is a mathematical operation that mixes two signals together. It has applications in areas like edge detection. The output of an LTI system
2013.06.18 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on nonlinear analysis of time series as part of the Summer School on Modern Statisitical Analysis and Computational Methods hosted by the Social Sciences Compuing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
Fourier analysis of signals and systemsBabul Islam
This document discusses Fourier analysis of signals and linear time-invariant (LTI) systems. It defines LTI systems and explains that they are mathematically easy to analyze due to properties like superposition. Fourier analysis is used to represent signals in the frequency domain using techniques like the Fourier series for periodic signals and the Fourier transform for aperiodic signals. The frequency response of an LTI system is its output when the input is an impulse, and the output of any LTI system is the convolution of the input signal and impulse response.
1) A system transforms an input signal into an output signal through interconnections between components. Systems can be continuous-time or discrete-time depending on if the input and output signals are continuous or discrete.
2) Systems are linear if they satisfy superposition, meaning the response to a weighted sum of inputs equals the sum of responses. Nonlinear systems do not satisfy superposition.
3) A system is time-invariant if a time shift in the input results in the same time shift in the output. Otherwise, it is time-varying.
This document provides an overview of a lecture on time domain analysis of first order systems. The key topics covered include:
- Modeling systems using state-space representation and defining state, state vector, and state-space.
- Analyzing systems in the time domain by applying standard test inputs like impulse, step, ramp, and parabolic signals to determine the system response over time.
- The time response of first order systems consists of transient and steady-state responses. It is determined by applying standard inputs and using the system transfer function and properties like the time constant T and gain K.
1) Control theory deals with analyzing and designing closed-loop control systems to achieve desired output behaviors.
2) The document provides examples of modeling control systems using transfer functions and state-space representations. These include modeling an RF control system and passive and active filter circuits.
3) State-space representation involves expressing higher-order differential equations as a set of first-order equations and representing the system using matrix equations that can be analyzed and simulated on computers. This allows visualization and analysis of dynamic systems.
The document outlines the syllabus for a course on digital signal processing. It includes 5 units: 1) Introduction to signals and systems, 2) Discrete time system analysis using z-transforms, 3) Discrete Fourier transforms and computation including fast Fourier transforms, 4) Design of digital filters including FIR and IIR filters, and 5) Digital signal processors and their architecture. It allocates a total of 45 periods to cover these topics. Textbooks recommended for the course provide further information on digital signal processing principles, algorithms, and applications.
Controllability of Linear Dynamical SystemPurnima Pandit
The document discusses linear dynamical systems and controllability of linear systems. It defines dynamical systems as mathematical models describing the temporal evolution of a system. Linear dynamical systems are ones where the evaluation functions are linear. Controllability refers to the ability to steer a system from any initial state to any final state using input controls. The document provides the definition of controllability for linear time-variant systems using the controllability Gramian matrix. It also gives the formula for the minimum-norm control input that can steer the system between any two states. An example of checking controllability for a time-invariant linear system is presented.
The document provides an overview of discrete time signal processing concepts including:
1) Signals can be classified as continuous, discrete, deterministic, random, periodic, and non-periodic. Systems can be linear, time-invariant, causal, stable/unstable, and recursive/non-recursive.
2) Digital signal processing has advantages over analog such as precision, stability and easy implementation of operations. It also has drawbacks like needing ADCs/DACs and being limited by sampling frequency.
3) Discrete time signals are only defined at discrete time instances while continuous time signals are defined for all time. Both can be represented graphically, functionally, through tables or sequences.
This document provides an overview of chapter 2 on introduction to systems from a signals and systems textbook. It defines different types of systems, including continuous-time and discrete-time systems. It outlines the key topics that will be covered, including system properties like linearity and time-invariance. It defines linear time-invariant systems and discusses how discrete-time signals and systems can be represented using impulse functions and the convolution sum. Examples are provided to illustrate different system properties.
This document provides an overview and instructions for generating real-time digital audio signals using filtering on an ARM FM4 processor. It introduces the Cypress FM4 hardware and development tools used, including Keil uVision and LabVIEW. It then covers generating sine waves of different frequencies on the FM4 by programming an interrupt service routine with a lookup table approach. Debugging and visualizing the generated audio signals using an oscilloscope is demonstrated. Designing and implementing finite impulse response (FIR) filters for audio processing on the FM4 is also outlined.
This document contains 7 problems related to analyzing discrete-time systems. Problem 1 involves determining the input-output relationship of a system formed by connecting two other systems in series. Problem 2 analyzes properties like linearity and time-invariance for a given system. Problems 3-4 check these properties for additional systems and statements. Problem 5 calculates the step response and response to other inputs for a given system. Problem 6 determines the steady state periodic response of a system to a periodic input using MATLAB code. Problem 7 involves finding controller parameters to place closed loop poles and determining the steady state output for a feedback system.
The document discusses various properties of systems including memory, linearity, shift-invariance, stability, causality, and how these properties relate. It defines each property and provides examples. Memory refers to whether a system's output depends only on the current input or on past inputs as well. Linearity means a system's output to two combined inputs is the combination of the outputs to the individual inputs. Shift-invariance means shifting the input shifts the output by the same amount. Stability refers to bounded inputs producing bounded outputs. Causality means the output cannot depend on future input values. The document also presents theorems about how these properties interact, such as that zero input must produce zero output for linear or homogeneous systems.
This lecture covers signal and systems analysis, including:
1) Definitions of signals, systems, and their properties like time-invariance, linearity, stability, causality, and memory.
2) Classification of signals as continuous-time vs discrete-time, analog vs digital, deterministic vs random, periodic vs aperiodic.
3) Concepts of orthogonality, correlation, autocorrelation as they relate to signal comparison.
4) Review of the Fourier series and Fourier transform as tools to represent signals in the frequency domain.
Skybuffer SAM4U tool for SAP license adoptionTatiana Kojar
Manage and optimize your license adoption and consumption with SAM4U, an SAP free customer software asset management tool.
SAM4U, an SAP complimentary software asset management tool for customers, delivers a detailed and well-structured overview of license inventory and usage with a user-friendly interface. We offer a hosted, cost-effective, and performance-optimized SAM4U setup in the Skybuffer Cloud environment. You retain ownership of the system and data, while we manage the ABAP 7.58 infrastructure, ensuring fixed Total Cost of Ownership (TCO) and exceptional services through the SAP Fiori interface.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
Fueling AI with Great Data with Airbyte WebinarZilliz
This talk will focus on how to collect data from a variety of sources, leveraging this data for RAG and other GenAI use cases, and finally charting your course to productionalization.
Salesforce Integration for Bonterra Impact Management (fka Social Solutions A...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on integration of Salesforce with Bonterra Impact Management.
Interested in deploying an integration with Salesforce for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
Skybuffer AI: Advanced Conversational and Generative AI Solution on SAP Busin...Tatiana Kojar
Skybuffer AI, built on the robust SAP Business Technology Platform (SAP BTP), is the latest and most advanced version of our AI development, reaffirming our commitment to delivering top-tier AI solutions. Skybuffer AI harnesses all the innovative capabilities of the SAP BTP in the AI domain, from Conversational AI to cutting-edge Generative AI and Retrieval-Augmented Generation (RAG). It also helps SAP customers safeguard their investments into SAP Conversational AI and ensure a seamless, one-click transition to SAP Business AI.
With Skybuffer AI, various AI models can be integrated into a single communication channel such as Microsoft Teams. This integration empowers business users with insights drawn from SAP backend systems, enterprise documents, and the expansive knowledge of Generative AI. And the best part of it is that it is all managed through our intuitive no-code Action Server interface, requiring no extensive coding knowledge and making the advanced AI accessible to more users.
Trusted Execution Environment for Decentralized Process MiningLucaBarbaro3
Presentation of the paper "Trusted Execution Environment for Decentralized Process Mining" given during the CAiSE 2024 Conference in Cyprus on June 7, 2024.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
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Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away
Nunit vs XUnit vs MSTest Differences Between These Unit Testing Frameworks.pdfflufftailshop
When it comes to unit testing in the .NET ecosystem, developers have a wide range of options available. Among the most popular choices are NUnit, XUnit, and MSTest. These unit testing frameworks provide essential tools and features to help ensure the quality and reliability of code. However, understanding the differences between these frameworks is crucial for selecting the most suitable one for your projects.
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Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.
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2. 2
Lecture 1: February 20, 2007
Topic:
1. Discrete-Time Signals and Systems
• signal classification -> signals to be applied in digital filter
theory within our course,
• some elementary discrete-time signals,
• discrete-time systems: definition, basic properties review,
discrete-time system classification, input-output model of
discrete-time systems -> system to be applied in digital filter
theory within our course,
• Linear discrete-time time-invariant system description in
time-, frequency- and transform-domain.
4. 4
1.1.1. Discrete and Digital Signals
1.1.1.1. Basic Definitions
Signals may be classified into four categories depending
on the characteristics of the time-variable and values
they can take:
• continuous-time signals (analogue signals),
• discrete-time signals,
• continuous-valued signals,
• discrete-valued signals.
5. 5
Continuous-time (analogue) signals:
Time: defined for every value of time ,
Descriptions: functions of a continuous variable t: ,
Notes: they take on values in the continuous
interval .
( )
f t
( ) ( , ) ,
f t a b for a b
t R
Note: ( )
( )
( , ) ( , )
,
f t C
f t j
a b and a b
a b
6. 6
Discrete-time signals:
Time: defined only at discrete values of time: ,
Descriptions: sequences of real or complex
numbers ,
Note A.: they take on values in the continuous
interval ,
Note B.: sampling of analogue signals:
• sampling interval, period: ,
• sampling rate: number of samples per
second,
• sampling frequency (Hz): .
( ) ( )
f nT f n
T
1/
S
f T
( ) ( , ) ,
f n a b for a b
t nT
7. 7
Continuous-valued signals:
Time: they are defined for every value of time or
only at discrete values of time,
Value: they can take on all possible values on
finite or infinite range,
Descriptions: functions of a continuous variable
or sequences of numbers.
8. 8
Discrete-valued signals:
Time: they are defined for every value of time or
only at discrete values of time,
Value: they can take on values from a finite set of
possible values,
Descriptions: functions of a continuous variable or
sequences of numbers.
9. 9
Digital filter theory:
Digital signals:
Definition and descriptions: discrete-time and
discrete-valued signals (i.e. discrete -time
signals taking on values from a finite set of
possible values),
Note: sampling, quatizing and coding process i.e.
process of analogue-to-digital conversion.
Discrete-time signals:
Definition and descriptions: defined only at discrete
values of time and they can take all possible
values on finite or infinite range (sequences of
real or complex numbers: ),
Note: sampling process, constant sampling period.
( )
f n
10. 10
1.1.1.2. Discrete-Time Signal Representations
A. Functional representation:
1 1,3
( ) 6 0,7
0
for n
x n for n
elsewhere
0 0
( ) 0,6 0,1, ,102
1 102
n
for n
y n for n
n
B. Graphical
representation
( )
x n
n
11. 11
D. Sequence representation:
( ) 0.12 2.01 1.78 5.23 0.12
x n
n … -2 -1 0 1 2
x(n) … 0.12 2.01 1.78 5.23 0.12
C. Tabular representation:
12. 12
1.1.1.3. Elementary Discrete-Time Signals
A. Unit sample sequence (unit sample, unit impulse,
unit impulse signal)
1 0
( )
0 0
for n
n
for n
( )
n
n
13. 13
B. Unit step signal (unit step, Heaviside step sequence)
1 0
( )
0 0
for n
u n
for n
n
( )
u n
14. 14
C. Complex-valued exponential signal
2 .
( ) , ( ) 1, arg ( ) 2 .
j nT
S
f n
x n e x n x n nT f nT
f
where
, , 1
R n N j is imaginary unit
and
T is sampling period and is sampling frequency.
S
f
(complex sinusoidal sequence, complex phasor)
15. 15
1.1.2. Discrete-Time Systems. Definition
A discrete-time system is a device or algorithm that
operates on a discrete-time signal called the input or
excitation (e.g. x(n)), according to some rule (e.g. H[.])
to produce another discrete-time signal called the output
or response (e.g. y(n)).
( ) ( )
y n H x n
This expression denotes also the transformation H[.]
(also called operator or mapping) or processing
performed by the system on x(n) to produce y(n).
16. 16
( )
x n
input signal
excitation
( )
y n
output signal
response
( ) ( )
H
x n y n
( ) ( )
y n H x n
.
H
discrete-time
system
Input-Output Model of Discrete-Time System
(input-output relationship description)
17. 17
1.1.3. Classification of Discrete-Time
Systems
1.1.3.1. Static vs. Dynamic Systems. Definition
A discrete-time system is called static or memoryless if its output
at any time instant n depends on the input sample at the same time,
but not on the past or future samples of the input. In the other case,
the system is said to be dynamic or to have memory.
If the output of a system at time n is completly determined by the
input samples in the interval from n-N to n ( ), the system is
said to have memory of duration N.
If , the system is static or memoryless.
If , the system is said to have finite memory.
If , the system is said to have infinite memory.
0
N
0
N
0 N
N
18. 18
Examples:
The static (memoryless) systems:
The dynamic systems with finite memory:
The dynamic system with infinite memory:
3
( ) ( ) ( )
y n nx n bx n
0
( ) ( ) ( )
N
k
y n h k x n k
0
( ) ( ) ( )
k
y n h k x n k
19. 19
1.1.3.2. Time-Invariant vs. Time-Variable Systems.
Definition
A discrete-time system is called time-invariant if its input-output
characteristics do not change with time. In the other case, the
system is called time-variable.
Definition. A relaxed system is time- or shift-invariant if
only if
implies that
for every input signal and every time shift k .
[.]
H
( ) ( )
H
x n y n
( ) ( )
H
x n k y n k
( )
x n
( ) ( )
y n H x n
( ) ( )
y n k H x n k
20. 20
Examples:
The time-invariant systems:
The time-variable systems:
3
( ) ( ) ( )
y n x n bx n
0
( ) ( ) ( )
N
k
y n h k x n k
3
( ) ( ) ( 1)
y n nx n bx n
0
( ) ( ) ( )
N
N n
k
y n h k x n k
21. 21
1.1.3.3. Linear vs. Non-linear Systems. Definition
A discrete-time system is called linear if only if it satisfies the linear
superposition principle. In the other case, the system is called non-
linear.
Definition. A relaxed system is linear if only if
for any arbitrary input sequences and , and any
arbitrary constants and .
The multiplicative (scaling) property of a linear system:
The additivity property of a linear system:
[.]
H
1 1 2 2 1 1 2 2
( ) ( ) ( ) ( )
H a x n a x n a H x n a H x n
1( )
x n 2 ( )
x n
1
a 2
a
1 1 1 1
( ) ( )
H a x n a H x n
1 2 1 2
( ) ( ) ( ) ( )
H x n x n H x n H x n
22. 22
Examples:
The linear systems:
The non-linear systems:
0
( ) ( ) ( )
N
k
y n h k x n k
2
( ) ( ) ( )
y n x n bx n k
3
( ) ( ) ( 1)
y n nx n bx n
0
( ) ( ) ( ) ( 1)
N
k
y n h k x n k x n k
23. 23
1.1.3.4. Causal vs. Non-causal Systems. Definition
Definition. A system is said to be causal if the output of the system
at any time n (i.e., y(n)) depends only on present and past inputs
(i.e., x(n), x(n-1), x(n-2), … ). In mathematical terms, the output of a
causal system satisfies an equation of the form
where is some arbitrary function. If a system does not satisfy
this definition, it is called non-causal.
( ) ( ), ( 1), ( 2),
y n F x n x n x n
[.]
F
24. 24
Examples:
The causal system:
The non-causal system:
0
( ) ( ) ( )
N
k
y n h k x n k
2
( ) ( ) ( )
y n x n bx n k
3
( ) ( 1) ( 1)
y n nx n bx n
10
10
( ) ( ) ( )
k
y n h k x n k
25. 25
1.1.3.5. Stable vs. Unstable of Systems. Definitions
An arbitrary relaxed system is said to be bounded input - bounded
output (BIBO) stable if and only if every bounded input produces
the bounded output. It means, that there exist some finite numbers
say and , such that
for all n. If for some bounded input sequence x(n) , the output y(n)
is unbounded (infinite), the system is classified as unstable.
x
M y
M
( ) ( )
x y
x n M y n M
26. 26
Examples:
The stable systems:
The unstable system:
0
( ) ( ) ( )
N
k
y n h k x n k
2
( ) ( ) 3 ( )
y n x n x n k
3
( ) 3 ( 1)
n
y n x n
27. 27
1.1.3.6. Recursive vs. Non-recursive Systems.
Definitions
A system whose output y(n) at time n depends on any number of the
past outputs values ( e.g. y(n-1), y(n-2), … ), is called a recursive
system. Then, the output of a causal recursive system can be
expressed in general as
where F[.] is some arbitrary function. In contrast, if y(n) at time n
depends only on the present and past inputs
then such a system is called nonrecursive.
( ) ( 1), ( 2), , ( ), ( ), ( 1), , ( )
y n F y n y n y n N x n x n x n M
( ) ( ), ( 1), , ( )
y n F x n x n x n M
28. 28
Examples:
The nonrecursive system:
The recursive system:
0
( ) ( ) ( )
N
k
y n h k x n k
0 1
( ) ( ) ( ) ( ) ( )
N N
k k
y n b k x n k a k y n k
30. 30
.
H
LTI system
unit impulse
( )
n
( ) ( )
h n H n
impulse response
LTI system description by convolution (convolution sum):
Viewed mathematically, the convolution operation satisfies the
commutative law.
( ) ( ) ( ) ( ) ( ) ( )* ( ) ( )* ( )
k k
y n h k x n k x k h n k h n x n x n h n
1.2.1.1 Impulse Response and Convolution
31. 31
1.2.1.2. Step Response
.
H
unit step
( )
u n
step response
unit-step
response
( ) ( )
g n H u n
( ) ( ) ( ) ( )
n
k k
g n h k u n k h k
These expressions relate the impulse response to the step response
of the system.
LTI system
32. 32
1.2.2. Impulse Response Property and
Classification of LTI Systems
1.2.2.1. Causal LTI Systems
A relaxed LTI system is causal if and only if its impulse response is
zero for negative values of n , i.e.
Then, the two equivalent forms of the convolution formula can be
obtained for the causal LTI system:
( ) 0 0
h n for n
0
( ) ( ) ( ) ( ) ( )
n
k k
y n h k x n k x k h n k
33. 33
1.2.2.2. Stable LTI Systems
A LTI system is stable if its impulse response is absolutely
summable, i.e.
2
( )
k
h k
34. 34
1.2.2.3. Finite Impulse Response (FIR) LTI Systems
and Infinite Impulse Response (IIR) LTI
Systems
Causal FIR LTI systems:
IIR LTI systems:
0
( ) ( ) ( )
N
k
y n h k x n k
0
( ) ( ) ( )
k
y n h k x n k
35. 35
1.2.2.4. Recursive and Nonrecursive LTI Systems
Causal nonrecursive LTI:
Causal recursive LTI:
LTI systems:
characterized by constant-coefficient difference equations
0
( ) ( ) ( )
N
k
y n h k x n k
0 1
( ) ( ) ( ) ( ) ( )
N M
k k
y n b k x n k a k y n k
36. 36
1.3. Frequency-Domain Representation of
Discrete Signals and LTI Systems
LTI system
( )
h n
complex-valued
exponencial
signal
( ) j n
x n e
impulse response
( ) ( ) ( )
k
y n h k x n k
LTI system output
( )
y n
37. 37
LTI system output:
( )
( ) ( ) ( ) ( )
( ) ( )
j n k
k k
j k j n j n j k
k k
y n h k x n k h k e
h k e e e h k e
( ) ( )
j n j
y n e H e
Frequency response: ( ) ( )
j j k
k
H e h k e
38. 38
( )
( ) ( )
j j j
H e H e e
( ) Re ( ) Im ( )
j j j
H e H e j H e
( ) ( )cos ( )sin
j
k k
H e h k k j h k k
Re ( ) ( )cos
j
k
H e h k k
Im ( ) ( )sin
j
k
H e h k k
39. 39
Magnitude response:
2 2
( ) Re ( ) Im ( )
j j j
H e H e H e
Im ( )
( ) arg ( )
Re ( )
j
j
j
H e
H e arctg
H e
Phase response:
( )
( )
d
d
Group delay function:
40. 40
1.3.1. Comments on relationship between the impulse
response and frequency response
The important property of the frequency response
is fact that this function is periodic with period .
2
2 2
( ) ( ) ( ) ( )
j l j l
j j k
k k
H e h k e h k e H e
( )
j
H e
( )
j
H e
1
( ) ( )
2
j j n
h n H e e d
In fact, we may view the previous expression as the exponential
Fourier series expansion for , with h(k) as the Fourier series
coefficients. Consequently, the unit impulse response h(k) is related
to through the integral expression
41. 41
1.3.2. Comments on symmetry properties
For LTI systems with real-valued impulse response, the magnitude
response, phase responses, the real component of and the imaginary
component of possess these symmetry properties:
The real component: even function of periodic with period
The imaginary component: odd function of periodic with period
( )
j
H e
2
2
Re ( ) Re ( )
j j
H e H e
Im ( ) Im ( )
j j
H e H e
42. 42
The magnitude response: even function of periodic with period
The phase response: odd function of periodic with period
2
( ) ( )
j j
H e H e
2
arg ( ) arg ( )
j j
H e H e
Consequence:
If we known and for , we can describe
these functions ( i.e. also ) for all values of .
( )
j
H e
( )
0
( )
j
H e
44. 44
1.3.3. Comments on Fourier Transform of Discrete Signals
and Frequency-Domain Description of LTI Systems
LTI system
( )
h n
impulse response
( )
j
H e
frequency response
input signal
( ), ( )
j
x n X e
output signal
( ), ( )
j
y n Y e
45. 45
The input signal x(n) and the spectrum of x(n):
( ) ( )
j j k
k
X e x k e
1
( ) ( )
2
j j n
x n X e e d
( ) ( )
j j k
k
Y e y k e
1
( ) ( )
2
j j n
y n Y e e d
The output signal y(n) and the spectrum of y(n):
( ) ( )
j j k
k
H e h k e
1
( ) ( )
2
j j n
h n H e e d
The impulse response h(n) and the spectrum of h(n):
Frequency-domain description of LTI system:
( ) ( ) ( )
j j j
Y e H e X e
46. 46
1.3.4. Comments on Normalized Frequency
It is often desirable to express the frequency response of an LTI
system in terms of units of frequency that involve sampling
interval T. In this case, the expressions:
( ) ( )
j j k
k
H e h k e
1
( ) ( )
2
j j n
h n H e e d
are modified to the form:
( ) ( )
j T j kT
k
H e h kT e
/
/
( ) ( )
2
T
j T j nT
T
T
h nT H e e d
47. 47
is periodic with period , where is
sampling frequency.
Solution: normalized frequency approach:
( )
j T
H e
2 / 2
T F
F
/2
F
/2 50
F kHz
50kHz
3
1 3
20 10 2
0.4
50 10 5
x
x
3
2 3
25 10
0.5
50 10 2
x
x
100
F kHz
1 20
f kHz
2 25
f kHz
Example:
49. 49
1.4.1. Z -Transform
Since Z – transform is an infinite power series, it exists
only for those values of z for which this series converges.
The region of convergence of X(z) is the set of all values
of z for which X(z) attains a finite value.
Definition: The Z – transform of a discrete-time signal x(n)
is defined as the power series:
( ) ( ) k
k
X z x n z
( ) [ ( )]
X z Z x n
where z is a complex variable. The above given relations
are sometimes called the direct Z - transform because
they transform the time-domain signal x(n) into its
complex-plane representation X(z).
50. 50
The procedure for transforming from z – domain to the
time-domain is called the inverse Z – transform. It can
be shown that the inverse Z – transform is given by
1
1
( ) ( )
2
n
C
x n X z z dz
j
1
( ) ( )
x n Z X z
where C denotes the closed contour in the region of
convergence of X(z) that encircles the origin.
51. 51
1.4.2. Transfer Function
0 1
( ) ( ) ( ) ( ) ( )
N M
k k
y n b k x n k a k y n k
Application of the Z-transform to this equation under
zero initial conditions leads to the notion of a transfer
function.
The LTI system can be described by means of a constant
coefficient linear difference equation as follows
52. 52
LTI System
( ) ( )
Y z Z y n
input signal
( )
x n
( ) ( )
X z Z x n
Transfer function: the ratio of the Z - transform of the
output signal and the Z - transform of the input signal of
the LTI system:
( ) [ ( )]
( )
( ) [ ( )]
Y z Z y n
H z
X z Z x n
output signal
( )
y n
( )
H z
( ) ( )
H z Z h n
( )
h n
53. 53
LTI system: the Z-transform of the constant coefficient
linear difference equation under zero initial
conditions:
0 1
( ) ( ) ( ) ( ) ( )
N M
k k
k k
Y z b k z X z a k z Y z
The transfer function of the LTI system:
0
1
( )
( )
( )
( )
1 ( )
N
k
k
M
k
k
b k z
Y z
H z
X z
a k z
0 1
( ) ( ) ( ) ( ) ( )
N M
k k
y n b k x n k a k y n k
H(z): may be viewed as a rational function of a complex
variable z (z-1).
54. 54
1.4.3. Poles, Zeros, Pole-Zero Plot
Let us assume that H(z) has been expressed in its
irreducible or so-called factorized form:
0
0 1
0
1 1
( )
( )
( )
1 ( ) ( )
N
N
k
k
N M
k k
M M
k
k
k k
z z
b k z
b
H z z
a
a k z z p
Pole-zero plot: the plot of the zeros and the poles of H(z)
in the z-plane represents a strong tool for LTI system
description.
Zeros of H(z): the set {zk} of z-plane for which H(zk)=0
Poles of H(z): the set {pk} of z -plane for which ( )
k
H p
55. 55
Example: the 4-th order Butterworth low-pass filter,
cut off frequency .
1 3
0
1
( )
( )
1 ( )
N
k
k
M
k
k
b k z
H z
a k z
z1= -1.0002, z2= -1.0000 + 0.0002j
z3= -1.0000 - 0.0002j, z4= -0.9998
0
1
( )
( )
1 ( )
N
k
k
M
k
k
b k z
H z
a k z
b =[ 0.0186 0.0743 0.1114 0.0743 0.0186 ]
a =[ 1.0000 -1.5704 1.2756 -0.4844 0.0762 ]
p1= 0.4488 + 0.5707j, p2= 0.4488 - 0.5707j
p3= 0.3364 + 0.1772j, p4= 0.3364 - 0.1772j
60. 60
1.4.4. Transfer Function and Stability of LTI Systems
Condition: LTI system is BIBO stable if and only if the
unit circle falls within the region of convergence of the
power series expansion for its transfer function. In the
case when the transfer function characterizes a causal LTI
system, the stability condition is equivalent to the
requirement that the transfer function H(z) has all of its
poles inside the unit circle.
61. 61
Example 1: stable system
Example 2: unstable system
2
1 2
1 0.16
( )
1 1.1 1.21
z
H z
z z
1 1 1
2 2 2
0.4 0.5500 0.9526 1.1 1
0.4 0.5500 0.9526 1.1 1
z p j p
z p j p
1 2
1 2
1 0.9 0.18
( )
1 0.8 0.64
z z
H z
z z
1 1 1
2 2 2
0.3 0.4000 0.6928 0.8 1
0.6 0.4000 0.6928 0.8 1
z p j p
z p j p
62. 62
Z – Domain:
0
1
( )
( )
1 ( )
N
k
k
M
k
k
b k z
H z
a k z
transfer function
Frequency – Domain:
0
1
( )
( )
1 ( )
N
j k
j k
M
j k
k
b k e
H e
a k e
frequency response
Time – Domain:
0 1
( ) ( ) ( ) ( ) ( )
N M
k k
y n b k x n k a k y n k
constant coefficient linear difference equation
1.4.5. LTI System Description. Summary
j j
z e e z
h(n)
Z
Z-1 FT-1
FT
63. 63
( )
H z
Z – Domain: transfer function
jw
j
z e
H z
H e
1
1
)
( ) (
2
n
C
H z
h n z dz
j
( )
h k
Time – Domain: impulse response
( ) ( )
j j k
k
H e e
h k
)
( (
) k
k
H z z
h k
Frequency – Domain: frequency response
( ) j
j
e z
e
H z H
(
2
)
) (
1 j k
j
H d
e
h k e
j
H e