This document discusses topics related to discrete-time signals and systems from the textbook Discrete-Time Signal Processing by Alan V. Oppenheim and Ronald W. Schafer. It covers sampling of continuous-time signals to obtain discrete-time signals, basic discrete-time sequences and operations, discrete-time systems including linear and time-invariant systems, and examples such as the ideal delay system and moving average filter. Frequency characteristics of discrete-time signals such as periodicity are also examined.
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.
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.
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.
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 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 discusses discrete-time systems and impulse responses. It begins with examples of different types of discrete-time systems, including averaging, differencing, and cosine matching. It then defines the impulse response as the output of a system when given a unit impulse as input. The remainder of the document provides examples of finding the impulse response for various systems, such as averaging, forward differencing, integration, and delay. Determining the impulse response involves inserting a unit impulse into the system equation.
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.
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.
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.
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.
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 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 discusses discrete-time systems and impulse responses. It begins with examples of different types of discrete-time systems, including averaging, differencing, and cosine matching. It then defines the impulse response as the output of a system when given a unit impulse as input. The remainder of the document provides examples of finding the impulse response for various systems, such as averaging, forward differencing, integration, and delay. Determining the impulse response involves inserting a unit impulse into the system equation.
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.
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.
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.
This document provides an overview of an advanced digital signal processing lecture. It discusses pre-requisites for the course including basic signals and communications knowledge and MATLAB proficiency. It outlines the course structure, including chapters covered, textbook references, and assessment breakdown. Key concepts from the first lecture are summarized such as characterizing signals as continuous or discrete, common signal representations including exponentials and sinusoids, and introducing linear time-invariant systems.
EC8553 Discrete time signal processing ssuser2797e4
This document contains a 10 question, multiple choice exam on discrete time signal processing. It covers topics like the discrete Fourier transform (DFT), finite word length effects, fixed point vs floating point representation, and FIR filter design. Specifically, it includes questions that calculate the 4 point DFT of a sequence, define twiddle factors, compare DIT and DIF FFT algorithms, and discuss stability and causality of systems.
Convolution discrete and continuous time-difference equaion and system proper...Vinod Sharma
This document discusses discrete time signals and discrete time convolution. Discrete time convolution describes the output of a linear, time-invariant system when its input is one discrete signal and its impulse response is another. The output is the sum of each input sample multiplied by the corresponding flipped and shifted impulse response. This allows characterization of a system by its impulse response. The document provides examples of discrete time convolution and discusses properties such as the number of samples in the convolved output depending on the lengths of the input and impulse response signals.
This document provides an introduction to discrete time signals and systems. It defines discrete time signals as functions of integer time and discusses various representations including graphical, functional, tabular, and sequence representations. It then describes common elementary discrete time signals like impulse, step, ramp, and exponential signals. The document also classifies signals as energy/power signals, periodic/aperiodic, even/odd, and discusses basic manipulations. Finally, it defines discrete time systems as those that transform an input signal to an output signal according to a rule and classifies systems as static/dynamic and with finite/infinite memory.
LTI System, Basic Types of Digital signals, Basic Operations, Causality, Stab...Waqas Afzal
This document discusses digital signal processing concepts including:
1) Basic types of digital signals such as unit impulse sequences and unit step sequences.
2) Linear time-invariant (LTI) systems and their properties including causality and stability. LTI systems can be represented using impulse responses.
3) Convolution, which is used to analyze LTI systems in the time domain by representing the output as a sum of delayed and scaled impulse responses of the input.
This document provides an overview of discrete-time signals and systems in signal processing. It begins with an introduction to discrete-time signals as sequences and sampling of continuous-time signals. Common basic discrete-time sequences like impulses, steps, and exponentials are described. Properties of linear, time-invariant discrete-time systems are introduced. The frequency domain representation of discrete-time signals using complex exponentials is also covered. Examples of periodic and non-periodic discrete-time sequences are provided to illustrate key concepts.
Digital Signal Processing (DSP) from basics introduction to medium level book based on Anna University Syllabus! This is just a share of worthfull book!
-Prabhaharan Ellaiyan
-prabhaharan429@gmail.com
-www.insmartworld.blogspot.in
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.
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 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.
This document provides an introduction to digital signal processing. It discusses how signals can be represented digitally by sampling analog signals and converting them to sequences of numbers. This allows signals to be processed using digital processors. Some key benefits of digital signal processing include accuracy, repeatability, flexibility, and easy implementation of nonlinear and time-varying operations in software. The document covers topics such as sampling, analog-to-digital conversion, reconstruction, discrete-time signals and systems, linearity, time-invariance, and examples of basic sequences like sinusoidal, exponential, and geometric sequences.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet 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
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.
This document outlines the syllabus and course objectives for the digital signal processing course ECE2006 being offered in the fall semester of 2021. The course aims to teach students concepts related to signals and systems in the time and frequency domains, design of analog and digital filters, and realization of digital filters using various structures. The syllabus is divided into 7 modules covering topics such as Fourier analysis, design of IIR and FIR filters, and realization of lattice filters. Students will be evaluated through continuous assessments, quizzes, assignments, and a final exam.
The document provides an overview of digital signal processing (DSP). It defines key concepts such as discrete-time signals and systems, linear time-invariant (LTI) systems, and impulse and step responses. DSP is concerned with representing signals as sequences and processing these sequences. Discrete-time signals are defined by values at discrete time instances. LTI systems are characterized by their impulse responses, and the input-output relationship is defined by convolution. Properties such as stability and causality are also discussed. The document outlines the course and provides references for further reading.
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms (DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of complex multiplication, for the signals of length = 2^, > 2.
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the
paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of
these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the
noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the
splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is
reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms
(DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are
applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of
complex multiplication, for the signals of length = 2^, > 2. .
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
The document presents a new method for signal denoising using a paired transform. The key points are:
1) The method splits a signal into multiple splitting-signals by applying a paired transform that represents the signal in both frequency and time.
2) Linear filtering for denoising can then be applied to each splitting-signal independently, reducing computational complexity compared to filtering the full signal.
3) Two models are proposed - one uses shifted discrete Fourier transforms on each splitting-signal, while the other uses non-shifted transforms, allowing different filter designs while achieving the same results.
This document discusses trends in integrated circuit applications and technologies. It focuses on artificial intelligence, internet of things, and system-on-chip designs. The document presents diagrams of typical system-on-chip architectures that integrate processors, memory, and other components. It also discusses challenges related to the growing gap between computing power and data storage, and approaches to address this like near-memory computing and computing in memory. The overall trends are driving the development of more complex, specialized integrated circuits.
This document is a two-part chapter about vector correlation and covariance from a textbook. It provides supplementary materials on the topics written by Chih-Wei Tang, who has a civil engineering degree from National Central University in Taiwan. The chapter references a probability and stochastic processes textbook for electrical and computer engineers as further reading.
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This document discusses digital signal processing concepts including:
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Digital Signal Processing (DSP) from basics introduction to medium level book based on Anna University Syllabus! This is just a share of worthfull book!
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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
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.
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This document outlines the syllabus and course objectives for the digital signal processing course ECE2006 being offered in the fall semester of 2021. The course aims to teach students concepts related to signals and systems in the time and frequency domains, design of analog and digital filters, and realization of digital filters using various structures. The syllabus is divided into 7 modules covering topics such as Fourier analysis, design of IIR and FIR filters, and realization of lattice filters. Students will be evaluated through continuous assessments, quizzes, assignments, and a final exam.
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noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the
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NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
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3) Two models are proposed - one uses shifted discrete Fourier transforms on each splitting-signal, while the other uses non-shifted transforms, allowing different filter designs while achieving the same results.
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The document discusses the z-transform, which is used to analyze discrete-time signals and sequences. It defines the z-transform and compares it to the Fourier transform. The z-transform allows analysis of signals that are not absolutely summable, unlike the Fourier transform. Examples are provided to demonstrate how to calculate the z-transform of different types of sequences, including right-sided, left-sided, two-sided, and finite-length sequences. The region of convergence is also discussed, which determines the values of z for which the z-transform converges.
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Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
VARIABLE FREQUENCY DRIVE. VFDs are widely used in industrial applications for...PIMR BHOPAL
Variable frequency drive .A Variable Frequency Drive (VFD) is an electronic device used to control the speed and torque of an electric motor by varying the frequency and voltage of its power supply. VFDs are widely used in industrial applications for motor control, providing significant energy savings and precise motor operation.
Discover the latest insights on Data Driven Maintenance with our comprehensive webinar presentation. Learn about traditional maintenance challenges, the right approach to utilizing data, and the benefits of adopting a Data Driven Maintenance strategy. Explore real-world examples, industry best practices, and innovative solutions like FMECA and the D3M model. This presentation, led by expert Jules Oudmans, is essential for asset owners looking to optimize their maintenance processes and leverage digital technologies for improved efficiency and performance. Download now to stay ahead in the evolving maintenance landscape.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
1. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
•Sampling:
2. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Application: Radar Detection
Application: Radar Detection
Application: Radar Detection
Application: Radar Detection
3. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Application: Sonar Detection
Application: Sonar Detection
Application: Sonar Detection
Application: Sonar Detection
6. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Application: Transmission of Speech Signals
Application: Transmission of Speech Signals
7. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Xa(t): Analog signal
X[n]: discrete signal
Cos(ωn)= cos(Ωt)|t=nT
ΩT
ω=ΩT
ω: frequency of discrete signal
Ω: frequency of analog signal
q y g g
T: sampling interval 1/T=f = sampling frequency
8. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Figure 2.2 (a) Segment of a continuous-time speech signal xa(t ). (b) Sequence of samples x[n] = xa(nT ) obtained
g ( ) g p g a( ) ( ) q p [ ] a( )
from the signal in part (a) with T = 125 µs.
9. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.1.1 Basic Sequences and Sequence Operations
Delayed Sequence: y[n] = x[n-n0],
where n0 is a integer representing the delay
⎧ 0
0
Unit sample sequence (Dirac delta function):
⎩
⎨
⎧
=
≠
=
δ
0
n
,
1
0
n
,
0
]
n
[
Expression of a sequence using delta function:
Expression of a sequence using delta function:
Expression of a sequence using delta function:
Expression of a sequence using delta function:
∑
∞
δ ]
k
[
]
k
[
]
[ ∑
−∞
=
−
δ
=
k
]
k
n
[
]
k
[
x
]
n
[
x
10. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Unit Step sequence:
The relation between unit function and delta function
The relation between unit function and delta function
]
[
]
1
[
]
[
]
[
]
1
[
]
1
[
]
[
∞
−
+
+
−
+
=
+
−
+
+
+
−∞
+
−∞
=
n
δ
n
δ
n
δ
n
δ
n
δ
δ
δ
K
K
∑
∑ −∞
=
∞
−∞
=
δ
=
−
⋅
δ
=
⎩
⎨
⎧
<
≥
=
n
k
k
]
k
[
]
k
n
[
u
]
k
[
0
n
,
0
0
n
,
1
]
n
[
u
equal)
are
equations
two
these
n,
number
finite
any
(for
If n-k ≥ 0, u[n-k]=1, then u[n-k]=1 exists when n≥k
Besides ]
1
n
[
u
]
n
[
u
]
n
[ −
−
=
δ
Besides, ]
1
n
[
u
]
n
[
u
]
n
[ =
δ
Impulse sequence:
Exponential and sinusoidal sequence (General form):
11. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Sinusoidal sequences: )
n
cos(
A
]
n
[
x 0 φ
+
ω
= , for all n
⎧ ≥
α 0
n
A n
Recall the exponential sequence
⎩
⎨
⎧
<
≥
α
=
0
n
,
0
0
n
,
A
]
n
[
x
if 0
j
e
|
| ω
α
=
α And
φ
j
e
A
A |
|
1 = then
( ) ]
[
2
1
]
[
]
[
2
1 0
0
2
1
n
ω
j
φ
j
n
ω
j
φ
j
e
Ae
e
Ae
n
x
n
x −
−
+
=
+
=
, if e
|
| α
α And e
A
A |
|
1 , then
|
|
|
|
]
[
)
(
1
1
0
e
α
e
A
α
A
n
x n
ω
j
n
φ
j
n
=
=
)
sin(
|
|
|
|
)
cos(
|
|
|
|
|
|
|
|
0
0
)
( 0
φ
n
ω
α
A
j
φ
n
ω
α
A
e
α
A
n
n
φ
n
ω
j
n
+
⋅
+
+
⋅
=
⋅
= +
Especially,| α| =1
)
i (
|
|
)
(
|
|
]
[ A
j
A
Frequency phase
)
sin(
|
|
)
cos(
|
|
]
[ 0
0
1 φ
n
ω
A
j
φ
n
ω
A
n
x +
+
+
=
Complex exponential sequence
)
sin(
)
cos(
]
[ 0
0
1
2
2
0
φ
n
ω
A
j
φ
n
ω
A
e
α
e
A
α
A
n
x
α
n
ω
j
n
φ
j
n
+
−
+
=
=
=
=
−
−
−
12. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
In discrete signals, time index n is an integer which results in many important
differences from continuous-time sequences
differences from continuous time sequences.
Frequency periodicity:
n
j
n
2
j
n
j
n
)
2
(
j 0
0
0
Ae
e
Ae
Ae
]
n
[
x ω
π
ω
π
+
ω
=
=
=
]
n
cos[
A
]
n
)
r
2
cos[(
A
]
n
[
x φ
+
ω
=
φ
+
π
+
ω
= ]
n
cos[
A
]
n
)
r
2
cos[(
A
]
n
[
x 0
0 φ
+
ω
=
φ
+
π
+
ω
=
Time periodicity:
Time periodicity holds only when the following relation exits,
p y y g ,
]
N
n
[
x
]
n
[
x +
=
For sinusoid signals, Acos(ω0n+φ)=Acos(ω0n+ω0N+φ)
ω0N=2πk
13. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
ω=ΩT
Ω↑ then ω↑
Example 2.1
In continuous signals, signals with higher frequency usually
have shorter repetition period but this doesn’t hold in
have shorter repetition period, but this doesn t hold in
discrete signals.
),
4
/
cos(
]
[
1 n
n
x π
= N=8
),
8
/
3
cos(
]
[
2 n
n
x π
= N=16
14. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.2 Discrete
2.2 Discrete-
-Time Systems
Time Systems
x[n] y[n]
{ }
⋅
T
]}
n
[
x
{
T
]
n
[
y =
Example 2 2 The ideal delay system
Example 2.2 The ideal delay system
y[n] = x[n-nd],
where n is a fixed positive integer called the delay of the system
where nd is a fixed positive integer called the delay of the system.
15. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.3 Moving Average
The general moving average system is defined by the equation
{ }
1
]
[
1
1
]
[
2
1
2
1
k
n
x
M
M
n
y
M
M
k
=
−
+
+
= ∑
−
=
{ }
]
[
]
[
]
1
[
]
[
1
1
2
1
1
2
1
M
n
x
n
x
M
n
x
M
n
x
M
M
−
+
+
+
+
−
+
+
+
+
+
= L
L
Figure 2.7 Sequence values involved in computing a moving average with M1 = 0 and M2 = 5.
16. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.2.1
2.2.1 Memoryless
Memoryless Systems
Systems
A system is referred to as memoryless if the output y[n] at every value of n
depends only on the input x[n] at the same value of n
Example 2.4
y[n] = (x[n])2 for all value of n
depends only on the input x[n] at the same value of n.
y[n] = (x[n]) , for all value of n
2.2.2 Linear Systems
2.2.2 Linear Systems
The class of linear systems is defined by the principle of superposition. If y1[n]
Additivity property
and y2[n] are the responses of a system when x1[n] and x2[n] are the respective
inputs, then the system is linear if and only if
]}
[
{
]}
[
{
]}
[
]
[
{ n
x
T
n
x
T
n
x
n
x
T +
+
Additivity property
Scaling or Homogeneity
property
]}
[
{
]}
[
{
]}
[
]
[
{ 2
1
2
1 n
x
T
n
x
T
n
x
n
x
T +
=
+
]
[
]}
[
{
]}
[
{ n
ay
n
x
aT
n
ax
T =
=
property
where a is an arbitrary constant. The first property is the additivity property, and
the second is the homogeneity or scaling property. These two properties together
comprise the principle of superposition stated as
comprise the principle of superposition, stated as
]}
[
{
]}
[
{
]}
[
]
[
{ 2
1
2
1 n
x
bT
n
x
aT
n
bx
n
ax
T +
=
+
17. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
For a linear system with arbitrary constants a and b, the
expression of generalized to the superposition of many inputs.
g y
]
n
[
x
a
]
n
[
x k
k
k
∑
=
Output of a linear system will be
General form:
]
n
[
y
b
]
n
[
y k
k
k
∑
=
p y
}
]
n
[
x
a
{
T
]
n
[
y
b
m k
k
k
m
m
∑ ∑
=
E l 2 5 Th l t t
n
Example 2.5 The accumulator system
∑
−∞
=
=
k
]
k
[
x
]
n
[
y Is linear?
Let x3 [n] = ax1[n]+bx2[n] and check the output is y3[n] = ay1[n]+by2[n] ?
])
k
[
bx
]
k
[
ax
(
]
k
[
x
]
n
[
y
n n
n
k
n
k
2
1
3
3 +
=
= ∑ ∑
−∞
= −∞
=
]
n
[
by
]
n
[
ay
]
k
[
x
b
]
k
[
x
a 2
1
k k
2
1 +
=
+
= ∑ ∑
−∞
= −∞
= Linear
18. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.6 A Nonlinear System
Consider the system defined by
|).
]
[
(|
log
]
[ 10 n
x
n
w =
This system is not linear. For x1[n]=1 and x2[n] = 10, then
.
1
)
10
(
log
)
1
(
log
)
11
(
log
)
10
1
(
log 10
10
10
10 =
+
≠
=
+
Also, we have x2[n] = 10 .x1[n], but
1
)
10
1
(
l
]
[
0
)
1
(
log
]
[ 10
1 ≠
=
=
n
w
.
1
)
10
1
(
log
]
[ 10
2 =
⋅
=
n
w
19. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.2.3 Time
2.2.3 Time-
-Invariant Systems
Invariant Systems
A time-invariant system (equivalently often referred to as a shift-invariant system)
is one for with a time shift or delay of the input sequence causes a corresponding
is one for with a time shift or delay of the input sequence causes a corresponding
shift in the output sequence. Specifically, suppose that a system transforms the
input sequence with values x[n] into the output sequence with values y[n]. The
system is said to be time-invariant if for all n0 the input sequence with values x1[n]
system is said to be time invariant if for all n0 the input sequence with values x1[n]
= x[n-n0] produces the output sequence with values y1[n] = y[n-n0].
Example 2.7 The Accumulator as a Time –Invariant System
Consider the accumulator from Example 2.5. We define x1[n]= x[n-n0]. To
show time invariance, we solve for both y[n-n0] and y1[n] and compare them
to see whether they are equal. Therefore, setting a system T{.}, we have
y q g y { }
.
]
[
]
[
]}
[
{
]
[
]}
[
{
]
[ 0
1
1
1 ∑
∑ −∞
=
−∞
=
−
=
=
=
=
n
k
n
k
n
k
x
k
x
n
x
T
n
y
and
n
x
T
n
y
,
]
[
]
[ 0
0
∑
−
−∞
=
+
=
=
−
n
0
n
n
k
then
n
k
k'
setting
and
k
x
n
n
y
g
Considerin
.
]
'
[
]
[
'
0
0 ∑
−∞
=
−
=
−
n
k
n
k
x
n
n
y
20. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.8 The Compressor System
Th t d fi d b th l ti
The system defined by the relation
with M a positive integer, is called a compressor. Specifically, it discards (M-1)
samples out of M; i e it creates the output sequence by selecting every Mth
.
n
-
Mn
x
n
y ∞
<
<
∞
= ],
[
]
[
samples out of M; i.e., it creates the output sequence by selecting every Mth
sample. The system is not time-invariant, the output of the system when the
input is x1[n] must be equal to y[n-n0]. The output y1[n] that results from the
input x1[n] can be directly computed to be
input x1[n] can be directly computed to be
)]
(
[
]
[
]
[
].
[
]
[
0
0
0
1
n
n
M
x
n
n
y
n
y
condition,
delay
output
to
Compared
n
Mn
x
n
y
have
we
condition,
delay
input
For
2 −
=
−
=
−
=
Time-invariant is not compressible.
].
[n
y1
≠
21. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.2.4 Causality
2.2.4 Causality
A system is causal if for every choice of n the output sequence value at the
A system is causal if, for every choice of n0, the output sequence value at the
index n = n0 depends only on the input sequence values for n ≤ n0.
For example, Example 2.4 is causal if –M1 ≥ 0 and M2 ≥ 0.
Example 2.9 The Forward and Backward Difference Systems
The forward difference system is defined by the relation
y y
y[n] = x[n+1] – x[n].
Obviously y[n] depends on x[n+1]; therefore, the system is noncausal.
However, the backward difference system, defined by
y[n] = x[n] – x[n-1]
is causal.
22. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.2.5 Stability
2.2.5 Stability
A system is stable in the bounded-input bounded-output (BIBO) sense if and
only if every bounded input sequence produces a bounded output sequence The
only if every bounded input sequence produces a bounded output sequence. The
input x[n] is bounded if there exits a fixed positive finite value Bx such that
|x[n]| ≦Bx ≦ ∞ for all n.
Stability requires that for every bounded input there exists a fixed positive finite
Stability requires that for every bounded input there exists a fixed positive finite
value By such that
|y[n]| ≦By ≦ ∞ for all n.
Example 2.10 Testing for Stability or Instability
(i) Ex 2.4. For |x[n]| ≦Bx , then |y[n]| = |x[n]|2≦ Bx2
(ii) E 2 6 F [ ] 0 [ ] l (| [ ]|)
(ii) Ex 2.6. For x[n] = 0, y[n] = log10(|x[n]|) = - ∞
(iii) Ex. 2.5. For x[n] = u[n] with Bx =1,
∑ ⎨
⎧ <
n
0
n
k
,
0
]
[
]
[
Ans: (i) stable
∑
−∞
=
⎩
⎨
≥
+
=
=
k
0
n
n
k
u
n
y
,
1
]
[
]
[
Ans: (i) stable
(ii) not stable
(iii) not stable
23. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.3 Linear Time
2.3 Linear Time-
-Invariant systems
Invariant systems
A particularly important class of systems consists of those that are linear and
time invariant. This class of systems has significant signal-processing
applications.
Let hk[n] be the response of the system to δ[n-k], an impulse occurring at
e k[ ] be e espo se o e sys e o δ[ ], a pu se occu g a
n=k.
∑
∞
−∞
=
−
δ
=
k
]}
k
n
[
]
k
[
x
{
T
]
n
[
y
Time invariant
∑ ∑
∑
∞
−∞
=
∞
−∞
=
∞
−∞
=
=
−
=
−
δ
=
k k
k
k
]
n
[
h
]
k
[
x
]
k
n
[
h
]
k
[
x
]}
k
n
[
{
T
]
k
[
x
h[n] with k delayed input
Linearity
h[n] with k delayed input
(T{.} is LTI system)
Define:
Convolution sum:
Convolution sum: y[n] = x[n]
y[n] = x[n] ∗
∗ h[n]
h[n]
24. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
y[n] = x[n]
y[n] = x[n] ∗
∗ h[n]
h[n]
Sum
Sum
∑
∞
−∞
=
−
⋅
=
k
]
k
n
[
h
]
k
[
x
]
n
[
y
volution
volution
of
Conv
of
Conv
example
example
An
e
An
e
Figure 2.8 Representation of the output of an LTI
system as the superposition of responses to
system as the superposition of responses to
individual samples of the input.
25. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
It is worthy to note that the term h[n-k] in convolution sum also can be
represented as h[-(k-n)]. Therefore, ∞
∞
( )
∑
∑
∞
∞
−
∞
−∞
=
−
−
⋅
=
−
⋅
= )]
n
k
(
[
h
]
k
[
x
]
k
n
[
h
]
k
[
x
]
n
[
y
k
h[k]
h[k]
h[
h[-
-k] = h[0
k] = h[0-
-k]
k]
h[
h[ k] h[0
k] h[0 k]
k]
h[n
h[n-
-k] = h[
k] = h[-
-(k
(k-
-n)]
n)]
Figure 2.9 Forming the sequence h[n − k]. (a) The sequence h[k] as a function of k. (b) The sequence h[−k] as a
function of k. (c) The sequence h[n − k] = h[ − (k − n)] as a function of k for n = 4.
26. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.11 Analytical Evaluation of Convolution sum
Example 2.11 Analytical Evaluation of Convolution sum
Consider a system with impulse response
y p p
⎩
⎨
⎧ −
≤
≤
=
−
−
=
otherwise
,
0
1
N
n
0
,
1
]
N
n
[
u
]
n
[
u
]
n
[
h
⎩
0 N-1
The input is ]
n
[
u
a
]
n
[
x n
=
∞
∑
∞
∞
−
−
−
⋅
= )]
n
k
(
[
h
]
k
[
x
]
n
[
y
27. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Ans:
Figure 2.10 Sequence involved in computing a discrete convolution.
(a)–(c) The sequences x[k] and h[n− k] as a function of k for different
values of n. (Only nonzero samples are shown.) (d) Corresponding
output sequence as a function of n.
28. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
29. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
2.4 Properties of Linear Time
2.4 Properties of Linear Time-
-Invariant Systems
Invariant Systems
1 Commutative:
With m = n-k
1. Commutative:
2. Distributive:
30. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Bounded Input is not guaranteed for Bounded Output
(The requirement for a stable system)
∞
(The requirement for a stable system)
Linear time-invariant systems are stable if and only if the impulse
response is absolutely summable, i.e., if
∑
∞
−∞
=
∞
<
=
k
|
]
k
[
h
|
S Sufficient
Sufficient condition for stability
condition for stability
∑
∑
∞
−∞
=
∞
−∞
=
−
≤
−
=
k
k
|
]
k
n
[
x
||
]
k
[
h
|
]
k
n
[
x
]
k
[
h
|
]
n
[
y
|
B
|
]
n
[
x
| ≤
If x[n] is bounded so that x
B
|
]
n
[
x
| ≤
If x[n] is bounded, so that
∑
∞
≤ x |
]
k
[
h
|
B
|
]
n
[
y
|
Consider:
Th [ ] i l l
−∞
=
k
⎪
⎨
⎧
≠
−
=
0
]
n
[
h
,
|
]
n
[
h
|
]
n
[
h
]
n
[
x
*
The sequence x[n] is clearly
bounded to unity. However,
the value of the output at
n=0 is:
⎪
⎩
⎨
=
−
=
0
]
n
[
h
,
0
|
]
n
[
h
|
]
n
[
x
∑ ∑
∞ ∞ 2
S
|
]
k
[
h
|
]
k
[
h
]
k
[
]
0
[
n=0 is:
Bounded input produce unbounded output
Bounded input produce unbounded output
∑ ∑
−∞
= −∞
=
=
=
−
=
k k
S
|
]
k
[
h
|
|
]
[
|
]
k
[
h
]
k
[
x
]
0
[
y
32. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
The expression of equivalent LTI systems (example 1)
noncausal
noncausal
causal
causal
Figure 2.12 (a) Cascade combination of two
LTI systems. (b) Equivalent cascade. (c)
]
1
[
])
[
]
1
[
(
]
[
h δ
δ
δ
y ( ) q ( )
Single equivalent system.
])
n
[
]
1
n
[
(
]
1
n
[
]
1
n
[
])
n
[
]
1
n
[
(
]
n
[
h
δ
−
+
δ
∗
−
δ
=
−
δ
∗
δ
−
+
δ
=
]
1
n
[
]
n
[ −
δ
−
δ
=
33. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
The expression of equivalent LTI systems (example 2)
Backward difference is an inverse system of accumulator.
Backward difference is an inverse system of accumulator.
How about convolute the accumulator with forward difference system?
How about convolute the accumulator with forward difference system?
34. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.5 Linear constant
2.5 Linear constant-
-coefficient Differenece Equations
coefficient Differenece Equations
Nth order linear constant coefficient difference equation:
Nth-order linear constant-coefficient difference equation:
∑ ∑
= =
−
=
−
N
0
k
M
0
m
m
k ]
m
n
[
x
b
]
k
n
[
y
a
∑ ∑
∞
−
⋅
=
=
= 1
n
1 ],
k
n
[
h
]
k
[
x
]
n
[
h
*
]
n
[
x
]
k
[
x
]
n
[
y
Example 2.12
Example 2.12
Consider:
0
k 0
m
∑
∑ ∑
−∞
=
∞
− ∞
−
δ
=
n
k
1
1
1
]
k
[
]
n
[
h
where
],
k
n
[
h
]
k
[
x
]
n
[
h
]
n
[
x
]
k
[
x
]
n
[
y
Consider:
∞
=
k
Input y[n] into a inverse system
Y[n] Inverse system ?
]
1
n
[
]
n
[
]
n
[
h2 −
δ
−
δ
=
Inverse system:
]
n
[
h
*
]
n
[
y
]
n
[
x ]
n
[
h
*
]
n
[
y
]
n
[
x 2
=
35. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
∑
=
n
k
]
k
[
x
]
n
[
y
Y[n]
X[n]
−∞
=
k
∑
−
=
−
1
n
]
k
[
x
]
1
n
[
y One-sample
Y[n]
[ ]
∑
−∞
=
k
]
[
]
[
y
∑
−1
n
One-sample
delay
Y[n-1]
∑
−∞
=
+
=
k
]
k
[
x
]
n
[
x
]
n
[
y
Y[n 1]
]
1
n
[
y
]
n
[
x
]
n
[
y −
+
= ]
n
[
x
]
1
n
[
y
]
n
[
y =
−
−
36. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.13 Difference equation representation of the moving
Example 2.13 Difference equation representation of the moving-
-average system
average system
])
1
M
n
[
u
]
n
[
u
(
)
1
M
(
1
]
n
[
h 2
2
−
−
−
+
=
Consider causal moving-average
system with M1=0,
∑ −
=
=
2
M
]
k
n
[
x
)
1
M
(
1
]
n
[
h
*
]
n
[
x
]
n
[
y ∑
=
+ 0
k
2 )
1
M
(
]}
n
[
u
*
])
1
M
n
[
]
n
[
{(
)
1
M
(
1
]
n
[
h 2
2
−
−
δ
−
δ
+
=
]
n
[
u
*
])
1
M
n
[
x
]
n
[
x
(
)
1
M
(
1
]
n
[
h
*
]
n
[
x
]
n
[
y 2
2
−
−
−
+
=
=
38. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.6 Frequency Domain Representation of Discrete
2.6 Frequency Domain Representation of Discrete-
-Time Signal and
Time Signal and Systems
Systems
2.6.1 Eigen functions for Linear Time
2.6.1 Eigen functions for Linear Time-
-Invariant Systems
Invariant Systems
Input sequence x[n] = ejwn is a set of eigen-function to represent the
frequency response of h[n]
e
]
k
[
h
]
k
n
[
x
]
k
[
h
]
n
[
x
*
]
n
[
h
]
n
[
y )
k
n
(
j
∑
∑
∞
−
ω
∞
=
=
=
}
e
]
k
[
h
{
e
e
]
k
[
h
]
k
n
[
x
]
k
[
h
]
n
[
x
*
]
n
[
h
]
n
[
y
k
j
n
j
k
)
(
j
k
∑
∑
∑
∞
ω
−
ω
−∞
=
−∞
=
=
=
−
⋅
=
=
}
e
]
k
[
h
{
e
k
∑
−∞
=
=
∑
∞
ω
−
ω
= k
j
j
e
]
k
[
h
)
e
(
H
If we define , then
n
j
j
e
)
e
(
H
]
n
[
y ω
ω
=
∑
−∞
=
k
]
[
)
(
If we define , t e )
(
]
[
y
Eigenfunction
j
j
j )
e
(
H
j
j
j j
|
)
(
H
|
)
(
H
ω
∠
ω
ω
Polar
Polar
form
form
)
e
(
jH
)
e
(
H
)
e
(
H j
I
j
R
j ω
ω
ω
+
=
)
e
(
H
j
j
j j
e
|
)
e
(
H
|
)
e
(
H ∠
ω
ω
=
form
form
39. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.14 Frequency response of the ideal delay system
Example 2.14 Frequency response of the ideal delay system
Consider a ideal delay system defined by y[n]= x[n-nd] , where nd is a fixed
integer. If we consider
x[n]=ejωn as input to this system, then we have
y[n] = ejω(n-nd) = e-jωnd ejωn.
The frequency response of the ideal delay is therefore
H(ejω) = e-jωnd
An alternative way to obtain the frequency response is to compute the H(ejω)
i F i t f
using Fourier transform
∑
∞
∞
−
ω
−
ω
−
δ
= n
j
d
j
e
]
n
n
[
)
e
(
H
From the Euler relation the real and imaginary parts are
From the Euler relation, the real and imaginary parts are
)
n
cos(
)
e
(
H d
j
R ω
=
ω Polar form
Polar form
j
|
)
e
(
H
| =
ω
1
)
n
sin(
)
e
(
H d
j
I ω
=
ω
d
j
n
)
e
(
H
|
)
(
|
ω
−
=
∠ ω
40. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
*
*For any input signal, x[n], if the input signal can be represented as
n
j
k
k
n
e
]
n
[
x ω
∑α
=
Then from the principle of superposition the corresponding output of a
Then, from the principle of superposition, the corresponding output of a
linear time-invariant system is
n
j
j
k
k
k
e
)
e
(
H
]
n
[
y ω
ω
∑α
=
k
k )
(
]
[
y ∑
Thus if we can find a representation of x[n] as a superposition of ocmplex
Thus, if we can find a representation of x[n] as a superposition of ocmplex
exponential sequences, then we can find the output as aforementioned
equation if we know the frequency response of the system.
41. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.15 Sinusoidal response of LTI system
Example 2.15 Sinusoidal response of LTI system
Since it is simple to express a sinusoid as a linear combination of complex
n
j
j
n
j
j
e
e
A
e
e
A
)
n
cos(
A
]
n
[
x 0
0
2
2
0
ω
−
φ
−
ω
φ
+
=
φ
+
ω
=
exponentials, let us consider a sinusoidal input
X1[n] X2[n]
The responses to x1[n] and x2[n] are y1[n] and y2[n].
A n
j
j
n
j
e
e
A
)
e
(
H
]
n
[
y 0
0
2
1 ω
φ
ω
=
n
j
j
n
j
e
e
A
)
e
(
H
]
n
[
y 0
0
2
2 ω
−
φ
−
ω
−
= )
(
]
[
y
2
]
e
e
)
e
(
H
e
e
)
e
(
H
[
A
]
n
[
y
]
n
[
y
]
n
[
y n
j
j
n
j
n
j
j
n
j 0
0
0
0
2
2
1 ω
−
φ
−
ω
−
ω
φ
ω
+
=
+
=
If h[ ] i l ill h l t th t H( j 0) H*( j 0) hi h i th t
If h[n] is real, we will show later that H(ejω0)=H*(e-jω0) which gives that
)
n
cos(
|
)
e
(
H
|
A
]
n
[
y n
j
θ
+
φ
+
ω
= ω
0
0
, where )
e
(
H j 0
ω
∠
=
θ
42. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
An important characteristic of discrete-time linear time-invariant
systems is of its periodicity of the variable ω with period 2π.
Consider
∑
∑
∞
−∞
=
ω
−
∞
−∞
=
π
+
ω
−
π
+
ω
=
=
n
n
j
n
n
)
2
(
j
n
)
2
(
j
e
]
n
[
h
e
]
n
[
h
)
e
(
H
n
j
n
2
j
n
j
n
)
2
(
j
e
e
e
e ω
−
π
−
ω
−
π
+
ω
−
=
=
)
(
)
( j
n
)
2
(
j ω
π
+
ω
)
e
(
H
)
e
(
H j
n
)
2
(
j ω
π
+
ω
=
Th f bt i H(
Th f bt i H( j
jω
ω) i i di ith i d
) i i di ith i d 2
2
Therefore, we obtain H(e
Therefore, we obtain H(ej
jω
ω) is periodic with period
) is periodic with period 2
2π
π.
.
43. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Ideal Highpass filter
Ideal Highpass filter
Ideal lowpass filter
Ideal lowpass filter
Ideal Highpass filter
Ideal Highpass filter
Ideal lowpass filter
Ideal lowpass filter
Ideal bandpass filter
Ideal bandpass filter
Figure. 2. 17
Figure. 2. 17
Figure. 2. 18
Figure. 2. 18
44. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.16 Frequency response of the moving
Example 2.16 Frequency response of the moving-
-averaging system
averaging system
⎧ ≤
≤ M
n
M
1 2
M
1
⎩
⎨
⎧ ≤
≤
−
= +
+
otherwise
,
0
M
n
M
,
]
n
[
h 2
1
1
M
M
1
2
1
∑
−
=
ω
−
ω
+
+
=
2
1
M
M
n
n
j
2
1
j
e
1
M
M
1
)
e
(
H
Frequency response
Frequency response
q y p
q y p
)
1
( 2
1
1 M
j
M
j
e
e +
− ω
ω )
(
2
1
2
1
1
1
1
)
( j
j
j
j
e
e
e
M
M
e
H −
−
−
+
+
= ω
ω
2
/
)
(
2
/
2
/
2
/
)
1
(
2
/
)
1
(
2
1
1
2
2
1
2
1
1
1 M
M
j
j
j
M
M
j
M
M
j
e
e
e
e
e
M
M
−
−
−
+
+
−
+
+
−
−
+
+
= ω
ω
ω
ω
ω
2
/
)
(
2
1
2
1
1
2
)
2
/
i (
]
2
/
)
1
(
sin[
1
1
1
M
M
j
e
M
M
M
M
e
e
M
M
−
−
+
+
=
+
+
ω
ω
2
1 )
2
/
sin(
1
M
M +
+ ω
45. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2
/
)
M
M
(
j
2
1
j 1
2
e
)
2
/
sin(
]
2
/
)
1
M
M
(
sin[
1
M
M
1
)
e
(
H −
ω
−
ω
ω
+
+
ω
+
+
=
2
1 )
2
/
sin(
1
M
M ω
+
+
Figure 2.19 (a) Magnitude and (b) phase of the frequency response
Figure 2.19 (a) Magnitude and (b) phase of the frequency response
of the moving
of the moving-
-average system for the case M1=0 and M2=4.
average system for the case M1=0 and M2=4.
46. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.6.2 Suddenly Applied Complex Exponential Inputs
2.6.2 Suddenly Applied Complex Exponential Inputs
In Sec 2 6 1 we have seen that complex inputs of the form e jωn for -∞<n<∞
In Sec. 2.6.1, we have seen that complex inputs of the form e jωn for ∞<n<∞
produces outputs of the form H(ejω)ejωn for causal LTI systems. If we change the
complex sinusoidal inputs as x[n] = ejωn u[n], we can have
⎧
⎪
⎪
⎪
⎪
⎨
⎧
≥
⎟
⎟
⎟
⎞
⎜
⎜
⎜
⎛
<
=
−
⋅
=
= ω
ω
−
∞
∞
=
∑
∑ 0
n
for
,
e
e
k
h
0
n
for
,
k
n
x
k
h
n
h
n
x
n
y n
j
n
k
j
k
]
[
0
]
[
]
[
]
[
*
]
[
]
[
⎪
⎩
⎟
⎠
⎜
⎝ =
−∞
=
∑
k
k
0
( ) [ ] [ ] [ ]
eq. 2.126 j k j n j k j n
y n h k e e h k e e
ω ω ω ω
∞ ∞
− −
⎛ ⎞ ⎛ ⎞
= −
⎜ ⎟ ⎜ ⎟
∑ ∑
For n≧0, it becomes
( ) [ ] [ ] [ ]
0 1
eq. 2.126
k k n
y n h k e e h k e e
= = +
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∑ ∑
( ) ( ) [ ]
eq. 2.127 j j n j k j n
H e e h k e e
ω ω ω ω
∞
−
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
∑
( ) ( ) [ ]
1
k n
= +
⎜ ⎟
⎝ ⎠
∑
]
[n
y
response
state
-
Steady ss ]
[n
y
response
Transient t
No transient response for n>M-1, if h[n] has finite
length (i.e, FIR filter) with M points.
47. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
h[n] has finite length
h[ ] h i fi it l th
Figure 2.20 Illustration of a real part of suddenly applied complex exponential input with (a) FIR and (b) IIR.
h[n] has infinite length
48. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.7 Representation of Sequences by Fourier Transforms
2.7 Representation of Sequences by Fourier Transforms
Fourier transform pair (Discrete nonperiodic signal):
Fourier transform pair (Discrete nonperiodic signal):
Fourier transform pair (Discrete nonperiodic signal):
Fourier transform pair (Discrete nonperiodic signal):
ω
π
= ω
π
π
−
ω
∫ d
e
)
e
(
X
2
1
]
n
[
x n
j
j
∑
∞
−∞
=
ω
−
ω
=
n
n
j
j
e
]
n
[
x
)
e
(
X
Discrete Fourier Transform (DTFT)
Discrete Fourier Transform (DTFT)
−∞
=
n
)
e
(
jH
)
e
(
H
)
e
(
H j
I
j
R
j ω
ω
ω
+
= I
R
)
e
(
H
j
j
j j
e
|
)
e
(
H
|
)
e
(
H
ω
∠
ω
ω
=
49. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.7 Representation of Sequences by Fourier Transforms
2.7 Representation of Sequences by Fourier Transforms
Fourier representation pair of discrete
Fourier representation pair of discrete-
-time signals.
time signals.
p p
p p g
g
ω
π
= ω
π
π
−
ω
∫ d
e
)
e
(
X
]
n
[
x n
j
j
2
1
∑
∞
ω
−
ω n
j
j
]
[
)
(
X ∑
∞
−
ω
−
ω
= n
j
j
e
]
n
[
x
)
e
(
X
j
The Fourier transform X(ejω) can be presented in
)
e
(
X
)
e
(
X
)
e
(
X j
I
j
R
j ω
ω
ω
+
=
jω
Rectangular form:
Rectangular form:
M it d S t
Phase Spectrum
)
e
(
X
j
j
j j
e
|
)
e
(
X
|
)
e
(
X
ω
∠
ω
ω
=
Polar form:
Polar form:
Magnitude Spectrum
p
50. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
∫−
n
j
j
d
e
e
X )
(
2
1
ω
π
π
π
ω
ω
∑ ∫
∫ ∑
∞
−∞
=
−
−
−
∞
−∞
=
−
⎞
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
=
m
m
n
j
n
j
m
m
j
d
e
m
x
d
e
e
m
x
2
1
]
[
]
]
[
[
2
1 )
(
ω
π
ω
π
π
π
ω
π
π
ω
ω
∑
∞
−∞
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
m m
n
m
n
m
x
)
(
)
(
sin
]
[
π
π
]
m
n
[
n
m
n
m
,
−
δ
=
⎩
⎨
⎧
≠
=
=
0
1
n
m
,
⎩ ≠
0
∑
∞
−
δ
= ]
m
n
[
]
m
[
x
]
n
[
x̂ ∑
−∞
=
m
Sufficient condition for
Sufficient condition for ]
n
[
x
]
n
[
x̂ = ∞
<
ω
|
)
e
(
X
| j
∑
∑
∞
−∞
=
ω
−
∞
−∞
=
ω
−
ω
≤
=
n
n
j
n
n
j
j
|
e
||
]
n
[
x
|
|
e
]
n
[
x
|
|
)
e
(
X
|
∑
∞
−∞
=
∞
≤
≤
n
|
]
n
[
x
|
52. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Any finite-length sequence
is absolutely summable and
y
thus will have a Fourier
transform representation.
53. Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example
Example 2.17 Absolute
2.17 Absolute Summability
Summability for
for A
A S
Sudden
udden-
-Applied Exponential
Applied Exponential
<
<
=
=
=
∑ ω
∞
ω
−
ω
1
|
a
|
or
ae
|
for
e
a
X(e
is
sequence
this
of
transform
Fourier
The
n
u
a
x[n]
Consider
j
-
n
j
n
j
n
1
|
1
)
].
[
<
<
<
−
=
= ∑ ω
−
=
i.e.,
x[n];
of
ty
summabili
absolute
the
for
condition
the
is
1
|
a
|
conidtion
the
Clearly,
1
|
a
|
or
ae
|
for
ae
e
a
X(e
j
n
1
|
1
)
0
∞
<
−
=
= ∑
∞
=
ω
a
a
X(e
n
n
j
|
|
1
1
|
|
)
0
54. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
⎧ ≤
|
|
1
Example 2.18 Square
Example 2.18 Square-
-summability
summability for the ideal
for the ideal lowpass
lowpass filter
filter
⎩
⎨
⎧
π
≤
ω
≤
ω
ω
≤
ω
=
ω
|
|
,
0
|
|
,
1
)
e
(
H
c
c
j
lp
π
=
ω
π
= ω
ω
−
ω
ω
ω
−
ω
∫ ]
e
[
jn
2
1
d
e
2
1
]
n
[
h n
j
n
j
lp
c
c
c
c
Ideal lowpass requires infinite
Ideal lowpass requires infinite
points
points
∞
<
<
−∞
π
ω
=
−
π
= ω
−
ω
n
,
n
n
sin
)
e
e
(
jn
2
1 c
n
j
n
j c
c
In real cases, the filter lengths can not be infinite
In real cases, the filter lengths can not be infinite
θ
θ
−
ω
θ
−
ω
+
π
=
π
ω
= ∫
∑
ω
ω
ω
−
−
=
ω
d
2
/
)]
sin[(
]
2
/
)
)(
1
M
2
sin[(
2
1
e
n
n
sin
)
e
(
H
c
c
n
j
M
M
n
c
j
M
)]
[(
56. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.19 Fourier Transform of a constant
Example 2.19 Fourier Transform of a constant
Consider x[n] =1 for all n. This sequence is neither absolutely summable nor
bl Th F i t f f th [ ] i th i di
square summable. The Fourier transform of the sequence x[n] is the periodic
impulse train
∑
∞
ω
π
+
ω
πδ
=
j
)
r
(
)
e
(
X 2
2
∑
−∞
=
r
)
(
)
(
Consider the inverse Fourier transform,
1
∫ ∑
∞
π
1
)
2
(
2
2
1
]
[ =
⋅
+
= ∫ ∑
−
−∞
=
ω
π
ω
πδ
π
ω
π
π
d
e
r
n
x n
j
r
Example 2.20 Fourier Transform of complex Exponential Sequences
Example 2.20 Fourier Transform of complex Exponential Sequences
∑
∞
−∞
=
ω
π
+
ω
−
ω
πδ
=
r
j
)
r
(
)
e
(
X 2
2 0
1 π ∞
∫
n
j
r
d
e
r
n
x 0
1
)
2
(
2
2
1
]
[ ω
π
π
ω
π
ω
ω
πδ
π
⋅
+
−
=
∞
−
−∞
=
∫ ∑
n
j
n
j
n
j
r
e
e
d
e
r 0
0
0 )
(
0 )
2
(
2
2
1 ω
ω
ω
ω
π
π
ω
π
ω
ω
πδ
π
=
⋅
⋅
+
−
= −
−
∞
−∞
=
∫ ∑
58. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Discrete-Time Fourier Transform of Unit Step Function
1
1/2
1/2
= +
1/2
-1/2
1
1 1/2
}
)
2
(
{
2
1
,
)
2
(
2
}
1
{
2
1
]
[
2
1
]
[
1
r
π
ω
πδ
F
r
π
ω
πδ
F
n
Sqn
n
u
+
=
∴
+
=
+
=
∞
−
∞
∑
∑
Q
2
1
2
1
]}
[
{
,
0
,
1
0
,
1
]
[
2
0
1
e
α
e
α
n
Sqn
F
n
n
n
Sqn
n
n
ω
j
n
n
n
ω
j
n
r
r
+
−
=
∴
⎩
⎨
⎧
<
−
≥
=
∞
=
−
−
−∞
=
−
−
−∞
=
−∞
=
∑
∑
∑
∑
1
1
1
1
1
1
1
1
2
1
)
(
1
1
1
2
1
2
1
)
(
1
2
1
1
0
0
'
1
'
'
e
α
e
α
e
α
e
α
ω
j
ω
j
ω
j
n
n
ω
j
n
n
n
ω
j
n
⎞
⎛
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
+
⎟
⎠
⎞
⎜
⎝
⎛
−
=
−
−
−
−
∞
=
−
∞
=
−
−
∑
∑
1
1
1
1
1
2
1
2
1
1
1
2
1
1
2
1
α
e
α
α
e
α
e
α
α
e
e
ω
j
ω
j
ω
j
ω
j
ω
j
+
=
−
+
−
−
=
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
= −
−
−
−
)
(
1
1
2
1
1
1
2
1
)}
(
{
,
1
1
2
2
ω
πδ
e
e
t
u
F
then
α
for
e
α
e
α
ω
j
ω
j
ω
j
ω
j
+
−
+
−
=
⇒
=
−
+
−
=
−
−
−
−
59. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.19 Fourier Transform of
Example 2.19 Fourier Transform of Complex Exponential Sequences
Complex Exponential Sequences
Consider a sequence x[n] whose Fourier transform is the periodic impulse train
∑
∞
ω
π
+
ω
−
ω
πδ
=
j
r
e
X )
2
(
2
)
( 0
−∞
=
r
Applying inverse Fourier transform, we have
( ) n
j
n
j
j
d
e
d
e
e
X
n
x
π
π
−
ω
π
π
−
ω
ω
ω
⋅
ω
−
ω
πδ
π
=
ω
⋅
π
=
∫
∫ 0
2
2
1
)
(
2
1
]
[
n
j 0
e ω
=
60. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.8 Symmetry Properties of The Fourier Transform
2.8 Symmetry Properties of The Fourier Transform
Definition
Definition
Definition
Definition
Conjugate
Conjugate-
-symmetric sequence:
symmetric sequence:
C j t
C j t ti t i
ti t i
]
n
[
x
])
n
[
*
x
]
n
[
x
(
]
n
[
x *
e
e −
=
−
+
=
2
1
]
[
])
[
*
]
[
(
]
[ *
1
]
n
[
x
]
n
[
x
]
n
[
x o
e +
=
Conjugate
Conjugate-
-antisymmetric sequence:
antisymmetric sequence: ]
n
[
x
])
n
[
*
x
]
n
[
x
(
]
n
[
x o
o −
−
=
−
−
=
2
Summation
Summation
Odd
Odd
Even sequence
Even sequence Odd sequence
Odd sequence
Definition
Definition
Conj gate
Conj gate s mmetric seq ence
s mmetric seq ence )
e
(
X
))
e
(
*
X
)
e
(
X
(
)
e
(
X j
*
j
j
j ω
−
ω
−
ω
ω
=
+
=
1
Conjugate
Conjugate-
-symmetric sequence:
symmetric sequence:
Conjugate
Conjugate-
-antisymmetric sequence:
antisymmetric sequence:
)
e
(
X
))
e
(
*
X
)
e
(
X
(
)
e
(
X e
e =
+
=
2
)
e
(
X
))
e
(
*
X
)
e
(
X
(
)
e
(
X j
*
o
j
j
j
o
ω
−
ω
−
ω
ω
−
=
−
=
2
1
Summation
Summation )
e
(
X
)
e
(
X
)
e
(
X j
o
j
e
j ω
ω
ω
+
=
Even spectrum
Even spectrum Odd spectrum
Odd spectrum
61. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
62. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example 2.21 Illustration of Symmetry Properties
Example 2.21 Illustration of Symmetry Properties
63. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.9 Fourier Transform Theorems
2.9 Fourier Transform Theorems
]}
n
[
x
{
F
)
e
(
X jω
=
)}
e
(
X
{
F
]
n
[
x
]}
n
[
x
{
F
)
e
(
X
F
jω
−
= 1
2.9.1 Linearity of the Fourier Transform
2.9.1 Linearity of the Fourier Transform
)
e
(
X
]
n
[
x j
F
ω
↔
)
e
(
X
]
n
[
x
F
j
F
ω
↔ 1
1
)
e
(
X
]
n
[
x j
F
ω
↔ 2
2
)
(
bX
)
(
X
]
[
b
]
[ j
j
F
ω
ω
)
e
(
bX
)
e
(
aX
]
n
[
bx
]
n
[
ax j
j ω
ω
+
↔
+ 2
1
2
1
64. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.9.2 Time Shifting and Frequency Shifting
2.9.2 Time Shifting and Frequency Shifting
F
)
e
(
X
]
n
[
x j
F
ω
↔
Time shifting
Time shifting )
e
(
X
e
]
n
n
[
x j
n
j
F
d ω
ω
−
⋅
↔
−
Time shifting
Time shifting )
e
(
X
e
]
n
n
[
x d ↔
Frequency shifting
Frequency shifting )
e
(
X
]
n
[
x
e )
(
j
F
n
j 0
0 ω
−
ω
ω
↔
2.9.3 Time Reversal
2.9.3 Time Reversal
j
F
ω
)
e
(
X
]
n
[
x jω
↔
X[n] is time reversed
X[n] is time reversed )
e
(
X
]
n
[
x j
F
ω
−
↔
− )
(
]
[
X[n] is real and time reversed
X[n] is real and time reversed )
e
(
X
]
n
[
x j
*
F
ω
↔
−
Conjugate-symmetric
65. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.9.4 Differentiation in Frequency
2.9.4 Differentiation in Frequency
F
)
e
(
X
]
n
[
x j
F
ω
↔
ω
)
(
dX j
F
ω
↔
ω
d
)
e
(
dX
j
]
n
[
nx
j
F
then
2.9.5 Parseval’s Theorem
2.9.5 Parseval’s Theorem
)
e
(
X
]
n
[
x j
F
ω
↔ )
e
(
X
]
n
[
x ↔
∫
∑
π
ω
∞
ω
=
= d
|
)
e
(
X
|
|
]
n
[
x
|
E j 2
2 1
then
∫
∑ π
−
−∞
= π
|
)
(
|
|
]
[
|
n 2
66. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.9.6 The Convolution Theorem
2.9.6 The Convolution Theorem
F
)
e
(
X
]
n
[
x j
F
ω
↔
)
(
H
]
[
h j
F
ω
↔ )
e
(
H
]
n
[
h jω
↔
∑
∞
=
−
= ]
n
[
h
*
]
n
[
x
]
k
n
[
h
]
k
[
x
]
n
[
y ∑
−∞
=
n
]
[
]
[
]
[
]
[
]
[
y
)
e
(
H
)
e
(
X
)
e
(
Y j
j
j ω
ω
ω
=
Then
Then
∑
∑
∑
∞
ω
−
∞
∞
ω
−
ω n
j
n
j
j
e
]
k
n
[
h
]
k
[
x
e
]
n
[
y
)
e
(
Y
)
e
(
H
)
e
(
X
)
e
(
Y j
j
j
=
Then
Then
∑ ∑
∑
∑
∑
∞ ∞
−∞
=
−∞
=
−∞
=
−
=
=
j
j
j
k
j
k
j
n
n
j
j
e
]
k
n
[
h
]
k
[
x
e
]
n
[
y
)
e
(
Y
∑ ∑
−∞
=
ω
ω
ω
−
−∞
=
ω
−
=
⋅
=
m
j
j
m
j
k
k
j
)
e
(
H
)
e
(
X
e
)
e
]
k
[
x
](
m
[
h
67. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
)
(
X
]
[ j
F
ω
↔
Consider ideal delay system
Consider ideal delay system
)
e
(
X
]
n
[
x jω
↔
d
n
j
F
d e
]
n
n
[ ω
−
↔
−
δ d ]
[
d
n
j
j
j
F
d e
)
e
(
X
)
e
(
Y
]
n
n
[
*
]
n
[
x
]
n
[
y ω
−
ω
ω
=
↔
−
δ
=
68. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
2.9.7 The Modulation or Windowing Theorem
2.9.7 The Modulation or Windowing Theorem
)
(
X
]
[ j
F ω
)
e
(
W
]
n
[
W
)
e
(
X
]
n
[
x
j
F
j
F
ω
ω
⎯→
←
⎯→
←
then ]
n
[
w
]
n
[
x
]
n
[
y =
∫
π
θ
θ )
(
j
j
1
∫
π
π
−
θ
−
ω
θ
θ
π
d
)
e
(
W
)
e
(
X )
(
j
j
2
1
C id n
j
]
[
]
[
]}
[
{
F ω
−
∞
∑
Consider n
j
e
]
n
[
w
]
n
[
x
]}
n
[
y
{
F ω
∞
−
∑
=
When ω =0,
0
0 =
ω
ω
−
∞
∞
−
=
ω = ∑ |
e
]
n
[
x
]
n
[
x
|
]}
n
[
y
{
F n
j
*
0
2
1
=
ω
ω
−
θ
π
π
−
θ
θ
π
= ∫ |
d
)
e
(
X
)
e
(
X )
(
j
*
j
Parseval’s Theorem
69. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
]
[
]
[
]
[
)
1
( n
n
n
n
u
a
n
u
na
n
u
a
n
Check +
=
+
1
1
1
1
1
1
ω
j
ω
j
ω
j
DTFT
ae
ae
ae
ω
d
d
j
−
−
−
−
+
⎥
⎦
⎤
⎢
⎣
⎡
−
⎯
⎯ →
⎯
( ) ( )2
2
1
1
1
1
1 ω
j
ω
j
ω
j
ae
ae
ae
ae
−
−
−
−
=
−
+
−
=
71. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example
Example 2.22
2.22
ω
−
ω
−
=
= ,
1
1
)
]
[
n
j
j
1
n
1
e
(e
X
transform
Fourier
its
with
n
u
a
[n]
x
For
−
= ].
5
[
]
[ n
A
n
u
a
n
x
of
transform
Fourier
the
find
please
( ) ω
−
ω
⋅
⎯→
⎯
− 0]
[
:
0
n
j
j
F
e
e
X
n
n
x
Answer
Q
{ } −
ω
−
−
=
⋅
=
∴
=
−
=
−
=
5
5
2
2
1
5
]}
[
{
1
]
[
].
[
]
5
[
]
5
[
]
[
j
j
n
5
-
n
x
a
F
e
n
x
F
n
x
n
x
n
u
a
n
x
a
and
{ }
{ } { } ω
−
ω
−
⋅
=
=
⇒
−
5
5
2
5
2
]
[
]
[
]}
[
{
1
]
[
j
j
j
e
a
n
x
F
a
n
x
F
a
{ } { } ω
−
−
2
1
]
[
]
[
j
a
72. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example
Example 2.23
2.23
f
T f
F i
i
th
fi d
Pl
( ) )
1
)(
1
(
1
be
ae
e
X
of
Transform
Fourier
inverse
the
find
Please
j
j
j
−
−
=
ω
−
ω
−
ω
:
)
1
)(
1
(
Answer
be
ae
( ) 1
)
/(
1
)
/(
)
1
)(
1
(
1
be
b
a
b
ae
b
a
a
be
ae
e
X
j
j
j
j
j
⎞
⎛
⎞
⎛
−
−
−
−
−
=
−
−
=
ω
−
ω
−
ω
−
ω
−
ω
]
[
]
[
]
[
1
n
u
b
b
a
b
n
u
a
b
a
a
n
x n
n
F
⎟
⎠
⎞
⎜
⎝
⎛
−
−
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⎯
⎯ →
⎯
−
73. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Chapter 2 Discrete
Chapter 2 Discrete-
-Time Signal and Systems
Time Signal and Systems
Example
Example 2.24
2.24
is
phase
linear
with
filter
highpass
a
of
response
frequency
The
0
|
|
)
(
c
c
n
j
j
|
|
,
,
e
e
H
p
f
g p
f
p
f q y
d
⎪
⎩
⎪
⎨
⎧
ω
<
ω
π
<
ω
<
ω
=
ω
−
ω
( ) )
(
)
(
1
)
( j
lp
n
j
n
j
j
lp
n
j
j
c
e
H
e
e
e
H
e
e
H
as
expressed
be
can
response
frequency
This
.
understood
is
2
of
period
a
where
d
d
d −
=
−
=
π
⎩
ω
ω
−
ω
−
ω
ω
−
ω
( )
{ }
)
(
sin
,
sin
)
(
d
j
c
j
lp
1
-
p
p
n
n
n
n
e
H
F
Since
ω
π
ω
=
ω
)
(
)
(
sin
]
[
)
(
d
d
c
d
j
n
n
n
n
n
n
e
H
−
π
−
ω
−
−
δ
=
∴ ω
74. Discrete-Time Signal Processing, 2/E by Alan V. Oppenheim and Ronald W. Schafer
Properties of
Properties of Fourier Representation
Fourier Representation
1 F F i
1. Four Fourier
representations:
Table 3.2
Table 3.2.
t
Ω
t
0
Ω
t
0
Ω
t
Ω
t
Ω
Ω
Ω
Ω
Ω
0
Ω
Ω
n
ω0
n
ω
n
ω
ω
ω
n
ω0
0
ω
n
ω
ω
ω