This document provides an introduction to signals and systems. It defines a signal as any physical quantity that varies over time, space, or other variables and carries information. A system is defined as a transformation that maps an input signal to an output signal. Common types of signals include continuous and discrete time signals, analog and digital signals, periodic and aperiodic signals, even and odd signals. Elementary operations on signals like time shifting, time scaling, and time reversal are also discussed.
Signals and Systems is an introduction to analog and digital signal processing, a topic that forms an integral part of engineering systems in many diverse areas, including seismic data processing, communications, speech processing, image processing, defense electronics, consumer electronics, and consumer products.
This document provides recommended problems related to signals and systems. Problem P2.1 involves determining the frequency and period of cosine signals with different phase shifts. Problem P2.2 is similar but with discrete-time cosine signals. Problem P2.3 involves sketching shifted and scaled versions of a sample discrete-time signal. The remaining problems involve determining whether signals are even, odd, or neither; evaluating sums; exploring periodicity of signal sums; properties of even and odd signals; and sampling a continuous-time signal.
This document provides recommended problems related to signals and systems. Problem P2.1 involves determining the frequency and period of cosine signals with different phase shifts. Problem P2.2 is similar but with discrete-time cosine signals. Problem P2.3 involves sketching shifted and scaled versions of a sample discrete-time signal. The remaining problems involve determining whether signals are even, odd, or neither; evaluating sums; exploring properties of periodic signals; and analyzing sampled continuous-time signals.
This document provides an introduction to signals and systems. It defines signals as functions that represent information over time and gives examples such as sound waves and stock prices. Systems are defined as generators or transformers of signals. Signal processing involves manipulating signals to extract useful information, often by converting them to electrical forms. The document then classifies different types of signals such as continuous-time vs discrete-time, analog vs digital, deterministic vs random, and energy vs power signals. It also introduces some basic continuous-time signals like the unit step function, unit impulse function, and complex exponential signals.
This document provides an overview of chapter 1 on signals from a textbook on signals and systems. It defines a signal as a function that varies over time or another independent variable. It classifies signals as continuous-time or discrete-time, even or odd, periodic or aperiodic, and energy or power signals. It also discusses transformations of signals including time shifting, time scaling, and time reversal. Exponential and sinusoidal signals are examined for both continuous-time and discrete-time cases. Finally, it introduces the unit impulse and unit step functions.
The document defines and classifies different types of signals including:
- Continuous-time and discrete-time signals
- Analog and digital signals
- Real and complex signals
- Deterministic and random signals
- Periodic and non-periodic signals
It also introduces important signal properties and functions including the unit-step function, unit-impulse (Dirac delta) function, and complex exponential and sinusoidal signals. Graphical representations and mathematical definitions are provided for key signals and functions.
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 contains a question bank with two mark questions and answers related to signals and systems. Some key topics covered include:
- Definitions of continuous and discrete time signals like unit step, unit impulse, ramp functions.
- Classifications of signals as periodic, aperiodic, even, odd, energy and power.
- Properties of Fourier series and transforms including Dirichlet conditions, time shifting property, Parseval's theorem.
- Definitions of causal, non-causal, static and dynamic systems.
- Calculations of Fourier and Laplace transforms of basic signals like impulse, step functions.
So in summary, this document provides a review of fundamental concepts in signals and systems along with practice
Signals and Systems is an introduction to analog and digital signal processing, a topic that forms an integral part of engineering systems in many diverse areas, including seismic data processing, communications, speech processing, image processing, defense electronics, consumer electronics, and consumer products.
This document provides recommended problems related to signals and systems. Problem P2.1 involves determining the frequency and period of cosine signals with different phase shifts. Problem P2.2 is similar but with discrete-time cosine signals. Problem P2.3 involves sketching shifted and scaled versions of a sample discrete-time signal. The remaining problems involve determining whether signals are even, odd, or neither; evaluating sums; exploring periodicity of signal sums; properties of even and odd signals; and sampling a continuous-time signal.
This document provides recommended problems related to signals and systems. Problem P2.1 involves determining the frequency and period of cosine signals with different phase shifts. Problem P2.2 is similar but with discrete-time cosine signals. Problem P2.3 involves sketching shifted and scaled versions of a sample discrete-time signal. The remaining problems involve determining whether signals are even, odd, or neither; evaluating sums; exploring properties of periodic signals; and analyzing sampled continuous-time signals.
This document provides an introduction to signals and systems. It defines signals as functions that represent information over time and gives examples such as sound waves and stock prices. Systems are defined as generators or transformers of signals. Signal processing involves manipulating signals to extract useful information, often by converting them to electrical forms. The document then classifies different types of signals such as continuous-time vs discrete-time, analog vs digital, deterministic vs random, and energy vs power signals. It also introduces some basic continuous-time signals like the unit step function, unit impulse function, and complex exponential signals.
This document provides an overview of chapter 1 on signals from a textbook on signals and systems. It defines a signal as a function that varies over time or another independent variable. It classifies signals as continuous-time or discrete-time, even or odd, periodic or aperiodic, and energy or power signals. It also discusses transformations of signals including time shifting, time scaling, and time reversal. Exponential and sinusoidal signals are examined for both continuous-time and discrete-time cases. Finally, it introduces the unit impulse and unit step functions.
The document defines and classifies different types of signals including:
- Continuous-time and discrete-time signals
- Analog and digital signals
- Real and complex signals
- Deterministic and random signals
- Periodic and non-periodic signals
It also introduces important signal properties and functions including the unit-step function, unit-impulse (Dirac delta) function, and complex exponential and sinusoidal signals. Graphical representations and mathematical definitions are provided for key signals and functions.
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 contains a question bank with two mark questions and answers related to signals and systems. Some key topics covered include:
- Definitions of continuous and discrete time signals like unit step, unit impulse, ramp functions.
- Classifications of signals as periodic, aperiodic, even, odd, energy and power.
- Properties of Fourier series and transforms including Dirichlet conditions, time shifting property, Parseval's theorem.
- Definitions of causal, non-causal, static and dynamic systems.
- Calculations of Fourier and Laplace transforms of basic signals like impulse, step functions.
So in summary, this document provides a review of fundamental concepts in signals and systems along with practice
Unit 1 -Introduction to signals and standard signalsDr.SHANTHI K.G
1) The document introduces various types of signals including continuous time signals, discrete time signals, standard signals like step signals, ramp signals, impulse signals, sinusoidal signals, and exponential signals.
2) Continuous time signals are defined for every instant in time while discrete time signals are defined for discrete instants in time. Common standard signals include unit step, ramp, parabolic, pulse, sinusoidal, and exponential signals.
3) Examples of applications of the standard signals are mentioned such as step signals being used for switching devices on and off, and sinusoidal signals being used to represent any sound signal.
1. The document discusses Fourier analysis techniques for representing signals, including Fourier series and the Fourier transform. It uses the example of a rectangular pulse train to illustrate these concepts.
2. A periodic signal like a rectangular pulse train can be represented by a Fourier series as a sum of sinusoids with frequencies that are integer multiples of the fundamental frequency.
3. The Fourier transform allows representing aperiodic signals as a sum of sinusoids of all possible frequencies, resulting in a continuous spectrum rather than a discrete line spectrum. The Fourier transform of a rectangular pulse is a sinc function.
1. The figure shows an electrical circuit driven by a heartbeat generator. Its output is associated with a recorder for later examination. The document discusses Fourier analysis of periodic and aperiodic signals from the circuit.
2. The document discusses Fourier analysis properties such as linearity, time shifting, differentiation, and integration that are applied to analyze signals from various systems like the stock market or a microphone.
3. The document discusses using Fourier analysis to transform voltage level signals from a microphone into sound waves for recording and communication. It also discusses properties of the continuous-time Fourier series such as linearity and time shifting that are applied to analyze the signals.
The document discusses Fourier analysis and its applications in signal processing. [1] It reviews the continuous and discrete Fourier transforms and their properties like aliasing. [2] Convolution is introduced as the output of a linear time-invariant system and is equivalent to multiplication in the frequency domain using the convolution theorem. [3] Filtering is described as a way to remove certain frequency components using window functions, while correlation decomposes a signal into its zero-phase spectrum and is used to find similarities between signals.
The document discusses Fourier analysis and signal processing. It introduces the continuous and discrete Fourier transforms which decompose a signal into weighted basis functions of cosine and sine. The continuous Fourier transform uses integrals while the discrete version uses sums, introducing aliasing effects. Convolution is also introduced, where a signal passed through a linear time-invariant system results in an output that is the convolution of the input signal and impulse response.
This document summarizes key aspects of designing an optimum receiver for binary data transmission presented in Chapter 5. It begins by representing signals using orthonormal basis functions to reduce the problem from waveforms to random variables. It then describes representing noise using a complete orthonormal set and how the noise coefficients are statistically independent Gaussian variables. Finally, it outlines how the optimum receiver works by projecting the received signal onto the basis functions, resulting in random variables that can be used for binary decision making to minimize the bit error probability.
1. The document provides a list of 2 mark questions and answers related to the Signals and Systems subject for the 3rd semester IT students.
2. It includes definitions of key terms like signal, system, different types of signals and their classifications. Properties of Fourier series and Fourier transforms are also covered.
3. The questions address topics ranging from periodic/aperiodic signals, even/odd signals, unit step and impulse functions, Fourier series, Fourier transforms, Laplace transforms, linear and time invariant systems.
1) Signals can be classified as continuous-time or discrete-time based on how they are defined over time. Continuous-time signals are defined for every instant in time while discrete-time signals are defined at discrete time instances.
2) A system is defined as a set of elements or devices that produce an output response to an input signal. The relationship between input and output signals is represented by a system operator.
3) Signals and systems can be further classified based on their properties, such as being deterministic or random, periodic or aperiodic, causal or non-causal, and more. Basic operations on signals include time scaling, time reversal, and time shifting.
1) Signals can be classified as continuous-time or discrete-time based on their definition over time. Continuous-time signals are defined for every instant in time while discrete-time signals are defined at discrete time instances.
2) A system is defined as a set of elements or devices that produce an output in response to an input signal. The relationship between input and output signals is represented by a system operator.
3) Signals and systems can be further classified based on their properties, such as being deterministic or random, periodic or aperiodic, causal or non-causal, and more. Basic operations on signals include time scaling, time reversal, and time shifting.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
The document provides homework problems and solutions related to signals processing. It includes problems on determining the frequency and period of signals, properties of even and odd signals, sampling of continuous signals, and periodicity of sums of signals. It also provides detailed solutions and explanations to each problem.
This document provides an overview of signals and systems. It defines key terms like signals, systems, continuous and discrete time signals, analog and digital signals, deterministic and probabilistic signals, even and odd signals, energy and power signals, periodic and aperiodic signals. It also classifies systems as linear/non-linear, time-invariant/variant, causal/non-causal, and with or without memory. Singularity functions like unit step, unit ramp and unit impulse are introduced. Properties of signals like magnitude scaling, time reflection, time scaling and time shifting are discussed. Energy and power of signals are defined.
1. The document discusses signals and systems, including continuous-time and discrete-time signals. It covers topics like transformations of signals, exponential and sinusoidal signals, and basic properties of systems.
2. Continuous-time signals are represented as functions of time t, while discrete-time signals are represented as sequences indexed by integer n. Exponential and sinusoidal signals can be represented using complex exponential functions.
3. The document provides examples and formulas for calculating energy, power, and other properties of signals. It also describes how signals can be transformed through operations like time shifting, scaling, reversal, and periodicity.
This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.
This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.
This document provides an introduction to signals and systems. It discusses various signal classifications including continuous-time vs discrete-time, and memory vs memoryless systems. Elementary signals such as unit step, impulse, and sinusoid functions are defined. Common signal operations including time reversal, time scaling, amplitude scaling and shifting are described. The relationships between the time and frequency domains are introduced. The document is intended to help students understand signal characteristics and operations in both the time and frequency domains.
The document provides notes on signals and systems from an EECE 301 course. It includes:
- An overview of continuous-time (C-T) and discrete-time (D-T) signal and system models.
- Details on chapters covering differentials/differences, convolution, Fourier analysis (both C-T and D-T), Laplace transforms, and Z-transforms.
- Examples of calculating the Fourier transform of specific signals like a decaying exponential and rectangular pulse. These illustrate properties of the Fourier transform.
The document outlines the goals and material to be covered in three upcoming classes on signals and systems. The classes will: (1) define different types of signals and explore the concept of a system, (2) examine linear, time-invariant systems and their representation in the time and frequency domains, and (3) review Fourier series/transforms and their practical applications including sampling, aliasing, and signal conversion.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Unit 1 -Introduction to signals and standard signalsDr.SHANTHI K.G
1) The document introduces various types of signals including continuous time signals, discrete time signals, standard signals like step signals, ramp signals, impulse signals, sinusoidal signals, and exponential signals.
2) Continuous time signals are defined for every instant in time while discrete time signals are defined for discrete instants in time. Common standard signals include unit step, ramp, parabolic, pulse, sinusoidal, and exponential signals.
3) Examples of applications of the standard signals are mentioned such as step signals being used for switching devices on and off, and sinusoidal signals being used to represent any sound signal.
1. The document discusses Fourier analysis techniques for representing signals, including Fourier series and the Fourier transform. It uses the example of a rectangular pulse train to illustrate these concepts.
2. A periodic signal like a rectangular pulse train can be represented by a Fourier series as a sum of sinusoids with frequencies that are integer multiples of the fundamental frequency.
3. The Fourier transform allows representing aperiodic signals as a sum of sinusoids of all possible frequencies, resulting in a continuous spectrum rather than a discrete line spectrum. The Fourier transform of a rectangular pulse is a sinc function.
1. The figure shows an electrical circuit driven by a heartbeat generator. Its output is associated with a recorder for later examination. The document discusses Fourier analysis of periodic and aperiodic signals from the circuit.
2. The document discusses Fourier analysis properties such as linearity, time shifting, differentiation, and integration that are applied to analyze signals from various systems like the stock market or a microphone.
3. The document discusses using Fourier analysis to transform voltage level signals from a microphone into sound waves for recording and communication. It also discusses properties of the continuous-time Fourier series such as linearity and time shifting that are applied to analyze the signals.
The document discusses Fourier analysis and its applications in signal processing. [1] It reviews the continuous and discrete Fourier transforms and their properties like aliasing. [2] Convolution is introduced as the output of a linear time-invariant system and is equivalent to multiplication in the frequency domain using the convolution theorem. [3] Filtering is described as a way to remove certain frequency components using window functions, while correlation decomposes a signal into its zero-phase spectrum and is used to find similarities between signals.
The document discusses Fourier analysis and signal processing. It introduces the continuous and discrete Fourier transforms which decompose a signal into weighted basis functions of cosine and sine. The continuous Fourier transform uses integrals while the discrete version uses sums, introducing aliasing effects. Convolution is also introduced, where a signal passed through a linear time-invariant system results in an output that is the convolution of the input signal and impulse response.
This document summarizes key aspects of designing an optimum receiver for binary data transmission presented in Chapter 5. It begins by representing signals using orthonormal basis functions to reduce the problem from waveforms to random variables. It then describes representing noise using a complete orthonormal set and how the noise coefficients are statistically independent Gaussian variables. Finally, it outlines how the optimum receiver works by projecting the received signal onto the basis functions, resulting in random variables that can be used for binary decision making to minimize the bit error probability.
1. The document provides a list of 2 mark questions and answers related to the Signals and Systems subject for the 3rd semester IT students.
2. It includes definitions of key terms like signal, system, different types of signals and their classifications. Properties of Fourier series and Fourier transforms are also covered.
3. The questions address topics ranging from periodic/aperiodic signals, even/odd signals, unit step and impulse functions, Fourier series, Fourier transforms, Laplace transforms, linear and time invariant systems.
1) Signals can be classified as continuous-time or discrete-time based on how they are defined over time. Continuous-time signals are defined for every instant in time while discrete-time signals are defined at discrete time instances.
2) A system is defined as a set of elements or devices that produce an output response to an input signal. The relationship between input and output signals is represented by a system operator.
3) Signals and systems can be further classified based on their properties, such as being deterministic or random, periodic or aperiodic, causal or non-causal, and more. Basic operations on signals include time scaling, time reversal, and time shifting.
1) Signals can be classified as continuous-time or discrete-time based on their definition over time. Continuous-time signals are defined for every instant in time while discrete-time signals are defined at discrete time instances.
2) A system is defined as a set of elements or devices that produce an output in response to an input signal. The relationship between input and output signals is represented by a system operator.
3) Signals and systems can be further classified based on their properties, such as being deterministic or random, periodic or aperiodic, causal or non-causal, and more. Basic operations on signals include time scaling, time reversal, and time shifting.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
The document provides homework problems and solutions related to signals processing. It includes problems on determining the frequency and period of signals, properties of even and odd signals, sampling of continuous signals, and periodicity of sums of signals. It also provides detailed solutions and explanations to each problem.
This document provides an overview of signals and systems. It defines key terms like signals, systems, continuous and discrete time signals, analog and digital signals, deterministic and probabilistic signals, even and odd signals, energy and power signals, periodic and aperiodic signals. It also classifies systems as linear/non-linear, time-invariant/variant, causal/non-causal, and with or without memory. Singularity functions like unit step, unit ramp and unit impulse are introduced. Properties of signals like magnitude scaling, time reflection, time scaling and time shifting are discussed. Energy and power of signals are defined.
1. The document discusses signals and systems, including continuous-time and discrete-time signals. It covers topics like transformations of signals, exponential and sinusoidal signals, and basic properties of systems.
2. Continuous-time signals are represented as functions of time t, while discrete-time signals are represented as sequences indexed by integer n. Exponential and sinusoidal signals can be represented using complex exponential functions.
3. The document provides examples and formulas for calculating energy, power, and other properties of signals. It also describes how signals can be transformed through operations like time shifting, scaling, reversal, and periodicity.
This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.
This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.
This document provides an introduction to signals and systems. It discusses various signal classifications including continuous-time vs discrete-time, and memory vs memoryless systems. Elementary signals such as unit step, impulse, and sinusoid functions are defined. Common signal operations including time reversal, time scaling, amplitude scaling and shifting are described. The relationships between the time and frequency domains are introduced. The document is intended to help students understand signal characteristics and operations in both the time and frequency domains.
The document provides notes on signals and systems from an EECE 301 course. It includes:
- An overview of continuous-time (C-T) and discrete-time (D-T) signal and system models.
- Details on chapters covering differentials/differences, convolution, Fourier analysis (both C-T and D-T), Laplace transforms, and Z-transforms.
- Examples of calculating the Fourier transform of specific signals like a decaying exponential and rectangular pulse. These illustrate properties of the Fourier transform.
The document outlines the goals and material to be covered in three upcoming classes on signals and systems. The classes will: (1) define different types of signals and explore the concept of a system, (2) examine linear, time-invariant systems and their representation in the time and frequency domains, and (3) review Fourier series/transforms and their practical applications including sampling, aliasing, and signal conversion.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
UNLOCKING HEALTHCARE 4.0: NAVIGATING CRITICAL SUCCESS FACTORS FOR EFFECTIVE I...amsjournal
The Fourth Industrial Revolution is transforming industries, including healthcare, by integrating digital,
physical, and biological technologies. This study examines the integration of 4.0 technologies into
healthcare, identifying success factors and challenges through interviews with 70 stakeholders from 33
countries. Healthcare is evolving significantly, with varied objectives across nations aiming to improve
population health. The study explores stakeholders' perceptions on critical success factors, identifying
challenges such as insufficiently trained personnel, organizational silos, and structural barriers to data
exchange. Facilitators for integration include cost reduction initiatives and interoperability policies.
Technologies like IoT, Big Data, AI, Machine Learning, and robotics enhance diagnostics, treatment
precision, and real-time monitoring, reducing errors and optimizing resource utilization. Automation
improves employee satisfaction and patient care, while Blockchain and telemedicine drive cost reductions.
Successful integration requires skilled professionals and supportive policies, promising efficient resource
use, lower error rates, and accelerated processes, leading to optimized global healthcare outcomes.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
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.
2. 1.3.1 Time Shifting
Consider a signal x(t). If it is time shifted by t0, the time-shifted version of x(t) is
represented by x(t t0). The two signals x(t) and x(t t0) are identical in shape but
time shifted relative to each other. If t0 is positive, the signal x(t) is delayed (right
shifted) by t0. If t0 is negative, the signal is advanced (left shifted) by t0. Signals
related in this fashion arise in applications such as sonar, seismic signal processing,
radar, and GPS. The time shifting operation is illustrated in Figure 1.2. If the signal x
(t) shown in Figure 1.2(a) is shifted by t0 ¼ 2 seconds, x(t 2) is obtained as shown
in Figure 1.2(b), i.e., x(t) is delayed (right shifted) by 2 seconds. If the signal is
advanced (left shifted) by 2 seconds, x(t þ 2) is obtained as shown in Figure 1.2(c),
i.e., x(t) is advanced (left shifted) by 2 seconds.
1.3.2 Time Scaling
The compression or expansion of a signal is known as time scaling. The time-scaling
operation is illustrated in Figure 1.3. If the signal x(t) shown in Figure 1.3(a) is
( )
Figure 1.1 Schematic representation of a system
0 2
1
-2 t
x(t)
4
0 2
1
-2 t
x(t-2)
1
-4 0 2
-2 t
x(t+2)
(a) (b)
(c)
Figure 1.2 Illustration of time shifting
2 1 Introduction
3. compressed in time by a factor 2, x(2t) is obtained as shown in Figure 1.3(b). If the
signal x(t) is expanded by a factor of 2, x(t/2) is obtained as shown in Figure 1.3(c).
1.3.3 Time Reversal
The signal x(t) is called the time reversal of the signal x(t). The x(t) is obtained
from the signal x(t) by a reflection about t ¼ 0. The time reversal operation is
illustrated in Figure 1.4. The signal x(t) is shown in Figure 1.4(a), and its time
reversal signal x(t) is shown in Figure 1.4(b).
(a) (b)
-4 -2
-2
2
2
4 t
x(t)
-1
1
-2
-2
2
2
t
x(2t)
-4 -2
-2
2
2
4 t
x(t/2)
Figure 1.3 Illustration of time scaling
t
-2 2
2
x(t)
(a) (b)
t
-2 2
2
x(-t)
Figure 1.4 Illustration of time reversal
1.3 Elementary Operations on Signals 3
4. Example 1.1 Consider the following signals x(t) and xi(t), i ¼ 1,2,3. Express them
using only x(t) and its time-shifted, time-scaled, and time-inverted version.
t
-2
2
x(t)
t
-2 2
2
t
4
4
4
t
2
2
Solution
0
0
0
0 t
2
2
x(t-2)
4
t
2
2
t
2
2
x(-t)
t
-2
2
x(t)
x1 t
ð Þ ¼ x t 2
ð Þ þ x t þ 2
ð Þ
t
2
2
x(-t)
t
-2
2
4 1 Introduction
5. x2 t
ð Þ ¼ x t 2
ð Þ þ x t 2
ð Þ
x3 t
ð Þ ¼ 2x
t
2
2
1.4 Classification of Signals
Signals can be classified in several ways. Some important classifications of signals
are:
1.4.1 Continuous-Time and Discrete-Time Signals
Continuous-time signals are defined for a continuous of values of the independent
variable. In the case of continuous-time signals, the independent variable t is
continuous as shown Figure 1.5(a).
Discrete-time signals are defined only at discrete times, and for these signals, the
independent variable n takes on only a discrete set of amplitude values as shown in
Figure 1.5(b).
1.4.2 Analog and Digital Signals
An analog signal is a continuous-time signal whose amplitude can take any value in
a continuous range. A digital signal is a discrete-time signal that can only have a
discrete set of values. The process of converting a discrete-time signal into a digital
signal is referred to as quantization.
−0.5
0.5
1.5
2.5
3.5
1
2
3
4
0
0 2 4 6 8
Time
a b
Time index n
Amplitude
−0.5
0.5
1.5
2.5
3.5
1
2
3
4
0
Amplitude
10 12 14 16 0 2 4 6 8 10 12 14 16
Figure 1.5 (a) Continuous-time signal, (b) discrete-time signal
1.4 Classification of Signals 5
6. 1.4.3 Periodic and Aperiodic Signals
A signal x(t) is said to be periodic with period T(a positive nonzero value), if it
exhibits periodicity, i.e., x(t þ T) ¼ x(t), for all values of t as shown in Figure 1.6(a).
Periodic signal has the property that it is unchanged by a time shift of T.
A signal that does not satisfy the above periodicity property is called an aperiodic
signal. The signal shown in Figure 1.6(b) is an example of an aperiodic signal.
Example 1.2 For each of the following signals, determine whether it is periodic or
aperiodic. If periodic, find the period.
(i) x(t) ¼ 5 sin(2πt)
(ii) x(t) ¼ 1 þ cos(4t þ 1)
(iii) x(t) ¼ e2t
(iv) x t
ð Þ ¼ ej 5tþπ
2
ð Þ
(v) x t
ð Þ ¼ ej 5tþπ
2
ð Þe2t
Solution
(i) It is periodic signal, period ¼ 2π
2π ¼ 1:
(ii) It is periodic, period ¼ 2π
4 ¼ π
2
(iii) It is aperiodic,
(iv) It is periodic. period ¼ 2π
5
(v) Since x(t) is a complex exponential multiplied by a decaying exponential, it is
aperiodic.
Example 1.3 If a continuous-time signal x(t) is periodic, for each of the following
signals, determine whether it is periodic or aperiodic. If periodic, find the period.
(i) x1(t) ¼ x(2t)
(ii) x2(t) ¼ x(t/2)
Solution Let T be the period of x(t). Then, we have
x t
ð Þ ¼ x t þ T
ð Þ
(a) (b)
0 1
1
2 Time
x(t)
0
−1
0
1
−0.5
0.5
0.05 0.1
Time (sec)
Amplitude
0.2
0.15
Figure 1.6 (a) Periodic signal, (b) aperiodic signal
6 1 Introduction
7. (i) For x1(t) to be periodic,
x 2t
ð Þ ¼ x 2t þ T
ð Þ
x 2t þ T
ð Þ ¼ x 2 t þ
T
2
¼ x1 t þ
T
2
Since x1 t
ð Þ ¼ x1 t þ T
2
, x1(t) is periodic with fundamental period T
2.
As x1(t) is compressed version of x(t) by half, the period of x1(t) is also com-
pressed by half.
(ii) For x2(t) to be periodic,
x t=2
ð Þ ¼ x
t
2
þ T
x
t
2
þ T
¼ x
1
2
t þ 2T
ð Þ
¼ x2 t þ 2T
ð Þ
Since x2(t) ¼ x2(t þ 2T), x2(t) is periodic with fundamental period 2T. As x2(t) is
expanded version of x(t) by two, the period of x2(t) is also twice the period of x(t).
Proposition 1.1 Let continuous-time signals x1(t) and x2(t) be periodic signals with
fundamental periods T1 and T2, respectively. The signal x(t) that is a linear combi-
nation of x1(t) and x2(t) is periodic if and only if there exist integers m and k such that
mT1 ¼ kT2 and
T1
T2
¼
k
m
¼ rational number ð1:2Þ
The fundamental period of x(t) is given by mT1 ¼ kT2 provided that the values of m
and k are chosen such that the greatest common divisor (gcd) between m and k is 1.
Example 1.4 For each of the following signals, determine whether it is periodic or
aperiodic. If periodic, find the period.
(i) x(t) ¼ 2 cos(4πt) þ 3 sin(3πt)
(ii) x(t) ¼ 2 cos(4πt) þ 3 sin(10t)
Solution
(i) Let x1(t) ¼ 2 cos(4πt) and x2(t) ¼ 3 sin(3πt).
The fundamental period of x1(t) is
T1 ¼
2π
4π
¼
1
2
1.4 Classification of Signals 7
8. The fundamental period of x2(t) is
T2 ¼
2π
3π
¼
2
3
The ratio T1
T2
¼ 1=2
2=3 ¼ 3
4 is a rational number. Hence, x(t) is a periodic signal.
The fundamental period of the signal x(t) is 4T1 ¼ 3T2 ¼ 2 seconds.
(ii) Let x1(t) ¼ 2 cos(4πt) and x2(t) ¼ 3 sin(10t).
The fundamental period of x1(t) is
T1 ¼
2π
4π
¼
1
2
The fundamental period of x2(t) is
T2 ¼
2π
10
¼
π
5
The ratio T1
T2
¼ 1=2
π=5 ¼ 5
2π is not a rational number. Hence, x(t) is an aperiodic signal.
Example 1.5 Consider the signals
x1 t
ð Þ ¼ cos
2πt
5
þ 2 sin
8πt
5
x2 t
ð Þ ¼ sin πt
ð Þ
Determine whether x3(t) ¼ x1(t)x2(t) is periodic or aperiodic. If periodic, find the
period.
Solution Decomposing signals x1(t) and x2(t) into sums of exponentials gives
x1 t
ð Þ ¼ 1
2 ej 2πt=5
ð Þ
þ 1
2 ej 2πt=5
ð Þ
þ ej 8πt=5
ð Þ
j ej 8πt=5
ð Þ
j
x2 t
ð Þ ¼
ej πt
ð Þ
2j
ej πt
ð Þ
2j
Then,
x3 t
ð Þ ¼
1
4j
ej 7πt=5
ð Þ
1
4j
ej 3πt=5
ð Þ
þ
1
4j
ej 3πt=5
ð Þ
1
4j
ej 7πt=5
ð Þ
ej 13πt=5
ð Þ
2
þ
ej 3πt=5
ð Þ
2
þ
ej 3πt=5
ð Þ
2
ej 13πt=5
ð Þ
2
It is seen that all complex exponentials are powers of ej(π/5)
. Hence, it is periodic.
Period is 2π
π=5 ¼ 10seconds:
8 1 Introduction
9. 1.4.4 Even and Odd Signals
The continuous-time signal is said to be even when x(t) ¼ x(t). The continuous-
time signal is said to be odd when x(t) ¼ x(t). Odd signals are also known as
nonsymmetrical signals. Examples of even and odd signals are shown in Figure 1.7
(a) and Figure 1.7(b), respectively.
Any signal can be expressed as sum of its even and odd parts as
x t
ð Þ ¼ xe t
ð Þ þ xo t
ð Þ ð1:3Þ
The even part and odd part of a signal are
xe t
ð Þ ¼
x t
ð Þ þ x t
ð Þ
2
ð1:3aÞ
xo t
ð Þ ¼
x t
ð Þ x t
ð Þ
2
ð1:3bÞ
Some important properties of even and odd signals are:
(i) Multiplication of an even signal by an odd signal produces an odd signal.
Proof Let y(t) ¼ xe(t)xo(t)
y t
ð Þ ¼ xe t
ð Þxo t
ð Þ
¼ xe t
ð Þxo t
ð Þ ¼ y t
ð Þ
ð1:4Þ
Hence, y(t) is an odd signal.
(ii) Multiplication of an even signal by an even signal produces an even signal.
Proof Let y(t) ¼ xe(t)xe(t)
y t
ð Þ ¼ xe t
ð Þxe t
ð Þ
¼ xe t
ð Þxe t
ð Þ ¼ y t
ð Þ
ð1:5Þ
Hence, y(t) is an even signal.
0 b
A
-b t
x(t)
(a) (b)
A
x(t)
0 b
-A
-b t
Figure 1.7 (a) Even signal, (b) odd signal
1.4 Classification of Signals 9
10. (iii) Multiplication of an odd signal by an odd signal produces an even signal.
Proof Let y(t) ¼ xo(t)xo(t)
y t
ð Þ ¼ x0 t
ð Þxo t
ð Þ
¼ xo t
ð Þ
ð Þ xo t
ð Þ
ð Þ
¼ xo t
ð Þxo t
ð Þ
¼ y t
ð Þ
Hence, y(t) is an even signal.
It is seen from Figure 1.7(a) that the even signal is symmetric about the vertical
axis, and hence
ð b
b
xe t
ð Þdt ¼ 2
ðb
0
xe t
ð Þdt ð1:6Þ
From Figure 1.6(b), it is also obvious that
ð b
b
xo t
ð Þdt ¼ 0 ð1:7Þ
Eqs. (1.6) and (1.7) are valid for no impulse or its derivative at the origin. These
properties are proved to be useful in many applications.
Example 1.6 Find the even and odd parts of x(t) ¼ ej2t
.
Solution From Eq. (1.3),
ej2t
¼ xe t
ð Þ þ xo t
ð Þ
where
xe t
ð Þ ¼
x t
ð Þ þ x t
ð Þ
2
¼
ej2t
þ ej2t
2
¼ cos 2t
ð Þ
xo t
ð Þ ¼
x t
ð Þ x t
ð Þ
2
¼
ej2t
ej2t
2
¼ j sin 2t
ð Þ:
Example 1.7 If xe(t) and xo(t) are the even and odd parts of x(t), show that
ð1
1
x2
t
ð Þdt ¼
ð1
1
x2
e t
ð Þdt þ
ð1
1
x2
o t
ð Þdt ð1:8Þ
10 1 Introduction
11. Solution ð1
1
x2
t
ð Þdt ¼
ð1
1
xe t
ð Þ þ xo t
ð Þ
ð Þ2
dt
¼
ð1
1
x2
e t
ð Þdt þ 2
ð1
1
xe t
ð Þxo t
ð Þdt þ
ð1
1
x2
o t
ð Þdt
¼
ð1
1
x2
e t
ð Þdt þ
ð1
1
x2
o t
ð Þdt
Since 2
ð1
1
xe t
ð Þxo t
ð Þdt ¼ 0
Example 1.8 For each of the following signals, determine whether it is even, odd,
or neither (Figure 1.8)
Solution By definition a signal is even if and only if x(t) ¼ x(t), while a signal is
odd if and only if x(t) ¼ x(t).
(a) It is readily seen that x(t) 6¼ x(t) for all t and x(t) 6¼ x(t) for all t; thus x(t) is
neither even nor odd.
(b) Since x(t) is symmetric about t ¼ 0, x(t) is even.
(c) Since x(t) ¼ x(t), x(t) is odd in this case.
-4 -2 2
2
4
x(t)
t
-2 2
2
x(t)
(a) (b)
(c)
-2
-2
2
2
x(t)
tt
Figure 1.8 Signals of example 1.8
1.4 Classification of Signals 11
12. 1.4.5 Causal, Noncausal, and Anticausal Signal
A causal signal is one that has zero values for negative time, i.e., t 0. A signal is
noncausal if it has nonzero values for both the negative and positive times. An
anticausal signal has zero values for positive time, i.e., t 0. Examples of causal,
noncausal, and anticausal signals are shown in Figure 1.9(a), 1.9(b), and 1.9(c),
respectively.
Example 1.9 Consider the following noncausal continuous-time signal. Obtain its
realization as causal signal.
-0.5
0.5
x(t)
1
1
-1
t
0
Solution
0
0.5
1.5
x(t)
2
1
-1
t
x(t)
t
x(t)
t
t
x(t)
(a) (b) (c)
Figure 1.9 (a) Causal signal, (b) noncausal signal, (c) anticausal signal
12 1 Introduction
13. 1.4.6 Energy and Power Signals
A signal x(t) with finite energy, which means that amplitude ! 0 as time ! 1, is
said to be energy signal, whereas a signal x(t) with finite and nonzero power is said to
be power signal. The instantaneous power p(t) of a signal x(t) can be expressed by
p t
ð Þ ¼ x2
t
ð Þ ð1:9Þ
The total energy of a continuous-time signal x(t) can be defined as
E ¼
ð1
1
x2
t
ð Þdt ð1:10aÞ
for a complex valued signal
E ¼
ð1
1
x t
ð Þ
j j2
dt ð1:10bÞ
Since the power is the time average of energy, the average power is defined as
P ¼ limT!1
1
T
ðT=2
T=2
x2
t
ð Þdt. The signal x(t) expressed by Eq. (1.11), which is
shown in Figure 1.10(a), is an example of energy signal.
x t
ð Þ ¼
t 0 t 1
1 1 t 2
(
ð1:11Þ
The energy of the signal is given by
E ¼
ð1
1
x2
t
ð Þdt ¼
ð1
0
t2
dt þ
ð2
1
1dt ¼
1
3
þ 1 ¼
4
3
The signal x(t) shown in Figure 1.10(b) is an example of a power signal. The
signal is periodic with period 2. Hence, averaging x2
(t) over infinitely large time
interval is the same as averaging over one period, i.e., 2. Thus, the average power P is
P ¼
1
2
ð1
1
x2
t
ð Þdt ¼
1
2
ð1
1
4t2
dt ¼
4
3
0 1
1
2 Time
x(t)
Figure 1.10(a) Energy
signal
1.4 Classification of Signals 13
14. Thus, an energy signal has finite energy and zero average power, whereas a power
signal has finite power and infinite energy.
Example 1.10 Compute energy and power for the following signals, and determine
whether each signal is energy signal, power signal, or neither.
(i) x(t) ¼ 4sin(2πt), 1 t 1.
(ii) x(t) ¼ 2e2|t|
, 1 t 1.
(iii) x t
ð Þ ¼
2
ffiffi
t
p t 1
0 t 1:
8
:
(iv) x(t) ¼ eat
for real value of a
(v) x(t) ¼ cos (t)
(vi) xðtÞ ¼ ej 2tþ
π
4Þ
ð
Solution
(i)
E ¼
ð1
1
x t
ð Þ
j j2
dt ¼
ð1
1
4 sin 2πt
ð Þ
j j2
dt
¼ 16
ð1
1
1 cos 4πt
ð Þ
2
dt
¼ 16
ð1
1
1
2
dt 8
ð1
1
cos 4πt
ð Þdt
¼ 1
P ¼ limT!1
1
T
ðT=2
T=2
x2
ðtÞdt ¼ limT!1
1
T
ðT=2
T=2
16 sin 2
ð2πtÞdt
¼ limT!1
1
T
ðT=2
T=2
16
1 cos ð4πtÞ
2
dt
¼ 16 limT!1
1
T
ðT=2
T=2
1
2
dt 16 limT!1
1
T
ðT=2
T=2
cos ð4πtÞ
2
dt
¼ 8
-4 -2
-2
2
2
4 t
x(t)
Figure 1.10(b) Power
signal
14 1 Introduction
15. The energy of the signal is infinite, and its average power is finite; x(t) is a power
signal.
(ii) x(t) ¼ 2e2|t|
E ¼
ð1
1
x t
ð Þ
j j2
dt ¼
ð1
1
2e2 t
j j 2
dt
¼ 4
ð0
1
e4t
dt þ 4
ð1
0
e4t
dt
¼
4
4
e4t 0
1
þ
4
4
e4t 1
0
¼
4
4
þ
4
4
¼ 2
P ¼ limT!1
1
T
ðT=2
T=2
x2
t
ð Þdt ¼ limT!1
1
T
ðT=2
T=2
2e2 t
j j 2
dt
¼ 4 limT!1
1
T
ð0
T=2
e4t
dt þ 4 limT!1
1
T
ðT=2
0
e4t
dt
¼
4
4
limT!1
1
T
e4t 0
T=2
þ
4
4
limT!1
1
T
e4t T=2
0
¼
4
4
limT!1
1
T
1 e2T
þ
4
4
limT!1
1
T
e2T
1
¼ 0 þ 0 ¼ 0
The energy of the signal is finite, and its average power is zero; x(t) is an energy
signal.
(iii) x t
ð Þ ¼
2
ffiffi
t
p t 1
0 t 1:
8
:
E ¼
ð1
1
x t
ð Þ
j j2
dt ¼
ð1
1
4
t
dt
¼ 4ln t
½ 1
1
¼ 1
1.4 Classification of Signals 15
16. P ¼ lim
T!1
1
T
ðT=2
T=2
x2
t
ð Þdt ¼ lim
T!1
1
T
ðT=2
1
4
t
dt
¼ 4 lim
T!1
1
T
ln t
½
1T=2
¼ 4 lim
T!1
1
T
ln
T
2
1
T
ln 1
½
¼ 4 lim
T!1
1
T
ln
T
2
¼ 4 lim
T!1
ln
T
2
T
0
B
B
@
1
C
C
A
Using L’Hospital’s rule, we see that the power of the signal is zero. That is
P ¼ 4 lim
T!1
ln T
2
T
¼ 4 lim
T!1
2
T
1
¼ 0
The energy of the signal is infinite and its average power is zero; x(t) is neither
energy signal nor power signal.
(iv) x(t) ¼ eat
for real value of a
E ¼
ð1
1
x t
ð Þ
j j2
dt ¼
ð1
1
eat
j j
2
dt ¼ 1,
P ¼ limT!1
1
T
ðT=2
T=2
x2
t
ð Þdt ¼ limT!1
1
T
ðT=2
T=2
e2at
dt
¼ lim
T!1
eaT
eaT
2aT
¼ lim
T!1
eaT
2aT
lim
T!1
eaT
2aT
¼ lim
T!1
eaT
2aT
0
Using L’Hospital’s rule, we see that the power of the signal is infinite. That is,
P ¼ lim
T!1
eaT
2aT
¼ lim
T!1
eaT
2
¼ 1
The energy of the signal is infinite and its average power is infinite; x(t) is neither
energy signal nor power signal.
(v) x(t) ¼ cos(t)
E ¼
ð1
1
x t
ð Þ
j j2
dt ¼
ð1
1
cos 2
t
ð Þdt ¼ 1,
16 1 Introduction
17. P ¼ limT!1
1
T
ðT=2
T=2
x2
t
ð Þdt ¼ limT!1
1
T
ðT=2
T=2
cos 2
t
ð Þdt
¼ limT!1
1
T
ðT=2
T=2
1 þ cos 2t
ð Þ
2
dt
¼ limT!1
1
T
ðT=2
T=2
1
2
dt þ limT!1
1
T
ðT=2
T=2
cos 2t
ð Þ
2
dt
¼
1
2
The energy of the signal is infinite and its average power is finite; x(t) is a power
signal.
(vi) x t
ð Þ ¼ ej 2tþπ
4
ð Þ, x t
ð Þ
j j ¼ 1.
E ¼
ð1
1
x t
ð Þ
j j2
dt ¼
ð1
1
dt ¼ 1,
P ¼ limT!1
1
T
ðT=2
T=2
x2
t
ð Þdt ¼ limT!1
1
T
ðT=2
T=2
1dt ¼ limT!11 ¼ 1
The energy of the signal is infinite and its average power is finite; x(t) is a power
signal.
Example 1.11 Consider the following signals, and determine the energy of each
signal shown in Figure 1.11. How does the energy change when transforming a
signal by time reversing, sign change, time shifting, or doubling it?
t
2
2
x(t)
t
-2
2
(t)
4
t
2
2
( )
t
2
-2
(t)
t
2
4
( )
Figure 1.11 Signals of example 1.11
1.4 Classification of Signals 17
18. Solution
xðtÞ ¼
(
t 0 t 2
0 otherwise
Ex ¼
ð1
1
jxðtÞj2
dt ¼
ð2
0
t2
dt ¼
t3
3
2
0
¼
8
3
x1ðtÞ ¼
(
t 2 t 0
0 otherwise
Ex1
¼
ð1
1
jx1ðtÞj2
dt ¼
ð0
2
t2
dt ¼
t3
3
0
2
¼
8
3
x2ðtÞ ¼
(
t 0 t 2
0 otherwise
Ex2
¼
ð1
1
jx2ðtÞj2
dt ¼
ð2
0
t2
dt ¼
t3
3
2
0
¼
8
3
x3ðtÞ ¼
(
ðt 2Þ 2 t 4
0 otherwise
Ex3
¼
ð1
1
jx3ðtÞj2
dt ¼
ð2
0
ðt 2Þ2
dt ¼
ð4
2
ðt2
4t þ 4Þdt
¼
t3
3
2t2
þ 4t
4
2
¼
8
3
x4ðtÞ ¼
(
2t 0 t 2
0 otherwise
Ex4
¼
ð1
1
jx4ðtÞj2
dt ¼
ð2
0
4t2
dt ¼ 4
t3
3
2
0
¼
32
3
The time reversal, sign change, and time shifting do not affect the signal energy.
Doubling the signal quadruples its energy. Similarly, it can be shown that the energy
of k x(t) is k2
Ex.
Proposition 1.2 The sum of two sinusoids of different frequencies is the sum of the
power of individual sinusoids regardless of phase.
Proof Let us consider a sinusoidal signal x(t) ¼ Acos(Ωt + θ). The power of x(t) is
given by
18 1 Introduction
19. P ¼ limT!1
1
T
ðT=2
T=2
x2
ðtÞdt ¼ limT!1
1
T
ðT=2
T=2
A2
cos 2
ðΩt þ θÞdt
¼ limT!1
1
2T
ðT=2
T=2
A2
½1 þ cos 2
ð2Ωt þ 2θÞdt
¼ limT!1
A2
2T
ðT=2
T=2
dt þ
ðT=2
T=2
cos 2Ωt þ 2θ
ð Þdt
#
¼
A2
2T
½T þ 0 ¼
A2
2
ð1:12Þ
Thus, a sinusoid signal of amplitude A has a power A2
2 regardless of the values of
its frequency Ω and phase θ.
Now, consider the following two sinusoidal signals:
x1 t
ð Þ ¼ A1 cos Ω1t þ θ1
ð Þ
x2 t
ð Þ ¼ A2 cos Ω2t þ θ2
ð Þ
Let xs t
ð Þ ¼ x1 t
ð Þ þ x2 t
ð Þ
The power of the sum of the two sinusoidal signals is given by
Ps ¼ limT!1
1
T
ðT
2
T
2
x2
s ðtÞ
¼ limT!1
1
T
ðT=2
T=2
½A1cos ðΩ1t þ θ1Þ þ A2cos ðΩ2t þ θ2Þ2
dt
¼ limT!1
1
T
ðT
2
T
2
A2
1cos 2
ðΩ1t þ θ1Þdt
þ limT!1
1
T
ðT
2
T
2
A2
2cos 2
ðΩ2t þ θ2Þdt
þ limT!1
2A1A2
T
ðT
2
T
2
cos ðΩ1t þ θ1Þcos ðΩ2t þ θ2Þdt
The first and second integrals on the right-hand side are the powers of the two
sinusoidal signals, respectively, and the third integral becomes zero since
cos Ω1t þ θ1
ð Þ cos Ω2t þ θ2
ð Þ ¼ cos Ω1 þ Ω2
ð Þt þ θ1 þ θ2
ð Þ
½
þ cos Ω1 Ω2
ð Þt þ θ1 θ2
ð Þ
½
Hence,
Ps ¼
A2
1
2
þ
A2
2
2
ð1:13Þ
1.4 Classification of Signals 19
20. It can be easily extended to sum of any number of sinusoids with distinct
frequencies
1.4.7 Deterministic and Random Signals
For any given time, the values of deterministic signal are completely specified as
shown in Figure 1.12(a). Thus, a deterministic signal can be described mathemati-
cally as a function of time. A random signal takes random statistically characterized
random values as shown in Figure 1.12(b) at any given time. Noise is a common
example of random signal.
1.5 Basic Continuous-Time Signals
1.5.1 The Unit Step Function
The unit step function is defined as
u t
ð Þ ¼
1 t 0
0 t 0
ð1:14Þ
which is shown in Figure 1.13.
It should be noted that u(t) is discontinuous at t ¼ 0.
1
u(t)
t
Figure 1.13 Unit step
function
1
1
−1
0.5
−0.5
0
0.8
0.6
0.4
0.2
0
0 50
0 0.05 0.15
0.1
Time (sec)
Amplitude
0.2
100 150
(a) (b)
Figure 1.12 (a) Deterministic signal, (b) random signal
20 1 Introduction
21. 1.5.2 The Unit Impulse Function
The unit impulse function also known as the Dirac delta function, which is often
referred as delta function is defined as
δ t
ð Þ ¼ 0, t 6¼ 0 ð1:15aÞ
ð1
1
δ t
ð Þ ¼ 1: ð1:15bÞ
The delta function shown in Figure 1.14(b) can be evolved as the limit of the
rectangular pulse as shown in Figure 1.14(a).
δ t
ð Þ ¼ lim
Δ!0
pΔ t
ð Þ ð1:16Þ
As the width Δ ! 0, the rectangular function converges to the impulse function
δ(t) with an infinite height at t ¼ 0, and the total area remains constant at one.
Some Special Properties of the Impulse Function
• Sampling property
If an arbitrary signal x(t) is multiplied by a shifted impulse function, the product is
given by
x t
ð Þδ t t0
ð Þ ¼ x t0
ð Þδ t t0
ð Þ ð1:17aÞ
implying that multiplication of a continuous-time signal and an impulse function
produces an impulse function, which has an area equal to the value of the
continuous-time function at the location of the impulse. Also, it follows that for
t0 ¼ 0,
x t
ð Þδ t
ð Þ ¼ x 0
ð Þδ t
ð Þ ð1:17bÞ
• Shifting property
ð1
1
x t
ð Þδ t t0
ð Þdt ¼ x t0
ð Þ ð1:18Þ
pΔ(t)
(a) (b)
1/Δ
Δ/2
−Δ/2
d(t)
t
t
0
Figure 1.14
(a) Rectangular pulse,
(b) unit impulse
1.5 Basic Continuous-Time Signals 21
22. • Scaling property
δ at þ b
ð Þ ¼
1
a
j j
δ t þ
b
a
ð1:19Þ
• The unit impulse function can be obtained by taking the derivative of the unit step
function as follows:
δ t
ð Þ ¼
du t
ð Þ
dt
ð1:20Þ
• The unit step function is obtained by integrating the unit impulse function as
follows:
u t
ð Þ ¼
ð t
1
δ t
ð Þdt ð1:21Þ
1.5.3 The Ramp Function
The ramp function is defined as
r t
ð Þ ¼
t t 0
0 t 0
ð1:22aÞ
which can also be written as
r t
ð Þ ¼ tu t
ð Þ ð1:22bÞ
The ramp function is shown in Figure 1.15.
1.5.4 The Rectangular Pulse Function
The continuous-time rectangular pulse function is defined as
Figure 1.15 The ramp
function
22 1 Introduction
23. x t
ð Þ ¼
1 t
j j T1
0 t
j j T1
ð1:23Þ
which is shown in Figure 1.16.
1.5.5 The Signum Function
The signum function also called sign function is defined as
sgn t
ð Þ ¼
1 t 0
0 t ¼ 0
1 t 0
8
:
ð1:24Þ
which is shown in Figure 1.17.
1.5.6 The Real Exponential Function
A real exponential function is defined as
x t
ð Þ ¼ Aeσt
ð1:25Þ
where both A and σ are real. If σ is positive, x(t) is a growing exponential signal. The
signal x(t) is exponentially decaying for negative σ. For σ ¼ 0, the signal x(t) is equal
to a constant. Exponentially decaying signal and exponentially growing signal are
shown in Figure 1.18(a) and (b), respectively.
0
1
1
-
1 t
x(t)
Figure 1.16 The
rectangular pulse function
-1
0
1
t
x(t)=
Figure 1.17 The signum
function
1.5 Basic Continuous-Time Signals 23
24. 1.5.7 The Complex Exponential Function
A real exponential function is defined as
x t
ð Þ ¼ Ae σþjΩ
ð Þt
ð1:26Þ
Hence
x t
ð Þ ¼ Aeσt
ejΩt
ð1:26aÞ
Using Euler’s identity
ejΩt
¼ cos Ωt
ð Þ þ j sin Ωt
ð Þ ð1:27Þ
Substituting Eq. (1.27) in Eq. (1.26a), we obtain
x t
ð Þ ¼ Aeσt
cos Ωt
ð Þ þ j sin Ωt
ð Þ
ð Þ ð1:28Þ
Real sine function and real cosine function can be expressed by the trigonometric
identities as
cos Ωt
ð Þ ¼ ejΩt
þejΩt
2 and sin Ωt
ð Þ ¼ ejΩt
ejΩt
2j
1.5.8 The Sinc Function
The continuous-time sinc function is defined as
Sinc t
ð Þ ¼
sin πt
ð Þ
πt
ð1:28Þ
which is shown in Figure 1.19
A
0
t
x(t)
A
0
t
x(t)
(a) (b)
Figure 1.18 Real exponential function. (a) Decaying, (b) growing
24 1 Introduction
25. Example 1.12 State whether the following signals are causal, anticausal, or
noncausal.
(a) x(t) ¼ e2t
u(t)
(b) x(t) ¼ tu(t) t(u(t 1) þ e(1t)
u(t 1))
(c) x(t) ¼ et
cos (2πt)u(1 t)
Solution (a)
1
0.25
x(t)
1
t
It is causal since x(t) ¼ 0 for t 0
(b)
1
1
t
x(t)
It is causal since x(t) ¼ 0 for t 0
Figure 1.19 The sinc function
1.5 Basic Continuous-Time Signals 25
26. (c)
(c )
It is non causal since for t0
u(1-t)
1
t
Example 1.13 Determine and plot the even and odd components of the following
continuous-time signal
x t
ð Þ ¼ tu t þ 2
ð Þ tu t 1
ð Þ
Solution
x t
ð Þ ¼ tu t þ 2
ð Þ tu t 1
ð Þ
x t
ð Þ ¼ tu t þ 2
ð Þ þ tu t þ 1
ð Þ
-2
-2 1
x(t)
1
t 2
-2
-2 1
x(-t)
1
t
xeðtÞ ¼
xðtÞ þ xðtÞ
2
¼
1
2
t
uðt þ 2Þ uðt 1Þ uðt þ 2Þ þ uðt þ 1Þ
26 1 Introduction
27. -0.5
2
-1
-2
1
1
)
(
t
xo t
ð Þ ¼
x t
ð Þ x t
ð Þ
2
¼
1
2
t u t þ 2
ð Þ u t 1
ð Þ þ u t þ 2
ð Þ u t þ 1
ð Þ
ð Þ
-0.5
2
-1
-2
1
1
t
Example 1.14 Simplify the following expressions:
(a) sin t
t2þ3
δ t
ð Þ
(b) 4þjt
3jt δ t 1
ð Þ
(c)
Ð1
1 4t 3
ð ÞÞδ t 1
ð Þdt
Solution
(a) sin 0
ð Þ
0þ3
δ t
ð Þ ¼ 0
(b) 4þjt
3jt δ t 1
ð Þ ¼ 4þj
3j δ t 1
ð Þ
(c)
Ð1
1 4 1
ð Þ 3
ð ÞÞδ t 1
ð Þdt ¼
Ð1
1 1δ t 1
ð Þdt ¼ 1:
1.5 Basic Continuous-Time Signals 27
28. 1.6 Generation of Continuous-Time Signals Using
MATLAB
An exponentially damped sinusoidal signal can be generated using the following
MATLAB command:
x(t) ¼ A ∗ sin (2 ∗ pi ∗ f0 ∗ t + θ) ∗ exp (a ∗ t)where a is positive for
decaying exponential.
Example 1.15 Write a MATLAB program to generate the following exponentially
damped sinusoidal signal.
x(t) ¼ 5 sin (2πt)e0.4t
10 t 10
Solution The following MATLAB program generates the exponentially damped
sinusoidal signal as shown in Figure 1.20.
MATLAB program to generate exponentially damped sinusoidal signal
clear all;clc;
x =inline('5*sin(2*pi*1*t).*exp(-.4*t)','t');
t = (-10:.01:10);
plot(t,x(t));
xlabel ('t (seconds)');
ylabel ('Ámplitude');
-10 -8 -6 -4 -2 0 2 4 6 8 10
-250
-200
-150
-100
-50
0
50
100
150
200
250
t (seconds)
Ámplitude
Figure 1.20 Exponentially damped sinusoidal signal with exponential parameter a ¼ 0.4.
28 1 Introduction
29. Example 1.16 Generate unit step function over [5,5] using MATLAB
Solution The following MATLAB program generates the unit step function over
[5,5] as shown in Figure 1.21.
MATLAB program to generate unit step function over [5,5]
clear all;clc;
u=inline('(t=0)','t');
t=-5:0.01:5;
plot(t,u(t))
xlabel ('t (seconds)');
ylabel ('Ámplitude')
axis([-5 5 -2 2])
Example 1.17 Generate the following rectangular pulse function rect(t) using
MATLAB:
rect t
10
¼
1, 5 t 5
0, elsewhere
Solution The following MATLAB program generates the rectangular pulse func-
tion as shown in Figure 1.22.
-5 -4 -3 -2 -1 0 1 2 3 4 5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
t (seconds)
Ámplitude
Figure 1.21 Unit step function
1.6 Generation of Continuous-Time Signals Using MATLAB 29
30. MATLAB program to generate rectangular pulse function
clear all;clc;
u=inline('(t=-5) (t5)','t');
t=-10:0.01:10;
plot(t,u(t))
xlabel ('t (seconds)');
ylabel ('Ámplitude')
axis([-10 10 -2 2])
1.7 Typical Signal Processing Operations
1.7.1 Correlation
Correlation of signals is necessary to compare one reference signal with one or more
signals to determine the similarity between them and to determine additional infor-
mation based on the similarity. Applications of cross correlation include cross-
spectral analysis, detection of signals buried in noise, pattern matching, and delay
measurements.
-10 -8 -6 -4 -2 0 2 4 6 8 10
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
t (seconds)
Ámplitude
Figure 1.22 Rectangular pulse function
30 1 Introduction
31. 1.7.2 Filtering
Filtering is basically a frequency domain operation. Filter is used to pass certain band
of frequency components without any distortion and to block other frequency
components. The range of frequencies that is allowed to pass through the filter is
called the passband, and the range of frequencies that is blocked by the filter is called
the stopband. A low-pass filter passes all low-frequency components below a certain
specified frequency Ωc, called the cutoff frequency, and blocks all high-frequency
components above Ωc. A high-pass filter passes all high-frequency components
above a certain cutoff frequency Ωc and blocks all low-frequency components
below Ωc. A band-pass filter passes all frequency components between two cutoff
frequencies Ωc1 and Ωc2 where Ωc1 Ωc2 and blocks all frequency components
below the frequency Ωc1 and above the frequency Ωc2. A band-stop filter blocks all
frequency components between two cutoff frequencies Ωc1 and Ωc2 where Ωc1 Ωc2
and passes all frequency components below the frequency Ωc1 and above the
frequency Ωc2. Notch filter is a narrow band-stop filter used to suppress a particular
frequency, called the notch frequency.
1.7.3 Modulation and Demodulation
Transmission media, such as cables and optical fibers, are used for transmission of
signals over long distances; each such medium has a bandwidth that is more suitable
for the efficient transmission of signals in the high-frequency range. Hence, for
transmission over such channels, it is necessary to transform the low-frequency
signal to a high-frequency signal by means of a modulation operation. The desired
low-frequency signal is extracted by demodulating the modulated high-frequency
signal at the receiver end.
1.7.4 Transformation
The transformation is the representation of signals in the frequency domain, and
inverse transform converts the signals from the frequency domain back to the time
domain. The transformation provides the spectrum analysis of a signal. From the
knowledge of the spectrum of a signal, the bandwidth required to transmit the signal
can be determined. The transform domain representations provide additional insight
into the behavior of the signal and make it easy to design and implement algorithms,
such as those for filtering, convolution, and correlation.
1.7 Typical Signal Processing Operations 31
32. 1.7.5 Multiplexing and Demultiplexing
Multiplexing is used in situations where the transmitting media is having higher
bandwidth, but the signals have lower bandwidth. Thus, multiplexing is the process
in which multiple signals, coming from different sources, are combined and trans-
mitted over a single channel. Multiplexing is performed by multiplexer placed at the
transmitter end. At the receiving end, the composite signal is separated by demul-
tiplexer performing the reverse process of multiplexing and routes the separated
signals to their corresponding receivers or destinations.
In electronic communications, the two basic forms of multiplexing are time-
division multiplexing (TDM) and frequency-division multiplexing (FDM). In
time-division multiplexing, transmission time on a single channel is divided into
non-overlapped time slots. Data streams from different sources are divided into units
with same size and interleaved successively into the time slots. In frequency-division
multiplexing (FDM), numerous low-frequency narrow bandwidth signals are com-
bined for transmission over a single communication channel. A different frequency
is assigned to each signal within the main channel. Code-division multiplexing
(CDM) is a communication networking technique in which multiple data signals
are combined for simultaneous transmission over a common frequency band.
1.8 Some Examples of Real-World Signals and Systems
1.8.1 Audio Recording System
An audio recording system shown in Figure 1.23(a) takes an audio or speech as input
and converts the audio signal into an electrical signal, which is recorded on a
magnetic tape or a compact disc. An example of recorded voice signal is shown in
Figure 1.23(b).
(a) (b)
Audio
output
signal
Audio
Recording
System
Figure 1.23 (a) Audio recording system, (b) the recorded voice signal “don’t fail me again”
32 1 Introduction