1) This document discusses frequency response and continuous-time Fourier series. It introduces the frequency response of linear time-invariant (LTI) systems, which describes how a system responds to multi-frequency inputs.
2) The frequency response is defined as the ratio of the output to input of an LTI system for exponential inputs. This allows determining the system's response to real sinusoidal and periodic signals through properties of linearity and time-invariance.
3) Fourier series are introduced as a way to represent periodic signals as a sum of sinusoids. The response of an LTI system to a periodic input can then be determined by applying the system's frequency response to each sinusoidal component of the Fourier series
This lecture covers signal and systems analysis, including:
1) Definitions of signals, systems, and their properties like time-invariance, linearity, stability, causality, and memory.
2) Classification of signals as continuous-time vs discrete-time, analog vs digital, deterministic vs random, periodic vs aperiodic.
3) Concepts of orthogonality, correlation, autocorrelation as they relate to signal comparison.
4) Review of the Fourier series and Fourier transform as tools to represent signals in the frequency domain.
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 outlines the content of a lecture on signals and systems. The key points are:
- Signals represent patterns of variation over time and can be continuous or discrete. Systems process input signals to produce output signals.
- The course will cover time and frequency domain analysis, Laplace transforms, Fourier transforms, sampling theory and z-transforms.
- Students will be assessed via exams, assignments and quizzes. Recommended reading materials are listed.
- The specific lecture will introduce signals, systems, their mathematical representations in continuous and discrete time, and properties like causality, linearity and time-invariance. Exercises are to read the first chapter of a referenced text.
This document summarizes a lecture on Fourier series and basis functions. It introduces Fourier series representation of periodic time functions using a basis of complex exponentials. A periodic signal can be expressed as a sum of these basis functions multiplied by coefficients. The coefficients can be determined by integrating the signal multiplied by basis functions over one period. Complex exponentials are eigenfunctions of linear time-invariant systems, and the corresponding eigenvalues can be used to determine the output of such systems when the input is an eigenfunction.
The document provides an overview of signals and systems topics to be covered in an EE 207 class, including detailed analysis of sinusoidal signals, phasor representation, frequency domain spectra, and practice problems. It defines a sinusoidal signal using amplitude, frequency, phase, and discusses representing the signal using a phasor or complex exponential. It also describes representing signals as the sum of complex conjugate signals, and plotting single-sided and double-sided frequency spectra. Practice problems cover determining if signals are periodic, calculating energy and power, representing signals using phasors, and sketching 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.
The following presentation is a part of the level 4 module -- Electrical and Electronic Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
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 lecture covers signal and systems analysis, including:
1) Definitions of signals, systems, and their properties like time-invariance, linearity, stability, causality, and memory.
2) Classification of signals as continuous-time vs discrete-time, analog vs digital, deterministic vs random, periodic vs aperiodic.
3) Concepts of orthogonality, correlation, autocorrelation as they relate to signal comparison.
4) Review of the Fourier series and Fourier transform as tools to represent signals in the frequency domain.
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 outlines the content of a lecture on signals and systems. The key points are:
- Signals represent patterns of variation over time and can be continuous or discrete. Systems process input signals to produce output signals.
- The course will cover time and frequency domain analysis, Laplace transforms, Fourier transforms, sampling theory and z-transforms.
- Students will be assessed via exams, assignments and quizzes. Recommended reading materials are listed.
- The specific lecture will introduce signals, systems, their mathematical representations in continuous and discrete time, and properties like causality, linearity and time-invariance. Exercises are to read the first chapter of a referenced text.
This document summarizes a lecture on Fourier series and basis functions. It introduces Fourier series representation of periodic time functions using a basis of complex exponentials. A periodic signal can be expressed as a sum of these basis functions multiplied by coefficients. The coefficients can be determined by integrating the signal multiplied by basis functions over one period. Complex exponentials are eigenfunctions of linear time-invariant systems, and the corresponding eigenvalues can be used to determine the output of such systems when the input is an eigenfunction.
The document provides an overview of signals and systems topics to be covered in an EE 207 class, including detailed analysis of sinusoidal signals, phasor representation, frequency domain spectra, and practice problems. It defines a sinusoidal signal using amplitude, frequency, phase, and discusses representing the signal using a phasor or complex exponential. It also describes representing signals as the sum of complex conjugate signals, and plotting single-sided and double-sided frequency spectra. Practice problems cover determining if signals are periodic, calculating energy and power, representing signals using phasors, and sketching 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.
The following presentation is a part of the level 4 module -- Electrical and Electronic Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
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.
1) A signal is a physical quantity that varies with respect to time, space, or other independent variables. Signals can be classified as discrete or continuous. 2) Unit impulse and unit step signals are defined for both discrete and continuous time. The discrete unit impulse is 1 at n=0 and 0 otherwise. The continuous unit impulse is 1 at t=0 and 0 otherwise. 3) Periodic signals repeat over a time period T, while aperiodic signals do not have this periodicity property. Even and odd signals satisfy certain symmetry properties when their argument is negated.
This document chapter discusses the characterization and representation of communication signals and systems. It describes how band-pass signals and systems can be represented by equivalent low-pass signals and systems using analytic signal representations and complex envelopes. It also discusses how the response of a band-pass system to a band-pass input signal can be determined from the equivalent low-pass representations. Key topics covered include the Fourier transform, Hilbert transform, and convolution properties used to relate band-pass and low-pass signal and system representations.
This lecture discusses Fourier series and Fourier transforms. Fourier series represent periodic signals as a sum of sinusoids, while Fourier transforms represent both periodic and non-periodic signals as a function of frequency. Examples of calculating the Fourier series and Fourier transform of common signals like sinusoids, step functions, and exponentials are provided. Exercises are suggested to practice calculating Fourier transforms and using them to analyze the frequency content of signals.
This document provides an overview of how linear systems analysis and Fourier transforms can be applied to analyze 2-dimensional optical images and optical systems. It explains that plane waves serve as the eigenfunctions for linear shift invariant optical systems, just as complex exponentials serve as the eigenfunctions for linear time invariant electrical systems. The Fourier transform can be used to decompose an optical image into its plane wave spectrum, and optical systems can be analyzed by multiplying the image spectrum by the system's optical transfer function and taking the inverse Fourier transform. As an example, it describes how a thin lens can be modeled as a phase shifting device and its optical transfer function calculated.
This document discusses concepts related to signals and systems. It begins by defining a signal as a time-varying quantity of information and a system as an entity that processes input signals to produce output signals. It then covers signal classification including continuous vs discrete time, analog vs digital, periodic vs aperiodic, deterministic vs random, and causal vs non-causal signals. Signal operations like time shifting, scaling, and inversion are described. Key concepts discussed in detail include signal size using energy and power, signal components and orthogonality, correlation as a measure of signal similarity, and trigonometric Fourier series. Worked examples are provided to illustrate various topics.
This document summarizes key concepts in digital and analog communications:
1) It defines source coding, channel encoding/decoding, digital modulation/demodulation, and how digital communication system performance is measured in terms of error probability.
2) Thermal noise in receivers is identified as the dominant source of noise limiting performance in VHF and UHF bands.
3) Storing data on magnetic/optical disks is analogous to transmitting a signal over a radio channel, with similar signal processing used for recovery.
4) Digital processing avoids signal degradation but requires more bandwidth, while analog processing is sensitive to variations but does not lose quality over time.
5) Fourier analysis is used to derive the
This document provides an overview of signals and systems. It defines key terms like signal, system, continuous and discrete time signals, analog and digital signals, periodic and aperiodic signals. It also discusses different types of signals like deterministic and probabilistic signals, energy and power signals. The document then classifies systems as linear/nonlinear, time-invariant/variant, causal/non-causal, and with/without memory. It provides examples of different signals and properties of signals like magnitude scaling, time shifting, reflection and scaling. Overall, the document introduces fundamental concepts in signals and systems.
The document discusses Fourier analysis techniques. It covers topics like line spectra and Fourier series, including periodic signals and average power. Key aspects covered include phasor representation of sinusoids, convergence conditions of Fourier series, and Parseval's power theorem relating signal power to Fourier coefficients.
The document discusses discrete Fourier series, discrete Fourier transform, and discrete time Fourier transform. It provides definitions and explanations of each topic. Discrete Fourier series represents periodic discrete-time signals using a summation of sines and cosines. The discrete Fourier transform analyzes a finite-duration discrete signal by treating it as an excerpt from an infinite periodic signal. The discrete time Fourier transform provides a frequency-domain representation of discrete-time signals and is useful for analyzing samples of continuous functions. Examples of applications are also given such as signal processing, image analysis, and wireless communications.
Instrumentation Engineering : Signals & systems, THE GATE ACADEMYklirantga
THE GATE ACADEMY's GATE Correspondence Materials consist of complete GATE syllabus in the form of booklets with theory, solved examples, model tests, formulae and questions in various levels of difficulty in all the topics of the syllabus. The material is designed in such a way that it has proven to be an ideal material in-terms of an accurate and efficient preparation for GATE.
Quick Refresher Guide : is especially developed for the students, for their quick revision of concepts preparing for GATE examination. Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
GATE QUESTION BANK : is a topic-wise and subject wise collection of previous year GATE questions ( 2001 – 2013). Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
Bangalore Head Office:
THE GATE ACADEMY
# 74, Keshava Krupa(Third floor), 30th Cross,
10th Main, Jayanagar 4th block, Bangalore- 560011
E-Mail: info@thegateacademy.com
Ph: 080-61766222
This document discusses network theory and Fourier analysis. It begins by introducing Fourier series, which represent periodic functions as the sum of sinusoidal waves. Both trigonometric and exponential forms of Fourier series are covered. It then discusses Fourier transforms, which extend the frequency spectrum concept to non-periodic functions by assuming an infinite period. Key topics include Fourier series coefficients, amplitude and phase spectra, waveform symmetries, and applications of Fourier analysis in network analysis. Fourier transforms represent the frequency spectrum of non-periodic signals through an integral transform analogous to Fourier series.
These notes were developed for use in the course Signals and Systems. The notes cover traditional, introductory concepts in the time domain and frequency domain analysis of signals and systems.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
This document provides an overview of time-domain analysis of linear time-invariant (LTI) systems. It discusses impulse response and unit step response, which are used to characterize the memory and stability of systems. Transient responses like rise time and settling time are also examined. Convolution is introduced as a way to calculate the output of LTI systems using the impulse response. Difference equations are presented as a method to model discrete-time linear shift-invariant (LSI) systems.
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 introduces signals and systems. It defines a signal as a function of time that conveys information and a system as a collection of items that transforms an input to an output. There are two types of signals: continuous-time signals that have a value for all points in time and discrete-time signals that have a value for only specific points in time, typically formed by sampling a continuous signal. Similarly, there are continuous-time and discrete-time systems. Most real systems are hybrid systems that use both continuous-time and discrete-time components. The purpose of system design is to create predictable input-output relationships, which are easiest to achieve if the system is linear and time-invariant.
Ch7 noise variation of different modulation scheme pg 63Prateek Omer
This document summarizes the noise performance of various modulation schemes. It begins by introducing a receiver model and defining figures of merit used to evaluate performance. It then analyzes the noise performance of coherent demodulation for DSB-SC and SSB modulation. The following key points are made:
1) Coherent detection of DSB-SC signals results in signal and noise being additive at both the input and output of the detector. The detector completely rejects the quadrature noise component.
2) For DSB-SC, the output SNR and reference SNR are equal, resulting in a figure of merit of 1.
3) Analysis of SSB modulation shows it achieves a 3 dB improvement in output SNR over
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 discusses signals and systems. It defines signals as functions that convey information about physical phenomena. Signals can be continuous or discrete, deterministic or random, even or odd, periodic or aperiodic. Systems process input signals to produce output signals. A system can be linear or nonlinear, time-invariant or time-variant, causal or non-causal, stable or unstable. Static systems are memoryless while dynamic systems possess memory. Common signal types include voltage, current, temperature, and vibration. Key system properties and classifications are discussed in detail.
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.
This document discusses signal transmission through linear systems. It begins by introducing linear time-invariant systems and their characterization in both the time and frequency domains using impulse response and transfer functions. It then covers various properties of ideal transmission lines including distortionless transmission where the output signal has the same shape as the input. The document also discusses concepts such as bandwidth, causality, stability, energy spectral density, and power spectral density as they relate to characterizing signals and linear systems.
1) A signal is a physical quantity that varies with respect to time, space, or other independent variables. Signals can be classified as discrete or continuous. 2) Unit impulse and unit step signals are defined for both discrete and continuous time. The discrete unit impulse is 1 at n=0 and 0 otherwise. The continuous unit impulse is 1 at t=0 and 0 otherwise. 3) Periodic signals repeat over a time period T, while aperiodic signals do not have this periodicity property. Even and odd signals satisfy certain symmetry properties when their argument is negated.
This document chapter discusses the characterization and representation of communication signals and systems. It describes how band-pass signals and systems can be represented by equivalent low-pass signals and systems using analytic signal representations and complex envelopes. It also discusses how the response of a band-pass system to a band-pass input signal can be determined from the equivalent low-pass representations. Key topics covered include the Fourier transform, Hilbert transform, and convolution properties used to relate band-pass and low-pass signal and system representations.
This lecture discusses Fourier series and Fourier transforms. Fourier series represent periodic signals as a sum of sinusoids, while Fourier transforms represent both periodic and non-periodic signals as a function of frequency. Examples of calculating the Fourier series and Fourier transform of common signals like sinusoids, step functions, and exponentials are provided. Exercises are suggested to practice calculating Fourier transforms and using them to analyze the frequency content of signals.
This document provides an overview of how linear systems analysis and Fourier transforms can be applied to analyze 2-dimensional optical images and optical systems. It explains that plane waves serve as the eigenfunctions for linear shift invariant optical systems, just as complex exponentials serve as the eigenfunctions for linear time invariant electrical systems. The Fourier transform can be used to decompose an optical image into its plane wave spectrum, and optical systems can be analyzed by multiplying the image spectrum by the system's optical transfer function and taking the inverse Fourier transform. As an example, it describes how a thin lens can be modeled as a phase shifting device and its optical transfer function calculated.
This document discusses concepts related to signals and systems. It begins by defining a signal as a time-varying quantity of information and a system as an entity that processes input signals to produce output signals. It then covers signal classification including continuous vs discrete time, analog vs digital, periodic vs aperiodic, deterministic vs random, and causal vs non-causal signals. Signal operations like time shifting, scaling, and inversion are described. Key concepts discussed in detail include signal size using energy and power, signal components and orthogonality, correlation as a measure of signal similarity, and trigonometric Fourier series. Worked examples are provided to illustrate various topics.
This document summarizes key concepts in digital and analog communications:
1) It defines source coding, channel encoding/decoding, digital modulation/demodulation, and how digital communication system performance is measured in terms of error probability.
2) Thermal noise in receivers is identified as the dominant source of noise limiting performance in VHF and UHF bands.
3) Storing data on magnetic/optical disks is analogous to transmitting a signal over a radio channel, with similar signal processing used for recovery.
4) Digital processing avoids signal degradation but requires more bandwidth, while analog processing is sensitive to variations but does not lose quality over time.
5) Fourier analysis is used to derive the
This document provides an overview of signals and systems. It defines key terms like signal, system, continuous and discrete time signals, analog and digital signals, periodic and aperiodic signals. It also discusses different types of signals like deterministic and probabilistic signals, energy and power signals. The document then classifies systems as linear/nonlinear, time-invariant/variant, causal/non-causal, and with/without memory. It provides examples of different signals and properties of signals like magnitude scaling, time shifting, reflection and scaling. Overall, the document introduces fundamental concepts in signals and systems.
The document discusses Fourier analysis techniques. It covers topics like line spectra and Fourier series, including periodic signals and average power. Key aspects covered include phasor representation of sinusoids, convergence conditions of Fourier series, and Parseval's power theorem relating signal power to Fourier coefficients.
The document discusses discrete Fourier series, discrete Fourier transform, and discrete time Fourier transform. It provides definitions and explanations of each topic. Discrete Fourier series represents periodic discrete-time signals using a summation of sines and cosines. The discrete Fourier transform analyzes a finite-duration discrete signal by treating it as an excerpt from an infinite periodic signal. The discrete time Fourier transform provides a frequency-domain representation of discrete-time signals and is useful for analyzing samples of continuous functions. Examples of applications are also given such as signal processing, image analysis, and wireless communications.
Instrumentation Engineering : Signals & systems, THE GATE ACADEMYklirantga
THE GATE ACADEMY's GATE Correspondence Materials consist of complete GATE syllabus in the form of booklets with theory, solved examples, model tests, formulae and questions in various levels of difficulty in all the topics of the syllabus. The material is designed in such a way that it has proven to be an ideal material in-terms of an accurate and efficient preparation for GATE.
Quick Refresher Guide : is especially developed for the students, for their quick revision of concepts preparing for GATE examination. Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
GATE QUESTION BANK : is a topic-wise and subject wise collection of previous year GATE questions ( 2001 – 2013). Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
Bangalore Head Office:
THE GATE ACADEMY
# 74, Keshava Krupa(Third floor), 30th Cross,
10th Main, Jayanagar 4th block, Bangalore- 560011
E-Mail: info@thegateacademy.com
Ph: 080-61766222
This document discusses network theory and Fourier analysis. It begins by introducing Fourier series, which represent periodic functions as the sum of sinusoidal waves. Both trigonometric and exponential forms of Fourier series are covered. It then discusses Fourier transforms, which extend the frequency spectrum concept to non-periodic functions by assuming an infinite period. Key topics include Fourier series coefficients, amplitude and phase spectra, waveform symmetries, and applications of Fourier analysis in network analysis. Fourier transforms represent the frequency spectrum of non-periodic signals through an integral transform analogous to Fourier series.
These notes were developed for use in the course Signals and Systems. The notes cover traditional, introductory concepts in the time domain and frequency domain analysis of signals and systems.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
This document provides an overview of time-domain analysis of linear time-invariant (LTI) systems. It discusses impulse response and unit step response, which are used to characterize the memory and stability of systems. Transient responses like rise time and settling time are also examined. Convolution is introduced as a way to calculate the output of LTI systems using the impulse response. Difference equations are presented as a method to model discrete-time linear shift-invariant (LSI) systems.
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 introduces signals and systems. It defines a signal as a function of time that conveys information and a system as a collection of items that transforms an input to an output. There are two types of signals: continuous-time signals that have a value for all points in time and discrete-time signals that have a value for only specific points in time, typically formed by sampling a continuous signal. Similarly, there are continuous-time and discrete-time systems. Most real systems are hybrid systems that use both continuous-time and discrete-time components. The purpose of system design is to create predictable input-output relationships, which are easiest to achieve if the system is linear and time-invariant.
Ch7 noise variation of different modulation scheme pg 63Prateek Omer
This document summarizes the noise performance of various modulation schemes. It begins by introducing a receiver model and defining figures of merit used to evaluate performance. It then analyzes the noise performance of coherent demodulation for DSB-SC and SSB modulation. The following key points are made:
1) Coherent detection of DSB-SC signals results in signal and noise being additive at both the input and output of the detector. The detector completely rejects the quadrature noise component.
2) For DSB-SC, the output SNR and reference SNR are equal, resulting in a figure of merit of 1.
3) Analysis of SSB modulation shows it achieves a 3 dB improvement in output SNR over
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 discusses signals and systems. It defines signals as functions that convey information about physical phenomena. Signals can be continuous or discrete, deterministic or random, even or odd, periodic or aperiodic. Systems process input signals to produce output signals. A system can be linear or nonlinear, time-invariant or time-variant, causal or non-causal, stable or unstable. Static systems are memoryless while dynamic systems possess memory. Common signal types include voltage, current, temperature, and vibration. Key system properties and classifications are discussed in detail.
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.
This document discusses signal transmission through linear systems. It begins by introducing linear time-invariant systems and their characterization in both the time and frequency domains using impulse response and transfer functions. It then covers various properties of ideal transmission lines including distortionless transmission where the output signal has the same shape as the input. The document also discusses concepts such as bandwidth, causality, stability, energy spectral density, and power spectral density as they relate to characterizing signals and linear systems.
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.
Vidyalankar final-essentials of communication systemsanilkurhekar
This document provides an overview of analog and digital communication systems. It discusses the basics of analog signals, frequency spectrum, and modulation. It then covers digital signals, terms, and performance metrics like data rate and bit error probability. Key concepts covered include Shannon capacity, signal energy and power, communication system blocks, filtering, and modulation. It also introduces concepts from probability theory and random processes used in analysis of communication systems like mean, autocorrelation, power spectral density, Gaussian processes, and noise. Examples of modulation techniques and noise sources in communication systems are briefly discussed.
The document provides an introduction to digital signal processing (DSP) algorithms and architecture. It discusses key topics including DSP systems, sampling processes, discrete Fourier transforms, linear time-invariant systems, digital filters, and decimation and interpolation. The learning objectives are to understand these fundamental DSP concepts and use MATLAB as an analysis and design tool. The contents cover DSP systems, sampling, the discrete Fourier transform and fast Fourier transform, linear time-invariant systems, digital filters, and decimation and interpolation. Textbooks and references on the subject are also listed.
This document contains lecture notes on signals and systems for a course at Chadalawada Ramanamma Engineering College. It includes:
1. An introduction to signals, systems, and some common elementary signals like the unit step, unit impulse, ramp, sinusoid, and exponential signals.
2. A classification of signals as continuous/discrete, deterministic/non-deterministic, even/odd, periodic/aperiodic, energy/power, and real/imaginary.
3. A discussion of basic operations on signals like amplitude scaling, addition, and subtraction.
Most electronic communication signals can be represented by sine and cosine waves. Sine and cosine waves are periodic and can be converted between each other using trigonometric identities. Fourier analysis techniques such as Fourier series and Fourier transforms are used to represent signals in the time and frequency domains.
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.
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.
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 discusses principles of communication and representation of signals. It begins with an introduction to the communication process and challenges involved. Signals exist in the time and frequency domains, and Fourier analysis using the Fourier series and Fourier transform helps characterize signals in the frequency domain. Periodic signals can be represented by a Fourier series which decomposes the signal into a sum of complex exponentials at discrete frequencies that are integer multiples of the fundamental frequency. Examples are provided to illustrate calculation of Fourier coefficients and representation of periodic signals in the exponential and trigonometric forms of the Fourier series. Spectral plots from a spectrum analyzer are also presented for various waveforms.
Fourier Transform in Signal and System of TelecomAmirKhan877722
This document provides an overview of key properties of the Fourier transform that can be used to simplify calculations and gain insights into applications. It discusses properties like linearity, time shifting, time scaling, time reversal, and multiplying signals by powers of time. Examples are provided to illustrate how these properties can be applied, such as using linearity to break a signal into simpler components before taking the Fourier transform. The document emphasizes that understanding these properties allows signal processing engineers to analyze and design systems like communications technology.
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.
I am John G. I am a Signals and Systems Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from Glasgow University. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signals and Systems.
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1. The document discusses the Fourier representation of signals and linear time-invariant (LTI) systems. It describes four classes of Fourier representations that are applicable to different types of signals based on their time properties - periodic or non-periodic, and discrete-time or continuous-time.
2. The Fourier series representation applies to continuous-time periodic signals, while the discrete Fourier transform applies to discrete-time periodic signals. Non-periodic signals are represented by the Fourier transform.
3. An example shows the calculation of discrete Fourier transform coefficients for a periodic discrete-time signal. The coefficients are plotted in both rectangular and polar coordinates in the frequency domain.
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.
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.
Sampling and Reconstruction (Online Learning).pptxHamzaJaved306957
1. Sampling and reconstruction of signals was analyzed using the impulse sampling math model.
2. The analysis showed that a bandlimited signal can be perfectly reconstructed from its samples as long as the sampling rate is at least twice the bandwidth of the signal.
3. If the sampling rate is lower than the minimum required rate, aliasing error occurs where frequency components fold back into the baseband.
TEST BANK For An Introduction to Brain and Behavior, 7th Edition by Bryan Kol...rightmanforbloodline
TEST BANK For An Introduction to Brain and Behavior, 7th Edition by Bryan Kolb, Ian Q. Whishaw, Verified Chapters 1 - 16, Complete Newest Versio
TEST BANK For An Introduction to Brain and Behavior, 7th Edition by Bryan Kolb, Ian Q. Whishaw, Verified Chapters 1 - 16, Complete Newest Version
TEST BANK For An Introduction to Brain and Behavior, 7th Edition by Bryan Kolb, Ian Q. Whishaw, Verified Chapters 1 - 16, Complete Newest Version
Here is the updated list of Top Best Ayurvedic medicine for Gas and Indigestion and those are Gas-O-Go Syp for Dyspepsia | Lavizyme Syrup for Acidity | Yumzyme Hepatoprotective Capsules etc
Cell Therapy Expansion and Challenges in Autoimmune DiseaseHealth Advances
There is increasing confidence that cell therapies will soon play a role in the treatment of autoimmune disorders, but the extent of this impact remains to be seen. Early readouts on autologous CAR-Ts in lupus are encouraging, but manufacturing and cost limitations are likely to restrict access to highly refractory patients. Allogeneic CAR-Ts have the potential to broaden access to earlier lines of treatment due to their inherent cost benefits, however they will need to demonstrate comparable or improved efficacy to established modalities.
In addition to infrastructure and capacity constraints, CAR-Ts face a very different risk-benefit dynamic in autoimmune compared to oncology, highlighting the need for tolerable therapies with low adverse event risk. CAR-NK and Treg-based therapies are also being developed in certain autoimmune disorders and may demonstrate favorable safety profiles. Several novel non-cell therapies such as bispecific antibodies, nanobodies, and RNAi drugs, may also offer future alternative competitive solutions with variable value propositions.
Widespread adoption of cell therapies will not only require strong efficacy and safety data, but also adapted pricing and access strategies. At oncology-based price points, CAR-Ts are unlikely to achieve broad market access in autoimmune disorders, with eligible patient populations that are potentially orders of magnitude greater than the number of currently addressable cancer patients. Developers have made strides towards reducing cell therapy COGS while improving manufacturing efficiency, but payors will inevitably restrict access until more sustainable pricing is achieved.
Despite these headwinds, industry leaders and investors remain confident that cell therapies are poised to address significant unmet need in patients suffering from autoimmune disorders. However, the extent of this impact on the treatment landscape remains to be seen, as the industry rapidly approaches an inflection point.
Does Over-Masturbation Contribute to Chronic Prostatitis.pptxwalterHu5
In some case, your chronic prostatitis may be related to over-masturbation. Generally, natural medicine Diuretic and Anti-inflammatory Pill can help mee get a cure.
These lecture slides, by Dr Sidra Arshad, offer a quick overview of the physiological basis of a normal electrocardiogram.
Learning objectives:
1. Define an electrocardiogram (ECG) and electrocardiography
2. Describe how dipoles generated by the heart produce the waveforms of the ECG
3. Describe the components of a normal electrocardiogram of a typical bipolar lead (limb II)
4. Differentiate between intervals and segments
5. Enlist some common indications for obtaining an ECG
6. Describe the flow of current around the heart during the cardiac cycle
7. Discuss the placement and polarity of the leads of electrocardiograph
8. Describe the normal electrocardiograms recorded from the limb leads and explain the physiological basis of the different records that are obtained
9. Define mean electrical vector (axis) of the heart and give the normal range
10. Define the mean QRS vector
11. Describe the axes of leads (hexagonal reference system)
12. Comprehend the vectorial analysis of the normal ECG
13. Determine the mean electrical axis of the ventricular QRS and appreciate the mean axis deviation
14. Explain the concepts of current of injury, J point, and their significance
Study Resources:
1. Chapter 11, Guyton and Hall Textbook of Medical Physiology, 14th edition
2. Chapter 9, Human Physiology - From Cells to Systems, Lauralee Sherwood, 9th edition
3. Chapter 29, Ganong’s Review of Medical Physiology, 26th edition
4. Electrocardiogram, StatPearls - https://www.ncbi.nlm.nih.gov/books/NBK549803/
5. ECG in Medical Practice by ABM Abdullah, 4th edition
6. Chapter 3, Cardiology Explained, https://www.ncbi.nlm.nih.gov/books/NBK2214/
7. ECG Basics, http://www.nataliescasebook.com/tag/e-c-g-basics
Rasamanikya is a excellent preparation in the field of Rasashastra, it is used in various Kushtha Roga, Shwasa, Vicharchika, Bhagandara, Vatarakta, and Phiranga Roga. In this article Preparation& Comparative analytical profile for both Formulationon i.e Rasamanikya prepared by Kushmanda swarasa & Churnodhaka Shodita Haratala. The study aims to provide insights into the comparative efficacy and analytical aspects of these formulations for enhanced therapeutic outcomes.
2. Recall course objectives
Main Course Objective:
Fundamentals of systems/signals interaction
(we’d like to understand how systems transform or affect signals)
Specific Course Topics:
-Basic test signals and their properties
-Systems and their properties
-Signals and systems interaction
Time Domain: convolution
Frequency Domain: frequency response
-Signals & systems applications:
audio effects, filtering, AM/FM radio
-Signal sampling and signal reconstruction
3. CT Signals and Systems in the FD -part I
Goals
I. Frequency Response of (stable) LTI systems
-Frequency Response, amplitude and phase definition
-LTI system response to multi-frequency inputs
II. (stable) LTI system response to periodic signals in the FD
-The Fourier Series of a periodic signal
-Periodic signal magnitude and phase spectrum
-LTI system response to general periodic signals
III. Filtering effects of (stable) LTI systems in the FD
- Noise removal and signal smoothing
4. Frequency Response of LTI systems
We have seen how some specific LTI system responses (the IR and
the step response) can be used to find the response to the system
to arbitrary inputs through the convolution operation.
However, all practical (periodic or pulse-like) signals that can be
generated in the lab or in a radio station can be expressed as
superposition of co-sinusoids with different frequencies, phases,
and amplitudes. (An oscillatory input is easier to reproduce in the
lab than an impulse delta, which has to be approximated.)
Because of this, it is of interest to study (stable) LTI system
responses to general multi-frequency inputs. This is what
defines the frequency response of the system.
We will later see how to use this information to obtain the
response of LTI systems to (finite-energy) signals (using
Fourier and Laplace transforms)
5. Response of Systems to Exponentials
Let a stable LTI system be excited by an exponential input
Here, and can be complex numbers!
From what we learned on the response of LTI systems:
The response to an exponential is another exponential
Problem: Determine
!
x(t) = Ae"t
!
B
!
!
"
!
A
!
y(t) = Be"t
!
x(t) = Ae"t
6. Response of Systems to Exponentials
Consider a linear ODE describing the LTI system as
Let and
Substituting in the ODE we see that
The proportionality constant is equal to the ratio:
B
A
=
bk"k
k= 0
M
#
ak"k
k= 0
N
#
=
bM "M
+ ....+ b2"2
+ b1" + b0
aN "N
+ ....+ a2"2
+ a1" + a0
!
ak
dk
y(t)
dtk
=
k= 0
N
" bk
dk
x(t)
dtk
k= 0
M
"
!
x(t) = Ae"t
!
y(t) = Be"t
!
Be"t
ak"k
=
k= 0
N
# Ae"t
bk"k
k= 0
M
#
!
"
!
B =
bk"k
k= 0
M
#
ak"k
k= 0
N
#
A
7. Response of Systems to Exponentials
Surprise #1:
The ratio
where
is the Transfer Function which we have obtained before
with Laplace Transforms!
!
B
A
= H(")
!
H(s) =
bM sM
+ ....+ b2s2
+ b1s + b0
aN sN
+ ....+ a2s2
+ a1s + a0
8. Response of Systems to Exponentials
Surprise #2:
If
using linearity
With bit of complex algebra we write
and the response
where
!
x(t) = Acos("t) = A e j"t
+ e# j"t
( )/2
!
y(t) = H( j")e j"t
+ H(#j")e# j"t
( )/2
!
H( j") = Be j#
B = H( j") " = #H( j$)
!
H("j#) = Be" j$
!
y(t) = Be j"t +#
+ Be$ j"t$#
( )/2 = Bcos( j"t + #)
9. Response of Systems to Exponentials
The ratio
is called the Frequency Response
Observe that is a complex function of
We can graph the FR by plotting:
its rectangular coordinates , against
or its polar coordinates , against
Polar coordinates are usually more informative
!
B
A
= H( j") =
bM ( j")M
+ ....+ b2( j")2
+ b1( j") + b0
aN ( j")N
+ ....+ a2( j")2
+ a1( j") + a0
!
H( j")
!
| H( j") |
!
"
!
H( j") = Re(H( j")) + jIm(H( j"))
!
Re H( j" )
!
Im H( j" )
!
"
!
"H( j# )
!
"
10. Frequency Response Plots
Compare rectangular (left) versus polar (right) plots
Polar: only low frequency co-sinusoids are passed, the shift in
phase is more or less proportional to frequency
Rectangular: contains the same information as polar
representation. More difficult to see what it does to co-
sinusoids. Mostly used in computer calculations
11. System classification according to FR
Depending on the plot of we will classify systems into:
Low-pass filters: for (*)
High-pass filters: for
Band-pass filters: for and
In order to plot the logarithmic scale is frequently used.
This scale defines decibel units
(here we use log_10)
(*) these plots correspond to the so-called ideal filters, because
they keep exactly a set of low, high, and band frequencies
!
| H(") |
!
| H( j") |# 0
!
|" |> K
!
| H( j") |# 0
!
|" |< K
!
| H( j") |# 0
!
|" |< K1
!
|" |> K2
!
| H( j") |
!
| H( j") |dB = 20log | H( j") |
!
| H(") |
!
| H(") |
!
| H(") |
!
"
!
"
!
"
12. dB representation of Frequency Response
A plot in dB makes it possible to see small values of the FR
magnitude . This is important when we want to
understand the quality of the filter. In particular we can
observe the critical values of the frequency for which the
character of the filter changes.
Example: Linear plot of the low-pass filter
!
| H( j") |
!
H( j") =
1
1+ j"
13. dB representation of Frequency Response
The same function in dB units has a plot:
14. dB representation of Frequency Response
Linear plot of the low-pass filter
!
H( j") =
1
(1+ j")(100 + j")
15. dB representation of Frequency Response
… and the plot of the same filter in dB:
16. Response of Systems to Co-sinusoids
The FR of an LTI system has the following properties:
Because of this, the response of the LTI system to real sinusoid
or co-sinusoid is as follows:
Knowledge of is enough to know how the system
responds to real co-sinusoidal signals
This property is what allows compute the FR experimentally
!
| H("j#) |=| H( j#) |
!
"H(# j$) = #"H( j$)
!
cos("t + #)
!
| H( j") |cos("t + # + $H( j"))
!
H( j")
17. Response to system to multi-frequency inputs
Use linearity to generalize the response of stable LTI
systems to multi-frequency inputs
The system acts on each cosine independently
!
y(t) = A1 | H( j"1) |cos("1t + #1 + $H( j"1))
+ A2 | H( j"2) |cos("2t + #2 + $H( j"2))
x(t) = A1 cos("1t + #1)
+ A2 cos("2t + #2)
18. CT Signals and Systems in the FD -part I
Goals
I. Frequency Response of (stable) LTI systems
-Frequency Response, amplitude and phase definition
-LTI system response to multi-frequency inputs
II. (stable) LTI system response to periodic signals in the FD
-The Fourier Series of a periodic signal
-Periodic signal magnitude and phase spectrum
-LTI system response to general periodic signals
III. Filtering effects of (stable) LTI systems in the FD
- Noise removal and signal smoothing
19. Signal decompositions in the TD and FD
In the Time Domain, a signal is decomposed as a “sum” of time
shifted and amplified rectangles
In the Frequency Domain, a signal is decomposed as a “sum” of time
shifted and amplified co-sinusoids
A Frequency Domain representation of a signal tells you how much
energy of the signal is distributed over each cosine/sine
The different ways of obtaining a FD representation of a signal
constitute the Fourier Transform
There are 4 categories of Fourier Transforms, and we will study 2:
Fourier Series: for periodic, analog signals
Fourier Transform: for aperiodic, analog signals
The other 2 categories are for discrete-time signals
20. Fourier Series of periodic signals
Let be a “nice” periodic signal with period
Using Fourier Series it is possible to write as
The coefficients are computed from the original
signal as
is the harmonic function, and
is the harmonic number
!
x(t)
!
T0
!
x(t) = X[k]e jk"0t
k=#$
$
%
!
x(t)
!
"0 =
2#
T0
!
X[k] =
1
T0
x(t)e" jk#0t
dt
0
T0
$!
X[k]
X[k]
!
k
24. “Proof” of Fourier Series
Fact #1: For any integer
if , and if
Proof: If a Fourier Series exists then
Multiply by on both sides
!
e" ji#0t
dt
0
T0
$ = 0
!
e" ji#0t
dt
0
T0
$ = T0
!
i = 0
!
i " 0
!
i
!
x(t) = X[i]e ji"0t
i=#$
$
%
x(t)e" jk#0t
= X[i]e ji#0t
e" jk#0t
=
i="$
$
% X[i]e j(i"k)#0t
i="$
$
%
!
e" jk#0t
25. “Proof” of Fourier Series
Proof (continued):
Integrate over a period on both sides
and use Fact #1 to show that
which leads to the Fourier formula
!
x(t)e" jk#0t
dt
0
T0
$ = X[i] e j(i"k)#0t
dt
0
T0
$
i="%
%
&
!
X[i] e j(i"k)#0t
dt
0
T0
$
i="%
%
& = T0X[k]
X[k] =
1
T0
x(t)e" jk#0t
dt
0
T0
$
26. Graphical description of a periodic signal as a
function of frequency
The Frequency Domain graphical representation of a periodic signal is a
plot of its Fourier coefficients (FC)
Since the coefficients are in general complex numbers, the (polar)
representation consists of:
1) a plot of for different
(the magnitude spectrum)
2)a plot of for different
(the phase spectrum)
The magnitude spectrum tells us
how many frequencies are
necessary to obtain
a good approximation of
the signal
Square wave: most of the signal can be approximated using low
frequencies, however discontinuities translate into lots of high
frequencies
!
| X[k] |
!
k
!
"X[k]
!
k
27. System response to periodic signal
To find the response of an (stable) LTI system on a
periodic signal of period :
We use the system FR and change each signal
frequency component as follows:
!
X[k] =
1
T0
x(t)e" jk#0t
dt
0
T0
$ , k = ...,"2,"1,0,1,2,...
!
y(t) = H(kj"0)X[k]e jk"0t
k=#$
$
% = H[k]X[k]e jk"0t
k=#$
$
%
!
x(t) = X[k]e jk"0t
k=#$
$
%
!
"0 =
2#
T0
= 2#f0
!
T0
y(t) = X[k] | H(kj"0) |e j(k"0t +#H(kj"0 ))
k=$%
%
& = | X[k] || H[k] |e j(k"0t +#H[k]+#X [k])
k=$%
%
&
28. Graphical signal/systems interaction in the FD
In this way, the interaction of periodic signals and systems in the FD
can be seen a simple vector multiplication/vector sum:
Once we have
computed the signal spectrum
and system FR, the magnitude spectrum
of the output can be computed
in a simple way (compare
it with convolution…)
The output vector phase corresponds to
a vector sum
!
| X[k] |
!
| H[k] |
!
|Y[k] |
!
"
!
=
"Y[k] = "X[k]+ "H[k]
29. CT Signals and Systems in the FD -part I
Goals
I. Frequency Response of (stable) LTI systems
-Frequency Response, amplitude and phase definition
-LTI system response to multi-frequency inputs
II. (stable) LTI system response to periodic signals in the FD
-The Fourier Series of a periodic signal
-Periodic signal magnitude and phase spectrum
-LTI system response to general periodic signals
III. Filtering effects of (stable) LTI systems in the FD
- Noise removal and signal smoothing
30. Noise Removal in the Frequency Domain
From our previous discussion on filters/multi-frequency inputs
we observe the following: rapid oscillations “on top” of slower
ones in signals can be “smoothed out” with low-pass filters
Example: In the signal the high frequency
component is while the low frequency component is
!
x(t) = cos(t /2) + cos("t)
!
cos("t)
!
cos(t /2)
31. Noise Removal in the Frequency Domain
The response of a low-pass filter to this signal becomes:
The output signal retains the slow oscillation of the input signal
while almost removing the high-frequency oscillation
!
y(t) = 0.89cos(t /2 " 26.5) + 0.303cos(#t " 72.3)
32. Noise Removal in the Frequency Domain
A low-frequency periodic signal subject to noise can be seen as
a low-frequency cosine superimposed with a high-frequency
oscillation. For example consider the following noisy signal:
33. Noise Removal in the Frequency Domain
The steady-state response of a low-pass filter (e.g. the RC low-
pass filter) to the signal is the following:
(This has exactly the same shape as the uncorrupted signal,
and we are able to remove the noise pretty well)
34. Noise Removal in the Frequency Domain
The use of low-pass filters for noise removal is a
widespread technique. Theoretically, we have
seen why this technique works for periodic signals
However, the same technique works for any
signal of finite energy. This is explained through
the theory of Fourier/Laplace transforms of
signals and systems that we will see next
Depending of the type of noise and signal, some
filters may work better than others. This leads
to the whole subject of “filter design” in signals and
systems (…)
35. Noise removal in the Frequency Domain
The problem is that as well as the noise we may find:
in audio signals: plenty of other high-frequency content
in image signals: edges contribute significantly to high-
frequency components
Thus an “ideal” low-pass filter tends to blur the data
E.g. edges in images can become blurred. Observe the
output to two different low-pass filters:
This illustrates that one has to be careful in the selection of filter (…)
36. Summary
Reasons why we study signals in the frequency domain (FD):
(a) Oscillatory inputs and sinusoids are easier to implement
and reproduce in a lab than an impulse signals (which usually we
need to approximate) and even unit steps
(b) A pulse-like signal can be expressed as a combination of
co-sinusoids, so if we know how to compute the response to a co-
sinusoid, then we can know what is the response to a pulse-like
signal (and to general periodic signals)
(c) Once signals and systems are in the FD, computations to
obtain outputs are very simple: we don’t need convolution
anymore! This is one of the reasons why signal processing is mainly
done in the FD. For example, deconvolution (inverting convolutions)
is more easily done in the FD
(d) The FD can be more intuitive than the TD. For example,
how noise removal works is easier to understand in the FD
37. Summary
Important points to remember:
1. The output of (stable) LTI systems to a complex exponential,
sinusoid or co-sinusoid, resp., is again another complex
exponential, sinusoid or co-sinusoid, resp.,
2. These outputs can be computed by knowing the magnitude and phase of
the Frequency Response H(jw) associated with the LTI system
3. Systems can be classified according to their Frequency Response as
low-pass, high-pass or band-pass filters
4. We can approximate a wide class of periodic signals as a sum of
complex exponentials or co-sinusoid via a Fourier Series (FS) expansion
5. The FS expansion allows us to see signals as functions of frequency
through its (magnitude and phase) spectrum
6. A (stable) LTI system acts on each frequency component of the FS of
a periodic signal independently. This leads to fast computations (compare
with a possible convolution… )
7. Low-pass filters are used for signal “smoothing” and noise removal