Lecture No 4: https://youtu.be/E3QT55J9uWs
Lecture No 5: https://youtu.be/pb7GdbcLnI0
Lecture No 6: https://youtu.be/aFXr1ufTF7Q
Lecture No 7: https://youtu.be/1Yt6ZCKhcYg
Lecture No 8: https://youtu.be/I8UWw3DC19Y
Lecture No 9: https://youtu.be/zRKFi3dotEc
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 10-15)Adnan Zafar
Lecture No 10: https://youtu.be/LIh9yo4rphU
Lecture No 11: https://youtu.be/rOpNHZiRxgg
Lecture No 12: https://youtu.be/sytUNcVKokY
Lecture No 13: https://youtu.be/YN0eAGYNWK4
Lecture No 14: https://youtu.be/OvCjohzmsPU
Lecture No 15: https://youtu.be/TBPeBhRoD90
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 31-39)Adnan Zafar
The document discusses sampling and analog-to-digital conversion. It explains the sampling theorem, which states that a signal can be reconstructed perfectly from samples taken at a minimum rate of twice the signal's bandwidth. It describes how sampling a signal results in multiplying it by an impulse train. It also discusses practical considerations in signal reconstruction using non-ideal interpolation filters and equalizers. Realizing reconstruction filters in hardware is challenging due to the non-realizability of ideal filters. Aliasing can also occur if the signal is not perfectly band-limited and the sampling rate is below the Nyquist rate.
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 16-21)Adnan Zafar
Lecture No 16: https://youtu.be/22XDP-_UKbg
Lecture No 17: https://youtu.be/CikQYWnvKdU
Lecture No 18: https://youtu.be/eT9sDYN4U30
Lecture No 19: https://youtu.be/7-jw3w9snik
Lecture No 20: https://youtu.be/kLmVgGSmfLE
Lecture No 21: https://youtu.be/Mm445diiQpM
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 22-30)Adnan Zafar
Lecture No 22: https://youtu.be/z3gia8eHEOo
Lecture No 23: https://youtu.be/tFZuaZ4i89I
Lecture No 24: https://youtu.be/BIcjuUxb6aE
Lecture No 25: https://youtu.be/ZPvO4CubmME
Lecture No 26: https://youtu.be/CxUWW4Uh5Gk
Lecture No 27: https://youtu.be/OZ2TwSXkeVw
Lecture No 28: https://youtu.be/HGYXtSvisRY
Lecture No 29: https://youtu.be/W1ehHa0AUnk
Lecture No 30: https://youtu.be/q5gh3tQ7aLk
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 1-3)Adnan Zafar
This document provides an overview of a communication systems course. It introduces the instructor, textbook, learning outcomes, and assessment criteria. The contents will cover communication systems fundamentals including analog and digital messages, modulation and detection techniques, source and error coding, and a brief history of telecommunications. Students will learn about signals, channels, modulation schemes like AM and FM, and analyze different transmission methods.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
Signal and System, CT Signal DT Signal, Signal Processing(amplitude and time ...Waqas Afzal
Signal and System(definitions)
Continuous-Time Signal
Discrete-Time Signal
Signal Processing
Basic Elements of Signal Processing
Classification of Signals
Basic Signal Operations(amplitude and time scaling)
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 10-15)Adnan Zafar
Lecture No 10: https://youtu.be/LIh9yo4rphU
Lecture No 11: https://youtu.be/rOpNHZiRxgg
Lecture No 12: https://youtu.be/sytUNcVKokY
Lecture No 13: https://youtu.be/YN0eAGYNWK4
Lecture No 14: https://youtu.be/OvCjohzmsPU
Lecture No 15: https://youtu.be/TBPeBhRoD90
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 31-39)Adnan Zafar
The document discusses sampling and analog-to-digital conversion. It explains the sampling theorem, which states that a signal can be reconstructed perfectly from samples taken at a minimum rate of twice the signal's bandwidth. It describes how sampling a signal results in multiplying it by an impulse train. It also discusses practical considerations in signal reconstruction using non-ideal interpolation filters and equalizers. Realizing reconstruction filters in hardware is challenging due to the non-realizability of ideal filters. Aliasing can also occur if the signal is not perfectly band-limited and the sampling rate is below the Nyquist rate.
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 16-21)Adnan Zafar
Lecture No 16: https://youtu.be/22XDP-_UKbg
Lecture No 17: https://youtu.be/CikQYWnvKdU
Lecture No 18: https://youtu.be/eT9sDYN4U30
Lecture No 19: https://youtu.be/7-jw3w9snik
Lecture No 20: https://youtu.be/kLmVgGSmfLE
Lecture No 21: https://youtu.be/Mm445diiQpM
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 22-30)Adnan Zafar
Lecture No 22: https://youtu.be/z3gia8eHEOo
Lecture No 23: https://youtu.be/tFZuaZ4i89I
Lecture No 24: https://youtu.be/BIcjuUxb6aE
Lecture No 25: https://youtu.be/ZPvO4CubmME
Lecture No 26: https://youtu.be/CxUWW4Uh5Gk
Lecture No 27: https://youtu.be/OZ2TwSXkeVw
Lecture No 28: https://youtu.be/HGYXtSvisRY
Lecture No 29: https://youtu.be/W1ehHa0AUnk
Lecture No 30: https://youtu.be/q5gh3tQ7aLk
Communication Systems_B.P. Lathi and Zhi Ding (Lecture No 1-3)Adnan Zafar
This document provides an overview of a communication systems course. It introduces the instructor, textbook, learning outcomes, and assessment criteria. The contents will cover communication systems fundamentals including analog and digital messages, modulation and detection techniques, source and error coding, and a brief history of telecommunications. Students will learn about signals, channels, modulation schemes like AM and FM, and analyze different transmission methods.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
Signal and System, CT Signal DT Signal, Signal Processing(amplitude and time ...Waqas Afzal
Signal and System(definitions)
Continuous-Time Signal
Discrete-Time Signal
Signal Processing
Basic Elements of Signal Processing
Classification of Signals
Basic Signal Operations(amplitude and time scaling)
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
EC8352- Signals and Systems - Unit 2 - Fourier transformNimithaSoman
This document discusses Fourier transforms and their applications. It begins by introducing Fourier transforms and noting that they are used widely in optics, image processing, speech processing, and medical signal processing. It then covers key topics such as:
- When periodic and aperiodic signals can be represented by Fourier series versus Fourier transforms
- Properties of continuous-time and discrete-time Fourier transforms
- Applications of Fourier transforms in filtering ECG signals, modeling diffractive gratings in optics, speech processing, and image processing
- Limitations of Fourier transforms in representing non-stable systems
The document provides an overview of Fourier transforms and their significance in decomposing signals into constituent frequencies, as well as examples of where they are applied in
Classification of signals
Deterministic and Random signals
Continuous time and discrete time signal
Even (symmetric) and Odd (Anti-symmetric) signal
Periodic and Aperiodic signal
Energy and Power signal
Causal and Non-causal signal
This document provides an overview of analog communication systems and modulation techniques. It discusses the basic components of communication systems including the transmitter, transmission channel, receiver, and transducers. It then describes analog modulation methods like amplitude modulation (AM) and frequency modulation (FM) and how they vary the amplitude or frequency of a carrier wave to transmit a baseband signal. Digital modulation techniques like amplitude-shift keying (ASK) and frequency-shift keying (FSK) are also introduced. Modems are defined as devices that enable data transfer over analog networks by modulating and demodulating signals.
This document provides an overview of communication basics and amplitude modulation. It discusses how communication involves transmitting and receiving information, and how modulation translates signals to higher frequencies for long-distance transmission. It then describes various amplitude modulation techniques like AM, DSB, and SSB. Key aspects covered include the AM envelope, frequency spectrum of AM waves, AM modulation indexes, and different AM modulation and demodulation methods.
The document provides an overview of digital signal processing (DSP). It defines DSP as the analysis, interpretation, and manipulation of signals that have been digitized. The document discusses the need for signal processing to remove noise, and categorizes signal processing as either analog or digital. It highlights advantages of digital over analog processing, describes common filters and their applications. The document also outlines different DSP processor architectures, applications of DSP, and recommendations books and resources to learn more about DSP.
This document provides an introduction to signals and systems. It defines a signal as a function that carries information about a physical phenomenon, and a system as an entity that processes signals to produce new outputs. Signals can be classified as continuous or discrete, deterministic or random, periodic or aperiodic, even or odd, energy-based or power-based, and causal or noncausal. The document discusses examples and properties of different signal types and how systems manipulate inputs to generate outputs. It covers key concepts like energy, power, periodicity, causality, and system modeling that are important foundations for signals and systems analysis.
Quadrature amplitude modulation (QAM) is a modulation technique that encodes data by changing both the amplitude and phase of carrier waves. It allows more data to be transmitted over a given bandwidth compared to techniques that only vary the amplitude or phase. QAM modulators use two carrier waves shifted in phase by 90 degrees that are modulated by separate data streams before being combined. Higher order QAM schemes use constellations with more points that allow more bits to be encoded per symbol. While this improves bandwidth efficiency, it also makes the system more susceptible to noise. QAM is widely used in technologies like DSL, wireless networks, cable TV, and microwave backhaul systems.
This document discusses digital filter design methods. It introduces IIR and FIR filters and their design techniques. The key methods covered are:
1. IIR filter design using impulse invariance, which samples the impulse response of an analog filter to obtain the discrete-time filter.
2. IIR filter design using bilinear transformation, which maps the continuous s-domain to the discrete z-domain to avoid aliasing.
3. FIR filter design using frequency sampling, which designs a linear phase FIR filter by sampling the desired frequency response and taking the inverse DFT.
A power amplifier is an electronic device that increases the power of an input signal so it can drive output devices like speakers or radio transmitters. It amplifies low-power signals to a higher power level needed to power external devices. Power amplifiers are used to boost signals to a level sufficient for driving loads such as speakers or transmitting antennas.
This document defines key concepts in signal processing including signals, systems, and digital signal processing. It provides examples of signals that vary with time or other variables and carry information. Characteristics of signals like amplitude, frequency, and phase are described. Systems are defined as physical devices that operate on signals, with examples of filters. Signal processing involves passing signals through systems to perform operations like filtering. A block diagram shows the basic components of a digital signal processing system including analog to digital conversion, processing, and digital to analog conversion. Finally, advantages of digital over analog signal processing are listed such as programmability, accuracy, storage, and lower cost.
The document discusses the objectives and outcomes of the Microwave Techniques course offered by the Department of Electronics and Communication Engineering at Matrusri Engineering College. The course aims to teach students about guided wave propagation, waveguides, microwave circuits, microwave tubes, and microwave solid state devices. The course outcomes include analyzing guided wave propagation, evaluating waveguide parameters, determining scattering parameters, understanding microwave tube operation, and analyzing microwave solid state devices. The document also provides the course syllabus, lesson plan, and textbook references.
These filters have properties that lie between those of the Butterworth and Chebyshev filters. So it is appropriate to call this kind of filters as transitional Butterworth-Chebyshev filters.
This document discusses various types of pulse modulation techniques used in analog and digital communication systems. It begins by defining pulse amplitude modulation (PAM) and describing how the amplitude of pulses varies proportionally to the message signal. It then discusses different types of PAM based on the sampling technique used - ideal, natural, and flat-top sampling. Flat-top sampling uses sample-and-hold circuits and can introduce amplitude distortion known as the aperture effect. The document also covers pulse width modulation (PWM), pulse position modulation (PPM), pulse code modulation (PCM), delta modulation (DM), and their advantages. It explains the sampling theorem and proves it through Fourier analysis. Finally, it discusses bandwidth requirements, transmission, drawbacks
This document discusses different types of waveguides, including rectangular waveguides, circular waveguides, coaxial lines, optical waveguides, and parallel-plate waveguides. It describes the different modes of wave propagation including TEM, TE, TM, and HE modes. Cutoff frequencies and wavelengths are defined for rectangular and parallel-plate waveguides. Dominant TE10 mode is described for rectangular waveguides.
This document provides an overview of Fourier analysis techniques for communication engineering experiments. It introduces Fourier series as a way to expand periodic signals into a sum of complex exponentials. The Fourier series coefficients represent the contribution of each harmonic frequency. MATLAB will be used to implement Fourier analysis and observe its applications in communication systems. Students are expected to review basic MATLAB commands and complete pre-lab exercises on vector operations and plotting signals before conducting the experiment.
Frequency shift keying (FSK) is a digital modulation technique that encodes digital information by shifting the frequency of a carrier wave. There are different types of FSK including binary FSK, which uses two discrete frequencies to represent binary 1 and 0, and double frequency shift keying (DFSK), which uses four frequencies to transmit two independent data streams simultaneously. FSK modulation can be demodulated using either FM detector demodulators, which treat the FSK signal as an FM signal, or filter-type demodulators, which use optimal filters matched to the FSK signal parameters. The filters are used to detect the mark and space frequencies, and a decision circuit then determines which was transmitted.
This document provides an overview of angle modulation techniques, specifically phase modulation (PM) and frequency modulation (FM). It defines angle modulation as a non-linear process where the modulated wave does not resemble the message wave but the amplitude remains constant. Basic concepts of PM and FM are explained, showing how the carrier signal's phase or frequency varies with the message signal. Equations are provided to define PM and FM. The bandwidth requirements for both techniques are also summarized, with Carson's rule stated for FM bandwidth.
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
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 provides an overview of signals and systems. It defines a signal as a physical quantity that varies with time and contains information. Signals are classified as deterministic or non-deterministic, periodic or aperiodic, even or odd, energy-based or power-based, and continuous-time or discrete-time. Systems are combinations of elements that process input signals to produce output signals. Key properties of systems include causality, linearity, time-invariance, stability, and invertibility. Applications of signals and systems are found in control systems, communications, signal processing, and more.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
EC8352- Signals and Systems - Unit 2 - Fourier transformNimithaSoman
This document discusses Fourier transforms and their applications. It begins by introducing Fourier transforms and noting that they are used widely in optics, image processing, speech processing, and medical signal processing. It then covers key topics such as:
- When periodic and aperiodic signals can be represented by Fourier series versus Fourier transforms
- Properties of continuous-time and discrete-time Fourier transforms
- Applications of Fourier transforms in filtering ECG signals, modeling diffractive gratings in optics, speech processing, and image processing
- Limitations of Fourier transforms in representing non-stable systems
The document provides an overview of Fourier transforms and their significance in decomposing signals into constituent frequencies, as well as examples of where they are applied in
Classification of signals
Deterministic and Random signals
Continuous time and discrete time signal
Even (symmetric) and Odd (Anti-symmetric) signal
Periodic and Aperiodic signal
Energy and Power signal
Causal and Non-causal signal
This document provides an overview of analog communication systems and modulation techniques. It discusses the basic components of communication systems including the transmitter, transmission channel, receiver, and transducers. It then describes analog modulation methods like amplitude modulation (AM) and frequency modulation (FM) and how they vary the amplitude or frequency of a carrier wave to transmit a baseband signal. Digital modulation techniques like amplitude-shift keying (ASK) and frequency-shift keying (FSK) are also introduced. Modems are defined as devices that enable data transfer over analog networks by modulating and demodulating signals.
This document provides an overview of communication basics and amplitude modulation. It discusses how communication involves transmitting and receiving information, and how modulation translates signals to higher frequencies for long-distance transmission. It then describes various amplitude modulation techniques like AM, DSB, and SSB. Key aspects covered include the AM envelope, frequency spectrum of AM waves, AM modulation indexes, and different AM modulation and demodulation methods.
The document provides an overview of digital signal processing (DSP). It defines DSP as the analysis, interpretation, and manipulation of signals that have been digitized. The document discusses the need for signal processing to remove noise, and categorizes signal processing as either analog or digital. It highlights advantages of digital over analog processing, describes common filters and their applications. The document also outlines different DSP processor architectures, applications of DSP, and recommendations books and resources to learn more about DSP.
This document provides an introduction to signals and systems. It defines a signal as a function that carries information about a physical phenomenon, and a system as an entity that processes signals to produce new outputs. Signals can be classified as continuous or discrete, deterministic or random, periodic or aperiodic, even or odd, energy-based or power-based, and causal or noncausal. The document discusses examples and properties of different signal types and how systems manipulate inputs to generate outputs. It covers key concepts like energy, power, periodicity, causality, and system modeling that are important foundations for signals and systems analysis.
Quadrature amplitude modulation (QAM) is a modulation technique that encodes data by changing both the amplitude and phase of carrier waves. It allows more data to be transmitted over a given bandwidth compared to techniques that only vary the amplitude or phase. QAM modulators use two carrier waves shifted in phase by 90 degrees that are modulated by separate data streams before being combined. Higher order QAM schemes use constellations with more points that allow more bits to be encoded per symbol. While this improves bandwidth efficiency, it also makes the system more susceptible to noise. QAM is widely used in technologies like DSL, wireless networks, cable TV, and microwave backhaul systems.
This document discusses digital filter design methods. It introduces IIR and FIR filters and their design techniques. The key methods covered are:
1. IIR filter design using impulse invariance, which samples the impulse response of an analog filter to obtain the discrete-time filter.
2. IIR filter design using bilinear transformation, which maps the continuous s-domain to the discrete z-domain to avoid aliasing.
3. FIR filter design using frequency sampling, which designs a linear phase FIR filter by sampling the desired frequency response and taking the inverse DFT.
A power amplifier is an electronic device that increases the power of an input signal so it can drive output devices like speakers or radio transmitters. It amplifies low-power signals to a higher power level needed to power external devices. Power amplifiers are used to boost signals to a level sufficient for driving loads such as speakers or transmitting antennas.
This document defines key concepts in signal processing including signals, systems, and digital signal processing. It provides examples of signals that vary with time or other variables and carry information. Characteristics of signals like amplitude, frequency, and phase are described. Systems are defined as physical devices that operate on signals, with examples of filters. Signal processing involves passing signals through systems to perform operations like filtering. A block diagram shows the basic components of a digital signal processing system including analog to digital conversion, processing, and digital to analog conversion. Finally, advantages of digital over analog signal processing are listed such as programmability, accuracy, storage, and lower cost.
The document discusses the objectives and outcomes of the Microwave Techniques course offered by the Department of Electronics and Communication Engineering at Matrusri Engineering College. The course aims to teach students about guided wave propagation, waveguides, microwave circuits, microwave tubes, and microwave solid state devices. The course outcomes include analyzing guided wave propagation, evaluating waveguide parameters, determining scattering parameters, understanding microwave tube operation, and analyzing microwave solid state devices. The document also provides the course syllabus, lesson plan, and textbook references.
These filters have properties that lie between those of the Butterworth and Chebyshev filters. So it is appropriate to call this kind of filters as transitional Butterworth-Chebyshev filters.
This document discusses various types of pulse modulation techniques used in analog and digital communication systems. It begins by defining pulse amplitude modulation (PAM) and describing how the amplitude of pulses varies proportionally to the message signal. It then discusses different types of PAM based on the sampling technique used - ideal, natural, and flat-top sampling. Flat-top sampling uses sample-and-hold circuits and can introduce amplitude distortion known as the aperture effect. The document also covers pulse width modulation (PWM), pulse position modulation (PPM), pulse code modulation (PCM), delta modulation (DM), and their advantages. It explains the sampling theorem and proves it through Fourier analysis. Finally, it discusses bandwidth requirements, transmission, drawbacks
This document discusses different types of waveguides, including rectangular waveguides, circular waveguides, coaxial lines, optical waveguides, and parallel-plate waveguides. It describes the different modes of wave propagation including TEM, TE, TM, and HE modes. Cutoff frequencies and wavelengths are defined for rectangular and parallel-plate waveguides. Dominant TE10 mode is described for rectangular waveguides.
This document provides an overview of Fourier analysis techniques for communication engineering experiments. It introduces Fourier series as a way to expand periodic signals into a sum of complex exponentials. The Fourier series coefficients represent the contribution of each harmonic frequency. MATLAB will be used to implement Fourier analysis and observe its applications in communication systems. Students are expected to review basic MATLAB commands and complete pre-lab exercises on vector operations and plotting signals before conducting the experiment.
Frequency shift keying (FSK) is a digital modulation technique that encodes digital information by shifting the frequency of a carrier wave. There are different types of FSK including binary FSK, which uses two discrete frequencies to represent binary 1 and 0, and double frequency shift keying (DFSK), which uses four frequencies to transmit two independent data streams simultaneously. FSK modulation can be demodulated using either FM detector demodulators, which treat the FSK signal as an FM signal, or filter-type demodulators, which use optimal filters matched to the FSK signal parameters. The filters are used to detect the mark and space frequencies, and a decision circuit then determines which was transmitted.
This document provides an overview of angle modulation techniques, specifically phase modulation (PM) and frequency modulation (FM). It defines angle modulation as a non-linear process where the modulated wave does not resemble the message wave but the amplitude remains constant. Basic concepts of PM and FM are explained, showing how the carrier signal's phase or frequency varies with the message signal. Equations are provided to define PM and FM. The bandwidth requirements for both techniques are also summarized, with Carson's rule stated for FM bandwidth.
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
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 provides an overview of signals and systems. It defines a signal as a physical quantity that varies with time and contains information. Signals are classified as deterministic or non-deterministic, periodic or aperiodic, even or odd, energy-based or power-based, and continuous-time or discrete-time. Systems are combinations of elements that process input signals to produce output signals. Key properties of systems include causality, linearity, time-invariance, stability, and invertibility. Applications of signals and systems are found in control systems, communications, signal processing, and more.
This document provides an overview of linear systems concepts covered in the MSE 280 Linear Systems course. It reviews continuous and discrete time signals, introduces the concepts of signal energy and power, and covers various transformations of signals including time shifts, reversals, scaling, and periodicity. Exponential and sinusoidal signals are also discussed, including how complex exponentials relate to sinusoidal signals. The objectives are to review signals and introduce key linear systems analysis tools and signal types.
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.
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.
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.
This document provides an introduction to basic system analysis concepts related to continuous time signals and systems. It defines key signal types such as continuous/discrete time signals, periodic/non-periodic signals, even/odd signals, deterministic/random signals, and energy/power signals. It also discusses important system concepts like linear/non-linear systems, causal/non-causal systems, time-invariant/time-variant systems, stable/unstable systems, and static/dynamic systems. Finally, it introduces common signal types like unit step, unit ramp, and delta/impulse functions as well as concepts like time shifting, scaling, and inversion of systems.
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 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.
This document discusses signals and systems. It begins with an introduction that signals arise in many areas like communications, circuit design, etc. and a signal contains information about some phenomenon. A system processes input signals to produce output signals.
It then discusses different types of signals like continuous-time and discrete-time signals. Deterministic signals can be written mathematically while stochastic signals cannot. Periodic signals repeat and aperiodic signals do not. Even and odd signals have specific properties related to their symmetry.
Operations on signals are also covered, including addition, multiplication by a constant, multiplication of two signals, time shifting which delays or advances a signal, and time scaling which compresses or expands a signal. Common signal models
This document provides an introduction to signals and systems. It defines different types of signals including continuous-time and discrete-time signals. It describes important elementary signals like sinusoidal, exponential, unit step, unit impulse, and ramp functions. It discusses operations that can be performed on signals like time shifting, time scaling, and time inversion. It also classifies signals as deterministic vs non-deterministic, periodic vs aperiodic, even vs odd, and energy vs power signals. Key properties of different signal types are covered.
This document discusses signals and systems. It begins by defining signals and examples, and systems as devices that process signals. Signals are classified as continuous-time or discrete-time, analog or digital, periodic or aperiodic, energy or power signals, and even or odd. Basic signals discussed include unit step, unit impulse, unit ramp, exponential, and sinusoidal. Common operations on signals like time reversal, time scaling, time shifting and amplitude transformations are described. The document then moves to discussing classifications of systems and properties of linear time-invariant continuous-time systems.
This document discusses signals and their classification. It defines signals, analog and digital signals, periodic and aperiodic signals. It also discusses representing signals in Matlab and Simulink. Key signal types covered include exponential, sinusoidal, unit impulse and step functions. Matlab is presented as a tool for programming and analyzing discrete signals while Simulink can be used to model and simulate continuous systems.
The document summarizes key concepts about even and odd signals from a lecture on signals and systems:
1) Even signals have the property that x(-t)=x(t), while odd signals have the property that x(-t)=-x(t). Any signal can be expressed as the sum of an even and an odd component.
2) Discrete time linear and time-invariant (LTI) systems can be characterized by their impulse response h[n]. The output of an LTI system is the convolution of the input signal with the impulse response.
3) Convolution is a mathematical operation that mixes two signals together. It has applications in areas like edge detection. The output of an LTI system
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.
Signals can be classified as continuous-time or discrete-time. Continuous-time signals have a value for all points in time, while discrete-time signals have values only at specific sample points. Common elementary signals include unit step, unit impulse, sinusoidal, and exponential functions. Signals can be further classified based on properties like periodicity, even/odd symmetry, and energy/power. Operations like time shifting, scaling, and inversion can be performed on signals. Discrete-time signals are often obtained by sampling continuous-time signals.
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basic concept of signals
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referred to as the "New Great Game." This research centres on the power struggle, considering
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Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
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Cooperation Organisation and the Belt and Road Economic Initiative.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
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.
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2. Contents
• Signals
• Classification of Signals
• Signal Comparison: Correlation, Cross Correlation Function
• Auto Correlation Function
• Orthogonal Signal Set
• Review of Fourier Series and Fourier Transform and their Properties
308201- Communication Systems 2
3. Signals
• A signal is a set of information or data.
• Examples
– a telephone or television signal,
– monthly sales of a corporation,
– the daily closing prices of a stock market.
• Signal is a function of a dependent variable against an
independent one.
– For our present course the independent variable is time.
• We deal exclusively with signals that are functions of time.
308201- Communication Systems 3
4. Classifications of Signals
• Continuous-time and discrete-time signals
• Analog and digital signals
• Periodic and aperiodic signals
• Energy and power signals
• Deterministic and probabilistic signals
• Other Classifications
– even / odd signals,
– One-dimensional / Multi-dimensional
308201- Communication Systems 4
5. Continuous-time and discrete-time
Signals
• A signal that is specified for every value of time t is a
continuous-time signal.
• A signal that is specified only at discrete values of t is a
discrete-time signals.
308201- Communication Systems 5
6. Continuous-time and discrete-time
Signals (continued)
• A discrete-time signal can be obtained by sampling a
continuous-time signal.
• In some cases, it is possible to 'undo' the sampling operation.
That is, it is possible to get back the continuous-time signal
from the discrete-time signal.
• How?
• Sampling Theorem
– The sampling theorem states that if the highest frequency in the signal
spectrum is 𝐵, the signal can be reconstructed from its samples taken
at a rate not less than 2𝐵 sample per second.
308201- Communication Systems 6
7. Analog and digital signals
• A signal whose amplitude can take
on any value in a continuous range is
an analog signal.
• A signal whose amplitude can take a
finite number of values is a digital
signal.
• The concept of analog and digital
signals is different from the concept
of continuous-time and discrete-
time signals.
• For example, we can have a digital
and continuous-time signal, or a
analog and discrete-time signal.
308201- Communication Systems 7
8. Periodic and Aperiodic Signals
• A signal 𝑔(𝑡) is said to be periodic if for some positive
constant 𝑇0,
𝑔 𝑡 = 𝑔(𝑡 + 𝑇0) for all 𝑡
• A signal is aperiodic if it is not periodic.
• Some famous periodic signals:
– sin 𝑤0𝑡
– cos 𝑤0𝑡
– 𝑒𝑗𝑤0𝑡 = cos 𝑤0𝑡 + j sin 𝑤0𝑡
• where 𝑤0 = 2𝜋
𝑇0
and 𝑇0 is the period of the function.
308201- Communication Systems 8
9. Periodic Signal
• A periodic signal 𝑔(𝑡) can be generated by periodic extension
of any segment of 𝑔(𝑡) of duration 𝑇0.
308201- Communication Systems 9
10. Energy and Power signal
• First, define energy.
– The signal energy 𝐸𝑔 of 𝑔(𝑡) is defined (for a real signal) as
𝐸𝑔 =
−∞
∞
𝑔2 𝑡 𝑑𝑡
– In the case of a complex valued signal 𝑔(𝑡), the energy is given by
𝐸𝑔 =
−∞
∞
𝑔∗ 𝑡 𝑔 𝑡 𝑑𝑡 =
−∞
∞
𝑔 𝑡 2𝑑𝑡
• A necessary condition for energy to be finite is that the
signal amplitude approaches zero as 𝑡 → ∞ otherwise the
above integral will not converge. If this condition doesn’t
hold the signal energy is infinite.
• A signal 𝑔(𝑡) is an energy signal if 𝐸𝑔 < ∞.
308201- Communication Systems 10
11. Energy and Power signal
• A power signal must have an infinite duration.
• In case of signals with infinite energy (e.g., periodic signals), a
more meaningful measure is the signal power.
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔2
(𝑡)𝑑𝑡
• For a complex signal 𝑔(𝑡), the signal power is given by
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔(𝑡) 2𝑑𝑡
• The square root of 𝑃
𝑔 is known as the rms value of 𝑔(𝑡)
• Note: A signal cannot be an energy and a power signal at the
same time.
308201- Communication Systems 11
12. Energy Signal
Example
• Determine the energy of the following signal.
• Signal Energy calculation
𝐸𝑔 =
−∞
∞
𝑔2 𝑡 𝑑𝑡 =
−1
0
(2)2𝑑𝑡 +
0
∞
(2𝑒
−𝑡
2)2𝑑𝑡 = 4 + 4 = 8
308201- Communication Systems 12
13. Power Signal
Example
• Why it is a power signal?
– Signal does not approach 0 as 𝑡 → ∞
– Signal is periodic
• Signal power calculation?
𝑃
𝑔 = lim
𝑇→∞
1
𝑇 −𝑇
2
𝑇
2
𝑔(𝑡) 2
𝑑𝑡 =
1
2 −1
1
𝑡2
𝑑𝑡 =
1
3
• Its rms value will be?
– Square root of 𝑃
𝑔 i.e., 1
3
308201- Communication Systems 13
14. Practice Problems
• Find the energy of the following signal.
• Find a suitable measure of the following signal.
308201- Communication Systems 14
15. Deterministic and Probabilistic Signals
• A signal whose physical description is known completely is a
deterministic signal.
– Mathematical form
– Graphical form
• A signal known only in terms of probabilistic descriptions is a
random signal.
– Mean value
– Mean square value
– Distributions
– Example: Noise, message signals
308201- Communication Systems 15
16. Other Classifications
• Even and Odd Signals
– A signal is even if 𝑥(𝑡) = 𝑥(−𝑡).
– A signal is odd if 𝑥(𝑡) = −𝑥(−𝑡)
• Examples:
– cos(𝑡) is an even signal.
– sin(𝑡) is an odd signal.
• One-dimensional & Multi-dimensional
– Speech varies as a function of time one-dimensional
– Image intensity varies as a function of (𝑥, 𝑦) coordinates multi-
dimensional
308201- Communication Systems 16
One dimensional
Two dimensional
17. Unit Impulse Function
• The unit impulse function or Dirac delta function is defined as
𝛿 𝑡 = 0 𝑡 ≠ 0
−∞
∞
𝛿 𝑡 𝑑𝑡 = 1
• Can be visualized as a tall, narrow, rectangular pulse of unit area.
• Multiplication of a function by an impulse – Sifting Property
𝑔 𝑡 𝛿 𝑡 = 𝑔 0 𝛿 𝑡
𝑔 𝑡 𝛿 𝑡 − 𝑇 = 𝑔 𝑇 𝛿 𝑡 − 𝑇
−∞
∞
𝑔(𝑡)𝛿 𝑡 − 𝑇 𝑑𝑡 = 𝑔(𝑇)
308201- Communication Systems 17
18. Unit step function
• Another useful signal is the unit step function 𝑢(𝑡), defined by
𝑢 𝑡 =
1 𝑡 ≥ 0
0 𝑡 < 0
• Observe that
−∞
𝑡
𝛿 𝜏 𝑑𝜏 =
1 𝑡 ≥ 0
0 𝑡 < 0
= 𝑢(𝑡)
• Which also results into
𝑑𝑢
𝑑𝑡
= 𝛿(𝑡)
308201- Communication Systems 18
19. Sinusoids
• Consider the sinusoid
𝑥 𝑡 = 𝐶 cos(2𝜋𝑓0𝑡 + 𝜃)
• 𝑓0 (measured in Hertz) is the frequency of the sinusoid and
𝑇0 = 1
𝑓0
is the period. 𝐶 is the amplitude and 𝜃 is the phase.
• Sometimes we use 𝑤0 (radians per second) to express 2𝜋𝑓0.
• Important Identities
𝑒±𝑗𝑥 = cos 𝑥 ± 𝑗 sin 𝑥
cos 𝑥 =
1
2
𝑒𝑗𝑥 + 𝑒−𝑗𝑥
sin 𝑥 =
1
2𝑗
𝑒𝑗𝑥 − 𝑒−𝑗𝑥
308201- Communication Systems 19
20. Signal Operations
308201- Communication Systems 20
Time Shifting Time Scaling Time Reversal or inversion
φ(𝑡) = g(𝑡 − 𝑇) represents “Delay” φ(𝑡) = g(2𝑡) represents
“Compression”
φ 𝑡 = g −𝑡
φ(𝑡) = g(𝑡 + 𝑇) represents “Advance” φ(𝑡) = g(𝑡/2) represents
“Expansion”
Mirror image on vertical axis i.e., y-axis
Note: −𝑥 𝑡 means mirror image on
horizontal axis i.e., x-axis
21. Signals and Vectors
• Signals and vectors are closely related. For
example,
– A vector has components,
– A signal has also its components.
• g is a certain vector.
– It is specified by its magnitude or length g and
direction.
• Consider a second vector x.
• Different ways to express g in term of vector x
– g = 𝑐x + 𝑒 = 𝑐1x + 𝑒1 = 𝑐2x + 𝑒2
– e is the error vector
– c is the magnitude of projection of g on x
– Choose c to minimize e i.e., e = g − cx
308201- Communication Systems 21
22. Inner product in vector spaces
• For convenience we define the dot (inner or scalar) product of two
vectors as
g, x = g x cos θ
• Therefore, x, x = x 2
• The length of the component of g along x is g cos 𝜃, but it is also
c 𝑥 .
c 𝑥 = g cos 𝜃
• Multiplying both sides by 𝑥
c 𝑥 2
= g x cos 𝜃 = g, x
c =
g, x
x, x
=
g. x
x 2
• When g, x = 0, we say that g and x are orthogonal to each other
i.e., geometrically, 𝜃 =
𝜋
2
308201- Communication Systems 22
23. Signals as vectors
• The same notion of inner product can be applied for signals.
• What is the useful part of this analogy?
– We can use some geometrical interpretation of vectors to understand
signals.
• Consider two (energy) signals 𝑦(𝑡) and 𝑥(𝑡).
• The inner product is defined by
𝑦(𝑡), 𝑥(𝑡) =
−∞
∞
𝑦 𝑡 𝑥 𝑡 𝑑𝑡
• For complex signals
𝑦(𝑡), 𝑥(𝑡) =
−∞
∞
𝑦 𝑡 𝑥∗ 𝑡 𝑑𝑡
• The two signals are orthogonal if
𝑦(𝑡), 𝑥(𝑡) = 0
308201- Communication Systems 23
24. Signals as vectors
• If a signal g(t) is approximated in terms of another real signal
x(t) over an interval (t1, t2), then the best approximation
would be the one that minimizes the size of error e(t) (Error
energy).
• Minimizing error energy would mean putting
• Which would eventually give
• which is optimum value of c that minimizes the energy of the
error signal in the approximation g(t)~cx(t).
308201- Communication Systems 24
2 2
1 1
2 2
( ) [ ( ) ( )]
t t
e
t t
E e t dt g t cx t dt
0
e
dE
dc
2
2
1
2
1
1
2
( ) ( )
1
( ) ( )
( )
t
t
t
t
x t
t
g t x t dt
c g t x t dt
E
x t dt
25. Signals as vectors
Example
308201- Communication Systems 25
For a square signal g(t) shown below find the component in g(t) of the form sin 𝑡.
𝐠(𝐭) ~ 𝐜 𝐬𝐢𝐧 𝐭
Objective: Select c so that the energy of the error signal is minimum. As we already know that
this condition holds for
c =
1
Ex t1
t2
g t x t dt
Let, x(t) = sin t
So, Ex = 0
2π
sin2
t dt = π
c =
1
𝜋 t1
t2
g t sin 𝑡 dt =
1
𝜋 0
π
sin 𝑡 dt +
1
𝜋 π
2𝜋
− sin 𝑡 dt =
4
𝜋
Therefor, g(t) ~
4
𝜋
sin t
26. Energy of orthogonal signals
• If vectors 𝑥 and 𝑦 are orthogonal, and if 𝑧 = 𝑥 + 𝑦
𝑧 2 = 𝑥 2+ 𝑦 2 (Pythagorean Theorem)
• If signals 𝑥(𝑡) and 𝑦(𝑡) are orthogonal and if 𝑧(𝑡) = 𝑥(𝑡) + 𝑦(𝑡) then
𝐸𝑧 = 𝐸𝑥 + 𝐸𝑦
• Proof?
308201- Communication Systems 26
27. Power of orthogonal signals
• The same concepts of orthogonality and inner product extend
to power signals.
• For example,
• 𝑔 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 = 𝐶1 cos(𝑤1𝑡 + 𝜃1) + 𝐶2 cos(𝑤2𝑡 + 𝜃2)
and 𝑤1 ≠ 𝑤2
𝑃𝑥 =
𝐶1
2
2
, 𝑃𝑦 =
𝐶2
2
2
• The signal 𝑥(𝑡) and 𝑦(𝑡) are orthogonal: 𝑦(𝑡), 𝑥(𝑡) = 0.
Therefore,
𝑃
𝑔 = 𝑃𝑥 + 𝑃𝑦 =
𝐶1
2
2
+
𝐶2
2
2
308201- Communication Systems 27
28. Signal Comparison: Correlation
• Why bother poor undergraduate students with correlation?
– Correlation is widely used in engineering.
• To design receivers in many communication systems
• To identify signals in radar systems
• For classifications.
308201- Communication Systems 28
29. Signal Comparison: Correlation
• If vectors 𝑥 and 𝑦 are given, we have the correlation measure as
𝑐𝑛 = cos 𝜃 =
𝑦, 𝑥
𝑦 𝑥
• Clearly, −1 ≤ 𝑐𝑛 ≤ 1 and is called correlation coefficient.
• In the case of energy signals:
𝑐𝑛 =
1
𝐸𝑦𝐸𝑥 −∞
∞
𝑦 𝑡 𝑥(𝑡) 𝑑𝑡
• Again, −1 ≤ 𝑐𝑛 ≤ 1
• Auto-Correlation
𝜓𝑔 𝜏 =
−∞
∞
𝑔 𝑡 𝑔(𝑡 + 𝜏) 𝑑𝑡
• It is a measure of the similarity of a signal with its displaced version.
308201- Communication Systems 29
30. Best friends, worst enemies and
complete strangers
• 𝑐𝑛 = 1. Best friends. This happens when g t = K x(t) and K
is positive.
– The signals are aligned, maximum similarity.
• 𝑐𝑛 = −1. Worst Enemies. This happens when g t = K x(t)
and K is negative.
– The signals are again aligned, but in opposite directions.
– The signals understand each other, but they do not like each other.
• 𝑐𝑛 = 0. Complete Strangers. The two signals are orthogonal.
– We may view orthogonal signals as unrelated signals.
308201- Communication Systems 30
31. Correlation Examples
• Find the correlation coefficients between:
– 𝑥 𝑡 = 𝐴0 cos 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1sin(𝑤1𝑡)
• 𝑐𝑥,𝑦 = 0
– 𝑥 𝑡 = 𝐴0 cos 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1cos(𝑤1𝑡) and 𝑤0 ≠ 𝑤1
• 𝑐𝑥,𝑦 = 0
– 𝑥 𝑡 = 𝐴0 cos 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1cos(𝑤0𝑡)
• 𝑐𝑥,𝑦 = 1
– 𝑥 𝑡 = 𝐴0 sin 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1sin(𝑤1𝑡) and 𝑤0 ≠ 𝑤1
• 𝑐𝑥,𝑦 = 0
– 𝑥 𝑡 = 𝐴0 sin 𝑤0𝑡 and 𝑦(𝑡) = 𝐴1sin(𝑤0𝑡)
• 𝑐𝑥,𝑦 = 1
– 𝑥 𝑡 = 𝐴0 sin 𝑤0𝑡 and 𝑦 𝑡 = −𝐴1 sin 𝑤0𝑡
• 𝑐𝑥,𝑦 = −1
308201- Communication Systems 31
32. Signal representation by
orthogonal signal sets
• Examine a way of representing a signal as a sum of orthogonal
signals.
• We know that a vector can be represented as the sum of
orthogonal vectors.
• Review the case of vectors and extend to signals.
308201- Communication Systems 32
33. Orthogonal vector space
• Consider a three-dimensional Cartesian vector space
described by three mutually orthogonal vectors, 𝑥1, 𝑥2 and
𝑥3.
𝑥𝑚, 𝑥𝑛 =
0 𝑚 ≠ 𝑛
𝑥𝑚
2 𝑚 = 𝑛
• Any three-dimensional vector can be expressed as a linear
combination of those three vectors:
𝑔 = 𝑐1𝑥1 + 𝑐2𝑥2 + 𝑐3𝑥3
• Where 𝑐𝑖 =
𝑔,𝑥𝑖
𝑥𝑖
2
• In this case, we say that this set of vectors i.e., 𝑥1, 𝑥2, 𝑥3 is a
complete set of orthogonal vectors in 3D space.
• Such vectors are known as a basis vector.
308201- Communication Systems 33
34. Orthogonal signal space
• Same notions of completeness extend to signals.
• A set of mutually orthogonal signals 𝑥1 𝑡 , 𝑥2 𝑡 , … , 𝑥𝑁(𝑡) is complete if it can
represent any signal belonging to a certain space. For example:
𝑔 𝑡 ≅ 𝑐1𝑥1(𝑡) + 𝑐2𝑥2(𝑡) + ⋯ + 𝑐𝑁𝑥𝑁(𝑡)
≅
𝑛=1
𝑁
𝑐𝑛𝑥𝑛(𝑡)
• If the approximation error is zero for any 𝑔(𝑡) then the set of signals
𝑥1 𝑡 , 𝑥2 𝑡 , … , 𝑥𝑁(𝑡) is complete. In general, the set is complete when 𝑁 → ∞.
• This would lead to the Generalized Fourier series of 𝑔(𝑡) as
𝑔 𝑡 =
𝑛=1
∞
𝑐𝑛𝑥𝑛(𝑡)
308201- Communication Systems 34
35. Parseval’s Theorem
• We have already established that the energy of the sum of
orthogonal signals is equal to the sum of their energies.
• The energy of 𝑔 𝑡 = 𝑛=1
∞
𝑐𝑛𝑥𝑛(𝑡) can be expressed as the sum of
the energies of the individual orthogonal components
𝐸𝑔 = 𝑐1
2
𝐸1 + 𝑐2
2
𝐸2 + ⋯
=
𝑛=1
∞
𝑐𝑛
2
𝐸𝑛
• In vector space, the square of the length of vector is equal to the
sum of the squares of the lengths of its orthogonal components.
– Parseval’s theorem is the statement of this fact when applied to
signals.
308201- Communication Systems 35
36. Trigonometric Fourier series
• Consider a signal set
{1, cos 𝑤0𝑡, cos 2𝑤0𝑡, … , cos 𝑛𝑤0𝑡, … , sin 𝑤0𝑡 , sin 2𝑤0𝑡 , … , sin 𝑛𝑤0𝑡}
• A sinusoid of frequency 𝑛𝑤0 is called the 𝑛𝑡ℎ harmonic of the
sinusoid, where 𝑛 is an integer.
• The sinusoid of frequency 𝑤0 is called the fundamental harmonic.
• This set is orthogonal over an interval of duration 𝑇0 = 2𝜋
𝑤0,
which is the period of the fundamental.
308201- Communication Systems 36
37. Trigonometric Fourier series
• The components of the following set are orthogonal.
{1, cos 𝑤0𝑡, cos 2𝑤0𝑡, … , cos 𝑛𝑤0𝑡, … , sin 𝑤0𝑡 , sin 2𝑤0𝑡 , … , sin 𝑛𝑤0𝑡}
𝑇0
cos 𝑛𝑤0𝑡 cos 𝑚𝑤0𝑡 𝑑𝑡 =
0 𝑚 ≠ 𝑛
𝑇0
2
𝑚 = 𝑛 ≠ 0
𝑇0
sin 𝑛𝑤0𝑡 sin 𝑚𝑤0𝑡 𝑑𝑡 =
0 𝑚 ≠ 𝑛
𝑇0
2
𝑚 = 𝑛 ≠ 0
𝑇0
sin 𝑛𝑤0𝑡 cos 𝑚𝑤0𝑡 𝑑𝑡 = 0 for all 𝑚 and 𝑛
• 𝑇0
means integral over an interval from 𝑡 = 𝑡1 to 𝑡 = 𝑡1 + 𝑇0 for
any value of 𝑡1.
308201- Communication Systems 37
38. Trigonometric Fourier series
• This set is also complete in 𝑇0. That is, any signal in an interval 𝑡1 ≤
𝑡 ≤ 𝑡1 + 𝑇0can be written as the sum of sinusoids. Or
𝑔 𝑡 = 𝑎0 + 𝑎1 cos 𝑤0𝑡 + 𝑎2 cos 2𝑤0𝑡+…+𝑏1 sin 𝑤0𝑡 + 𝑏2 sin 2𝑤0𝑡 + ⋯
= 𝑎0 +
𝑛=1
∞
𝑎𝑛 cos 𝑛𝑤0𝑡 + 𝑏𝑛 sin 𝑛𝑤0𝑡
• Series coefficients
𝑎𝑛 =
𝑔 𝑡 , cos 𝑛𝑤0𝑡
cos 𝑛𝑤0𝑡 , cos 𝑛𝑤0𝑡
𝑏𝑛 =
𝑔 𝑡 , sin 𝑛𝑤0𝑡
sin 𝑛𝑤0𝑡 , sin 𝑛𝑤0𝑡
308201- Communication Systems 38
39. Trigonometric Fourier Coefficients
• Therefore
𝑎𝑛 =
𝑡1
𝑡1+𝑇0
𝑔 𝑡 cos 𝑛𝑤0𝑡 𝑑𝑡
𝑡1
𝑡1+𝑇0
cos2𝑛𝑤0𝑡𝑑𝑡
As
𝑡1
𝑡1+𝑇0
cos2𝑛𝑤0𝑡𝑑𝑡 = 𝑇0
2 and 𝑡1
𝑡1+𝑇0
sin2𝑛𝑤0𝑡𝑑𝑡 = 𝑇0
2
We get
𝑎0 =
1
𝑇0 𝑡1
𝑡1+𝑇0
𝑔(𝑡)𝑑𝑡
𝑎𝑛 =
2
𝑇0 𝑡1
𝑡1+𝑇0
𝑔(𝑡) cos 𝑛𝑤0𝑡 𝑑𝑡 𝑛 = 1,2,3
𝑏𝑛 =
2
𝑇0 𝑡1
𝑡1+𝑇0
𝑔(𝑡) sin 𝑛𝑤0𝑡 𝑑𝑡 𝑛 = 1,2,3
308201- Communication Systems 39
40. Exponential Fourier Series
• Consider a set of exponentials
𝑒𝑗𝑛𝑤0𝑡 𝑛 = 0, ±1, ±2, …
• The components of this set are orthogonal.
• A signal 𝑔 𝑡 can be expressed as an exponential series over
an interval 𝑇0 as follows
𝑔 𝑡 =
𝑛=−∞
∞
𝐷𝑛𝑒𝑗𝑛𝑤0𝑡 𝐷𝑛 =
1
𝑇0 𝑇0
𝑔(𝑡)𝑒−𝑗𝑛𝑤0𝑡 𝑑𝑡
308201- Communication Systems 40
41. Trigonometric and exponential
Fourier series
• Trigonometric and exponential Fourier series are related. In
fact, a sinusoid in the trigonometric series can be expressed
as a sum of two exponentials using Euler's formula.
𝐶𝑛 cos(𝑛𝑤0𝑡 + 𝜃𝑛) =
𝐶𝑛
2
𝑒𝑗(𝑛𝑤0𝑡+𝜃𝑛) + 𝑒−𝑗(𝑛𝑤0𝑡+𝜃𝑛)
=
𝐶𝑛
2
𝑒𝑗𝜃𝑛 𝑒𝑗𝑛𝑤0𝑡
+
𝐶𝑛
2
𝑒−𝑗𝜃𝑛 𝑒−𝑗𝑛𝑤0𝑡
= 𝐷𝑛𝑒𝑗𝑛𝑤0𝑡 + 𝐷−𝑛𝑒−𝑗𝑛𝑤0𝑡
Where
𝐷𝑛 =
𝐶𝑛
2
𝑒𝑗𝜃𝑛 and 𝐷−𝑛 =
𝐶𝑛
2
𝑒−𝑗𝜃𝑛
308201- Communication Systems 41
42. Parseval's Theorem
• Exponential Fourier series representation
𝑔 𝑡 =
𝑛=−∞
∞
𝐷𝑛𝑒𝑗𝑛𝑤0𝑡
• Where
𝐷𝑛 =
1
𝑇0 𝑇0
𝑔(𝑡)𝑒−𝑗𝑛𝑤0𝑡
𝑑𝑡
• Power for the exponential representation is
𝑃
𝑔 =
𝑛=−∞
∞
𝐷𝑛
2
308201- Communication Systems 42
43. Example
• Find the exponential Fourier series for the following signal
308201- Communication Systems 43