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 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 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 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 4-9)Adnan Zafar
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 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.
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
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 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 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 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 4-9)Adnan Zafar
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 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.
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
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)
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
Pulse modulation techniques can encode an analog signal for transmission. This document discusses several techniques including:
- Pulse-amplitude modulation (PAM) which varies pulse amplitudes based on sample values of the message signal.
- Pulse code modulation (PCM) which assigns a binary code to each analog sample. PCM is commonly used in digital communications systems.
- Delta modulation which transmits one bit per sample indicating if the current sample is more positive or negative than the previous. It requires higher sampling rates than PCM for equal quality.
This document discusses frequency modulation (FM). It begins by defining the angle of a carrier signal and how that angle can be varied to achieve FM or phase modulation. It then provides key details about FM, including that the message signal controls the carrier frequency. The FM signal equation is presented using Bessel functions. Important parameters like modulation index and frequency deviation are defined. Signal waveforms are shown for different input signals. The spectrum of an FM signal is discussed, including the Bessel coefficients and significant sidebands. Narrowband and wideband FM are differentiated. An example of VHF/FM radio transmission parameters is provided. Finally, power in FM signals is addressed.
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 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.
- The document discusses linear time-invariant (LTI) systems and their representations in the time domain.
- It covers various properties of LTI systems including parallel and cascade connections, causality, stability, and memory.
- Methods for representing LTI systems using impulse responses, differential/difference equations, and step responses are presented.
- Solving techniques for determining the homogeneous and particular solutions of LTI systems described by differential or difference equations are outlined.
PCM is an important method of analog-to-digital conversion where an analog signal is converted into an electrical waveform of two or more levels. The essential operations in a PCM transmitter are sampling, quantizing, and coding the analog signal. In the receiver, the operations are regeneration, decoding, and demodulation of the quantized samples. Regenerative repeaters are used to reconstruct the transmitted sequence of coded pulses and perform equalization, timing, and decision making functions. While PCM systems allow for regeneration and multiplexing, they are more complex than analog methods and increase channel bandwidth requirements.
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.
Pulse modulation techniques encode information by manipulating pulse characteristics such as amplitude, width, and position. Common pulse modulation types include pulse amplitude modulation (PAM), pulse width modulation (PWM), pulse position modulation (PPM), and pulse code modulation (PCM). PCM is the most widely used technique, where an analog signal is sampled, quantized into discrete levels, and encoded into a binary digital signal for transmission.
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.
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
Digital data is represented as variations in the amplitude of a carrier wave in amplitude-shift keying (ASK), a type of modulation.
In an ASK system, a symbol, representing one or more bits, is sent by transmitting a fixed-amplitude carrier wave at a fixed frequency for a specific time duration.
A simple form of ASK modulation is considered that amplitude modulates a carrier based on a direct mapping of the source data bits to the waveform symbol. The most rudimentary form of ASK is given the special name On–Off Keying (OOK) modulation.
Fm demodulation using zero crossing detectormpsrekha83
This document discusses FM demodulation using a zero crossing detector. It contains a block diagram of a zero crossing detector system consisting of a zero crossing detector, pulse generator, DC block, and low pass filter. It explains that the zero crossing detector operates by measuring the time difference between adjacent zero crossings of the FM wave, which is related to the instantaneous frequency and can be used to recover the message signal. It notes the advantages and disadvantages of this technique.
The document provides an overview of adaptive filters. It discusses that adaptive filters are digital filters that have self-adjusting characteristics to changes in input signals. They have two main components: a digital filter with adjustable coefficients and an adaptive algorithm. Common adaptive algorithms are LMS and RLS. Adaptive filters are used for applications like noise cancellation, system identification, channel equalization, and signal prediction. The key aspects of adaptive filter theory and algorithms like LMS, RLS, Wiener filters are also covered.
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
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 discusses the design of FIR filters using window functions. It begins by explaining that windows are used to modify the impulse response of filters to reduce ripples and achieve a smooth transition from passband to stopband. It then provides examples of common window functions, including rectangular, Hanning, Hamming, and Blackman windows. It concludes by showing the design of a low-pass FIR filter using a Hamming window to meet specific specifications for cutoff frequency and transition width.
Lecture Notes on Adaptive Signal Processing-1.pdfVishalPusadkar1
Adaptive filters are time-variant, nonlinear, and stochastic systems that perform data-driven approximation to minimize an objective function. The chapter discusses adaptive filter applications like system identification, inverse modeling, linear prediction, and noise cancellation. It also covers stochastic signal models, optimum linear filtering techniques like Wiener filtering, and solutions to the Wiener-Hopf equations. Numerical techniques like steepest descent are discussed for minimizing the mean square error function in adaptive filters. Stability and convergence analysis is presented for the steepest descent approach.
This document summarizes key concepts in sampling and pulse code modulation (PCM). It discusses:
- The sampling theorem which states that a signal can be reconstructed from samples if sampled at twice the bandwidth or higher.
- How PCM works by sampling an analog signal, quantizing the samples into digital values, coding the values into binary numbers, and transmitting the digital data.
- Key advantages of PCM include using inexpensive digital circuits, enabling all-digital transmission and further digital processing.
- Techniques like companding, differential PCM, and increasing bit depth can improve the quality and efficiency of PCM systems.
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.
Signal, Sampling and signal quantizationSamS270368
Signal sampling is the process of converting a continuous-time signal into a discrete-time signal by capturing its amplitude at regularly spaced intervals of time. This is typically done using an analog-to-digital converter (ADC). The rate at which samples are taken is called the sampling frequency, often denoted as Fs, and is measured in hertz (Hz). The Nyquist-Shannon sampling theorem states that to accurately reconstruct a signal from its samples, the sampling frequency must be at least twice the highest frequency component present in the signal (the Nyquist frequency). Sampling at a frequency below the Nyquist frequency can result in aliasing, where higher frequency components are incorrectly interpreted as lower frequency ones.
This document provides a lecture summary on modeling digital control systems. It discusses the sampling theorem, analog-to-digital converter (ADC) and digital-to-analog converter (DAC) models, and how to combine these models. The sampling theorem establishes the minimum sampling rate needed to reconstruct a band-limited signal. The ADC and DAC are modeled as an ideal sampler and zero-order hold, respectively. Their combination transfer function is derived as (1 - z-1) times the z-transform of the analog subsystem transfer function divided by s. Examples are provided to illustrate selecting suitable sampling rates and deriving combined digital system transfer functions.
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
Pulse modulation techniques can encode an analog signal for transmission. This document discusses several techniques including:
- Pulse-amplitude modulation (PAM) which varies pulse amplitudes based on sample values of the message signal.
- Pulse code modulation (PCM) which assigns a binary code to each analog sample. PCM is commonly used in digital communications systems.
- Delta modulation which transmits one bit per sample indicating if the current sample is more positive or negative than the previous. It requires higher sampling rates than PCM for equal quality.
This document discusses frequency modulation (FM). It begins by defining the angle of a carrier signal and how that angle can be varied to achieve FM or phase modulation. It then provides key details about FM, including that the message signal controls the carrier frequency. The FM signal equation is presented using Bessel functions. Important parameters like modulation index and frequency deviation are defined. Signal waveforms are shown for different input signals. The spectrum of an FM signal is discussed, including the Bessel coefficients and significant sidebands. Narrowband and wideband FM are differentiated. An example of VHF/FM radio transmission parameters is provided. Finally, power in FM signals is addressed.
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 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.
- The document discusses linear time-invariant (LTI) systems and their representations in the time domain.
- It covers various properties of LTI systems including parallel and cascade connections, causality, stability, and memory.
- Methods for representing LTI systems using impulse responses, differential/difference equations, and step responses are presented.
- Solving techniques for determining the homogeneous and particular solutions of LTI systems described by differential or difference equations are outlined.
PCM is an important method of analog-to-digital conversion where an analog signal is converted into an electrical waveform of two or more levels. The essential operations in a PCM transmitter are sampling, quantizing, and coding the analog signal. In the receiver, the operations are regeneration, decoding, and demodulation of the quantized samples. Regenerative repeaters are used to reconstruct the transmitted sequence of coded pulses and perform equalization, timing, and decision making functions. While PCM systems allow for regeneration and multiplexing, they are more complex than analog methods and increase channel bandwidth requirements.
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.
Pulse modulation techniques encode information by manipulating pulse characteristics such as amplitude, width, and position. Common pulse modulation types include pulse amplitude modulation (PAM), pulse width modulation (PWM), pulse position modulation (PPM), and pulse code modulation (PCM). PCM is the most widely used technique, where an analog signal is sampled, quantized into discrete levels, and encoded into a binary digital signal for transmission.
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.
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
Digital data is represented as variations in the amplitude of a carrier wave in amplitude-shift keying (ASK), a type of modulation.
In an ASK system, a symbol, representing one or more bits, is sent by transmitting a fixed-amplitude carrier wave at a fixed frequency for a specific time duration.
A simple form of ASK modulation is considered that amplitude modulates a carrier based on a direct mapping of the source data bits to the waveform symbol. The most rudimentary form of ASK is given the special name On–Off Keying (OOK) modulation.
Fm demodulation using zero crossing detectormpsrekha83
This document discusses FM demodulation using a zero crossing detector. It contains a block diagram of a zero crossing detector system consisting of a zero crossing detector, pulse generator, DC block, and low pass filter. It explains that the zero crossing detector operates by measuring the time difference between adjacent zero crossings of the FM wave, which is related to the instantaneous frequency and can be used to recover the message signal. It notes the advantages and disadvantages of this technique.
The document provides an overview of adaptive filters. It discusses that adaptive filters are digital filters that have self-adjusting characteristics to changes in input signals. They have two main components: a digital filter with adjustable coefficients and an adaptive algorithm. Common adaptive algorithms are LMS and RLS. Adaptive filters are used for applications like noise cancellation, system identification, channel equalization, and signal prediction. The key aspects of adaptive filter theory and algorithms like LMS, RLS, Wiener filters are also covered.
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
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 discusses the design of FIR filters using window functions. It begins by explaining that windows are used to modify the impulse response of filters to reduce ripples and achieve a smooth transition from passband to stopband. It then provides examples of common window functions, including rectangular, Hanning, Hamming, and Blackman windows. It concludes by showing the design of a low-pass FIR filter using a Hamming window to meet specific specifications for cutoff frequency and transition width.
Lecture Notes on Adaptive Signal Processing-1.pdfVishalPusadkar1
Adaptive filters are time-variant, nonlinear, and stochastic systems that perform data-driven approximation to minimize an objective function. The chapter discusses adaptive filter applications like system identification, inverse modeling, linear prediction, and noise cancellation. It also covers stochastic signal models, optimum linear filtering techniques like Wiener filtering, and solutions to the Wiener-Hopf equations. Numerical techniques like steepest descent are discussed for minimizing the mean square error function in adaptive filters. Stability and convergence analysis is presented for the steepest descent approach.
This document summarizes key concepts in sampling and pulse code modulation (PCM). It discusses:
- The sampling theorem which states that a signal can be reconstructed from samples if sampled at twice the bandwidth or higher.
- How PCM works by sampling an analog signal, quantizing the samples into digital values, coding the values into binary numbers, and transmitting the digital data.
- Key advantages of PCM include using inexpensive digital circuits, enabling all-digital transmission and further digital processing.
- Techniques like companding, differential PCM, and increasing bit depth can improve the quality and efficiency of PCM systems.
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.
Signal, Sampling and signal quantizationSamS270368
Signal sampling is the process of converting a continuous-time signal into a discrete-time signal by capturing its amplitude at regularly spaced intervals of time. This is typically done using an analog-to-digital converter (ADC). The rate at which samples are taken is called the sampling frequency, often denoted as Fs, and is measured in hertz (Hz). The Nyquist-Shannon sampling theorem states that to accurately reconstruct a signal from its samples, the sampling frequency must be at least twice the highest frequency component present in the signal (the Nyquist frequency). Sampling at a frequency below the Nyquist frequency can result in aliasing, where higher frequency components are incorrectly interpreted as lower frequency ones.
This document provides a lecture summary on modeling digital control systems. It discusses the sampling theorem, analog-to-digital converter (ADC) and digital-to-analog converter (DAC) models, and how to combine these models. The sampling theorem establishes the minimum sampling rate needed to reconstruct a band-limited signal. The ADC and DAC are modeled as an ideal sampler and zero-order hold, respectively. Their combination transfer function is derived as (1 - z-1) times the z-transform of the analog subsystem transfer function divided by s. Examples are provided to illustrate selecting suitable sampling rates and deriving combined digital system transfer functions.
This document discusses sampling of continuous-time signals. The key points are:
- Sampling is the process of converting a continuous-time analog signal into a discrete-time signal by taking samples at regular time intervals. This allows the signal to be processed digitally.
- The Nyquist sampling theorem states that a signal must be sampled at a rate at least twice the highest frequency component to avoid aliasing during reconstruction.
- Ideal sampling involves sampling the signal using impulse trains at the sampling intervals. In the frequency domain, sampling results in shifting the spectrum of the original continuous-time signal.
- Practical considerations in analog-to-digital and digital-to-analog conversion include quantization
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingAmr E. Mohamed
This document discusses multi-rate digital signal processing and concepts related to sampling continuous-time signals. It begins by introducing discrete-time processing of continuous signals using an ideal continuous-to-discrete converter. It then covers the Nyquist sampling theorem and relationships between continuous and discrete Fourier transforms. It discusses ideal and practical reconstruction using zero-order hold and anti-imaging filters. Finally, it introduces the concepts of downsampling and upsampling in multi-rate digital signal processing systems.
Pulse modulation schemes aim to transfer an analog signal over an analog channel as a two-level signal by modulating a pulse wave. Some schemes also allow digital transfer of the analog signal with a fixed bit rate. Pulse modulation includes analog-over-analog methods like PAM, PWM, and PPM as well as analog-over-digital methods like PCM, DPCM, ADPCM, DM, and delta-sigma modulation. Sampling is the reduction of a continuous signal to a discrete signal by taking values at points in time. The Nyquist-Shannon sampling theorem states that a bandlimited signal can be perfectly reconstructed from samples if the sampling rate is at least twice the highest frequency in the signal.
This document discusses sampling and related concepts in signal processing. It begins by introducing the need to convert analog signals to discrete-time signals for digital processing. It then covers the sampling theorem, which states that a band-limited signal can be reconstructed if sampled at twice the maximum frequency. The document describes three main sampling methods: ideal (impulse), natural (pulse), and flat-top sampling. It also discusses aliasing, which occurs when a signal is under-sampled. The key aspects of sampling covered are the sampling rate, reconstruction of sampled signals, and anti-aliasing filters.
The document discusses sampling theory and its applications. It introduces the sampling theorem, which states that a bandlimited continuous-time signal can be perfectly reconstructed from its samples if sampled above the Nyquist rate. It then covers signal representation using impulse functions, the Fourier analysis of sampled signals, and techniques for signal reconstruction, interpolation, downsampling, and oversampling. The document aims to explain how sampling works and its role in analog-to-digital conversion.
This document discusses the sampling theorem and its applications. The sampling theorem states that a continuous-time signal that is bandlimited can be perfectly reconstructed from its samples if it is sampled at or above the Nyquist rate. The document covers key aspects of the sampling theorem including signal reconstruction using sinc functions, aliasing, and applications such as downsampling, upsampling, and oversampling.
This document discusses analog to digital conversion and signal sampling and reconstruction. It begins by explaining the advantages of digital over analog communication and the process of analog to digital conversion using pulse code modulation. It then covers the sampling theorem, which states that a bandlimited signal can be reconstructed from samples taken at or above the Nyquist rate. The document provides the mathematical representations of sampling and discusses how the spectrum of a sampled signal consists of copies of the original spectrum spaced at integer multiples of the sampling frequency. It describes how reconstruction involves the use of an interpolation filter, such as the sinc function, to filter out copies and reconstruct the original signal from its samples. Finally, it discusses practical considerations in sampling and reconstruction.
Lecture Notes: EEEC6440315 Communication Systems - Spectral AnalysisAIMST University
1) The document discusses Fourier transforms and their applications in signal analysis and communication systems. It defines the direct and inverse Fourier transforms and their use in relating time and frequency domain representations of signals.
2) Modulation is described as a way to shift signal spectra to different frequency bands, allowing multiple signals to be transmitted simultaneously without interference. Common modulation techniques are discussed along with their effects on signal spectra.
3) Characteristics of linear time-invariant systems are outlined, defining how they affect input signals in both time and frequency domains. Distortion effects of non-linear systems are also covered.
4) Concepts of energy, energy spectral density, and essential bandwidth are defined in relation to signal frequency spectra.
The document discusses vibration isolation of a LEGO platform using low-cost instrumentation as part of an open source project. It summarizes using inertial mass actuators to apply skyhook damping control to actively isolate the platform from environmental vibrations. Preliminary measurements identify the platform's modal properties. A real implementation is presented using 4 symmetric actuators. Control design applies skyhook damping in modal coordinates to optimally isolate each mode.
Frequency division multiplexing (FDM) divides the available bandwidth into non-overlapping frequency bands, with each band carrying a separate signal. Time division multiplexing (TDM) divides signals into different time slots to share the same frequency channel. TDM samples voice frequency signals at 8 kHz and quantizes each sample to 8 bits, requiring 64 kbps per voice channel. A 32-channel PCM system using TDM requires 2 Mbps of bandwidth and can carry more voice channels than FDM with the same bandwidth.
Electrocardiography is a medical topic that has piqued engineers' curiosity. One of the most essential signals observed in heart patients is the electrocardiogram (ECG). The electrocardiogram, or ECG, is an extremely valuable medical device. The objective of an ECG is to assist clinicians in quickly diagnosing human or animal heart activity and detecting aberrant heart activities. The heart's job is to contract rhythmically, pump blood to the lungs for oxygenation, and then return this oxygenated blood to the rest of the body. The spread of electrical signals created by the heart pacemaker, the Sinoatrial (SA) node, maintains and signals this precise rhythm. Detecting such electrical activity in the heart can aid in the detection of a variety of cardiac problems
Spectral Analysis of Sample Rate ConverterCSCJournals
The aim of digital sample rate conversion is to bring a digital audio signal from one sample frequency to another. The distortion of the audio signal introduced by the sample rate converter should be as low as possible. The generation of the output samples from the input samples may be performed by the application of various methods. In this paper, a new technique of digital sample-rate converter is proposed. We perform the spectral analysis of proposed digital sample rate converter.
Digital Instrumentation
The document discusses digital instrumentation and data acquisition systems. It covers topics like analog and digital signals, data sampling, the sampling theorem, anti-aliasing, sample and hold circuits, data acquisition systems, and interfacing with computers. Key points include:
- Analog signals are continuous while digital signals are discrete. Sampling converts a continuous signal to a discrete one.
- The sampling theorem states the sampling rate must be at least twice the highest frequency component to avoid aliasing.
- Sample and hold circuits create and store samples of an input voltage to digitize analog signals for data conversion.
- Data acquisition systems collect data from sensors, condition signals, convert to digital, and transfer to computers or
This document provides an overview of equalizer design in digital communication systems. It discusses the need for equalization to address inter-symbol interference caused by channel limitations. It describes two main equalizer designs: zero-forcing equalizers that apply the inverse channel response and minimum mean square error equalizers that minimize the error between the equalized signal and desired signal. It explains how the tap coefficients of these equalizers can be calculated using linear algebra methods like solving sets of equations. The document concludes by noting that equalization is a key technique in modern communications to compensate for channel distortions.
1) The document discusses various pulse modulation techniques including pulse amplitude modulation (PAM), pulse width modulation (PWM), and pulse position modulation (PPM).
2) It provides details on the generation and detection of PAM and PWM signals, explaining the use of sampling, comparators, sawtooth waves, and filters.
3) The document compares different sampling techniques for PAM including natural sampling, flat top sampling, and discusses the need for analog to digital conversion in communication systems.
1) The document discusses various topics related to digital communication including sampling theory, analog to digital conversion, pulse code modulation, quantization, coding, and time division multiplexing.
2) In analog to digital conversion, an analog signal is sampled, quantized by assigning it to discrete amplitude levels, and coded by mapping each level to a binary sequence.
3) The Nyquist sampling theorem states that a signal must be sampled at a rate at least twice its highest frequency to avoid aliasing when reconstructing the original signal.
The Presentation includes Basics of Non - Uniform Quantization, Companding and different Pulse Code Modulation Techniques. Comparison of Various PCM techniques is done considering various Parameters in Communication Systems.
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3. Sampling Theorem
• Why do we need sampling?
• Analog signals can be digitized though sampling and
quantization.
– A/D Converter
• In A/D converter, the sampling rate must be large enough to
permit the analog signal to be reconstructed from the
samples with sufficient accuracy.
• The sampling theorem defines the basis for determining the
proper (lossless) sampling rate for a given signal.
308201- Communication Systems 3
4. Sampling Theorem
• A consider a signal 𝑔(𝑡) whose spectrum is band limited to
𝐵 𝐻𝑧, that is
𝐺 𝑓 = 0 for 𝑓 > 𝐵
• We can reconstruct the signal exactly (without any error) from
its discrete time samples (samples taken uniformly at a rate of
𝑅 samples per second).
– Condition for 𝑅?
– The condition is that 𝑅 > 2𝐵
• In other words, the minimum sampling frequency for perfect
signal recovery is 𝑓𝑠 = 2𝐵 𝐻𝑧
308201- Communication Systems 4
5. Sampling Theorem
• Consider a signal 𝑔(𝑡) whose spectrum is band limited to
𝐵 𝐻𝑧
• Sampling at a rate 𝑓𝑠 means we take 𝑓𝑠 uniform samples per
second.
• This uniform sampling can be accomplished by multiplying
𝑔(𝑡) by an impulse train 𝛿𝑇𝑠
(𝑡)
• The pulse train consist of unit impulses repeating periodically
every 𝑇𝑠 seconds. 308201- Communication Systems 5
6. Sampling Theorem
• This result in the sampled signal 𝑔(𝑡) as follows
• The relationship between sampled signal 𝑔(𝑡) and the original signal 𝑔(𝑡)
is
𝑔 𝑡 = 𝑔 𝑡 𝛿𝑇𝑠
𝑡 =
𝑛
𝑔(𝑛𝑇𝑠)𝛿(𝑡 − 𝑛𝑇𝑠)
• Since the impulse train is a periodic signal of period 𝑇𝑠, it can be expressed
as an exponential Fourier series i.e.,
𝛿𝑇𝑠
𝑡 =
1
𝑇𝑠
−∞
∞
𝑒𝑗𝑛𝑤𝑠𝑡 𝑤𝑠 =
2𝜋
𝑇𝑠
= 2𝜋𝑓𝑠
𝑔 𝑡 =
1
𝑇𝑠
−∞
∞
𝑔 𝑡 𝑒𝑗𝑛𝑤𝑠𝑡
308201- Communication Systems 6
7. • The Fourier transform of 𝑔 𝑡 will be as follows
𝐺 𝑓 =
1
𝑇𝑠
−∞
∞ 𝐺 𝑓 − 𝑛𝑓𝑠
• This mean that the spectrum 𝐺 𝑓 consist of 𝐺 𝑓 , scaled by
a constant
1
𝑇𝑠
, repeating periodically with period 𝑓𝑠 =
1
𝑇𝑠
𝐻𝑧
• Can 𝑔(𝑡) be reconstructed from 𝑔 𝑡 without any loss or
distortion?
– If yes then we should be able to recover 𝐺(𝑓) from 𝐺 𝑓
– Only possible if there is no overlap among replicas in 𝐺 𝑓
Sampling Theorem
308201- Communication Systems 7
8. Sampling Theorem
• It can be seen that for no overlap among replicas in 𝐺 𝑓
𝑓𝑠 > 2𝐵
• Also since 𝑇𝑠 =
1
𝑓𝑠
so, 𝑇𝑠 <
1
2𝐵
• As long as the sampling frequency 𝑓𝑠 is greater than twice the
signal bandwidth 𝐵 , 𝐺 𝑓 will have nonoverlapping
repetitions of 𝐺 𝑓 .
• 𝑔(𝑡) can be recovered using an ideal low pass filter of
bandwidth 𝐵 𝐻𝑧.
• The minimum sampling rate 𝑓𝑠 = 2𝐵 required to recover 𝑔(𝑡)
from 𝑔 𝑡 is called the Nyquist Rate and the corresponding
sampling interval 𝑇𝑠 =
1
2𝐵
is called the Nyquist Interval.
308201- Communication Systems 8
9. Signal Reconstruction from
Uniform Samples
• The process of reconstructing a continuous time signal 𝑔(𝑡)
from its samples is also known as interpolation.
• We have already established that uniform sampling at above
the Nyquist rate preserves all the signal information.
– Passing the sampled signal through an ideal low pass filter of
bandwidth 𝐵 𝐻𝑧 will reconstruct the message signal.
• The sampled message signal can be represented as
𝑔 𝑡 =
𝑛
𝑔(𝑛𝑇𝑠)𝛿(𝑡 − 𝑛𝑇𝑠)
• The low pass filter of bandwidth 𝐵 𝐻𝑧 and gain 𝑇𝑠 will have
the following transfer function
𝐻 𝑓 = 𝑇𝑠𝛱
𝑤
4𝜋𝐵
= 𝑇𝑠𝛱
𝑓
2𝐵
308201- Communication Systems 9
10. Signal Reconstruction from
Uniform Samples
• To recover the analog signal from its uniform samples, the
ideal interpolation filter transfer function and impulse
response is as follows
𝐻 𝑓 = 𝑇𝑠𝛱
𝑓
2𝐵
⟺ ℎ 𝑡 = 2𝐵𝑇𝑠 sinc 2𝜋𝐵𝑡
• Assuming the use of Nyquist sampling rate i.e., 2𝐵𝑇𝑠 = 1
ℎ 𝑡 = sinc 2𝜋𝐵𝑡
308201- Communication Systems 10
11. Signal Reconstruction from
Uniform Samples
• It can be observed that ℎ(𝑡) = 0 at all Nyquist sampling
instants i.e., 𝑡 = ±𝑛
2𝐵 except 𝑡 = 0
• When the sampled signal 𝑔 𝑡 is applied at the input of this
filter, each sample in 𝑔 𝑡 , being an impulse, generate a sinc
pulse of height equal to the strength of the sample.
𝑔 𝑡 =
𝑘
𝑔 𝑘𝑇𝑠 ℎ(𝑡 − 𝑘𝑇𝑠) =
𝑘
𝑔 𝑘𝑇𝑠 sinc(2𝜋𝐵𝑡 − 𝑘𝜋)
308201- Communication Systems 11
12. Example
• Determine the Nyquist interval and sampling rate for the
signal 𝑔1(𝑡), 𝑔2 𝑡 , 𝑔1
2
(𝑡), 𝑔2
𝑚
(𝑡) and 𝑔1 𝑡 𝑔2 𝑡 when the
spectra of 𝐺1 𝑤 and 𝐺2 𝑤 is given as follows
• Bandwidth of 𝑔1 𝑡 is 100𝐾𝐻𝑧 and 𝑔2 𝑡 is 150𝐾𝐻𝑧
– Nyquist sample rate for 𝑔1 𝑡 is 200𝐾𝐻𝑧 and 𝑔2 𝑡 is 300𝐾𝐻𝑧
• Bandwidth of 𝑔1
2
(𝑡) is twice the bandwidth of 𝑔1 𝑡 i.e.,
200𝐾𝐻𝑧 and 𝑔2
𝑚
(𝑡) will have bandwidth 𝑚 × 150𝐾𝐻𝑧
– Nyquist sample rate for 𝑔1
2
(𝑡) is 400𝐾𝐻𝑧 and 𝑔2
𝑚
(𝑡) is 𝑚 × 300𝐾𝐻𝑧
• Bandwidth of 𝑔1 𝑡 𝑔2 𝑡 is 250𝐾𝐻𝑧
– Nyquist sample rate for 𝑔1 𝑡 𝑔2 𝑡 is 500𝐾𝐻𝑧
308201- Communication Systems 12
13. Example
• Determine the Nyquist sampling rate and the Nyquist
sampling interval for the signal
– sin𝑐 100𝜋𝑡
– sinc2 100𝜋𝑡
sin𝑐 100𝜋𝑡 ⟺ 0.01𝛱
𝑤
200𝜋
• Bandwidth is 50𝐻𝑧 and the Nyquist sample rate 100𝐻𝑧
sinc2 100𝜋𝑡 ⟺ 0.01∆
𝑤
400𝜋
• Bandwidth is 100𝐻𝑧 and the Nyquist sample rate 200𝐻𝑧
308201- Communication Systems 13
14. Practical Signal Reconstruction
(Interpolation)
• Signal reconstruction discussed earlier require an ideal low
pass filter.
– Unrealizable
– Noncausal
– This also lead to a infinitely long nature of sinc reconstruction pulse.
• For practical applications, we need a realizable signal
reconstruction system from the uniform signal samples.
• For practical implementation, the reconstruction pulse must
be easy to generate.
308201- Communication Systems 14
15. Practical Signal Reconstruction
(Interpolation)
• We need to analyze the accuracy of the reconstructed signal when
using a non-ideal interpolation pulse 𝑝(𝑡).
𝑔 𝑡 =
𝑛
𝑔 𝑛𝑇𝑠 𝑝(𝑡 − 𝑛𝑇𝑠)
𝑔 𝑡 = 𝑝 𝑡 ∗
𝑛
𝑔 𝑛𝑇𝑠 𝛿(𝑡 − 𝑛𝑇𝑠)
𝑔 𝑡 = 𝑝 𝑡 ∗ 𝑔 𝑡
• In frequency domain, the relationship between the reconstructed
and the original analog signal is given as
𝐺 𝑓 = 𝑃(𝑓)
1
𝑇𝑠
−∞
∞
𝐺 𝑓 − 𝑛𝑓𝑠
• The reconstructed signal 𝑔 𝑡 using pulse 𝑝(𝑡) consist of multiple
replicas of 𝐺(𝑓) shifted to 𝑛𝑓𝑠 and filtered by 𝑃(𝑓).
308201- Communication Systems 15
16. Practical Signal Reconstruction
(Interpolation)
• So to fully recover the 𝑔(𝑡) from 𝑔 𝑡 , we need further
processing/filtering.
– Such filters are often referred to as equalizers
• Let us denote the equalizer function as 𝐸(𝑓), so now distortionless
reconstruction will require
𝐺 𝑓 = 𝐸(𝑓)𝐺 𝑓
𝐺 𝑓 = 𝐸(𝑓)𝑃(𝑓)
1
𝑇𝑠
−∞
∞
𝐺 𝑓 − 𝑛𝑓𝑠
• The equalizer must remove all the shifted replicas 𝐺 𝑓 − 𝑛𝑓𝑠
except the low pass term with 𝑛 = 0 i.e.,
𝐸 𝑓 𝑃 𝑓 = 0 𝑓 > 𝑓𝑠 − 𝐵
also
𝐸 𝑓 𝑃 𝑓 = 𝑇𝑠 𝑓 < 𝐵
308201- Communication Systems 16
17. Practical Signal Reconstruction
(Interpolation)
• The equalizer 𝐸(𝑓) must be
– lowpass in nature to stop all frequency components above 𝑓𝑠 − 𝐵
– Inverse of 𝑃(𝑓) within the signal bandwidth of 𝐵 𝐻𝑧
• A practical signal reconstruction system can be represented as
• Let us consider a very simple interpolating pulse generator
that generate short pules i.e.,
308201- Communication Systems 17
18. Practical Signal Reconstruction
(Interpolation)
• The short pulse can be expressed as
𝑝 𝑡 = ∏
𝑡 − 0.5𝑇𝑝
𝑇𝑝
• The reconstruction will first generate 𝑔 𝑡 as
𝑔 𝑡 =
𝑛
𝑔 𝑛𝑇𝑠 ∏
𝑡 − 0.5𝑇𝑝 − 𝑛𝑇𝑠
𝑇𝑝
• The Fourier transform of 𝑝(𝑡) is
𝑃 𝑓 = 𝑇𝑝 sinc 𝜋𝑓𝑇𝑝 𝑒−𝑗𝜋𝑓𝑇𝑝
• The equalizer frequency response should satisfy
𝐸 𝑓 =
𝑇𝑠
𝑃(𝑓) 𝑓 ≤ 𝐵
Flexible 𝐵 < 𝑓 <
1
𝑇𝑠
− 𝐵
0 𝑓 ≥
1
𝑇𝑠
− 𝐵
308201- Communication Systems 18
19. Practical Issues in Signal
Sampling and Reconstruction
• Realizability of Reconstruction Filters
• The Treachery of Aliasing
– Defectors Eliminated: The antialiasing Filter
• Sampling forces non-bandlimited signals to appear
band limited
308201- Communication Systems 19
20. Realizability of Reconstruction Filters
• If a signal is sampled at the Nyquist rate 𝑓𝑠 = 2𝐵 𝐻𝑧, the
spectrum 𝐺 𝑓 consists of repetitions of 𝐺(𝑓) without any
gaps between successive cycles.
• To recover g(t) from 𝑔 𝑡 , we need to pass the sampled signal
through an ideal low pass filter.
– Ideal Filter is practically unrealizable
– Solution?
308201- Communication Systems 20
21. Realizability of Reconstruction Filters
• A practical solution to this problem is to sample the signal at a
rate higher than the Nyquist rate i.e., 𝑓𝑠 > 2𝐵 or 𝑤𝑠 > 4𝜋𝐵
• This yield 𝐺 𝑓 , consisting of repetitions of G(f) with a finite
band gap between successive cycles.
• We can recover 𝑔(𝑡) from 𝑔 𝑡 using a low pass filter with a
gradual cut-off characteristics.
• There is still an issue with this approach: Guess?
• Despite the gradual cut-off characteristics, the filter gain is
required to be zero beyond the first cycle of 𝐺(𝑓).
308201- Communication Systems 21
22. The Treachery of Aliasing
• Another practical difficulty in reconstructing the signal from
its samples lies in the fundamental theory of sampling
theorem.
• The sampling theorem was proved on the assumption that the
signal 𝑔(𝑡) is band-limited.
• All practical signals are time limited.
– They have finite duration or width.
– They are non-band limited.
– A signal cannot be time limited and band limited at the same time.
308201- Communication Systems 22
23. The Treachery of Aliasing
• In this case, the spectrum of 𝐺 𝑓 will consist of overlapping
cycles of 𝐺(𝑓) repeating every 𝑓𝑠 𝐻𝑧
• Because of non-bandlimited spectrum, the spectrum overlap
is unavoidable, regardless of the sampling rate.
– Higher sampling rates can reduce but cannot eliminate the overlaps
• Due to the overlaps, 𝐺 𝑓 no longer has complete
information of 𝐺 𝑓 . 308201- Communication Systems 23
24. The Treachery of Aliasing
• If we pass 𝑔 𝑡 even through an ideal low pass filter with cut-
off frequency
𝑓𝑠
2
, the output will not be 𝐺(𝑓) but as 𝐺𝑎(𝑓)
• This 𝐺𝑎(𝑓) is distorted version of 𝐺(𝑓) due to
– The loss of tail of 𝐺(𝑓) beyond 𝑓 > 𝑓𝑠
2
– The reappearance of this tail inverted or folded back into the spectrum
• This tail inversion is known as spectral folding or aliasing.
308201- Communication Systems 24
25. Defectors Eliminated:
The Antialiasing Filter
• In the process of aliasing, we are not only losing all the
frequency components above the folding frequency i.e.,
𝑓𝑠
2 𝐻𝑧 but these components reappear (aliased) as lower
frequency components.
– Aliasing is destroying the integrity of the frequency components below
the folding frequency.
• Solution?
– The frequency components beyond the folding frequency are
suppressed from 𝑔(𝑡) before sampling.
• The higher frequencies can be suppressed using a low pass
filter with cut off 𝑓𝑠
2 𝐻𝑧 known as antialiasing filter.
308201- Communication Systems 25
26. Defectors Eliminated:
The Antialiasing Filter
• The antialiasing filter is used before sampling
• The sampled signal spectrum and the reconstructed signal
𝐺𝑎𝑎(𝑓) will be as follows
308201- Communication Systems 26
27. Sampling forces non-bandlimited
signals to appear band limited
• The above spectrum consist of overlapping cycles of 𝐺 𝑓
– This also mean that 𝑔 𝑡 are sub-Nyquist samples of 𝑔(𝑡)
• We can also view the spectrum as the spectrum of 𝐺𝑎(𝑓) repeating
periodically every 𝑓𝑠 𝐻𝑧 without overlap.
– The spectrum 𝐺𝑎(𝑓) is band limited to 𝑓𝑠
2 𝐻𝑧
• The sub-Nyquist samples of 𝑔(𝑡) can also be viewed as Nyquist
samples of 𝑔𝑎 𝑡 .
• Sampling a non-bandlimited signal 𝑔(𝑡) at a rate 𝑓𝑠 𝐻𝑧 make the
samples appear to be Nyquist samples of some signal 𝑔𝑎 𝑡 band
limited to 𝑓𝑠
2 𝐻𝑧
308201- Communication Systems 27
28. Pulse Code Modulation
• Analog signal is characterised by an amplitude that can take
on any value over a continuous range.
– It can take on an infinite number of values
• Digital signal amplitude can take on only a finite number of
values.
• Pulse code modulation (PCM) is a tool for converting an
analog signal into a digital signal (A/D conversion).
– Sampling and Quantization
308201- Communication Systems 28
29. Example
(Digital Telephone)
• The audio signal bandwidth is about 15𝐾𝐻𝑧.
• For speech, subjective tests show that signal articulation
(intelligibility) is not affected if all the components above 3400𝐻𝑧
are suppressed.
• Since the objective in telephone communication is intelligibility
rather than high fidelity, the components above 3400𝐻𝑧 are
eliminated by a low pass filter.
• The resulting bandlimited signal is sampled at a rate of
8000 𝐻𝑧 (8000 samples per second).
– Why higher sampling rate than Nyquist sample rate of 6.8𝐾𝐻𝑧 ?
• Each sample is then quantized into 256 levels 𝐿 = 256 .
– 8 bits required to sample each pulse.
• Hence, a telephone signal requires 8 × 8000 = 64𝐾 binary pulses
per second (64𝐾𝑏𝑝𝑠).
308201- Communication Systems 29
30. Quantization
• To transmit analog signals over a digital communication link, we must
discretize both time and values.
• Quantization spacing is
2𝑚𝑝
𝐿
; sampling interval is 𝑇, not shown in figure.
308201- Communication Systems 30
31. Quantization
• For quantization, we limit the amplitude of the message signal
𝑚(𝑡) to the range (−𝑚𝑝, 𝑚𝑝).
– Note that 𝑚𝑝 is not necessarily the peak amplitude of 𝑚 𝑡 .
– The amplitudes of 𝑚(𝑡) beyond ±𝑚𝑝 are simply chopped off.
• 𝑚𝑝 is not a parameter of the signal 𝑚(𝑡); rather it is the limit
of the quantizer.
• The amplitude range −𝑚𝑝, 𝑚𝑝 is divided into 𝐿 uniformly
spaced intervals, each of width ∆𝑣 =
2𝑚𝑝
𝐿
.
• A sample value is approximated by the midpoint of the
interval in which it lies.
– Quantization Error!
308201- Communication Systems 31
32. Quantization Error
• If 𝑚(𝑘𝑇𝑠) is the 𝑘𝑡ℎ sample of the signal 𝑚(𝑡), and if 𝑚(𝑘𝑇𝑠) is the
corresponding quantized sample, then from the interpolation formula
𝑚 𝑡 =
𝑘
𝑚 𝑘𝑇𝑠 sinc(2𝜋𝐵𝑡 − 𝑘𝜋)
and
𝑚 𝑡 =
𝑘
𝑚 𝑘𝑇𝑠 sinc(2𝜋𝐵𝑡 − 𝑘𝜋)
• Where 𝑚(t) is the signal reconstructed from the quantized samples.
• The distortion component 𝑞(𝑡) in the reconstructed signal is
𝑞 𝑡 = 𝑚 𝑡 − 𝑚(𝑡)
𝑞 𝑡 =
𝑘
𝑞 𝑘𝑇𝑠 sinc(2𝜋𝐵𝑡 − 𝑘𝜋)
• 𝑞 𝑘𝑇𝑠 is the quantization error in the kth sample.
• The signal 𝑞 𝑡 is the undesired signal and act as noise i.e., quantization
noise.
308201- Communication Systems 32
33. Quantization Error
• We can calculate the power or mean square value of 𝑞 𝑡 as
𝑃𝑞 = 𝑞2(𝑡) = lim
𝑇→∞
1
𝑇
−𝑇
2
𝑇
2
𝑞2
(𝑡) 𝑑𝑡
= lim
𝑇→∞
1
𝑇
−𝑇
2
𝑇
2
𝑘
𝑞 𝑘𝑇𝑠 sinc(2𝜋𝐵𝑡 − 𝑘𝜋)
2
𝑑𝑡
• Since the signal sinc(2𝜋𝐵𝑡 − 𝑚𝜋) and sinc(2𝜋𝐵𝑡 − 𝑛𝜋) are
orthogonal i.e.,
−∞
∞
sinc(2𝜋𝐵𝑡 − 𝑚𝜋) sinc(2𝜋𝐵𝑡 − 𝑛𝜋) 𝑑𝑡 =
0 𝑚 ≠ 𝑛
1
2𝐵
𝑚 = 𝑛
• See Prob. 3.7-4 for proof.
308201- Communication Systems 33
34. Quantization Error
• As we know
𝑃𝑞 = 𝑞2(𝑡) = lim
𝑇→∞
1
𝑇
−𝑇
2
𝑇
2
𝑘
𝑞2
𝑘𝑇𝑠 sinc2
(2𝜋𝐵𝑡 − 𝑘𝜋) 𝑑𝑡
= lim
𝑇→∞
1
𝑇
𝑘
𝑞2
𝑘𝑇𝑠
−𝑇
2
𝑇
2
sinc2
(2𝜋𝐵𝑡 − 𝑘𝜋) 𝑑𝑡
• Due to the orthogonality condition, the cross product terms will
vanish and we get
𝑃𝑞 = 𝑞2(𝑡) = lim
𝑇→∞
1
2𝐵𝑇
𝑘
𝑞2 𝑘𝑇𝑠
• Since the sampling rate is 2𝐵, the total number of sample over
averaging interval 𝑇 is 2𝐵𝑇.
– The above relation is the average or the mean of the square of the
quantization error.
308201- Communication Systems 34
35. Quantization Error
• Since the quantum levels are separated by ∆𝑣 =
2𝑚𝑝
𝐿 and the
sample value is approximated by the midpoint of the subinterval (of
height ∆𝑣).
– The maximum quantization error is ± ∆𝑣
2
– The quantization error lies in the range (− ∆𝑣
2 , ∆𝑣
2)
• Assuming that the quantization error is equally likely to lie
anywhere in the range (− ∆𝑣
2 , ∆𝑣
2) , the mean square of
quantizing error 𝑞2 is given by
𝑞2 =
1
∆𝑣
−∆𝑣
2
∆𝑣
2
𝑞2 𝑑𝑞 =
∆𝑣 2
12
=
𝑚2
𝑝
3𝐿2
• Hence, the quantization noise can be represented as
𝑁𝑞 =
𝑚2
𝑝
3𝐿2
308201- Communication Systems 35
36. Quantization Error
• The reconstructed signal 𝑚(𝑡) at the receiver output is
𝑚 𝑡 = 𝑚 𝑡 + 𝑞 𝑡
• Can we determine the signal to noise ratio of the reconstructed
signal?
• The power of the message signal 𝑚(𝑡) is 𝑃𝑚 = 𝑚2(𝑡), then
𝑆0 = 𝑚2(𝑡)
and
𝑁0 = 𝑁𝑞 =
𝑚2
𝑝
3𝐿2
• Signal to noise ratio will be
𝑆0
𝑁0
= 3𝐿2
𝑚2(𝑡)
𝑚2
𝑝
308201- Communication Systems 36
38. Differential Pulse Code Modulation
• PCM is not very efficient system because it generates so many
bits and requires so much bandwidth to transmit.
• We can exploit the characteristics of the source signal to
improve the encoding efficiency of the A/D converter.
– Differential Pulse Code Modulation (DPCM)
• In analog messages we can make a good guess about a sample
value from knowledge of past sample values.
– High Nyquist sample rate.
• What can be done to exploit this characteristic of analog
messages?
– Instead of transmitting the sample values, we transmit the different
between the successive sample values.
308201- Communication Systems 38
39. Differential Pulse Code Modulation
• If 𝑚[𝑘] is the 𝑘𝑡ℎ sample, instead of transmitting 𝑚[𝑘], we transmit
the difference
𝑑 𝑘 = 𝑚 𝑘 − 𝑚[𝑘 − 1]
• At the receiver, knowing 𝑑[𝑘] and several previous sample value
𝑚 𝑘 − 1 , we can reconstruct 𝑚[𝑘].
• How this can improve the efficiency of the A/D converter?
• The difference between successive samples is generally much
smaller than the sample values.
– The peak amplitude 𝑚𝑝 of the transmitted values is reduced considerably.
– This will reduce the quantization interval as ∆𝑣 ∝ 𝑚𝑝
– This will reduce the quantization noise as 𝑁𝑞 ∝ ∆𝑣2
– For a given 𝑛 (or transmission bandwidth), we can increase 𝑆𝑁𝑅
– For a given 𝑆𝑁𝑅, we can reduce 𝑛 (or transmission bandwidth)
308201- Communication Systems 39
40. Differential Pulse Code Modulation
• We can further improve this scheme by estimating
(predicting) the 𝑘𝑡ℎ sample 𝑚[𝑘] from knowledge of several
previous sample values i.e., 𝑚[𝑘].
• We now transmit the difference (prediction error) i.e.,
𝑑 𝑘 = 𝑚 𝑘 − 𝑚[𝑘]
• At the receiver, we determine the estimate 𝑚[𝑘] from the
previous sample values and then generate 𝑚[𝑘] as
𝑚 𝑘 = 𝑑 𝑘 − 𝑚[𝑘]
• How this is an improved scheme?
– The estimated value 𝑚[𝑘] will be close to 𝑚 𝑘 and their difference
(prediction error), 𝑑 𝑘 will be even smaller than the difference
between successive samples.
– Differential PCM.
– Previous method is a special case of DPCM i.e., 𝑚 𝑘 = 𝑚[𝑘 − 1]
308201- Communication Systems 40
41. Estimation of 𝑚[𝑘]
• A message signal sample 𝑚(𝑡 + 𝑇𝑠) can be expressed as a Taylor series
expansion i.e.,
𝑚 𝑡 + 𝑇𝑠 = 𝑚 𝑡 + 𝑇𝑠𝑚 𝑡 +
𝑇𝑠
2
2!
𝑚 𝑡 +
𝑇𝑠
3
3!
𝑚 𝑡 + ⋯
≈ 𝑚 𝑡 + 𝑇𝑠𝑚 𝑡 for small 𝑇𝑠
• From the knowledge of the signal and its derivatives at instant 𝑡, we can
predict a future signal value at 𝑡 + 𝑇𝑠
– Even with just the first derivative we can approximate the future value
• Let the 𝑘𝑡ℎ
sample of 𝑚(𝑡) be 𝑚[𝑘] i.e., putting 𝑡 = 𝑘𝑇𝑠 in 𝑚(𝑡)
𝑚 𝑘𝑇𝑠 = 𝑚 𝑘 and 𝑚 𝑘𝑇𝑠 ± 𝑇𝑠 = 𝑚[𝑘 ± 1]
• The future sample can be predicted as
𝑚(𝑘𝑇𝑠 + 𝑇𝑠) = 𝑚(𝑘𝑇𝑠) + 𝑇𝑠
𝑚(𝑘𝑇𝑠) − 𝑚(𝑘𝑇𝑠 − 𝑇𝑠)
𝑇𝑠
𝑚 𝑘 + 1 = 2𝑚 𝑘 − 𝑚[𝑘 − 1]
• We can predict the (𝑘 + 1)𝑡ℎ sample from the two previous samples.
• How the predicted value can be improved?
– Consider more terms in the Taylor series.
308201- Communication Systems 41
42. Analysis of DPCM
• In DPCM, we transmit not the present sample 𝑚[𝑘], but 𝑑 𝑘 =
𝑚 𝑘 − 𝑚 𝑘 . At receiver, we generate 𝑚 𝑘 from the past samples
and add the received 𝑑[𝑘] to get 𝑚[𝑘].
• There is one difficulty associated with this scheme?
– At receiver, instead of the past samples i.e., 𝑚 𝑘 − 1 , 𝑚 𝑘 − 2 , …, we have
their quantized versions i.e., 𝑚𝑞 𝑘 − 1 , 𝑚𝑞 𝑘 − 2 , …
– We cannot determine 𝑚 𝑘 , we can only determine 𝑚𝑞[𝑘] i.e., estimation
of the quantized sample 𝑚𝑞[𝑘] using 𝑚𝑞 𝑘 − 1 , 𝑚𝑞 𝑘 − 2 , …
– This will increase the error in reconstruction
• What could be a better strategy in this case?
– Determine 𝑚𝑞[𝑘], the estimate of 𝑚𝑞[𝑘] (instead of 𝑚[𝑘]), at the
transmitter from the previous quantized samples 𝑚𝑞 𝑘 − 1 , 𝑚𝑞 𝑘 − 2 , …
– The difference 𝑑 𝑘 = 𝑚 𝑘 − 𝑚𝑞[𝑘] is now transmitted
– At receiver, we generate 𝑚𝑞 𝑘 , and from the received 𝑑[𝑘], we
can reconstruct 𝑚𝑞 𝑘
308201- Communication Systems 42
43. Analysis of DPCM
• The 𝑞[𝑘] at the receiver output is the quantization noise associated with
difference signal 𝑑[𝑘], which is generally much smaller than 𝑚[𝑘].
308201- Communication Systems 43
DPCM Receiver
DPCM Transmitter
𝑑 𝑘 = 𝑚 𝑘 − 𝑚𝑞[𝑘]
𝑑𝑞 𝑘 = 𝑑 𝑘 + 𝑞[𝑘]
𝑚𝑞 𝑘 = 𝑚𝑞 𝑘 + 𝑑𝑞 𝑘
= 𝑚 𝑘 + 𝑞[𝑘]
𝑚𝑞 𝑘 = 𝑚 𝑘 + 𝑞 𝑘
44. SNR Improvement using DPCM
• Let 𝑚𝑝 and 𝑑𝑝 be the peak amplitudes of 𝑚(𝑡) and 𝑑(𝑡)
respectively
• If we use the same value of 𝐿 in both cases, the quantization step
∆𝑣 in DPCM is reduced by the factor
𝑑𝑝
𝑚𝑝
• As the quantization noise 𝑁𝑝 =
(∆𝑣)2
12
, the quantization noise in case
of DPCM is reduced by a factor
𝑚𝑝
𝑑𝑝
2
, and the SNR is increased by
the same factor.
• SNR improvement in DPCM due to prediction will be
𝐺𝑝 =
𝑃𝑚
𝑃𝑑
• Where 𝑃𝑚 and 𝑃𝑑 are the powers of 𝑚(𝑡) and 𝑑(𝑡) respectively.
308201- Communication Systems 44
45. Delta Modulation
• Sample correlation used in DPCM is further exploited in delta
modulation by over sampling (typically four times the
Nyquist rate) the baseband signal.
– Small prediction error
– Encode using 1 bit (i.e., 𝐿=2)
• Delta Modulation (DM) is basically a 1-bit DPCM.
• In DM, we use a first order predictor, which is just a time
delay of 𝑇𝑠.
• The DM transmitter (modulator) and receiver (demodulator)
are identical to DPCM with a time delay as a predictor.
308201- Communication Systems 45
46. Delta Modulation
𝑚𝑞 𝑘 = 𝑚𝑞 𝑘 − 1 + 𝑑𝑞 𝑘
• So
𝑚𝑞 𝑘 − 1 = 𝑚𝑞 𝑘 − 2 + 𝑑𝑞[𝑘 − 1]
• We get
𝑚𝑞 𝑘 = 𝑚𝑞 𝑘 − 2 + 𝑑𝑞 𝑘 − 1 + 𝑑𝑞 𝑘
• Preceding iteratively we get
𝑚𝑞 𝑘 =
𝑚=0
𝑘
𝑑𝑞[𝑚]
• The receiver is just an accumulator (adder).
• If 𝑑𝑞 𝑘 is represented by impulses, then the accumulator (receiver) may be
realized by an integrator. 308201- Communication Systems 46
47. Delta Modulation
• Note that in DM, the modulated signal carries information about the
difference between the successive samples.
– If difference is positive or negative, a positive or negative pulse is generated in
modulated signal 𝑑𝑞[𝑘]
– DM carries the information about the derivative of 𝑚(𝑡)
– That’s why integration of DM signal yield approximation of 𝑚(𝑡) i.e., 𝑚𝑞(𝑡)
308201- Communication Systems 47