This document provides an overview of digital signal processing (DSP). It discusses:
1. The differences between analog and digital signals, and how analog signals are converted to digital signals through sampling and analog-to-digital conversion.
2. The basic operations in DSP - addition, multiplication, and delay. A digital processor performs these operations on digitized signals.
3. The advantages of DSP over analog signal processing, including greater accuracy, integration capabilities, and exact linear phase systems. DSP is also less sensitive to variations and easier to adjust through software.
4. Some disadvantages of DSP like increased complexity, limited frequency range due to sampling rates, and higher power consumption than analog circuits.
Analog signals are continuous with infinite values while digital signals are discrete with a finite set of values. Analog signals can represent values more exactly but are more difficult to process, while digital signals are less exact but easier to process. Examples of analog signals include audio and video, while digital signals include text and integers. Analog transmission is unaffected by content but prone to distortion over long distances, while digital transmission recovers and retransmits signals to achieve greater distances. Applications of analog include thermometers and audio tapes, while digital includes computers, phones and more complex systems.
This document provides an overview of analog and digital signals. It defines analog signals as continuous with an infinite range of values, while digital signals are discrete with a finite set of possible values. Though analog signals can represent values more exactly, digital signals are easier to process electronically. An oscilloscope is used to visualize both analog and digital signals over time. Key components of signals like amplitude, period, frequency, rise and fall times are defined. Examples of sine wave, square wave and logic level signals are shown.
This document discusses digital technology and how analog signals are converted to digital format for use in computers and storage. It explains that:
1. Computers use binary digits (bits) represented as 0s and 1s rather than decimal digits. Bits are grouped into bytes of 8 bits each to represent numbers.
2. Analog signals like sound are converted to digital format by sampling the amplitude of the signal at regular intervals and assigning it a numeric value.
3. Increasing the sampling rate and precision of this process reduces sampling errors when converting back to an analog signal, improving fidelity of the reproduced sound.
>Introduction to Digital Signal Processing
>Analog Signal Processing Versus Digital Signal Processing
>Classification Of Signals
>Comparison Between Continuous-Time & Discrete-Time Sinusoids
>Characteristics Of Discrete-time Sinusoids
>A/D and D/A Conversion
1. Analog-to-digital conversion (ADC) allows computers to interact with analog signals by sampling and quantizing analog signals from devices like CD players.
2. During recording, an ADC converts an analog audio signal into a digital format by repeatedly measuring and assigning a binary number to the signal's amplitude at set intervals defined by the sample rate.
3. During playback, a digital-to-analog converter (DAC) reconverts the digital numbers back into an analog signal by combining the amplitude information from each sample to rebuild the original wave.
The document discusses the analog to digital conversion process. It explains that sounds are analog waves but computers are digital so an conversion is needed. The sound card contains an analog-to-digital converter (ADC) that samples sounds and converts them to binary digits and a digital-to-analog converter (DAC) that converts the digital signals back to analog waves for playback. The key parameters for conversion are the sampling rate, which must be over twice the highest frequency to avoid quality issues, and the bit depth, which determines the number of possible values and thus the resolution/quality. Higher rates and depths allow for better quality recordings.
Introduction to digital signal processing 2Hossam Hassan
The document discusses digital signal processing. It begins by listing the objectives, which include explaining how analog signals are converted to digital form through sampling and analog-to-digital conversion. It then covers digital signal processing basics, how analog signals are converted to digital via sampling and ADCs, different types of ADCs, digital signal processors and their applications, and digital-to-analog conversion.
The document provides an overview of analog communication systems. It defines analog signals as continuous signals that contain time-varying quantities, unlike digital signals which have discrete values. It then discusses the basic components of a communication system including the information signal, digitally compressed data, carrier signal, modulated carrier, transmitted signal, received signal, amplified signal, and recovered information signal. Finally, it covers transmission media such as radio waves, microwaves, and infrared transmission and their role in analog communication.
Analog signals are continuous with infinite values while digital signals are discrete with a finite set of values. Analog signals can represent values more exactly but are more difficult to process, while digital signals are less exact but easier to process. Examples of analog signals include audio and video, while digital signals include text and integers. Analog transmission is unaffected by content but prone to distortion over long distances, while digital transmission recovers and retransmits signals to achieve greater distances. Applications of analog include thermometers and audio tapes, while digital includes computers, phones and more complex systems.
This document provides an overview of analog and digital signals. It defines analog signals as continuous with an infinite range of values, while digital signals are discrete with a finite set of possible values. Though analog signals can represent values more exactly, digital signals are easier to process electronically. An oscilloscope is used to visualize both analog and digital signals over time. Key components of signals like amplitude, period, frequency, rise and fall times are defined. Examples of sine wave, square wave and logic level signals are shown.
This document discusses digital technology and how analog signals are converted to digital format for use in computers and storage. It explains that:
1. Computers use binary digits (bits) represented as 0s and 1s rather than decimal digits. Bits are grouped into bytes of 8 bits each to represent numbers.
2. Analog signals like sound are converted to digital format by sampling the amplitude of the signal at regular intervals and assigning it a numeric value.
3. Increasing the sampling rate and precision of this process reduces sampling errors when converting back to an analog signal, improving fidelity of the reproduced sound.
>Introduction to Digital Signal Processing
>Analog Signal Processing Versus Digital Signal Processing
>Classification Of Signals
>Comparison Between Continuous-Time & Discrete-Time Sinusoids
>Characteristics Of Discrete-time Sinusoids
>A/D and D/A Conversion
1. Analog-to-digital conversion (ADC) allows computers to interact with analog signals by sampling and quantizing analog signals from devices like CD players.
2. During recording, an ADC converts an analog audio signal into a digital format by repeatedly measuring and assigning a binary number to the signal's amplitude at set intervals defined by the sample rate.
3. During playback, a digital-to-analog converter (DAC) reconverts the digital numbers back into an analog signal by combining the amplitude information from each sample to rebuild the original wave.
The document discusses the analog to digital conversion process. It explains that sounds are analog waves but computers are digital so an conversion is needed. The sound card contains an analog-to-digital converter (ADC) that samples sounds and converts them to binary digits and a digital-to-analog converter (DAC) that converts the digital signals back to analog waves for playback. The key parameters for conversion are the sampling rate, which must be over twice the highest frequency to avoid quality issues, and the bit depth, which determines the number of possible values and thus the resolution/quality. Higher rates and depths allow for better quality recordings.
Introduction to digital signal processing 2Hossam Hassan
The document discusses digital signal processing. It begins by listing the objectives, which include explaining how analog signals are converted to digital form through sampling and analog-to-digital conversion. It then covers digital signal processing basics, how analog signals are converted to digital via sampling and ADCs, different types of ADCs, digital signal processors and their applications, and digital-to-analog conversion.
The document provides an overview of analog communication systems. It defines analog signals as continuous signals that contain time-varying quantities, unlike digital signals which have discrete values. It then discusses the basic components of a communication system including the information signal, digitally compressed data, carrier signal, modulated carrier, transmitted signal, received signal, amplified signal, and recovered information signal. Finally, it covers transmission media such as radio waves, microwaves, and infrared transmission and their role in analog communication.
Digital signal processing is a specialized microprocessor with its architecture optimized for operational needs of digital signal processing
Application's of DSP like STFT and Wavelet transform has been explained in detail with images.
digital to analog (DAC) & analog to digital converter (ADC) Asif Iqbal
This document summarizes different types of digital to analog converters (DACs). It discusses the basic concept of converting digital data to analog signals by using a circuit that can produce analog outputs. It then describes several DAC specializations:
1) Binary weighted DAC which uses a reference voltage and weighted resistors to produce analog outputs corresponding to the digital input bits.
2) Flash type ADC which uses a voltage divider network and parallel comparators to directly convert an entire digital word to an analog voltage very quickly.
3) Successive approximation ADC which uses a comparator and feedback loop in a step-wise process to iteratively approximate the analog output voltage, providing a tradeoff between speed and circuit complexity.
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.
Real-Time Signal Processing: Implementation and Applicationsathish sak
This document discusses real-time signal processing, including what it means, why it is used, and platforms for implementation. Real-time signal processing allows signals to be collected, analyzed, and modified in real-time as they occur. It is used to avoid time and money lost when collecting and processing data separately. Common platforms include software/PC, hardware like FPGAs, and firmware/hardware like DSPs, each with their own benefits and drawbacks relating to flexibility, speed, cost, and practicality. The document focuses on DSPs as a popular "middle ground" option and discusses code generation applications and the Embedded Target for TI's C6711 DSP.
This document is a thesis written by Edgar Daniel Chaguaro Aldaz titled "Digital Signal Processing Filtering Algorithm Audio Equalization Using Matlab". It is 40 pages long plus 3 appendices and was submitted on May 7, 2015 for a Bachelor of Engineering degree. The thesis analyzes different digital signal processing filtering algorithms and implements a basic graphic equalizer using Matlab. It describes digital signal processing concepts such as sampling, quantization, filters, and audio equalization. The equalizer application allows processing audio signals using filters and displaying the results graphically in Matlab.
Digital signal processing involves representing and processing signals in the form of discrete numeric values. It has various applications including radar, biomedical monitoring, speech recognition, communications, image processing, and multimedia. Key aspects of digital signal processing implementation are analog to digital conversion, digital processing, and digital to analog conversion. Limitations include information loss due to sampling, aliasing effects, limited frequency resolution, and quantization error. However, digital signal processing provides advantages such as reprogrammability, accuracy control, easy storage and transport of signals, and ability to implement sophisticated algorithms.
Signal processing involves the analysis, interpretation, and manipulation of signals like sound, images, and sensor data. It is used to retrieve the original signal from one that has been contaminated with noise during transmission. There are two main categories: analog signal processing, which uses electronic circuits like filters on un-digitized signals; and digital signal processing, which uses computers or specialized chips to process digitized signals. The digital signal processing block diagram consists of an anti-aliasing filter, analog-to-digital converter, digital filter, digital-to-analog converter, and reconstruction filter. Digital signal processing has advantages over analog like greater accuracy, flexibility, ease of storage and operation, and ability to multiplex signals.
1. The document discusses digital signal processing (DSP) and provides an overview of key concepts such as analog and digital signals, sampling, quantization, coding, and the sampling theorem.
2. It also covers DSP applications in various fields like medical, military, industrial and more. Common signal types like deterministic, random, even, odd and sinusoidal signals are defined.
3. Real-time DSP considerations are discussed, noting the need to ensure processing time meets operational requirements for applications that must operate in real-time.
This document discusses different types of signals and their representations. It describes:
- Analog and digital signals as the two main types.
- Periodic signals that repeat over a measurable time frame called a period. Aperiodic signals do not repeat.
- Continuous signals having time as a continuum, while discrete signals are obtained by sampling continuous signals at discrete time intervals.
- Even signals being identical to their time-reversed counterparts, odd signals changing sign when reversed, and orthogonal signals having no relationship between components.
Lecture: Digital Signal Processing Batch 2009ubaidis
Here are the solutions to the questions:
(a) y[nT] = 3(−0.2)n u[n − 3], T = 2 ms
Energy: E = 9(0.2)3/(1−0.2) = 9
Power: P = 0 (since the signal decays to zero as n increases)
This is an energy signal.
(b) z[nT] = 4(1.1)n u[n + 1], T = 0.02 s
Energy: E = ∞ (since (1.1)n increases without bound)
Power: P = 4.4
This is a power signal.
(c
Ee341 dsp1 1_sv_chapter1_hay truy cap vao trang www.mientayvn.com de tai them...www. mientayvn.com
This document provides an introduction to digital signal processing. It discusses signals, analog versus digital signals, sampling, quantization, coding, and analog-to-digital and digital-to-analog conversion. The key advantages of digital signal processing are also summarized, such as flexibility, reliability, size/power benefits, and suitability for sophisticated applications. Common digital signal processing applications are then outlined, including radar, biomedical processing, speech/audio, communications, image processing, and more.
This document provides an overview of digital signal processing (DSP). It begins by defining an analog signal and a digital signal. It then describes the basic components of a DSP system, which includes an analog-to-digital converter (ADC) to convert the analog input signal to digital, a digital signal processor to process the digital signal, and a digital-to-analog converter (DAC) to reconstruct the analog output signal. Finally, it discusses some advantages and limitations of DSP systems compared to analog systems and provides examples of DSP applications.
The document discusses various techniques for converting digital and analog signals for transmission through different mediums. It explains that digital data needs to be converted to either digital or analog signals depending on whether the transmission medium is digital or analog. Key techniques discussed include:
- Digital to digital encoding like Manchester encoding for digital mediums
- Analog to digital conversion like PCM for converting analog signals to digital
- Digital to analog modulation like FSK for converting digital signals to analog for analog mediums
- Analog to analog conversion like AM/FM for analog signal transmission.
REAL TIME SPECIAL EFFECTS GENERATION AND NOISE FILTRATION OF AUDIO SIGNAL USI...ijcsa
Digital signal processing is being increasingly used for audio processing applications. Digital audio effects
refer to all those algorithms that are used for enhancing sound in any of the steps of a processing chain of
music production. Real time audio effects generation is a highly challenging task in the field of signal
processing. Now a day, almost every high end multimedia audio device does digital signal processing in
one form or another. For years musicians have used different techniques to give their music a unique
sound. Earlier, these techniques were implemented after a lot of work and experimentation. However, now
with the emergence of digital signal processing this task is simplified to a great extent. In this article, the
generations of special effects like echo, flanging, reverberation, stereo, karaoke, noise filtering etc are
successfully implemented using MATLAB and an attractive GUI has been designed for the same.
Digital Signal Processing[ECEG-3171]-Ch1_L01Rediet 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.
#Africa#Ethiopia
This document provides an introduction to digital signal processing (DSP). It defines signal processing and distinguishes between analog signal processing (ASP) and DSP. For DSP, analog signals are first converted to digital using analog-to-digital converters before processing, while for ASP entire processing is done in analog domain. Some key advantages of DSP over ASP include more compact size, accuracy, flexibility, easy storage and modification of digital signals. Additional complexity of analog-to-digital and digital-to-analog conversion is a disadvantage of DSP.
Advanced Topics In Digital Signal ProcessingJim Jenkins
This four-day course from Applied Technology Institute examines advanced digital signal processing techniques used in modern fourth generation modems. The course will cover topics such as digital filters, channelizers, filter design techniques, digital baseband transmission, signal conditioning, sigma-delta converters, carrier centered modulation and demodulation, synchronization, and adaptive filters. Students will learn how to size and design efficient digital filters, understand multirate signal processing, and limitations of DSP-based solutions. The instructor, Dr. Fred Harris, is an expert in DSP and its applications in communication systems.
This document discusses digital signal processing and multirate digital signal processing. It covers topics like sampling rate conversion using interpolation and decimation filters, polyphase filters, and applications of multirate DSP systems. It also describes digital signal processors, focusing on architectures like Von Neumann, Harvard, and SHARC that are optimized for digital signal processing tasks through features like separate data/program memories, pipelining, and multiplier-accumulator units.
This document describes the implementation of a real-time signal processing system using the overlap-save algorithm. It outlines the four main stages of implementation: 1) algorithm design using MATLAB, 2) emulating the DSP hardware constraints, 3) testing the program on a DSP simulator, and 4) running the program on the target DSP hardware. It also provides code for designing an FIR filter kernel and constructing an initial Simulink model of the overlap-save algorithm to process input blocks of length 512 with an output of 400 samples per block.
This document provides an introduction and overview of digital signal processing. It defines signals and how they can be classified as continuous or discrete, continuous or discrete valued. It describes how analog signals are converted to digital signals through sampling and quantization. It explains that digital signal processing involves passing signals through systems to process the signals. It outlines some key advantages of digital signal processing over analog processing, such as flexibility, accuracy, and easy storage. Finally, it provides some examples of applications of digital signal processing like voice recognition, telecommunications, consumer electronics, imaging, and more.
Digital signal processing is a specialized microprocessor with its architecture optimized for operational needs of digital signal processing
Application's of DSP like STFT and Wavelet transform has been explained in detail with images.
digital to analog (DAC) & analog to digital converter (ADC) Asif Iqbal
This document summarizes different types of digital to analog converters (DACs). It discusses the basic concept of converting digital data to analog signals by using a circuit that can produce analog outputs. It then describes several DAC specializations:
1) Binary weighted DAC which uses a reference voltage and weighted resistors to produce analog outputs corresponding to the digital input bits.
2) Flash type ADC which uses a voltage divider network and parallel comparators to directly convert an entire digital word to an analog voltage very quickly.
3) Successive approximation ADC which uses a comparator and feedback loop in a step-wise process to iteratively approximate the analog output voltage, providing a tradeoff between speed and circuit complexity.
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.
Real-Time Signal Processing: Implementation and Applicationsathish sak
This document discusses real-time signal processing, including what it means, why it is used, and platforms for implementation. Real-time signal processing allows signals to be collected, analyzed, and modified in real-time as they occur. It is used to avoid time and money lost when collecting and processing data separately. Common platforms include software/PC, hardware like FPGAs, and firmware/hardware like DSPs, each with their own benefits and drawbacks relating to flexibility, speed, cost, and practicality. The document focuses on DSPs as a popular "middle ground" option and discusses code generation applications and the Embedded Target for TI's C6711 DSP.
This document is a thesis written by Edgar Daniel Chaguaro Aldaz titled "Digital Signal Processing Filtering Algorithm Audio Equalization Using Matlab". It is 40 pages long plus 3 appendices and was submitted on May 7, 2015 for a Bachelor of Engineering degree. The thesis analyzes different digital signal processing filtering algorithms and implements a basic graphic equalizer using Matlab. It describes digital signal processing concepts such as sampling, quantization, filters, and audio equalization. The equalizer application allows processing audio signals using filters and displaying the results graphically in Matlab.
Digital signal processing involves representing and processing signals in the form of discrete numeric values. It has various applications including radar, biomedical monitoring, speech recognition, communications, image processing, and multimedia. Key aspects of digital signal processing implementation are analog to digital conversion, digital processing, and digital to analog conversion. Limitations include information loss due to sampling, aliasing effects, limited frequency resolution, and quantization error. However, digital signal processing provides advantages such as reprogrammability, accuracy control, easy storage and transport of signals, and ability to implement sophisticated algorithms.
Signal processing involves the analysis, interpretation, and manipulation of signals like sound, images, and sensor data. It is used to retrieve the original signal from one that has been contaminated with noise during transmission. There are two main categories: analog signal processing, which uses electronic circuits like filters on un-digitized signals; and digital signal processing, which uses computers or specialized chips to process digitized signals. The digital signal processing block diagram consists of an anti-aliasing filter, analog-to-digital converter, digital filter, digital-to-analog converter, and reconstruction filter. Digital signal processing has advantages over analog like greater accuracy, flexibility, ease of storage and operation, and ability to multiplex signals.
1. The document discusses digital signal processing (DSP) and provides an overview of key concepts such as analog and digital signals, sampling, quantization, coding, and the sampling theorem.
2. It also covers DSP applications in various fields like medical, military, industrial and more. Common signal types like deterministic, random, even, odd and sinusoidal signals are defined.
3. Real-time DSP considerations are discussed, noting the need to ensure processing time meets operational requirements for applications that must operate in real-time.
This document discusses different types of signals and their representations. It describes:
- Analog and digital signals as the two main types.
- Periodic signals that repeat over a measurable time frame called a period. Aperiodic signals do not repeat.
- Continuous signals having time as a continuum, while discrete signals are obtained by sampling continuous signals at discrete time intervals.
- Even signals being identical to their time-reversed counterparts, odd signals changing sign when reversed, and orthogonal signals having no relationship between components.
Lecture: Digital Signal Processing Batch 2009ubaidis
Here are the solutions to the questions:
(a) y[nT] = 3(−0.2)n u[n − 3], T = 2 ms
Energy: E = 9(0.2)3/(1−0.2) = 9
Power: P = 0 (since the signal decays to zero as n increases)
This is an energy signal.
(b) z[nT] = 4(1.1)n u[n + 1], T = 0.02 s
Energy: E = ∞ (since (1.1)n increases without bound)
Power: P = 4.4
This is a power signal.
(c
Ee341 dsp1 1_sv_chapter1_hay truy cap vao trang www.mientayvn.com de tai them...www. mientayvn.com
This document provides an introduction to digital signal processing. It discusses signals, analog versus digital signals, sampling, quantization, coding, and analog-to-digital and digital-to-analog conversion. The key advantages of digital signal processing are also summarized, such as flexibility, reliability, size/power benefits, and suitability for sophisticated applications. Common digital signal processing applications are then outlined, including radar, biomedical processing, speech/audio, communications, image processing, and more.
This document provides an overview of digital signal processing (DSP). It begins by defining an analog signal and a digital signal. It then describes the basic components of a DSP system, which includes an analog-to-digital converter (ADC) to convert the analog input signal to digital, a digital signal processor to process the digital signal, and a digital-to-analog converter (DAC) to reconstruct the analog output signal. Finally, it discusses some advantages and limitations of DSP systems compared to analog systems and provides examples of DSP applications.
The document discusses various techniques for converting digital and analog signals for transmission through different mediums. It explains that digital data needs to be converted to either digital or analog signals depending on whether the transmission medium is digital or analog. Key techniques discussed include:
- Digital to digital encoding like Manchester encoding for digital mediums
- Analog to digital conversion like PCM for converting analog signals to digital
- Digital to analog modulation like FSK for converting digital signals to analog for analog mediums
- Analog to analog conversion like AM/FM for analog signal transmission.
REAL TIME SPECIAL EFFECTS GENERATION AND NOISE FILTRATION OF AUDIO SIGNAL USI...ijcsa
Digital signal processing is being increasingly used for audio processing applications. Digital audio effects
refer to all those algorithms that are used for enhancing sound in any of the steps of a processing chain of
music production. Real time audio effects generation is a highly challenging task in the field of signal
processing. Now a day, almost every high end multimedia audio device does digital signal processing in
one form or another. For years musicians have used different techniques to give their music a unique
sound. Earlier, these techniques were implemented after a lot of work and experimentation. However, now
with the emergence of digital signal processing this task is simplified to a great extent. In this article, the
generations of special effects like echo, flanging, reverberation, stereo, karaoke, noise filtering etc are
successfully implemented using MATLAB and an attractive GUI has been designed for the same.
Digital Signal Processing[ECEG-3171]-Ch1_L01Rediet 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.
#Africa#Ethiopia
This document provides an introduction to digital signal processing (DSP). It defines signal processing and distinguishes between analog signal processing (ASP) and DSP. For DSP, analog signals are first converted to digital using analog-to-digital converters before processing, while for ASP entire processing is done in analog domain. Some key advantages of DSP over ASP include more compact size, accuracy, flexibility, easy storage and modification of digital signals. Additional complexity of analog-to-digital and digital-to-analog conversion is a disadvantage of DSP.
Advanced Topics In Digital Signal ProcessingJim Jenkins
This four-day course from Applied Technology Institute examines advanced digital signal processing techniques used in modern fourth generation modems. The course will cover topics such as digital filters, channelizers, filter design techniques, digital baseband transmission, signal conditioning, sigma-delta converters, carrier centered modulation and demodulation, synchronization, and adaptive filters. Students will learn how to size and design efficient digital filters, understand multirate signal processing, and limitations of DSP-based solutions. The instructor, Dr. Fred Harris, is an expert in DSP and its applications in communication systems.
This document discusses digital signal processing and multirate digital signal processing. It covers topics like sampling rate conversion using interpolation and decimation filters, polyphase filters, and applications of multirate DSP systems. It also describes digital signal processors, focusing on architectures like Von Neumann, Harvard, and SHARC that are optimized for digital signal processing tasks through features like separate data/program memories, pipelining, and multiplier-accumulator units.
This document describes the implementation of a real-time signal processing system using the overlap-save algorithm. It outlines the four main stages of implementation: 1) algorithm design using MATLAB, 2) emulating the DSP hardware constraints, 3) testing the program on a DSP simulator, and 4) running the program on the target DSP hardware. It also provides code for designing an FIR filter kernel and constructing an initial Simulink model of the overlap-save algorithm to process input blocks of length 512 with an output of 400 samples per block.
This document provides an introduction and overview of digital signal processing. It defines signals and how they can be classified as continuous or discrete, continuous or discrete valued. It describes how analog signals are converted to digital signals through sampling and quantization. It explains that digital signal processing involves passing signals through systems to process the signals. It outlines some key advantages of digital signal processing over analog processing, such as flexibility, accuracy, and easy storage. Finally, it provides some examples of applications of digital signal processing like voice recognition, telecommunications, consumer electronics, imaging, and more.
This document discusses how analog signals can be processed using digital systems. It explains that analog signals from the real world are first converted to digital signals using an analog-to-digital converter. The digital signals are then processed by a digital system, such as a computer or calculator. The processed digital signals are then converted back to analog signals using a digital-to-analog converter to interface with the analog world. The digital system is composed of sub-systems, modules, and basic logic gate units that perform logical operations on inputs.
The document discusses analog vs digital signals in Arduino. It defines analog signals as continuously variable signals like temperature, while digital signals can only have discrete values like 1s and 0s. It explains that Arduino uses an analog-to-digital converter (ADC) to convert analog sensor output to digital values readable by the microcontroller. The ADC divides the analog range into discrete steps, returning a number between 0-1023. Functions like analogRead() are used to read analog pin values as digital numbers.
Intorduction to information theory and applications copyBikash Thapa
This document provides an introduction to analog and digital signals. It discusses:
- Analog signals are continuous over time and can assume an infinite number of values. Examples include audio and video. Digital signals are discrete and can only assume a finite number of values.
- The key differences between analog and digital signals - analog signals are continuous while digital signals are discrete, analog has theoretically infinite resolution while digital has finite resolution.
- Applications of each type of signal - analog is used in devices like thermometers while digital is used in computers and mobile phones.
This document provides an introduction and overview of a course on Digital Signal Processing taught by Professor S.C. Dutta Roy at the Indian Institute of Technology, Delhi. The professor introduces himself and provides an outline of the course contents, which will include a review of signals and systems, discrete time signals and systems, sampling, the discrete Fourier transform, Z-transforms, digital filter design, and implementation considerations. Examples of signals that will be discussed are also provided, such as electrocardiogram signals which are often corrupted with power line interference that digital signal processing can help remove.
Digital signal processing (DSP) involves converting analog signals to digital signals and manipulating the digital signals using software algorithms. DSP systems use analog-to-digital conversion to convert analog signals to digital signals represented as sequences of numbers. They then process the digital signals using a digital signal processor and convert them back to analog signals using digital-to-analog conversion. Key techniques in DSP include decomposing signals into simple components, processing the components individually, and then combining the results.
Simple description about the analog and digital signals
and a description about analog to digital conversion &
digital to analog conversion..............
Biomedical digital signal processing : Digital Hearing Aid Pooja Yadav
This document will cover the application of digital signal processing in hearing aid.
Contents:
1. Introduction to Digital Signal Processing(DSP) in hearing aid
2. Advantages of DSP
3. Limitations of DSP
4. Analysis on its physical advantages
5. Conclusion
1. The document outlines a digital logic and design course, including lecture topics on analog and digital signals and data, representing continuous signals digitally, and number systems.
2. Distribution of marks is outlined, with exams, quizzes, assignments, and labs accounting for marks.
3. Analog data and signals are continuous, while digital data and signals are discrete and can have a limited number of defined values represented using voltages. Continuous signals can be represented digitally by taking samples.
DIGITAL SIGNAL PROCESWSING AND ITS APPLICATIONLokeshBanarse
Digital signal processing (DSP) involves using digital technology to process analog signals. It converts analog signals into digital data that can be manipulated and analyzed. DSP has applications in areas like audio processing, image processing, radar, and mobile phones. The key components of DSP systems are program memory, data memory, a compute engine, and input/output interfaces. DSP emerged in the 1960s and was initially used for applications like radar, sonar, and space exploration. It later expanded into commercial uses with the growth of personal computers and consumer electronics.
This document discusses signal processing and digital signal processing. It defines a signal as something that conveys information, like a person's voice. A system manipulates signals to perform functions. Signal processing converts signals into forms suitable for further processing. Digital signal processing offers advantages like flexibility, accuracy, and lower power consumption compared to analog processing. It has applications in areas like telephones, imaging, filtering, and medical equipment. A typical digital signal processing system involves analog to digital conversion, digital processing, and digital to analog conversion. The Nyquist sampling theorem states that a signal must be sampled at least twice as fast as its highest frequency to perfectly reconstruct it.
The application wavelet transform algorithm in testing adc effective number o...ijcsit
In evaluating Analog to Digital Convertors, many parameters are checked for performance and error rate.
One of these parameters is the device Effective Number of Bits. In classical testing of Effective Number of
Bits, testing is based on signal to noise components ratio (SNR), whose coefficients are driven via
frequency domain (Fourier Transform) of ADC’s output signal. Such a technique is extremely sensitive to
noise and require large number of data samples. That is, longer and more complex testing process as the
device under test increases in resolutions. Meanwhile, a new time – frequency domain approach (known as
Wavelet transform) is proposed to measure and analyze Analog-to-Digital Converters parameter of
Effective Number of Bits with less complexity and fewer data samples.
The document discusses digital signal processing (DSP). It begins by defining analog and digital signals, noting that analog signals are continuous while digital signals are discrete. It then outlines the basic components of a DSP system, including an analog-to-digital converter (ADC) to digitize the input signal, a digital signal processor to perform operations on the digital signal, and a digital-to-analog converter (DAC) to reconstruct the analog output signal. Finally, it lists some advantages and limitations of DSP systems compared to analog systems, and provides examples of DSP applications.
The document discusses analog to digital conversion. It defines analog and digital signals and explains the conversion process.
1) Analog signals are continuously variable, while digital signals represent information as discrete numbers.
2) An analog-to-digital converter (ADC) samples an analog signal by taking regular snapshots and assigning the closest digital value.
3) A digital-to-analog converter (DAC) reconstructs the analog signal from the digital values for playback. Increasing the sampling rate and precision reduces errors between the original and reconstructed signals.
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Lec2
1. Digital Signal Processing
Prof: S. C. Dutta Roy
Department of Electrical Engineering
Indian Institute of Technology, Delhi
Lecture - 2
Digital Signal Processing Introduction (Contd…)
This is the second lecture on DSP and the topic of today is Introduction to Digital Signal
Processing, continued from the last lecture, and we shall also discuss digital signals. In the last
lecture we talked about the course contents, the books that are recommended, viz Mitra for DSP
and Oppenheim and Schafer for DTSP. We talked about definition of a signal. A signal is a
function of one or more variables. And accordingly it can be a one dimensional signal, two
dimensional signal or multi dimensional.
(Refer Slide Time: 02:19 - 03:19)
We also said that basically there are two kinds of signals: analog and digital. Analog can be
continuous time (CT) or discrete time (DT). A discrete time signal is not a digital signal; a
discrete time signal is one in which the amplitude is a continuum, and not discretized. On the
1
2. other hand, if the time is discretized, this signal is still analog. Only when a discrete time signal
is passed through an A to D converter, it becomes a digital signal. To repeat, in analog domain,
we can have two kinds of signals-continuous time and discrete time. And then we also
emphasized the need for processing: why should one process a signal? This is to obtain a better
quality signal than the original one. It could be for example, noise filtering or improving the
quality of a picture in terms of brightness or contrast or whatever it is.
(Refer Slide Time: 03:57 – 04:48)
We also said that signal processing that one can make can be of three types. You can have analog
signal processing or Digital Signal Processing or mixed signal processing. We also emphasized
that analog signal processing dominated prior to 70s. Between 1970 and 2000 it is basically
Digital Signal Processing which got dominance over analog. The current trend is to have mixed
signal processing, namely a combination of digital and analog. The next question that we
elaborate on is, why Digital Signal Processing? We can do analog signal processing only,
because most of the signals in practice are analog. But DSP should be the natural choice if the
signal is digital, for example, the rainfall data at Delhi over the last 20 years on which is based
the prediction for the next year. Predictions sometimes fail miserably, as it has this year, but the
data that you collect is a digital data: 1980 so many inches, 1981 so many inches and so on. Now
2
3. the amplitude can be a continuum so it is a discrete time signal, but the amplitude or the number
of inches of rainfall can also be coded, to make it a digital signal.
(Refer Slide Time: 05:56 - 07:14)
If you so desire, if you want to do DSP then the amplitude or the height of rainfall can also be
coded. But you see the time is discretized: 1980, 1981, 1982 etc and there is nothing between 80
and 81, there is nothing like 1980.5 or 1980.41. So, the natural choice would be DSP and this is
what is done. This data is fed with many other parameters, controlling monsoon or rainfall to a
super computer which basically does DSP. What is DSP? Addition, multiplication and delay or
recalling of past signals are the basic operations in DSP. Even if the signal is analog, one prefers
to use DSP. The reasons are many. DSP has many advantages over ASP or Analog Signal
Processing. And many a times it happens to be a less costly proposition. Digital IC’s are
available off the shelf at very low cost and therefore in many cases, analog signal processing
gives way to Digital Signal Processing even if the original signal is analog.
3
4. (Refer Slide Time: 07:49 - 11:08)
How it is done is represented by this block diagram where the input is analog CT; then we have a
sample and hold device which basically consists of a switch which samples the analog signal and
holds the samples for some time so that it can be converted to a digital form, because conversion
from analog to digital requires time. During that time the signal must be held. After the A to D,
you get a coded signal, whose usual form is binary. This is then processed by a digital processor.
What basically the DSP does is to combine signals by multiplication, addition and by recalling
past signals, through delays.
There are basically three operations: delay, multiplication, and addition. So this is what the
digital processor does. It is a very simple device, it simply performs 3 operations. Multiplication
is repeated addition, with only two signals at a time; subtraction is also a form of addition with
one of these signals negated or the sign bit changed. So, there is no integration, there is no
differentiation; there are no complicated operations which limits the usefulness of analog
processing. The output of the digital processor is also digital, a sequence of numbers, and in
order that output will be useful, you must convert it back to analog form. So you have a D to A
or digital to analog converter. The output of the digital to analog converter is in the form of stair
cases.
4
5. The signal changes from one level to the next and then stays. The abrupt change gives rise to
high frequencies in the signal. These high frequencies present in the signal have to be gotten rid
of by an analog low pass filter, and finally this analog output is the output that is to be used. It
could be, for example, a piece of music which has to be filtered from surrounding noise and
perhaps mixed with other signals, and finally it has to be played. It has to be heard on the loud
speaker and therefore you do require an analog output. Pictorially this slide represents the basic
operations.
(Refer Slide Time: 11:23 - 13:19)
The first picture is that of an analog input which you wish to process by a digital processor and
this is what happens when you sample it. The signal is still sampled at 0, 2 microseconds, 4
microseconds, 6 microsecond, 8 microseconds. This is uniform sampling and at each point, 6
microsecond for example, you pick up the value of the signal and then hold it till the next
sampling comes. This holding is required for A to D conversion, which requires time. After the
A to D conversion, the signal may look like a sequence like this, where 1 stands for positive and
0 stands for negative.
5
6. There are many other choices of coding but this is a typical coding where any pulse which is
positive is taken as 1; for example this part of the signal is 1, 0. Then after processing by the DSP
which carries out the operations of multiplication, addition and delay, may be the output of the
DSP is something like this, which now has to be converted to analog form. So you have a D to A
converter which again produces stair case waveforms like this. That is, these codes are converted
to amplitudes, volts or amperes and a typical D to A output looks like this, which, as I said,
contains high frequencies which have to be got rid of. Therefore you pass this through an analog
low pass filter to get the desired output which may typically look like this. This picture shows
exactly the operations which are involved in processing an analog signal by digital means. The
advantages that accrue out of the use of DSP, compared to ASP, are many.
(Refer Slide Time: 14:07 - 17:17)
First, the digital signal processor is less sensitive to component tolerances and environmental
changes. In analog signal processing change in any element, an inductance or a capacitance, or a
small shift in the power supply causes a change in the performance. This also happens in digital
circuits but since you are coding high amplitude by 1 and low amplitude by 0, even if the
amplitude changes by 10% it does not matter, because 1 remains 1 and 0 remains 0. Unless the
amplitude drops down to the noise margin level you have nothing to worry. Therefore digital
6
7. circuits are less sensitive to component tolerances and environmental changes. Digital circuits
allow volume production without the need for adjustment during construction or use. Once the
process is set, the timings and the steps, nothing has to be changed. Digital circuits are of course
amenable to full integration. This is not possible for analog circuits because of difficulty of
integrating inductances and transformers.
A transformer cannot be made into an integrated chip; same is the case with an inductance of a
respectable value. Inductances of very small values (Pico-Henry) can be made by IC’s. But an
inductance of a respectable value like micro-Henry or milli-Henry is very difficult to make
because inductance basically requires a large volume. Inductance is flux per unit current. In order
to generate flux you require space. You cannot generate a large amount of flux in a small space
unless the magnetic permeability is astronomically high. Very high permeability materials are
not available and therefore analog circuits are not amenable to full integration; Inductances and
transformers have to be outside the chip.
In Digital Signal Processing, accuracy can be increased almost indefinitely by increasing the
word length. Of course the cost also rises; as you put more number of bits you have to pay more
but the accuracy increases. In an analog signal processor the dynamic range is limited by the
power supply. In an operational amplifier (op amp) for example, the output cannot exceed + Vcc
or – Vcc; it gets saturated. In Digital Signal Processing since we are handling numbers there is no
question of saturation. We can handle as large numbers as possible and if fixed point does not do
the job, use floating point with characteristic and mantissa stored in separate registers. You can
then achieve very large dynamic range. This is not possible in analog signal processing.
The same digital signal processor can do multiple jobs. For example, use the same processor to
time share two different operations and the following picture shows how this is done.
7
8. (Refer Slide Time: 18:23 - 19:26)
The upper signal is one which exists only at 0 seconds, 2 microseconds, 4 microseconds, 6
microseconds and so on. In the space in between you can put in another signal (this is time
multiplexing) at 1 microsecond, 3 microseconds, 5 microseconds and so on. These two signals
are there mixed and the composite signal looks like this. So if there is a processing occurring at
odd values of time then it handles only the first signal. The same processor can also perform
another operation taking signals at the even values of time. So the same processor can perform
operations on two different signals which have been time multiplexed and then at the output you
can separate the two. Use of the same processor simultaneously for different operations is a great
advantage of using a digital signal processor. Time multiplexing is one of the great advantages of
DSP; you cannot do it with ASP. If you mix two continuous time or discrete time signals, there is
no way you can separate them.
Continuing the advantages of DSP, a digital signal processor can be very easily adjusted for
different characteristics by simply changing some numbers, which form the coefficients. This
can also be done in analog signal processing by varying a resistor, or a capacitor or an inductor
but changing a number is much simpler then changing R, L or C. Varying an inductance for
example is a very elaborate procedure. Varying a capacitance is not a problem but then you have
8
9. problems of contact. You also have problems of environmental changes and so on and so forth.
Similarly for resistors, you know that the potentiometer contact gradually wears off due to
mechanical friction. There is nothing like this in DSP.
(Refer Slide Time: 22:14 - 22:50)
You simply change a number which is one of the coefficients and then you get a different
characteristic. This is required in adaptive filters or variable cutoff filters.
One of the major and dominant advantages of DSP is that exact linear phase systems can be
designed. This is not possible in analog signal processing and has been a dream for analog
communication engineers. Exact linear phase means there is no delay distortion. We shall come
back to the importance of linear phase in a later lecture. But this is one of the major advantages
of DSP that it can attain exact linear phase and therefore it permits processing without any delay
distortion.
One of the advantages of DSP is also that different parts of the signal processor can work at
different rates. That is, the sampling rate can be different in different parts of the processor.
9
10. There are well established techniques for changing one sampling rate to another. You can sample
up or sample down without any problem. This facility is not possible in analog signal processing.
(Refer Slide Time: 23:50 - 24:58)
When two amplifiers are cascaded, ideally the overall gain is the product of the two gains.
However, this is not invariably the case in practice because one amplifier loads the other. As
soon as you put a loud speaker at the output of your power amplifier, unless it is perfectly
matched, the power amplifier output goes down. There is no such problem in cascading two or
more digital circuits provided you do not exceed the fan out capability of the circuits. In short
there are no loading problems due to cascading in digital circuits. Digital signals can easily be
stored in magnetic tapes and disks, including optical disks, as long as one likes. There will be no
problem in storage.
In DSP, very low frequency signal processing is possible with ease. But in analog signal
processing, inductance poses a problem. There is no such problem in Digital Signal Processing at
very low frequencies. However, all are not roses with Digital Signal Processing, there are
disadvantages also. The major disadvantage, of course, is increased complexity. You have to use
more devices: sample and hold, A to D, D to A, low pass filters etc. Another major disadvantage
10
11. is that the frequency range for operation of a DSP is limited due to the limited rate at which you
can sample and hold. Also in A to D conversion, the speed is limited.
(Refer Slide Time: 26:02 - 27:24)
In practice you cannot use a sampling frequency, with today’s state of the art, which is more than
10 mega hertz. And therefore if the sampling frequency is 10 mega hertz, naturally signal
processing cannot be done for more than 5 mega hertz. The other disadvantage is that, if you
have a purely passive circuit consisting of inductance, resistance and capacitance to do the
analog signal processing, there is power dissipation only in the resistance. But it is much less
compared to the power drawn by digital circuits, main dissipation being in the active devices, the
transistors. A typical AT and T DSP32C processor which contains 405,000 transistors dissipates
1 watt. There are no such problems with purely LC circuits. But even then one does use DSP
because of its dominant advantages. This picture is very rapidly changing, however, power
dissipation being one of the major challenges for DSP engineers.
One of the aims of TI Texas Instruments is to make digital circuits which will require no power
supply, particularly for devices which are to be implanted in live bodies like the human body.
The heart transplant or the hearing aids would not, if DSP engineers are successful in future,
11
12. require a battery. Battery is a major problem; to make a small battery is a severe problem. If you
want to put it inside the human body, space as well as life of the battery are the problems. So
they are aiming at devices which will work from body temperature. As long as the person is alive
and his body temperature is between two limits, the device will work. We now come back to
digital signals.
(Refer Slide Time: 29:12 - 33:09)
A digital signal is obtained from an analog signal by sampling. What you get here after sampling
xa(t) at regular intervals of T is xa(nT); this is not a digital signal but an analog discrete time
signal. It exists only at capital T, 2T and so on. But its amplitude can be a continuum. Then you
feed it to an A to D and what you get at the output is a truly digital signal for which we give the
symbol x(n); the subscript a is lost because it becomes digital and capital T in the process is also
lost, in the sense that when small n becomes a variable, it can take integer values only- positive
or negative. And each integer value of n in the corresponding discrete time signal corresponds to
the integer multiplied by the sampling interval. So T is lost in the digital signal but if this signal
was derived from an analog signal, then T has to be kept in mind. One way of interpreting x(n) is
that capital T has been normalized to 1.
12
13. However, it is a beauty of DSP and Discrete Time Signal Processing (DTSP) that the
mathematical techniques for handling discrete time signals and digital signals are the same. They
are: difference equations, z transforms, Fourier transforms and discrete Fourier transforms. So
whether the dependent variable is a continuum, like xa(nT), or of discrete amplitude, like x(n),
the techniques are the same. And therefore we shall use xa(nT) and x(n) interchangeably. We
shall always take that the signal has been discretized in time as well as amplitude. This sampling
interval T is obviously the reciprocal of the sampling frequency fs. We shall see later that our
signal processing must be limited to half of the sampling frequency, we cannot go beyond that.
Obviously then, digital signal is a sequence of numbers and the sequence sometimes is simply
represented as x(n) where n takes values theoretically from – infinity to + infinity but at intervals
of 1. To show that it is a sequence we sometimes give the symbol of x(n) within curly brackets.
(Refer Slide Time: 33:19 - 36:16)
For example, we can have a signal like 0, –1, 1, 2, – 2 and so on. I am writing it in ordinary
arithmetic but I have taken them to be coded, that is, quantized into integer numbers. In this
sequence how is n indicated? n = 0 is indicated through an arrow and one writes n = 0 below it;
in the first x(n), signal x(n) is equal to 2 at n = + 1, at n = + 2 the value is – 2, for n = – 1 it is 0,
for n = – 2 it is – 1 and so on. So this is the complete description and representation of the signal.
13
14. We shall not write binary numbers here, we shall write the integer equivalent of the number. This
sequence is not necessarily real; it can be complex. What you see is a real sequence, but we can
have a sequence like 1 + j, 2 – j 2 and so on. For example, if you consider an analytic signal, it
has a real part and an imaginary part; it simply means that there are two real signals which are
orthogonal to each other.
Therefore a digital signal can be real or complex. The length of the sequence is the total number
of samples in the sequence. If n goes from – infinity to + infinity, then obviously it is an infinite
signal. But suppose we have a sequence 1, 2, 3, 4 with 1 at n = 0; then obviously it is a finite
length sequence .The length of this sequence is equal to the number of samples in the sequence.
That is, the length of this signal is 4 although n takes values 0, 1, 2, 3. If the signal extends from
n = 0 to N – 1, the length is N because there is a sample at n = 0.
(Refer Slide Time: 36:25 - 37:58)
Consider a sequence that exists from some value of n, say n = N1, and then goes to the right up to
any value of n, but on the left this sequence is all 0, where N1 can be anything like 0, –1, or +10.
This signal exists to the right only and therefore it is called a right sided signal. On the other
hand, if this signal exists only on the left side and is 0 on the right side, then this is called a left
14
15. sided signal. A signal in general can be combination of left sided and right sided signals. If it is
of finite length, then we can always view it as right sided or left sided. If it is of infinite length,
then obviously it is a combination of right sided and left sided signals. A signal or a sequence
can be even or odd or a mixture of even and odd.
(Refer Slide Time: 38:09 - 39:12)
An even sequence is one whose left side is the mirror image of the right side. If centering at n =
0, you have the same amplitude at n = 1 and n = – 1, same amplitude at n = 2 and n = – 2, and so
on, then it is an even signal. In other words, an even signal xe(n) is the same as xe(– n). On the
other hand, an odd signal is one in which the signal xo(n) is the negative of xo(– n).
15
16. (Refer Slide Time: 39:23 - 41:08)
For an old signal, what should be the value at n = 0? Because xo(0) should be negative of x0(– 0),
it means that xo(0) must be 0. An even signal can have any value at n = 0, while an odd signal
must have a 0 value at n = 0. In general, a signal is neither even nor odd but it can be
decomposed into two parts an even part xe(n) and an odd part xo(n). And if you want to find out
the even part, it is very easy. xe(n) is just half of x(n) + x(– n). When you add the two, the odd
parts will cancel. In a similar manner, the odd part is given by x0(n) = half of [x(n) – x(– n)].
When you subtract x(– n) from x(n), the even parts cancel.
16
17. (Refer Slide Time: 41:13 - 43:33)
There are quite a few problems in the book on decomposition of signals into even and odd parts;
try to work out as many as you can.
We introduce now a periodic sequence or periodic digital signal. A signal x(n) is said to be
periodic if x(n) = x(n + k N) where N is the smallest positive integer; it means that x(n) repeats
after every N samples. And the smallest N for the validity of this relation is called the period.
Therefore the period has no dimension, it is not in seconds or micro seconds, it is a pure number.
That is, it indicates the number of samples after which the sequence repeats.
Why is it the smallest N? Suppose 20 is the period then repetition will occur after 40 samples, 60
samples and so on, but the repetition will not occur for 10. So the smallest value of N for which
this relationship is valid is called the period of the periodic signal or sequence. We will show a
little later that while cos(ωt) or sin(ωt) is always periodic, this is not true about cos(ω0n + φ) or
sin(ω1n + φ).
17
18. (Refer Slide Time: 44:01 - 45:26)
We now define what a bounded sequence is. A sequence x(n), which may be real (positive or
negative) or complex, whose magnitude is bounded, i.e. less than or equal to some number M,
which is less than infinity, is said to be a bounded sequence.
If the summation of magnitude of x(n) over its total length, in general from – infinity to +
infinity, is bounded, i.e. less than equal to some number P, which is less than infinity, then x(n) is
said to be an absolutely summable sequence. You can similarly define a square summable
sequence.
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19. (Refer Slide Time: 45:35 - 47:23)
A square summable sequence is one in which the magnitude squared summed up from n = –
infinity to + infinity, is less than or equal to some number Q, which is less than infinity. Suppose
you have a sequence 1/n, n = 1 to ∞. Is it absolutely summable? No, because the summation n =
1 to ∞ |x(n)| does not converge. Is it square summable? Yes, it is. You notice that this represents
the energy of the sequence, just as integral of the square of voltage across a 1 ohm resistance is
energy, or integral of the square of current through a 1 ohm resistance is also energy.
The energy of a sequence is defined as |x(n)| the whole square summed up over all the samples.
So a square summable sequence can also be called, alternatively, a finite energy sequence. A
sequence may not be square summable, just as an analog signal may not be an energy signal.
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20. (Refer Slide Time: 47:38 - 49:28)
We now define average power. We have defined what energy is. In analog domain, power is
energy per unit time. Similarity, if we find out the energy of a sequence over a certain length, say
from n = – K to + K, then this represents energy over the length, 2K + 1. If we divide this by the
length 2K + 1, then this will represent average power over the same length. Now we allow K to
go to infinity so that we cover the total length of the signal, so then we would get the average
power of the signal over its total length because it is a limiting process, it may not be possible to
go on adding for infinite length sequences. So what you do is to take sufficiently large value of
K, then increment it by 1, 2, 3 and so on and see whether the average power changes. Usually the
sequences trail off at very large values of n. Therefore average power is a meaningful definition.
If you have a periodic signal, then your job is very simple.
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21. (Refer Slide Time: 49:36 - 51:29)
For a periodic signal of period N, all you have to do is to take the average over one period,
because every period is identical replica of any other period. What you do is to sum up |x(n)| the
whole square from n = 0 to N – 1 (because at n = N it starts repeating) and then divide by N. So
this is the average power of a periodic signal. We have defined absolutely summable sequence,
square summable sequence which is the same as finite energy sequence, and then average power
in a sequence. Now look at some typical elementary digital signals. The most elementary signal
is the δ(n) and it is defined as one for n = 0, and 0 for n not equal to 0. In other words its diagram
shall be just one sample of amplitude 1 occurring at n = 0 and it is 0 on either side; this is the
most elementary signal that one can think of and is called a digital impulse. In comparison, the
analog impulse δ(t) is quite complicated. How is δ (t) defined?
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22. (Refer Slide Time: 51:43 - 52:22)
It exists at t = 0 only and the amplitude there is infinitely large which does not mean anything.
You must define infinity otherwise the function will be useless for us; so δ(t) is defined by ∫ δ(t)
dt from 0 – to 0 +, that is the area under the curve, is 1. δ(t) is not a function in the ordinary
sense, it is an unlimited function whose amplitude cannot be defined. δ(t) can be defined only in
terms of the area. However there is no such problem in a digital signal; δ(n) is 1 at n = 0 and 0
everywhere else.
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23. (Refer Slide Time: 52:44 - 53:46)
A combination of such impulse functions give rise to a digital unit step u(n); u(n) is a signal
which is 1 for n greater than or equal to 0 and 0 otherwise. Obviously u(n) can be written in
terms of δ(n); it is summation of δ(n-k), k going from 0 to infinity. δ(n-1) describes an impulse
with one sample delay, δ(n-2) is an impulse with two samples delay, and so on. So u(n) is very
simply related to δ(n – k). δ(n) can also be related to u(n).
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24. (Refer Slide Time: 53:58 - 54:40)
δ(n) is simply one signal of amplitude 1 at n = 0 and the rest all are 0. Obviously, for δ(n), you
subtract from u(n) all that occurs to the right of n = 0, which is simply u(n – 1). Therefore δ(n)
can be replaced by u(n) – u(n – 1). I think our time has come to an end and therefore we will stop
here today.
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