This document provides an overview of frequency analysis techniques for signals and systems, including the Fourier series, Fourier transform, discrete-time Fourier series (DTFS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). It discusses properties and applications of these techniques, such as analyzing periodic and aperiodic signals. Examples are provided to illustrate calculating the Fourier series and transform of simple signals. The document also covers sampling theory and the Nyquist criterion for proper reconstruction of signals from samples.
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
The document discusses decimation in time (DIT) and decimation in frequency (DIF) fast Fourier transform (FFT) algorithms. DIT breaks down an N-point sequence into smaller DFTs of even and odd indexed samples, recursively computing smaller and smaller DFTs until individual points remain. DIF similarly decomposes the computation but by breaking the frequency domain spectrum into smaller DFTs. Both algorithms reduce the computational complexity of computing the discrete Fourier transform from O(N^2) to O(NlogN) operations.
This document discusses the decimation-in-time (DIT) algorithm for computing the discrete Fourier transform (DFT) in a more efficient manner than directly calculating all N points. DIT works by splitting the input sequence into smaller sequences, computing smaller DFTs, and recombining the results. Specifically, it separates the input into even and odd samples, computes partial DFTs on each, and combines them using a "butterfly" structure. This structure implements each step of the algorithm with one multiplication, reducing the complexity from N^2 operations to N^2/2 + N operations. The document provides an example of an 8-point DFT computed using the DIT algorithm and butterfly structure.
3.Frequency Domain Representation of Signals and SystemsINDIAN NAVY
This document provides an overview of frequency domain representation of signals and systems. It defines key concepts such as the Fourier transform, which converts a signal from the time domain to the frequency domain. The frequency spectrum shows the distribution of frequencies within a signal. Periodic signals can be represented using Fourier series, while aperiodic signals use the Fourier transform. Properties of the Fourier transform such as linearity, time shifting, and the convolution theorem are also covered.
This document discusses digital filter design methods. It introduces IIR and FIR filters and their design techniques. The key methods covered are:
1. IIR filter design using impulse invariance, which samples the impulse response of an analog filter to obtain the discrete-time filter.
2. IIR filter design using bilinear transformation, which maps the continuous s-domain to the discrete z-domain to avoid aliasing.
3. FIR filter design using frequency sampling, which designs a linear phase FIR filter by sampling the desired frequency response and taking the inverse DFT.
This document discusses fast Fourier transform (FFT) algorithms. It provides an overview of FFTs and how they are more efficient than direct computation of the discrete Fourier transform (DFT). It describes decimation-in-time and decimation-in-frequency FFT algorithms and how they exploit properties of the DFT. The document also gives an example of calculating an 8-point DFT using the radix-2 decimation-in-frequency algorithm.
This document summarizes key aspects of the discrete Fourier transform (DFT). It defines the DFT, provides the formula for calculating it, and explains that the DFT transforms a discrete-time signal from the time domain to the frequency domain. It also outlines several important properties of the DFT, including linearity, shift property, duality, symmetry, and circular convolution. Examples are provided to illustrate duality and symmetry. References for further information on the discrete Fourier transform are also included.
The document discusses digital filters and their design. It begins with an introduction to filters and their uses in signal processing applications. It then covers linear time-invariant filters and their transfer functions. It discusses the differences between non-recursive (FIR) and recursive (IIR) filters. The document presents various filter structures for implementation, including direct form I and direct form II structures. It also discusses designing FIR and IIR filters as well as issues in their implementation.
The document discusses decimation in time (DIT) and decimation in frequency (DIF) fast Fourier transform (FFT) algorithms. DIT breaks down an N-point sequence into smaller DFTs of even and odd indexed samples, recursively computing smaller and smaller DFTs until individual points remain. DIF similarly decomposes the computation but by breaking the frequency domain spectrum into smaller DFTs. Both algorithms reduce the computational complexity of computing the discrete Fourier transform from O(N^2) to O(NlogN) operations.
This document discusses the decimation-in-time (DIT) algorithm for computing the discrete Fourier transform (DFT) in a more efficient manner than directly calculating all N points. DIT works by splitting the input sequence into smaller sequences, computing smaller DFTs, and recombining the results. Specifically, it separates the input into even and odd samples, computes partial DFTs on each, and combines them using a "butterfly" structure. This structure implements each step of the algorithm with one multiplication, reducing the complexity from N^2 operations to N^2/2 + N operations. The document provides an example of an 8-point DFT computed using the DIT algorithm and butterfly structure.
3.Frequency Domain Representation of Signals and SystemsINDIAN NAVY
This document provides an overview of frequency domain representation of signals and systems. It defines key concepts such as the Fourier transform, which converts a signal from the time domain to the frequency domain. The frequency spectrum shows the distribution of frequencies within a signal. Periodic signals can be represented using Fourier series, while aperiodic signals use the Fourier transform. Properties of the Fourier transform such as linearity, time shifting, and the convolution theorem are also covered.
This document discusses digital filter design methods. It introduces IIR and FIR filters and their design techniques. The key methods covered are:
1. IIR filter design using impulse invariance, which samples the impulse response of an analog filter to obtain the discrete-time filter.
2. IIR filter design using bilinear transformation, which maps the continuous s-domain to the discrete z-domain to avoid aliasing.
3. FIR filter design using frequency sampling, which designs a linear phase FIR filter by sampling the desired frequency response and taking the inverse DFT.
This document discusses fast Fourier transform (FFT) algorithms. It provides an overview of FFTs and how they are more efficient than direct computation of the discrete Fourier transform (DFT). It describes decimation-in-time and decimation-in-frequency FFT algorithms and how they exploit properties of the DFT. The document also gives an example of calculating an 8-point DFT using the radix-2 decimation-in-frequency algorithm.
This document summarizes key aspects of the discrete Fourier transform (DFT). It defines the DFT, provides the formula for calculating it, and explains that the DFT transforms a discrete-time signal from the time domain to the frequency domain. It also outlines several important properties of the DFT, including linearity, shift property, duality, symmetry, and circular convolution. Examples are provided to illustrate duality and symmetry. References for further information on the discrete Fourier transform are also included.
The Fourier transform decomposes a signal into its constituent frequencies, representing it in the frequency domain rather than the spatial domain, which can make certain operations and analyses easier to perform; it has both magnitude and phase components that provide information about the frequency content and relative phases of the signal. The discrete Fourier transform (DFT) is a sampled version of the continuous Fourier transform that is useful for digital signal and image processing applications.
This document provides an overview of discrete time systems and their representations. It discusses key concepts such as:
- The difference between continuous and discrete time systems
- Representing discrete time systems using difference equations and block diagrams
- Classifying systems as static/dynamic, time-variant/invariant, linear/nonlinear, causal/non-causal, and stable/unstable
- Examples are provided to illustrate different system types.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
IIR filter realization using direct form I & IISarang Joshi
The document discusses IIR filter realization using Direct Form I and Direct Form II structures. It presents the difference equation and transfer function for an IIR filter. It also provides examples of implementing IIR filters using Direct Form I and Direct Form II structures based on a given difference equation or transfer function.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
DSP_2018_FOEHU - Lec 08 - The Discrete Fourier TransformAmr E. Mohamed
The document provides an overview of the Discrete Fourier Transform (DFT). It begins by discussing limitations of the discrete-time Fourier transform (DTFT) and z-transform in that they are defined for infinite sequences and continuous variables. The DFT avoids these issues by being a numerically computable transform for finite discrete-time signals. It works by taking a finite signal, making it periodic, and computing its discrete Fourier transform which is a discrete frequency spectrum. This makes the DFT highly suitable for digital signal processing. The document then provides details on computation of the DFT and its relationship to the DTFT and z-transform.
Fir filter design using Frequency sampling methodSarang Joshi
The document discusses the design of a finite impulse response (FIR) filter using the frequency sampling technique. It describes how to determine the impulse response of an FIR filter of length 7 to meet specific frequency response specifications. Frequency samples are taken at points and the desired response is defined. The discrete-time Fourier transform (DTFT) and inverse DTFT are used to calculate the impulse response coefficients that produce the desired filter frequency response. Equations for the impulse response h(n) are provided.
DSP_2018_FOEHU - Lec 03 - Discrete-Time Signals and SystemsAmr E. Mohamed
The document discusses discrete-time signals and systems. It defines discrete-time signals as sequences represented by x[n] and discusses important sequences like the unit sample, unit step, and periodic sequences. It then defines discrete-time systems as devices that take a discrete-time signal x(n) as input and produce another discrete-time signal y(n) as output. The document classifies systems as static vs. dynamic, time-invariant vs. time-varying, linear vs. nonlinear, and causal vs. noncausal. It provides examples to illustrate each classification.
Circular convolution is performed on two signals x1 and x2.
x1 and x2 are periodic signals with period 4. The circular convolution sums the product of the signals at each time offset.
The convolution is computed for different time offsets from 0 to 3. The results of the convolution at each offset are 34, 36, 34, 28, forming the output signal y(m).
This document discusses digital signal processing and the design of finite impulse response (FIR) filters using the window method. It begins with an introduction to FIR filters, noting their advantages over infinite impulse response (IIR) filters such as being easily designed with linear phase and being unconditionally stable. The document then covers FIR filter design concepts like phase delay, linear phase response, and filter specifications. It presents the window method approach to FIR filter coefficient calculation and discusses filter design considerations like coefficient calculation methods and filter structure selection.
This document provides an overview of digital filter design. It introduces finite impulse response (FIR) and infinite impulse response (IIR) filters. FIR filters are designed using window techniques like rectangular, Hamming, and Kaiser windows. IIR filters are designed using approximation methods like Butterworth, Chebyshev I, and Chebyshev II. MATLAB code is provided to design low pass, high pass, and other filters using different window and approximation techniques. Pros and cons of FIR and IIR filters are discussed along with references.
The document discusses 11 properties of the Fourier transform: (1) Linearity and superposition, (2) Time scaling, (3) Time shifting, (4) Duality or symmetry, (5) Area under the time domain function equals the Fourier transform at f=0, (6) Area under the Fourier transform equals the time domain function at t=0, (7) Frequency shifting, (8) Differentiation in the time domain, (9) Integration in the time domain, (10) Multiplication in the time domain becomes convolution in the frequency domain, and (11) Convolution in the time domain becomes multiplication in the frequency domain. Each property is explained briefly.
Fir filter design (windowing technique)Bin Biny Bino
The window design technique for FIR filters involves choosing an ideal frequency-selective filter with the desired passband and stopband characteristics, and then multiplying or "windowing" its infinite impulse response with an appropriate window function to make it causal and finite. This windowing in the time domain corresponds to convolution in the frequency domain. Common window functions are used to truncate the ideal filter response while maintaining desirable filtering properties. MATLAB code can be used to implement windowed FIR filters.
This document discusses the process of sampling in signal processing. It defines key terms like analog and digital signals, sampling frequency, and samples. It explains how sampling works by taking regular measurements of a continuous signal's amplitude over time. This converts it into a discrete-time signal. It discusses applications of sampling like audio sampling, where signals are typically sampled above 20 kHz. It also discusses video sampling rates and speech sampling rates. The document contains examples and diagrams to illustrate these concepts.
The document discusses sampling theory and analog-to-digital conversion. It begins by explaining that most real-world signals are analog but must be converted to digital for processing. There are three steps: sampling, quantization, and coding. Sampling converts a continuous-time signal to a discrete-time signal by taking samples at regular intervals. The sampling theorem states that the sampling frequency must be at least twice the highest frequency of the sampled signal to avoid aliasing. Finally, it provides an example showing how to calculate the minimum sampling rate, or Nyquist rate, given the highest frequency of a signal.
This document discusses different realization structures for causal IIR digital filters, including direct form I and II structures, cascade structures, and parallel form I and II structures. It provides examples of implementing a 3rd order IIR transfer function using each type of structure. Direct form I structures realize the coefficients directly from the transfer function. Cascade structures decompose the transfer function into lower order sections. Parallel forms use partial fraction expansions to decompose into parallel signal flows.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
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
This document provides an overview of signals and systems in digital signal processing. It defines what a signal and system are, provides examples of common discrete-time signals like impulse functions and exponential functions. It also discusses signal operations such as addition, delaying, time reversing and rate changing. The document classifies signals as periodic/aperiodic, even/odd, energy/power signals. It also classifies systems as continuous/discrete-time, time-variant/invariant, linear/non-linear, stable/unstable systems. In addition, it provides representations of systems using impulse response, difference equations and transfer functions.
The Fourier transform decomposes a signal into its constituent frequencies, representing it in the frequency domain rather than the spatial domain, which can make certain operations and analyses easier to perform; it has both magnitude and phase components that provide information about the frequency content and relative phases of the signal. The discrete Fourier transform (DFT) is a sampled version of the continuous Fourier transform that is useful for digital signal and image processing applications.
This document provides an overview of discrete time systems and their representations. It discusses key concepts such as:
- The difference between continuous and discrete time systems
- Representing discrete time systems using difference equations and block diagrams
- Classifying systems as static/dynamic, time-variant/invariant, linear/nonlinear, causal/non-causal, and stable/unstable
- Examples are provided to illustrate different system types.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
IIR filter realization using direct form I & IISarang Joshi
The document discusses IIR filter realization using Direct Form I and Direct Form II structures. It presents the difference equation and transfer function for an IIR filter. It also provides examples of implementing IIR filters using Direct Form I and Direct Form II structures based on a given difference equation or transfer function.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
DSP_2018_FOEHU - Lec 08 - The Discrete Fourier TransformAmr E. Mohamed
The document provides an overview of the Discrete Fourier Transform (DFT). It begins by discussing limitations of the discrete-time Fourier transform (DTFT) and z-transform in that they are defined for infinite sequences and continuous variables. The DFT avoids these issues by being a numerically computable transform for finite discrete-time signals. It works by taking a finite signal, making it periodic, and computing its discrete Fourier transform which is a discrete frequency spectrum. This makes the DFT highly suitable for digital signal processing. The document then provides details on computation of the DFT and its relationship to the DTFT and z-transform.
Fir filter design using Frequency sampling methodSarang Joshi
The document discusses the design of a finite impulse response (FIR) filter using the frequency sampling technique. It describes how to determine the impulse response of an FIR filter of length 7 to meet specific frequency response specifications. Frequency samples are taken at points and the desired response is defined. The discrete-time Fourier transform (DTFT) and inverse DTFT are used to calculate the impulse response coefficients that produce the desired filter frequency response. Equations for the impulse response h(n) are provided.
DSP_2018_FOEHU - Lec 03 - Discrete-Time Signals and SystemsAmr E. Mohamed
The document discusses discrete-time signals and systems. It defines discrete-time signals as sequences represented by x[n] and discusses important sequences like the unit sample, unit step, and periodic sequences. It then defines discrete-time systems as devices that take a discrete-time signal x(n) as input and produce another discrete-time signal y(n) as output. The document classifies systems as static vs. dynamic, time-invariant vs. time-varying, linear vs. nonlinear, and causal vs. noncausal. It provides examples to illustrate each classification.
Circular convolution is performed on two signals x1 and x2.
x1 and x2 are periodic signals with period 4. The circular convolution sums the product of the signals at each time offset.
The convolution is computed for different time offsets from 0 to 3. The results of the convolution at each offset are 34, 36, 34, 28, forming the output signal y(m).
This document discusses digital signal processing and the design of finite impulse response (FIR) filters using the window method. It begins with an introduction to FIR filters, noting their advantages over infinite impulse response (IIR) filters such as being easily designed with linear phase and being unconditionally stable. The document then covers FIR filter design concepts like phase delay, linear phase response, and filter specifications. It presents the window method approach to FIR filter coefficient calculation and discusses filter design considerations like coefficient calculation methods and filter structure selection.
This document provides an overview of digital filter design. It introduces finite impulse response (FIR) and infinite impulse response (IIR) filters. FIR filters are designed using window techniques like rectangular, Hamming, and Kaiser windows. IIR filters are designed using approximation methods like Butterworth, Chebyshev I, and Chebyshev II. MATLAB code is provided to design low pass, high pass, and other filters using different window and approximation techniques. Pros and cons of FIR and IIR filters are discussed along with references.
The document discusses 11 properties of the Fourier transform: (1) Linearity and superposition, (2) Time scaling, (3) Time shifting, (4) Duality or symmetry, (5) Area under the time domain function equals the Fourier transform at f=0, (6) Area under the Fourier transform equals the time domain function at t=0, (7) Frequency shifting, (8) Differentiation in the time domain, (9) Integration in the time domain, (10) Multiplication in the time domain becomes convolution in the frequency domain, and (11) Convolution in the time domain becomes multiplication in the frequency domain. Each property is explained briefly.
Fir filter design (windowing technique)Bin Biny Bino
The window design technique for FIR filters involves choosing an ideal frequency-selective filter with the desired passband and stopband characteristics, and then multiplying or "windowing" its infinite impulse response with an appropriate window function to make it causal and finite. This windowing in the time domain corresponds to convolution in the frequency domain. Common window functions are used to truncate the ideal filter response while maintaining desirable filtering properties. MATLAB code can be used to implement windowed FIR filters.
This document discusses the process of sampling in signal processing. It defines key terms like analog and digital signals, sampling frequency, and samples. It explains how sampling works by taking regular measurements of a continuous signal's amplitude over time. This converts it into a discrete-time signal. It discusses applications of sampling like audio sampling, where signals are typically sampled above 20 kHz. It also discusses video sampling rates and speech sampling rates. The document contains examples and diagrams to illustrate these concepts.
The document discusses sampling theory and analog-to-digital conversion. It begins by explaining that most real-world signals are analog but must be converted to digital for processing. There are three steps: sampling, quantization, and coding. Sampling converts a continuous-time signal to a discrete-time signal by taking samples at regular intervals. The sampling theorem states that the sampling frequency must be at least twice the highest frequency of the sampled signal to avoid aliasing. Finally, it provides an example showing how to calculate the minimum sampling rate, or Nyquist rate, given the highest frequency of a signal.
This document discusses different realization structures for causal IIR digital filters, including direct form I and II structures, cascade structures, and parallel form I and II structures. It provides examples of implementing a 3rd order IIR transfer function using each type of structure. Direct form I structures realize the coefficients directly from the transfer function. Cascade structures decompose the transfer function into lower order sections. Parallel forms use partial fraction expansions to decompose into parallel signal flows.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
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
This document provides an overview of signals and systems in digital signal processing. It defines what a signal and system are, provides examples of common discrete-time signals like impulse functions and exponential functions. It also discusses signal operations such as addition, delaying, time reversing and rate changing. The document classifies signals as periodic/aperiodic, even/odd, energy/power signals. It also classifies systems as continuous/discrete-time, time-variant/invariant, linear/non-linear, stable/unstable systems. In addition, it provides representations of systems using impulse response, difference equations and transfer functions.
Digital Signal Processing Tutorial: Chapt 4 design of digital filters (FIR) Chandrashekhar Padole
This document discusses digital filters and the design of finite impulse response (FIR) filters. It covers topics such as the design of FIR filters using windows, properties of FIR filters including their linear phase characteristics, and comparisons between FIR and infinite impulse response (IIR) filters. MATLAB code is provided to demonstrate the effects of linear phase characteristics on filtered signals. Linear phase filters are shown to preserve signal shape while non-linear phase filters can distort signals.
This document provides an overview of the z-transform and its properties. It defines the z-transform as a power series representation of a discrete-time signal and explains that it exists only when the series converges within a region of convergence (ROC). Several examples of calculating the z-transform of finite and infinite duration signals are shown along with the corresponding ROCs. Common methods for taking the inverse z-transform like synthetic division, partial fraction expansion, and Cauchy's integration method are also briefly mentioned.
This document provides information about sound and hearing. It defines sound as vibrations that have frequencies between 20-20,000 Hertz that can be detected by the human ear. It describes hearing as the ability to perceive sound through detecting vibrations in the air, liquid or solid medium through an organ such as the ear. It explains that sound travels through variations in pressure that push surrounding material like water or air, and discusses key characteristics of sound waves such as pitch, frequency, amplitude and speed of sound. The document also describes how we are able to hear sound, with our ears converting vibrations into nerve signals sent to the brain for interpretation.
As technology demands on logistics services providers (LSPs) become more intense, organizations are seeking to integrate or consolidate their third-part logistics (3PL) providers' solutions for tasks such as warehousing, inventory management, shipment management, cross-docking, order management, bar coding, analytics and far more. We offer a roadmap for selecting whether to make such a transition in logistics systems via a big bang or phased/pilot approach.
Mit appkind erhältst du ein Marketinginstrument um deine Kunden mobil, auf dem Smartphone, zu erreichen.
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Mit den News-Streams kannst du über vorhandene Kommunikationskanäle Neuigkeiten auf die Smartphones deiner App-Nutzer bringen und sie mit einer Push-Nachricht darüber informieren.
Im Rahmen der App-Gestaltung und der Erstellung der Inhalte lieferst du uns einfach Basis-Informationen zu deinem Unternehmen oder Produkten und unser Team erledigt den Rest. Unsere App-Profis wissen, worauf es ankommt und erstellen eine kommunikationsstarke App.
Weitere Informationen findest du auf www.appkind.de
El documento propone 51 medidas para reformar la Constitución española, el modelo territorial del Estado, la ley electoral y el sistema económico. Algunas de las principales propuestas son establecer un modelo federal cooperativo con competencias claramente definidas, devolver competencias de educación, sanidad y justicia al Estado, eliminar los privilegios fiscales del País Vasco y Navarra, y fusionar municipios pequeños para racionalizar la administración local. También se proponen medidas para mejorar la independencia del poder judicial y luchar contra la corrupción.
This document provides a classification for the 2016 PEDALES DE HIERRO cycling race. It lists the top 85 male finishers, their times, positions, and clubs. The first place finisher was CÁRDENAS GONZÁLEZ with a time of 1:55:58.
El documento describe los beneficios del bilingüismo para el cerebro humano. Ser bilingüe ayuda a desarrollar mejor los procesos cognitivos, genera una mayor concentración, y puede atrasar la aparición del Alzheimer hasta 5 años. Aprender un segundo idioma involucra varias áreas del cerebro y mejora la atención, la memoria y la capacidad de resolución de problemas.
Dc Bar Social Media Twitter Shaun Dakin Dc Bar PresentationSuzanne Turner
This document provides an overview of Twitter and how lawyers can use it. It discusses what Twitter is, including that it allows microblogging and instant messaging of up to 140 character messages. It provides basics of Twitter like how to tweet, follow others, and use hashtags. It encourages lawyers to start using Twitter to engage in conversations, share content, and get their message out. Examples are given of law firms and government agencies using Twitter along with tools to help manage Twitter accounts.
Cynthia Perez has over 5 years of experience in administrative, customer service, and reception roles. She is fluent in Spanish and proficient in Microsoft Office programs including Word, Excel, PowerPoint, and Outlook. Perez has worked in property management, healthcare, software, and catering. Her most recent roles include serving as a member services representative at Tufts Health Plan, where she assisted customers with accounts, and as an administrative assistant intern at Winn Residential, where she oversaw the reception area and created work orders.
mls is improving list broking in Spain. we have introduced in this market some international lists and now we would like to improve the local market. In this document we have summarised the main rules, players and penalties.
El documento resume la evolución del escudo del Real Madrid a lo largo de su historia desde 1902 hasta la actualidad. El primer escudo consistía en las iniciales del club sobre un fondo blanco, pero para partidos oficiales se incluía el escudo de la ciudad de Madrid. En 1908 se adoptó un diseño más estilizado con la M más prominente. En 1920 el Rey concedió el título de Real y se añadió la corona real al escudo. Durante la Segunda República se eliminó la corona, y en 1941 se recuperó tras la Guerra Civil. En 2001 se
The document introduces the Global University Entrepreneurial Spirit Students' Survey (GUESSS) project. GUESSS is an international research project that surveys entrepreneurial intentions of students worldwide. It has surveyed over 93,000 students from 26 countries. The 2013 survey will focus on entrepreneurial career intentions and the effectiveness of university entrepreneurship programs. Country representatives coordinate survey distribution and receive benefits including national reports and access to the international dataset.
This document provides information about an upcoming Homecoming football game between the Highland Park Giants and Niles West Indians. It discusses the importance of the game and good sportsmanship. It also lists upcoming basketball and football events, including ticket prices and locations. Homecoming committee members and their roles are outlined.
Describes Signal Processing in Radar Systems,
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
I recommend to see the presentation on my website under RADAR Folder, Signal Processing Subfolder.
DSP, Differences between Fourier series ,Fourier Transform and Z transform Naresh Biloniya
The document compares and contrasts the z-transform, Fourier series, and Fourier transform.
1) The z-transform is used for discrete-time signals, Fourier series is used for continuous periodic signals, and Fourier transform can be used for both discrete and continuous signals.
2) The z-transform converts difference equations to algebraic equations. Fourier series expands periodic functions as an infinite sum of sines and cosines. The Fourier transform provides a frequency representation of signals.
3) The inverse of the z-transform and Fourier transform are defined mathematically, while there is no inverse of a Fourier series since it does not change the domain of the original signal.
This document discusses using a C8051 MCU to calculate FFTs in real-time for audio spectrum analysis. It introduces:
(1) Key techniques used in the FFT software to optimize efficiency and accuracy, including avoiding multiplications, 16-bit integer storage, and using the MCU's on-chip PLL.
(2) The FFT algorithm and techniques like bit-reversal, windowing and anti-aliasing filtering.
(3) Details of the software design and implementation, including interrupt service routines to read ADC samples, windowing functions, bit-reversal indexing, and FFT computation loops.
(4) Experimental results analyzing the performance under different configurations and analyzing properties like
The document provides notes on signals and systems from an EECE 301 course. It includes:
- An overview of continuous-time (C-T) and discrete-time (D-T) signal and system models.
- Details on chapters covering differentials/differences, convolution, Fourier analysis (both C-T and D-T), Laplace transforms, and Z-transforms.
- Examples of calculating the Fourier transform of specific signals like a decaying exponential and rectangular pulse. These illustrate properties of the Fourier transform.
A Simple Communication System Design Lab #3 with MATLAB SimulinkJaewook. Kang
This document outlines the schedule and topics for a series of labs on communication system design using MATLAB Simulink. The upcoming Lab #3 will cover phase splitting, which extracts the real and imaginary components from a complex baseband signal, and up/down conversion, which shifts signals between baseband and intermediate frequencies. The lab is scheduled for April 1st from 1-4pm and will be instructed by Jaewook Kang. Previous and future labs will cover topics like OFDM, S-function design, channel modeling, and subsystem implementation.
This document discusses the discrete Fourier transform (DFT) and its inverse, the inverse discrete Fourier transform (IDFT). It defines the DFT and IDFT formulas, introduces twiddle factors, and shows how to represent DFTs and IDFTs using matrices. Examples are provided to calculate DFTs and IDFTs both directly from the definition and using matrix representations.
This document provides an outline for a textbook on advanced digital signal processing. It covers topics such as sampling theory, the discrete Fourier transform and fast Fourier transform, digital filters, multirate digital signal processing, and spectral estimation. Sampling theory concepts discussed include the sampling theorem, which states that a continuous-time signal can be reconstructed from its samples if it is bandlimited and sampled at a rate at least twice its highest frequency. Non-ideal effects of sampling such as aliasing are also covered.
This document provides an outline and introduction to the key concepts in advanced digital signal processing. It discusses sampling of continuous-time signals to generate discrete-time signals, including the sampling theorem which establishes conditions for perfect reconstruction of a signal from its samples. It also introduces digital signal processors and discusses common operations in digital signal processing like the discrete Fourier transform.
This document discusses Nyquist's criterion for distortionless transmission of binary signals over a baseband channel. It states that intersymbol interference (ISI) can be eliminated by choosing a transmit filter response P(f) that satisfies the Nyquist criterion. An ideal rectangular pulse shape meets the criterion but is physically unrealizable. A more practical raised cosine pulse is proposed, which introduces a rolloff factor to trade off excess bandwidth for slower decay. The full-cosine case provides additional zero-crossings that aid synchronization but doubles the bandwidth.
This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.
This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
EC8553 Discrete time signal processing ssuser2797e4
This document contains a 10 question, multiple choice exam on discrete time signal processing. It covers topics like the discrete Fourier transform (DFT), finite word length effects, fixed point vs floating point representation, and FIR filter design. Specifically, it includes questions that calculate the 4 point DFT of a sequence, define twiddle factors, compare DIT and DIF FFT algorithms, and discuss stability and causality of systems.
Performance evaluations of grioryan fft and cooley tukey fft onto xilinx virt...csandit
A large family of signal processing techniques consist of Fourier-transforming a signal,
manipulating the Fourier-transformed data in a simple way, and reversing the transformation.
We widely use Fourier frequency analysis in equalization of audio recordings, X-ray
crystallography, artefact removal in Neurological signal and image processing, Voice Activity
Detection in Brain stem speech evoked potentials, speech processing spectrograms are used to
identify phonetic sounds and so on. Discrete Fourier Transform (DFT) is a principal
mathematical method for the frequency analysis. The way of splitting the DFT gives out various
fast algorithms. In this paper, we present the implementation of two fast algorithms for the DFT
for evaluating their performance. One of them is the popular radix-2 Cooley-Tukey fast Fourier
transform algorithm (FFT) [1] and the other one is the Grigoryan FFT based on the splitting by
the paired transform [2]. We evaluate the performance of these algorithms by implementing
them on the Xilinx Virtex-II pro [3] and Virtex-5 [4] FPGAs, by developing our own FFT
processor architectures. Finally we show that the Grigoryan FFT is working fatser than
Cooley-Tukey FFT, consequently it is useful for higher sampling rates. Operating at higher
sampling rates is a challenge in DSP applications.
PERFORMANCE EVALUATIONS OF GRIORYAN FFT AND COOLEY-TUKEY FFT ONTO XILINX VIRT...cscpconf
A large family of signal processing techniques consist of Fourier-transforming a signal,manipulating the Fourier-transformed data in a simple way, and reversing the transformation.We widely use Fourier frequency analysis in equalization of audio recordings, X-ray crystallography, artefact removal in Neurological signal and image processing, Voice Activity Detection in Brain stem speech evoked potentials, speech processing spectrograms are used to identify phonetic sounds and so on. Discrete Fourier Transform (DFT) is a principal mathematical method for the frequency analysis. The way of splitting the DFT gives out various fast algorithms. In this paper, we present the implementation of two fast algorithms for the DFT for evaluating their performance. One of them is the popular radix-2 Cooley-Tukey fast Fourier transform algorithm (FFT) [1] and the other one is the Grigoryan FFT based on the splitting by the paired transform [2]. We evaluate the performance of these algorithms by implementing
them on the Xilinx Virtex-II pro [3] and Virtex-5 [4] FPGAs, by developing our own FFT processor architectures. Finally we show that the Grigoryan FFT is working fatser than
Cooley-Tukey FFT, consequently it is useful for higher sampling rates. Operating at higher
sampling rates is a challenge in DSP applications
The document discusses properties of the discrete-time Fourier transform (DTFT) and discrete Fourier transform (DFT), including:
1) Key properties of the DTFT such as linearity, time-shifting, differentiation, and the convolution theorem for computing linear convolution in the frequency domain. MATLAB functions like freqz are introduced for computing DTFT values.
2) The energy density spectrum of a sequence and how it relates to the total energy via Parseval's theorem.
3) The definition of the DFT as a sampled version of the DTFT, allowing transformation between time and frequency domains for finite-length sequences. The DFT and inverse DFT (IDFT) are defined.
This document chapter discusses the characterization and representation of communication signals and systems. It describes how band-pass signals and systems can be represented by equivalent low-pass signals and systems using analytic signal representations and complex envelopes. It also discusses how the response of a band-pass system to a band-pass input signal can be determined from the equivalent low-pass representations. Key topics covered include the Fourier transform, Hilbert transform, and convolution properties used to relate band-pass and low-pass signal and system representations.
This document discusses linear algebra concepts for digital filter design. It begins with definitions of filters and digital filters, then covers topics like FIR and IIR filter design. Key points include:
- A filter removes unwanted things from an item of interest passed through a structure. A digital filter removes unwanted frequency components from a discretized signal passed through a structure with delay, multiplier, and summer elements.
- FIR filter design involves choosing filter coefficients to satisfy desired frequency response specifications. IIR filter design similarly involves solving systems of equations to determine coefficients for structures using feedback.
- Matlab code for FIR filter design is provided in slide 17 for appreciation. A presentation on slides 3-4 and 17 is assigned.
Sampling and Reconstruction (Online Learning).pptxHamzaJaved306957
1. Sampling and reconstruction of signals was analyzed using the impulse sampling math model.
2. The analysis showed that a bandlimited signal can be perfectly reconstructed from its samples as long as the sampling rate is at least twice the bandwidth of the signal.
3. If the sampling rate is lower than the minimum required rate, aliasing error occurs where frequency components fold back into the baseband.
The document discusses Fourier series and Fourier transforms. Some key points:
- Any periodic function can be expressed as the sum of an infinite number of sine and cosine waves of different frequencies, known as a Fourier series.
- The Fourier transform decomposes both periodic and non-periodic signals into the frequencies they contain. It represents the frequencies that make up the signal.
- The Fourier series is used for periodic signals and results in discrete frequency spectra. The Fourier transform is used for non-periodic signals and results in continuous frequency spectra.
- Examples are provided to demonstrate how Fourier analysis can be used to decompose signals into their frequency components and reconstruct them.
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Chapter wise All Notes of First year Basic Civil Engineering
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Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
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Chapter 5
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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Digital Signal Processing Tutorial:Chapt 3 frequency analysis
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Chapt 03 & 04
Frequency Analysis of Signals
and System
& DFT
Digital Signal Processing
PrePrepared by
IRDC India
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Chapt 3 : Frequency Analysis of Signals &
Systems
Contents:
• Frequency Analysis: CTS and DTS
• Properties of the Fourier Transform for DTS
• Frequency domain characteristics of LTI systems
• LTI system as a frequency selective filter
• Inverse systems and de-convolution
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Frequency Analysis: CTS and DTS
• Analysis tools
• Fourier Series
• Fourier Transform
• DTFS
• DTFT
• DFT
Jean Baptiste Joseph
Fourier (1768 - 1830).
Fourier was a French
mathematician, who was
taught by Lagrange and
Laplace
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e-TECHNote
This PPT is sponsored by
IRDC India
www.irdcindia.com
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Fourier Series
•Continuous time variable t
•Discrete frequency variable k
•Signal to be analyzed has to
be periodic
• Fourier series for continuous-time periodic signal
•
tTp
∑
∞
−∞=
=
k
tkFj
k eCtx 02
)( π
dtetx
T
C tkFj
Tp
k
p
02
)(
1 π−
∫=
Tp-> is the Fundamental Period of Signal
where
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•Continuous time variable t
•Continuous frequency variable F
•Signal to be analyzed is aperiodic
Fourier Transform
• Fourier Transform for continuous-time periodic
signal
t
∫
∞
∞−
= dFeFXtx Ftj π2
)()(
Tp-> is the Fundamental Period of Signal
where
∫
∞
∞−
−
= dtetxFX Ftj π2
)()(
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• Fourier Series Periodic signal
• Fourier Transform Aperiodic signal
When Fourier series is applied to the aperiodic signal by assuming
its period = , then it is called as Fourier transform
Fourier Series & Fourier Transform
Fourier
Series
t
Tp F
Fourier
Series
Tp=∞
∆F=1/Tp
F
∆F=1/Tp=0
∞
)(lim)( txtx T
T ∞→
=
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•Discrete time variable t
•Discrete frequency variable k
•Signal to be analyzed has to
be periodic
DTFS
• Fourier series for discrete-time periodic signal
∑
−
=
=
1
0
2
)(
N
k
k
N
knj
eCnx
π
∑
−
=
−
=
1
0
2
)(
1 N
n
k
N
knj
enx
N
C
π
Tp-> is the Fundamental Period of Signal
where
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•Discrete time variable t
•Continuous frequency variable F
•Signal to be analyzed is aperiodic
DTFT
• Fourier Transform for discrete-time periodic
signal
∫=
π
ω
ωω
π 2
)(
2
1
)( deXnx nj
Tp is the Fundamental Period of Signal
where
∑
∞
−∞=
−
=
n
jwn
enxX )()(ω
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Dirichlet Conditions
• To exist Fourier transform of the signal, it should satisfy
Dirichlet Conditions
1. The signal has finite number of discontinuities
2. It has finite number of maxima and minima
3. The signal should be absolutely integral
i.e.
∞<∫
∞
∞−
dttx )(
∑
∞
−∞=
∞<
n
dtnx )(
or
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Problem(1)
• Obtain exponential Fourier series for the waveform shown in fig
•
0 2Π 4Π ωt
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Solution (1)
dtetx
T
C tjk
T
k
0
0
)(
1
0
ω−
∫=
As per Fourier series
π20 =T ttx 0
2
5
)( ω
π
= for π20 ≤≤ wt
tdwet
T
C tjkw
k 0
2
0
02
5
0
0
1 −
∫=
π
π ω tdwteC tjkw
k 0
2
0
0
0
2
5
2
1
∫
−
=
π
ω
ππ
( )
π
π
2
0
022
1
)()2(
5 0
−−
−
=
−
tjkw
jk
e tjkw
k
jCk
π2
5
=
LLL ++++−−= −− twjtwjtjwtwj
ejejejejtx 0000 2
4
52
2
5
2
5
2
52
4
5
)( ππππ
0
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Problem (2)
Obtain Fourier transform for the gate function ( rectangular pulse)
as shown in fig
0-T/2 T/2 t
,1)( =tf 22
TT
t ≤≤−
=0 otherwise
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Solution (2)
We know
∫
∞
∞−
−
= dtetfwF jwt
)()(
∫−
−
=
2/
2/
.1
T
T
jwt
dte
2/
2/
T
T
jwt
jw
e
−
−
−
= [ ]2/2/1 jwTjwT
ee
jw
−
−
= −
jw
ee jwTjwT 2/2/
−
=
−
w
wT )2/sin(2
=
)2/(sin
2/
)2/sin(
wTcT
wt
wT
T ==
Amplitude
Phase
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Problem (3)
A finite duration sequence of Length L is given as
Determine the N-point DFT of this sequence for N≥L
,1)( =nx 10 −≤≤ Ln
=0 otherwise
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Solution (3)
The Fourier transform (DTFT) is given by
∑
∞
−∞=
−
=
n
jwn
enxX )()(ω
∑
−
=
−
=
1
0
.1
L
n
jwn
e ∑
−
=
−
=
1
0
)(
L
n
njw
e
We know summation formula
∑=
+
−
−
=
n
k
n
k
a
a
a
0
1
1
1
jw
jwL
e
e
X −
−
−
−
=
1
1
)(ω 2/2/2/2/
2/2/2/2/
jwjwjwjw
jwLjwLjwLjwL
eeee
eeee
−−
−−
−
−
=
)(
)(
2/2/2/
2/2/2/
jwjwjw
jwLjwLjwL
eee
eee
−−
−−
−
−
=
)2/sin(
)2/sin(2/)1(
w
wL
e Ljw −−
=
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Contd..
DFT of x(n) can be obtained by sampling X(w) at N
equally spaced frequencies
N
k
wk
π2
=∴ 1,.......1,0 −= Nk
Nkj
NkLj
e
e
kX /2
/2
1
1
)( π
π
−
−
−
−
= 1,.......1,0 −= Nk
,)( LkX = 0=KIf N=L,
0= 1.......2,1 −= LK
•If N>L, computational point of view sequence x(n) is extended by
appending N-L zeroes(zero padding).
• DFT would approach to DTFT as no of points in DFT tends to infinity
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Nyquist Criterion – Sampling Theorem
• Continuous time domain( analog ) signal is converted into discrete time
signal by sampling process. It should be sampled in such a way that the original
signal can be reconstructed from the samples
• Nyquist criterion or sampling theorem suggest the minimum sampling
frequency by which signal should be sampled in order to have proper reconstruction
if required.
Mathematically, sampled signal xs(t) is obtained by multiplying sampling
function gT(t) with original signal x(t)
)1.......().........()()( tgtxtx Ts =
Where gT(t) continuous train of pulse with period T (sampling period)
)(
1
)(
Triodsamplingpe
fequencysamplingfr s =
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Contd..
The Fourier series of periodic signal gT(t) can be given as
dtetg
T
C tnfj
T
T
n
sπ2
2/
2/
)(
1 −
−
∫=
)2........(..........)( 2
∑
∞
−∞=
=
n
tnfj
n
s
eCtg π
Where
gT(t)
x(t)
t
t
Thus, from eq(1) and eq(2), we get
∑
∞
−∞=
=
n
tnfj
ns
s
eCtxtx π2
)()(
)3.........(..........)( 2
∑
∞
−∞=
=
n
tnfj
n
s
etxC π
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Contd..
Taking Fourier transform of xs(t), we get
∫
∞
∞−
−
= dtetxfX ftj
ss
π2
)()(
∫ ∑
∞
∞−
−
∞
−∞=
= dteetxC ftj
n
tnfj
n
s ππ 22
)(
Interchanging the order of integration and summation , we get
∑ ∫
∞
−∞=
−−
∞
∞−
=
n
tnffj
n dtetxC s )(2
)( π
∑
∞
−∞=
−=
n
sns nffXCfX )()(
By definition
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Contd..
Thus, spectrum of sampled signal is the spectrum of x(t) plus the spectrum of
x(t) translated to each harmonic fo the sampling frequency as shown in figure
X(f)
f
-fh fh
-fh fh fs-fh fs+fhfs 2fs-fh 2fs+fh2fs
-2fs-fh -2fs+fh-2fs -fs-fh -fs
-fs+fh
There are three cases when sampling frequency fs is compared with
highest frequency present in original signal
Xs(f)
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Contd..
Case I: fs > 2fh
In this case all spectrums would be isolated from each other . This
leads to proper reconstruction of original signal
Case II: fs = 2fh
-fh fh fs-fh fs+fhfs 2fs-fh 2fs+fh2fs
-2fs-fh -2fs+fh-2fs -fs-fh -fs
-fs+fh
Xs(f)
Case III: fs < 2fh
-fh fh fs 2fs
-2fs -fs
Xs(f)
Overlapping of spectrum causes aliasing effect which distorts in
reconstruction of original signal
24. N-points in time
domain will give
N-points in DFT
and vice-versa
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DFT
DTFT has continuous frequency domain variable which makes
impossible to store X(w) with digital device. Thus continuous
frequency variable (f or w) from DTFT is sampled to get
discrete frequencies (wk).
DTFT calculated for discrete frequencies (wk) is called as
Discrete Fourier Transform
N
k
wk
π2
=
∑
−
=
−
=
1
0
/2
)()(
N
n
Nknj
enxkX π
1,.......1,0 −= Nk
∑
−
=
=
1
0
/2
)(
1
)(
N
k
Nknj
ekX
N
nx π
1,.......1,0 −= Nn
DFT
Inverse DFT ( IDFT)
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Physical Significance
• Signal to be analyzed is 64 points in length
x(n)
analog discrete
0 10 20 30 40 50 60 70
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60 70
-0 .5
-0 .4
-0 .3
-0 .2
-0 .1
0
0 .1
0 .2
0 .3
0 .4
0 .5
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k=31
)/2sin()/2cos(
2
NnkjNnke N
nk
j
ππ
π
+=
−
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
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Relation bet VA and DFT
Vector algebra -2/3D space DFT
No of analysis vectors 2/3 N
No of elements in each
vector
2/3 N
Projection measurement
method
Inner Product Inner Product
37. 10/7/2009
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Problem(2)
Calculate DFT of x(n)= {1 2 2 1}. From calculated DFT of x(n),
determine x(n) again using IDFT. Verify your answer.
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Solution(2)
}2221{)( =nx
We know
∑
−
=
−
=
1
0
/2
)()(
N
n
Nknj
enxkX π
1,.......1,0 −= Nk
For k=0
∑∑ ==
==
3
0
3
0
0
)()()0(
nn
nxenxX
61221 =+++=
For k=1
∑=
−
=
3
0
4/
)()1(
n
nj
enxX π
4/3
2/4/0
)3(
)2()1()0(
π
ππ
j
jj
ex
exexex
−
−−
+
++=
N=4
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Contd..
2/3
2/0
)3(
)2()1()0(
π
ππ
j
jj
ex
exexex
−
−−
+
++=
jj .1)1.(2).(21 +−+−+=
jX −−= 1)1(
Similarly, calculate X(2) & X(3)
Then use IDFT to calculate x(n)
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Twiddle Factors
Twiddle Factor N
j
N eW
π2
=
( ) 1
00 2
==∴ N
j
N eW
π
( ) NN
jj
N eeW
ππ 22 11
==∴
( ) NN
jj
N eeW
ππ 42 22
==∴
So on…….
For N=4
For N=8
W4
0
W4
3
W4
2
W4
1
W8
0
W8
7
W8
6
W8
5
W8
4
W8
3
W8
2
W8
1
( ) N
l
N
jljl
N eeW
ππ 22
==
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Contd..
1
4
)5(4mod
4
5
4 WWW ==
Proof:
( ) 4
10
4
2 55
4
ππ jj
eeW ==∴
πππ 2
1
2
1
.2)2( jjj
eee ==
+
4
1.2
4
2
.1
ππ jj
ee ==
1
4W= N
ljl
N eW
π2
=Q
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Contd..
3
4
6
8 WW =
Proof:
( ) 8
12
8
2 66
8
ππ jj
eeW ==∴ 4
3.2
4
6 ππ jj
ee ==
3
4W= N
ljl
N eW
π2
=Q
2
8
6
8 WW −=
Proof:
( ) 8
12
8
2 66
8
ππ jj
eeW ==∴ 8
4
8
4 )1( π
ππ jjj
eee ==
+
2
8)1( W−= N
ljl
N eW
π2
=Q
2
8W−=
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Matrix Method for DFT calculation
∑
−
=
−
=
1
0
/2
)()(
N
n
Nknj
enxkX π
1,.......1,0 −= Nk
∑
−
=
=∴
1
0
)()(
N
n
kn
NWnxkX 1,.......1,0 −= Nk
We know
N
j
N eW
π2−
=and
In matrix form ]][[][ xWX kn
N=
=
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
9
4
6
4
3
4
0
4
6
4
4
4
2
4
0
4
3
4
2
4
1
4
0
4
0
4
0
4
0
4
0
4
x
x
x
x
WWWW
WWWW
WWWW
WWWW
X
X
X
X4-point DFT
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Shuffled sequence
The input sequence placement at FFT flow graph is not in order but in
shuffled order.
Perfect shuffling can be obtained by using bit reversal algorithm.
( to be used in FFT implementation)
2 point FFT
0
1
4 point FFT
0
1
2
3
0 2
1 3
0 2 1 3
8 point FFT
0
1
2
3
4
5
6
7
0 4
1 5
2 6
3 7
0 4 2 6
1 5 3 7
0 1 0 4 2 6 1 5 3 7
16-point shuffled sequence ?????
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Bit Reversal Algorithm
2 point FFT 4 point FFT 8 point FFT
0
1
0
1
2
3
00
01
10
11
00
10
01
11
0
2
1
3
0
1
2
3
4
5
6
7
000
001
010
011
100
101
110
111
000
100
010
110
001
101
011
111
0
4
2
6
1
5
3
7
Bit Reversal
Algorithm
Bit Reversal
Algorithm
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Properties of DFT
1) Periodicity
and x[n] is periodic such that x[n+N]=x[n] for all n
If x[n] X(k) ,
Then, X[k+N] = X(k) for all k
i.e. DFT of periodic sequence is also periodic with same period
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2) Linearity
][][
][][
22
11
kXnx
kXnx
DFT
DFT
→←
→←If
and
then for any real-valued or complex valued constants a1 and a2 ,
][][][][ 22112211 kXakXanxanxa DFT
+ →←+
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Problem
a) Find the DFT of x1(n)={ 0 1 2 3} and x2(n)= { 1 2 2 1}.
b) Calculate the DFT of x3(n)={ 2 5 6 5} using results obtained
in a) otherwise not.
}1,0,1,6{)(2 jjkX +−−−=
}22,2,22,6{)(1 jjkX −−−+−=
Since x1(n) + 2x2(n)= x3(n) , X3(k)= X1(k) + 2X2(k)
}1,0,1,6{2}22,2,22,6{)(3 jjjjkX +−−−+−−−+−=
}4,2,4,18{)(3 −−−=kX
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3) Circular time shift, x((n-k))N
x(n)={1 2 3 4 }
xp(n)
xp(n)
Circular delay by
1,x((n-1))
Circular advance
by 1 ,x((n+1))
x(0)=1x(2)=3
x(1)=2
x(3)=4
x(1)=2x(3)=4
x(2)=3
x(0)=1
x(n)
x((n+1))4
x(3)=4x(1)=2
x(0)=3
x(2)=3
x((n-1))4
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Contd..
Circular shift property
N
kljDFT
N
DFT
ekXlnx
kXnx
π2
].[))((
][][
−
→←−
→←If
then
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Problem
If x(n)= { 1 2 3 4} ,find X(k). Also using this result find the DFT of
h(n)={ 3 4 1 2}.
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4) Time Reversal
)())(()())((
][][
kNXkXnNxnx
kXnx
N
DFT
N
DFT
−=− →←−=−
→←If
then
i.e. reversing the N-point sequence in time domain is
equivalent to reversing the DFT sequence
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Problem
If x(n)= { 1 2 3 4} ,find X(k). Also using this result find the DFT of
h(n)={ 1 4 3 2}. Verify your answer with DFT calculation
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5) Circular frequency shift
N
DFTj
DFT
lkXenx
kXnx
N
nl
))(()(
][][
2
− →←
→←
π
If
then
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Problem
If x(n)= { 1 2 3 4} ,find X(k). Also using this result find the DFT of
h(n)={ 1 -2 3 -4}. Verify your answer with DFT calculation
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6) Symmetry properties
)()()()()( njxnjxnxnxnx o
I
e
I
o
R
e
R +++=
)()()()()( kjXkjXkXkXkX o
I
e
I
o
R
e
R +++=
EvenalEvenal DFT
,Re,Re →←
EvenaginaryEvenaginary DFT
,Im,Im →←
OddaginaryOddal DFT
,Im,Re →←
OddalOddaginary DFT
,Re,Im →←
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7)Parseval’s Relation
∑∑
−
=
−
=
=
1
0
2
1
0
2
|)(|
1
|)(|
N
k
N
n
kX
N
nx
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Problem
2) Determine the missing value from following sequence
x(n)= { 1 3 _ 2 } if its DFT is X(k)={ 8 -1-j -2 -1+j}
1) x(n)= {1 2 3 1}. Prove parseval’s relation for this
sequence and its DFT .
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8) Circular Convolution
][][
][][
22
11
kXnx
kXnx
DFT
DFT
→←
→←If
and
then
][].[][][ 2121 kXkXnxnx DFT
→←⊗
where
∑
−
=
−==⊗
1
0
21321 ))(()(][][][
N
k
Nknxkxnxnxnx
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Circular convolution of two sequences
Circular convolution of sequences x1(n)={2 1 2 1} and x2(n)={1 2 3 4 }
x(0)=2x(2)=2
x(1)=1
x(3)=1
x1(n)
x(0)=1x(2)=3
x(1)=2
x(3)=4
x2(n)
x2(0)=1x2(2)=3
x2(3)=4
x2(1)=2
x2((-n)) 26
4
2
x1(n)x2((-n))
x3(0)=2+4+6+2
= 14
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Problem
Find the circular convolution of sequences x1(n)={2 1 2 1} and
x2(n)={1 2 3 4 } using DFT .
Steps:
1. Find DFT of both sequences
2. Multiply both DFT’s
3. Take inverse DFT of product
( DFT product IDFT )
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Contd..
X3(0)=60
X3(2)=-4
x3(0)=14
x3(1)=16
X3(1)=0
X3(3)=0
x3(2)=14
x3(3)=16
1
j
1−
j−1−
1−
X3(k)= X1(k).X2(k)= {60 0 -4 0}=
60-4=56
60+4=64
0+0=0
0-0=0
56+0=56
64-0=64
56-0=56
64+0=64
16)3(,14)2(,16)1(,14)0( 3333 ==== xxxx
IDFT( Twiddle factors in reverse
order and divide by N at the end)
1
--
4
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Linear convolution using Circular convolution
Steps:
1. Append zeros to both sequence such that both sequence
will have same length of N1+N2-1
2. Number of zeros to be appended in each sequence is
addition of length of both sequences minus (one plus
length of respective sequence)
3. As both sequences are of same length equal to the length
of linear convoluted signal, apply circular convolution to
both modified signals using DFT ( DFT product IDFT )
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Problem
Calculate the linear convolution of following signals by using DFT
method only. x(n)={ 1 2 3 2 -2} and h(n)={ 1 2 2 1}
Verify your answer with tabulation method.
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Linear Filtering
If the system has frequency response H(w) and input
signal spectrum is X(w)
H(w)
X(w) Y(w)
Then , out spectrum of the system given by
Y(w)=H(w).X(w)
•In application, convolution would be used to calculate the
output of the system or DFT would be used to calculate the
spectrum of output.
•Even, DFT can be used for convolution
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Contd..
•Instead of taking DFT/convolution for whole input sequence,
DFT/convolution can be applied to smaller blocks of input
sequence. This would yields two advantages
•DFT/convolution size would be smaller and hence
computational complexity
• In online filtering delay can be kept small as only small
number of points will required to store in buffer for
DFT/convolution calculations
•If the input length is very large as compared to impulse
response of the system, then computational complexity of
the DFT/convolution would be more.
•There are two methods to do linear filtering by braking up input
sequence into smaller blocks
• Overlap add method
• Overlap save method
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Overlap-add Method
•In the overlap-add method the input x(n) is broken up into consecutive
non-overlapping blocks xi(n)
•The output yi(n) for each input xi(n) is computed separately by convolving
(non-cyclic/linear) xi(n) with h(n).
•The output blocks yi(n) would be lager than corresponding input blocks
xi(n)
•Hence, the each output block yi(n) will be overlapped with next and
previous blocks to get y(n)
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Contd..
h(n)xi(n)
yi(n)
x(n)
y(n)
Adding overlapped points
Overlapping length=M-1
Length of each block= N
Length of impulse
response = M
Length of
each block=
N+M-1
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Problem
An FIR digital filter has the unit impulse response sequence h(n)={ 2 2 1}.
Determine the output sequence in response to the input sequence x(n)= {3 0
-2 0 2 1 0 -2 -1 0}
93. Overlapping length=M-1
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Overlap-save Method
h(n)xi(n)
yi(n)
x(n)
y(n)
Discard overlapped points
Overlapping length=M-1
Length of each block= N
Length of impulse
response = M
Length of
each block=
N+M-1
M-1 zeros
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Problem
An FIR digital filter has the unit impulse response sequence h(n)={ 2 2 1}.
Determine the output sequence in response to the input sequence x(n)= {3 0
-2 0 2 1 0 -2 -1 0}