A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms (DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are applied. Such simplified model for splitting the filtration allows for saving 2N − 4(r + 1) operations of complex multiplication, for the signals of length N = 2^r, r > 2.
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms (DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of complex multiplication, for the signals of length = 2^, > 2.
Course 10 example application of random signals - oversampling and noise sh...wtyru1989
1. The document reviews quantization error in analog-to-digital conversion. It discusses assumptions about the quantization error process and how the error varies with bit depth.
2. Oversampling is described as a technique to reduce quantization error by increasing the sampling rate before quantization. This spreads the quantization noise over a wider bandwidth, lowering its power within the signal bandwidth.
3. Delta-sigma modulation is presented as a method to shape the quantization noise power spectrum through feedback. The noise is concentrated at higher frequencies, improving noise performance for a given bit depth.
This lecture covers signal and systems analysis, including:
1) Definitions of signals, systems, and their properties like time-invariance, linearity, stability, causality, and memory.
2) Classification of signals as continuous-time vs discrete-time, analog vs digital, deterministic vs random, periodic vs aperiodic.
3) Concepts of orthogonality, correlation, autocorrelation as they relate to signal comparison.
4) Review of the Fourier series and Fourier transform as tools to represent signals in the frequency domain.
This document discusses different types of signals and how they can be classified. It describes signals as being either continuous or discrete in time, periodic or aperiodic, analog or digital, deterministic or random, and having either finite energy or finite power. Common deterministic signals like impulse, step, ramp and parabolic signals are defined mathematically. The document provides an introduction to different signal classifications that are important for signal processing.
This document summarizes key concepts in digital and analog communications:
1) It defines source coding, channel encoding/decoding, digital modulation/demodulation, and how digital communication system performance is measured in terms of error probability.
2) Thermal noise in receivers is identified as the dominant source of noise limiting performance in VHF and UHF bands.
3) Storing data on magnetic/optical disks is analogous to transmitting a signal over a radio channel, with similar signal processing used for recovery.
4) Digital processing avoids signal degradation but requires more bandwidth, while analog processing is sensitive to variations but does not lose quality over time.
5) Fourier analysis is used to derive the
Convolution discrete and continuous time-difference equaion and system proper...Vinod Sharma
This document discusses discrete time signals and discrete time convolution. Discrete time convolution describes the output of a linear, time-invariant system when its input is one discrete signal and its impulse response is another. The output is the sum of each input sample multiplied by the corresponding flipped and shifted impulse response. This allows characterization of a system by its impulse response. The document provides examples of discrete time convolution and discusses properties such as the number of samples in the convolved output depending on the lengths of the input and impulse response signals.
The document discusses linear time-invariant (LTI) systems. It explains that:
1) The response of an LTI system to any input can be found by convolving the system's impulse response with the input. This is done using a convolution sum in discrete time and a convolution integral in continuous time.
2) Discrete-time signals and continuous-time signals can both be represented as weighted sums or integrals of shifted impulse functions.
3) For LTI systems, the impulse responses are simply time-shifted versions of the same underlying function, allowing the system to be fully characterized by its impulse response.
Digital signal processing involves processing digital signals using digital computers and software. There are several types of signals that can be classified based on properties like being continuous or discrete in time and value, deterministic or random, and single or multichannel. Common signals include unit impulse, unit step, and periodic sinusoidal waves. Signals can also be categorized as energy signals with finite energy, power signals with finite power, and even/odd based on their symmetry. Digital signal processing is used in applications like speech processing, image processing, and more.
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms (DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of complex multiplication, for the signals of length = 2^, > 2.
Course 10 example application of random signals - oversampling and noise sh...wtyru1989
1. The document reviews quantization error in analog-to-digital conversion. It discusses assumptions about the quantization error process and how the error varies with bit depth.
2. Oversampling is described as a technique to reduce quantization error by increasing the sampling rate before quantization. This spreads the quantization noise over a wider bandwidth, lowering its power within the signal bandwidth.
3. Delta-sigma modulation is presented as a method to shape the quantization noise power spectrum through feedback. The noise is concentrated at higher frequencies, improving noise performance for a given bit depth.
This lecture covers signal and systems analysis, including:
1) Definitions of signals, systems, and their properties like time-invariance, linearity, stability, causality, and memory.
2) Classification of signals as continuous-time vs discrete-time, analog vs digital, deterministic vs random, periodic vs aperiodic.
3) Concepts of orthogonality, correlation, autocorrelation as they relate to signal comparison.
4) Review of the Fourier series and Fourier transform as tools to represent signals in the frequency domain.
This document discusses different types of signals and how they can be classified. It describes signals as being either continuous or discrete in time, periodic or aperiodic, analog or digital, deterministic or random, and having either finite energy or finite power. Common deterministic signals like impulse, step, ramp and parabolic signals are defined mathematically. The document provides an introduction to different signal classifications that are important for signal processing.
This document summarizes key concepts in digital and analog communications:
1) It defines source coding, channel encoding/decoding, digital modulation/demodulation, and how digital communication system performance is measured in terms of error probability.
2) Thermal noise in receivers is identified as the dominant source of noise limiting performance in VHF and UHF bands.
3) Storing data on magnetic/optical disks is analogous to transmitting a signal over a radio channel, with similar signal processing used for recovery.
4) Digital processing avoids signal degradation but requires more bandwidth, while analog processing is sensitive to variations but does not lose quality over time.
5) Fourier analysis is used to derive the
Convolution discrete and continuous time-difference equaion and system proper...Vinod Sharma
This document discusses discrete time signals and discrete time convolution. Discrete time convolution describes the output of a linear, time-invariant system when its input is one discrete signal and its impulse response is another. The output is the sum of each input sample multiplied by the corresponding flipped and shifted impulse response. This allows characterization of a system by its impulse response. The document provides examples of discrete time convolution and discusses properties such as the number of samples in the convolved output depending on the lengths of the input and impulse response signals.
The document discusses linear time-invariant (LTI) systems. It explains that:
1) The response of an LTI system to any input can be found by convolving the system's impulse response with the input. This is done using a convolution sum in discrete time and a convolution integral in continuous time.
2) Discrete-time signals and continuous-time signals can both be represented as weighted sums or integrals of shifted impulse functions.
3) For LTI systems, the impulse responses are simply time-shifted versions of the same underlying function, allowing the system to be fully characterized by its impulse response.
Digital signal processing involves processing digital signals using digital computers and software. There are several types of signals that can be classified based on properties like being continuous or discrete in time and value, deterministic or random, and single or multichannel. Common signals include unit impulse, unit step, and periodic sinusoidal waves. Signals can also be categorized as energy signals with finite energy, power signals with finite power, and even/odd based on their symmetry. Digital signal processing is used in applications like speech processing, image processing, and more.
This document discusses signals and their classification. It defines signals, analog and digital signals, periodic and aperiodic signals. It also discusses representing signals in Matlab and Simulink. Key signal types covered include exponential, sinusoidal, unit impulse and step functions. Matlab is presented as a tool for programming and analyzing discrete signals while Simulink can be used to model and simulate continuous systems.
The document discusses sampling of continuous-time signals to create discrete-time signals. It explains that for perfect reconstruction, the sampling frequency must be greater than twice the maximum frequency of the original continuous-time signal, as specified by the Nyquist rate. A common method for sampling is to use an impulse train, and then reconstruct the signal by passing it through a low-pass filter. Often a zero-order hold is used to sample and communicate the signal, which simply holds each value until the next sample, and this provides a sufficiently accurate reconstructed continuous-time signal.
This document covers key concepts about signals including:
1) It defines continuous-time and discrete-time signals, and discusses the concepts of energy and power for both types of signals.
2) It provides the mathematical definitions of total energy, average power, and characterizes signals based on whether they have finite or infinite total energy and average power.
3) It discusses properties of exponential and sinusoidal signals, including that they have infinite total energy but finite average power.
4) It introduces common basic signals like the unit impulse and unit step signals in both continuous and discrete time.
This document presents a new DFT-based approach for detecting and correcting gain mismatch in time-interleaved ADCs. It introduces the gain mismatch problem in TI-ADCs and how it reduces the spurious free dynamic range. The proposed method uses the discrete Fourier transform to detect the gain mismatch between ADC sub-channels based on the difference between the ideal DFT and actual DFT. It then introduces a feedback system using this difference signal to iteratively correct the gain mismatch. Simulation results show the approach improves SFDR by more than 30dB by correcting a ±2% gain mismatch in a two-channel TI-ADC.
These notes were developed for use in the course Signals and Systems. The notes cover traditional, introductory concepts in the time domain and frequency domain analysis of signals and systems.
This document discusses the discrete Fourier transform (DFT) and fast Fourier transform (FFT). It begins by contrasting the frequency and time domains. It then defines the DFT, showing how it samples the discrete-time Fourier transform (DTFT) at discrete frequency points. It provides an example 4-point DFT calculation. It discusses the computational complexity of the direct DFT algorithm and how the FFT reduces this to O(N log N) by decomposing the DFT into smaller transforms. It explains the decimation-in-time FFT algorithm using butterfly operations across multiple stages. Finally, it notes that the inverse FFT can be computed using the FFT along with conjugation and scaling steps.
This document provides solutions to problems related to waves and diffraction in solids. It covers topics such as Fourier analysis, group velocity, dispersion, physical examples of waves including phonons and lattice vibrations, Maxwell's equations, x-ray scattering, experimental diffraction methods, quantum behavior of phonons, and statistical mechanics. The document contains detailed solutions to problems in each of these areas of waves and diffraction in solids.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
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.
1. The document discusses signals and systems, including continuous-time and discrete-time signals. It covers topics like transformations of signals, exponential and sinusoidal signals, and basic properties of systems.
2. Continuous-time signals are represented as functions of time t, while discrete-time signals are represented as sequences indexed by integer n. Exponential and sinusoidal signals can be represented using complex exponential functions.
3. The document provides examples and formulas for calculating energy, power, and other properties of signals. It also describes how signals can be transformed through operations like time shifting, scaling, reversal, and periodicity.
Instrumentation Engineering : Signals & systems, THE GATE ACADEMYklirantga
THE GATE ACADEMY's GATE Correspondence Materials consist of complete GATE syllabus in the form of booklets with theory, solved examples, model tests, formulae and questions in various levels of difficulty in all the topics of the syllabus. The material is designed in such a way that it has proven to be an ideal material in-terms of an accurate and efficient preparation for GATE.
Quick Refresher Guide : is especially developed for the students, for their quick revision of concepts preparing for GATE examination. Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
GATE QUESTION BANK : is a topic-wise and subject wise collection of previous year GATE questions ( 2001 – 2013). Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
Bangalore Head Office:
THE GATE ACADEMY
# 74, Keshava Krupa(Third floor), 30th Cross,
10th Main, Jayanagar 4th block, Bangalore- 560011
E-Mail: info@thegateacademy.com
Ph: 080-61766222
This document provides an overview of signals and systems. It defines key terms like signal, system, continuous and discrete time signals, analog and digital signals, periodic and aperiodic signals. It also discusses different types of signals like deterministic and probabilistic signals, energy and power signals. The document then classifies systems as linear/nonlinear, time-invariant/variant, causal/non-causal, and with/without memory. It provides examples of different signals and properties of signals like magnitude scaling, time shifting, reflection and scaling. Overall, the document introduces fundamental concepts in signals and systems.
The document discusses various properties of signals including:
- Analog signals can have an infinite number of values while digital signals are limited to a set of values.
- Phase describes the position of a waveform relative to a reference point in time.
- Total energy and average power of continuous and discrete signals can be calculated through integrals and sums.
- Periodic, even, odd, exponential, and sinusoidal signals are described.
- Unit impulse and step signals are defined for both discrete and continuous time domains.
- A signal's frequency spectrum shows the collection of component frequencies and bandwidth is the range of these frequencies.
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.
Describes Pulse Compression 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.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
This document discusses various operations that can be performed on signals. It was prepared by Dishant Patel, Vishal Gohel, Jay Panchal, and Manthan Panchal, and guided by Prof. Hardik Patel. The key operations discussed are time shifting, time scaling, time inversion/folding, amplitude scaling, addition, subtraction, and multiplication of signals. These basic operations are important for analyzing and manipulating signals for different purposes.
This document provides an overview of discrete-time signals and systems in digital signal processing (DSP). It discusses key concepts such as:
1) Discrete-time signals which are represented by sequences of numbers and how common signals like impulses and steps are represented.
2) Discrete-time systems which take a discrete-time signal as input and produce an output signal through a mathematical algorithm, with the impulse response characterizing the system.
3) Important properties of linear time-invariant (LTI) systems including superposition, time-shifting of inputs and outputs, and representation using convolution sums or difference equations.
The document defines and classifies different types of signals including:
- Continuous-time and discrete-time signals
- Analog and digital signals
- Real and complex signals
- Deterministic and random signals
- Periodic and non-periodic signals
It also introduces important signal properties and functions including the unit-step function, unit-impulse (Dirac delta) function, and complex exponential and sinusoidal signals. Graphical representations and mathematical definitions are provided for key signals and functions.
This document discusses pulse code modulation (PCM) for analog to digital conversion. PCM involves sampling an analog signal, quantizing the sample values, and encoding the quantized levels into binary codes. The sampling rate must be at least twice the bandwidth of the analog signal. More bits per code provide higher quality but require more bandwidth. PCM is used in telephone systems with 8-bit coding at 8 kHz sampling, and in compact discs with 16-bit coding at 44.1 kHz sampling.
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the
paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of
these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the
noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the
splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is
reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms
(DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are
applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of
complex multiplication, for the signals of length = 2^, > 2. .
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
The document presents a new method for signal denoising using a paired transform. The key points are:
1) The method splits a signal into multiple splitting-signals by applying a paired transform that represents the signal in both frequency and time.
2) Linear filtering for denoising can then be applied to each splitting-signal independently, reducing computational complexity compared to filtering the full signal.
3) Two models are proposed - one uses shifted discrete Fourier transforms on each splitting-signal, while the other uses non-shifted transforms, allowing different filter designs while achieving the same results.
Wavelet packet transforms were used to extract features from acoustic emission signals for tool wear monitoring. The acoustic emission signals were decomposed into different frequency bands using wavelet packet transforms. The root mean square values of the decomposed signals in each frequency band were extracted as features. Seven features were found to be most sensitive to tool wear based on analysis of experimental data. By dividing the features by cutting speed, the sensitivity of the features to changes in cutting conditions was reduced, providing effective monitoring of tool wear under different conditions using wavelet packet analysis of acoustic emission signals.
This document discusses signals and their classification. It defines signals, analog and digital signals, periodic and aperiodic signals. It also discusses representing signals in Matlab and Simulink. Key signal types covered include exponential, sinusoidal, unit impulse and step functions. Matlab is presented as a tool for programming and analyzing discrete signals while Simulink can be used to model and simulate continuous systems.
The document discusses sampling of continuous-time signals to create discrete-time signals. It explains that for perfect reconstruction, the sampling frequency must be greater than twice the maximum frequency of the original continuous-time signal, as specified by the Nyquist rate. A common method for sampling is to use an impulse train, and then reconstruct the signal by passing it through a low-pass filter. Often a zero-order hold is used to sample and communicate the signal, which simply holds each value until the next sample, and this provides a sufficiently accurate reconstructed continuous-time signal.
This document covers key concepts about signals including:
1) It defines continuous-time and discrete-time signals, and discusses the concepts of energy and power for both types of signals.
2) It provides the mathematical definitions of total energy, average power, and characterizes signals based on whether they have finite or infinite total energy and average power.
3) It discusses properties of exponential and sinusoidal signals, including that they have infinite total energy but finite average power.
4) It introduces common basic signals like the unit impulse and unit step signals in both continuous and discrete time.
This document presents a new DFT-based approach for detecting and correcting gain mismatch in time-interleaved ADCs. It introduces the gain mismatch problem in TI-ADCs and how it reduces the spurious free dynamic range. The proposed method uses the discrete Fourier transform to detect the gain mismatch between ADC sub-channels based on the difference between the ideal DFT and actual DFT. It then introduces a feedback system using this difference signal to iteratively correct the gain mismatch. Simulation results show the approach improves SFDR by more than 30dB by correcting a ±2% gain mismatch in a two-channel TI-ADC.
These notes were developed for use in the course Signals and Systems. The notes cover traditional, introductory concepts in the time domain and frequency domain analysis of signals and systems.
This document discusses the discrete Fourier transform (DFT) and fast Fourier transform (FFT). It begins by contrasting the frequency and time domains. It then defines the DFT, showing how it samples the discrete-time Fourier transform (DTFT) at discrete frequency points. It provides an example 4-point DFT calculation. It discusses the computational complexity of the direct DFT algorithm and how the FFT reduces this to O(N log N) by decomposing the DFT into smaller transforms. It explains the decimation-in-time FFT algorithm using butterfly operations across multiple stages. Finally, it notes that the inverse FFT can be computed using the FFT along with conjugation and scaling steps.
This document provides solutions to problems related to waves and diffraction in solids. It covers topics such as Fourier analysis, group velocity, dispersion, physical examples of waves including phonons and lattice vibrations, Maxwell's equations, x-ray scattering, experimental diffraction methods, quantum behavior of phonons, and statistical mechanics. The document contains detailed solutions to problems in each of these areas of waves and diffraction in solids.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
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.
1. The document discusses signals and systems, including continuous-time and discrete-time signals. It covers topics like transformations of signals, exponential and sinusoidal signals, and basic properties of systems.
2. Continuous-time signals are represented as functions of time t, while discrete-time signals are represented as sequences indexed by integer n. Exponential and sinusoidal signals can be represented using complex exponential functions.
3. The document provides examples and formulas for calculating energy, power, and other properties of signals. It also describes how signals can be transformed through operations like time shifting, scaling, reversal, and periodicity.
Instrumentation Engineering : Signals & systems, THE GATE ACADEMYklirantga
THE GATE ACADEMY's GATE Correspondence Materials consist of complete GATE syllabus in the form of booklets with theory, solved examples, model tests, formulae and questions in various levels of difficulty in all the topics of the syllabus. The material is designed in such a way that it has proven to be an ideal material in-terms of an accurate and efficient preparation for GATE.
Quick Refresher Guide : is especially developed for the students, for their quick revision of concepts preparing for GATE examination. Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
GATE QUESTION BANK : is a topic-wise and subject wise collection of previous year GATE questions ( 2001 – 2013). Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions
Bangalore Head Office:
THE GATE ACADEMY
# 74, Keshava Krupa(Third floor), 30th Cross,
10th Main, Jayanagar 4th block, Bangalore- 560011
E-Mail: info@thegateacademy.com
Ph: 080-61766222
This document provides an overview of signals and systems. It defines key terms like signal, system, continuous and discrete time signals, analog and digital signals, periodic and aperiodic signals. It also discusses different types of signals like deterministic and probabilistic signals, energy and power signals. The document then classifies systems as linear/nonlinear, time-invariant/variant, causal/non-causal, and with/without memory. It provides examples of different signals and properties of signals like magnitude scaling, time shifting, reflection and scaling. Overall, the document introduces fundamental concepts in signals and systems.
The document discusses various properties of signals including:
- Analog signals can have an infinite number of values while digital signals are limited to a set of values.
- Phase describes the position of a waveform relative to a reference point in time.
- Total energy and average power of continuous and discrete signals can be calculated through integrals and sums.
- Periodic, even, odd, exponential, and sinusoidal signals are described.
- Unit impulse and step signals are defined for both discrete and continuous time domains.
- A signal's frequency spectrum shows the collection of component frequencies and bandwidth is the range of these frequencies.
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.
Describes Pulse Compression 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.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
This document discusses various operations that can be performed on signals. It was prepared by Dishant Patel, Vishal Gohel, Jay Panchal, and Manthan Panchal, and guided by Prof. Hardik Patel. The key operations discussed are time shifting, time scaling, time inversion/folding, amplitude scaling, addition, subtraction, and multiplication of signals. These basic operations are important for analyzing and manipulating signals for different purposes.
This document provides an overview of discrete-time signals and systems in digital signal processing (DSP). It discusses key concepts such as:
1) Discrete-time signals which are represented by sequences of numbers and how common signals like impulses and steps are represented.
2) Discrete-time systems which take a discrete-time signal as input and produce an output signal through a mathematical algorithm, with the impulse response characterizing the system.
3) Important properties of linear time-invariant (LTI) systems including superposition, time-shifting of inputs and outputs, and representation using convolution sums or difference equations.
The document defines and classifies different types of signals including:
- Continuous-time and discrete-time signals
- Analog and digital signals
- Real and complex signals
- Deterministic and random signals
- Periodic and non-periodic signals
It also introduces important signal properties and functions including the unit-step function, unit-impulse (Dirac delta) function, and complex exponential and sinusoidal signals. Graphical representations and mathematical definitions are provided for key signals and functions.
This document discusses pulse code modulation (PCM) for analog to digital conversion. PCM involves sampling an analog signal, quantizing the sample values, and encoding the quantized levels into binary codes. The sampling rate must be at least twice the bandwidth of the analog signal. More bits per code provide higher quality but require more bandwidth. PCM is used in telephone systems with 8-bit coding at 8 kHz sampling, and in compact discs with 16-bit coding at 44.1 kHz sampling.
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the
paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of
these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the
noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the
splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is
reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms
(DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are
applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of
complex multiplication, for the signals of length = 2^, > 2. .
NEW METHOD OF SIGNAL DENOISING BY THE PAIRED TRANSFORMmathsjournal
The document presents a new method for signal denoising using a paired transform. The key points are:
1) The method splits a signal into multiple splitting-signals by applying a paired transform that represents the signal in both frequency and time.
2) Linear filtering for denoising can then be applied to each splitting-signal independently, reducing computational complexity compared to filtering the full signal.
3) Two models are proposed - one uses shifted discrete Fourier transforms on each splitting-signal, while the other uses non-shifted transforms, allowing different filter designs while achieving the same results.
Wavelet packet transforms were used to extract features from acoustic emission signals for tool wear monitoring. The acoustic emission signals were decomposed into different frequency bands using wavelet packet transforms. The root mean square values of the decomposed signals in each frequency band were extracted as features. Seven features were found to be most sensitive to tool wear based on analysis of experimental data. By dividing the features by cutting speed, the sensitivity of the features to changes in cutting conditions was reduced, providing effective monitoring of tool wear under different conditions using wavelet packet analysis of acoustic emission signals.
Cyclostationary analysis of polytime coded signals for lpi radarseSAT Journals
This document discusses cyclostationary analysis of polytime coded signals for low probability of intercept (LPI) radars. It begins with an introduction to LPI radars and their modulation and detection techniques, focusing on polytime codes. It then describes cyclostationary signal processing methods like the direct frequency smoothing method (DFSM) and fast Fourier transform accumulation method (FAM) that can be used to extract parameters from polytime coded signals. The document analyzes example polytime coded signals with and without noise using these cyclostationary techniques and accurately extracts key parameters like carrier frequency, bandwidth, and code rate. It finds the FAM method has better computational efficiency than DFSM for long signals.
Chaotic signals denoising using empirical mode decomposition inspired by mult...IJECEIAES
The document describes a new method for denoising chaotic signals corrupted by additive noise using empirical mode decomposition (EMD) inspired by multivariate denoising. EMD is used to decompose the noisy chaotic signal into intrinsic mode functions (IMFs), which are then thresholded using a multivariate denoising algorithm combining wavelet transforms and principal component analysis. This proposed EMD-MD method is compared to other techniques using metrics like root mean square error and signal-to-noise ratio gain. Simulation results on Lorenz, Chen and Rossler chaotic systems show the EMD-MD method achieves the best denoising performance compared to conventional methods.
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.
On The Fundamental Aspects of DemodulationCSCJournals
When the instantaneous amplitude, phase and frequency of a carrier wave are modulated with the information signal for transmission, it is known that the receiver works on the basis of the received signal and a knowledge of the carrier frequency. The question is: If the receiver does not have the a priori information about the carrier frequency, is it possible to carry out the demodulation process? This tutorial lecture answers this question by looking into the very fundamental process by which the modulated wave is generated. It critically looks into the energy separation algorithm for signal analysis and suggests modification for distortionless demodulation of an FM signal, and recovery of sub-carrier signals
Analysis the results_of_acoustic_echo_cancellation_for_speech_processing_usin...Venkata Sudhir Vedurla
This document presents an analysis of acoustic echo cancellation for speech processing using the LMS adaptive filtering algorithm. It begins with an abstract that outlines the challenges of conventional echo cancellation techniques and the need for a computationally efficient, rapidly converging algorithm. It then provides background on acoustic echo, the principles of echo cancellation, discrete time signals, speech signals, and an overview of the LMS adaptive filtering algorithm and its application to echo cancellation. The document analyzes the performance of the LMS algorithm for echo cancellation by examining how the step size parameter affects convergence and steady state error. It concludes that the LMS algorithm is well-suited for echo cancellation due to its computational simplicity, though the step size must be carefully selected for optimal performance
The document describes an experiment to verify the Nyquist sampling theorem using MATLAB. It discusses sampling a continuous time signal at frequencies below, equal to, and above twice the maximum frequency of the signal. The results show aliasing when sampling below the Nyquist rate, no aliasing when sampling at the Nyquist rate, and perfect reconstruction when sampling above the Nyquist rate. The experiment generates a sinusoidal signal, samples it at different rates, and plots the discrete and reconstructed continuous signals to demonstrate the sampling theorem.
Electrocardiogram Denoised Signal by Discrete Wavelet Transform and Continuou...CSCJournals
One of commonest problems in electrocardiogram (ECG) signal processing is denoising. In this paper a denoising technique based on discrete wavelet transform (DWT) has been developed. To evaluate proposed technique, we compare it to continuous wavelet transform (CWT). Performance evaluation uses parameters like mean square error (MSE) and signal to noise ratio (SNR) computations show that the proposed technique out performs the CWT.
RESOLVING CYCLIC AMBIGUITIES AND INCREASING ACCURACY AND RESOLUTION IN DOA ES...csandit
A method to resolve cyclic ambiguities and increase the accuracy and the resolution in the
direction-of-arrival (DOA) estimation using the Estimation of Signal Parameters via Rotational
Invariance Technique (ESPRIT)algorithm is proposed. It is based on rotating the array and
sampling the received signal at multiple positions. Using this approach, the gain in accuracy
and resolution is addressed as function of the mean and variance of the DOA. Simulations
results are provided as a means of verifying this analysis.
The document describes an experiment on linear time invariant systems. The objectives are to: 1) Convolve a signal with an impulse response, 2) Find step responses using the impulse response for rectangular, exponential and sinusoidal inputs, 3) Show stable and unstable conditions using pole-zero plots, 4) Apply filtering to an image using circular convolution with overlap add and save methods. Background topics discussed are aliasing, impulse/step inputs, and even/odd signals. Questions involve plotting a signal, its Fourier transform, and filtering an image.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
ANALYSIS OF FRACTIONALLY SPACED WIDELY LINEAR EQUALIZATION OVER FREQUENCY SEL...ijwmn
This paper deals with the performance analysis of both linear (LE) and widely linear (WLE) equalization
processes studied in the symbol spaced mode (SSE) as well as in the fractionally spaced one (FSE). This
analysis is evaluated - using Matlab software- in terms of bit error rate (BER) and mean square error (MSE)
in a system using a rectilinear modulation of type Pulse Amplitude Modulation (PAM) over a frequency
selective channel and corrupted by multiple interferences. Moreover, the impact of the number of external
interferers on the behavior of the different equalizers is studied. Thus, simulation results show the outstanding performance of the fractionally spaced mode when compared to the symbol spaced one besides the
out-performance of the widely linear processing in both modes. Furthermore, results show the performance
degradation for the different studied equalizers when the number of interferences increases.
Performance of MMSE Denoise Signal Using LS-MMSE TechniqueIJMER
This paper presents performance of mmse denoises signal using consistent cycle spinning
(ccs) and least square (LS) techniques. In the past decade, TV denoise technique is used to reduced the
noisy signal. The main drawback is the low quality signal and high MMSE signal. Presently, we
proposed the CCS-MMSE and LS-MMSE technique .The CCS-MMSE technique consists of two steps.
They are wavelet based denoise and consistent cycle spinning. The wavelet denoise is powerful
decorrelating effect on many signal domains. The consistent cycle spinning is used to estimation the
MMSE in the signal domain. The LS-MMSE is better estimation of MMSE signal domain compare to
CCS-MMSE.The experimental result shows the average MMSE reduction using various techniques.
The document discusses lossy compression techniques such as quantization and transform coding. It explains that lossy compression achieves higher compression rates by introducing some loss or distortion to the reconstructed data compared to the original. It describes various distortion measures used to evaluate the quality of lossy compression, including mean squared error and peak signal-to-noise ratio. It also discusses the discrete cosine transform, a widely used transform coding technique that decomposes images or signals into different frequency components for more efficient compression.
Performance of MMSE Denoise Signal Using LS-MMSE TechniqueIJMER
This paper presents performance of mmse denoises signal using consistent cycle spinning (ccs) and least square (LS) techniques. In the past decade, TV denoise technique is used to reduced the noisy signal. The main drawback is the low quality signal and high MMSE signal. Presently, we
proposed the CCS-MMSE and LS-MMSE technique .The CCS-MMSE technique consists of two steps. They are wavelet based denoise and consistent cycle spinning. The wavelet denoise is powerful decorrelating effect on many signal domains. The consistent cycle spinning is used to estimation the
MMSE in the signal domain. The LS-MMSE is better estimation of MMSE signal domain compare to
CCS-MMSE.The experimental result shows the average MMSE reduction using various techniques.
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.
Intersymbol interference caused by multipath in band limited frequency selective time dispersive channels distorts the transmitted signal, causing bit error at receiver. ISI is the major obstacle to high speed data transmission over wireless channels. Channel estimation is a technique used to combat the intersymbol interference. The objective of this paper is to improve channel estimation accuracy in MIMO-OFDM system by using modified variable step size leaky Least Mean Square (MVSSLLMS) algorithm proposed for MIMO OFDM System. So we are going to analyze Bit Error Rate for different signal to noise ratio, also compare the proposed scheme with standard LMS channel estimation method.
This document discusses dynamics of structures with uncertainties. It begins with an introduction to stochastic single degree of freedom systems and how natural frequency variability can be modeled using probability distributions. It then discusses how to extend this approach to stochastic multi degree of freedom systems using stochastic finite element formulations and modal projections. Key challenges with statistical overlap of eigenvalues are noted. The document provides mathematical models of equivalent damping in stochastic systems and examples of stochastic frequency response functions.
Similar to New Method of Signal Denoising by the Paired Transform (20)
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵mathsjournal
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
Modernization of radar technology and improved signal processing techniques are necessary to improve detection systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints
Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Qua...mathsjournal
Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by thegenetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.
An Application of Assignment Problem in Laptop Selection Problem Using MATLABmathsjournal
The assignment – selection problem used to find one-to- one match of given “Users” to “Laptops”, the main objective is to minimize the cost as per user requirement. This paper presents satisfactory solution for real assignment – Laptop selection problem using MATLAB coding.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
Cubic Response Surface Designs Using Bibd in Four Dimensionsmathsjournal
Response Surface Methodology (RSM) has applications in Chemical, Physical, Meteorological, Industrial and Biological fields. The estimation of slope response surface occurs frequently in practical situations for the experimenter. The rates of change of the response surface, like rates of change in the yield of crop to various fertilizers, to estimate the rates of change in chemical experiments etc. are of
interest. If the fit of second order response is inadequate for the design points, we continue the
experiment so as to fit a third order response surface. Higher order response surface designs are sometimes needed in Industrial and Meteorological applications. Gardiner et al (1959) introduced third order rotatable designs for exploring response surface. Anjaneyulu et al (1994-1995) constructed third order slope rotatable designs using doubly balanced incomplete block designs. Anjaneyulu et al (2001)
introduced third order slope rotatable designs using central composite type design points. Seshu babu et al (2011) studied modified construction of third order slope rotatable designs using central composite
designs. Seshu babu et al (2014) constructed TOSRD using BIBD. In view of wide applicability of third
order models in RSM and importance of slope rotatability, we introduce A Cubic Slope Rotatable Designs Using BIBD in four dimensions.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
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under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
Table of Contents - September 2022, Volume 9, Number 2/3mathsjournal
Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
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.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
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
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
New Method of Signal Denoising by the Paired Transform
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
1
NEW METHOD OF SIGNAL DENOISING BY THE
PAIRED TRANSFORM
Artyom M. Grigoryan and Sree P.K. Devieni
Department of Electrical Engineering, University of Texas at San Antonio, USA
ABSTRACT
A parallel restoration procedure obtained through a splitting of the signal into multiple signals by the
paired transform is described. The set of frequency-points is divided by disjoint subsets, and on each of
these subsets, the linear filtration is performed separately. The method of optimal Wiener filtration of the
noisy signal is considered. In such splitting, the optimal filter is defined as a set of sub filters applied on the
splitting-signals. Two new models of filtration are described. In the first model, the traditional filtration is
reduced to the processing separately the splitting-signals by the shifted discrete Fourier transforms
(DFTs). In the second model, the not shifted DFTs are used over the splitting-signals and sub filters are
applied. Such simplified model for splitting the filtration allows for saving 2 − 4( + 1) operations of
complex multiplication, for the signals of length = 2^, 2. .
KEYWORDS
Time-frequency analysis, signal reconstruction, paired transform, Wiener filtration
1. INTRODUCTION
Denoising of a time series signal on multiple channels is possible by splitting it into many signals
which can be processed independently. First and foremost available tool is the Fourier integral
transformation which represents a time signal in frequency bins/objects where time information is
lost completely. Suppressing the noisy frequency objects is the common approach in this method.
The wavelet transformation represents a time signal as scale-time signals where the information
of scale (referred to as “frequency”) and time is preserved. These signals offer faster and better
edge preservation algorithms than the Fourier method. Denoising as classified among a class of
restoration problems has the objective to reconstruct the signal as close as possible to the actual
signal in some distance measure. Restoration of a signal from garbled/incomplete information is
the classical Wiener problem [1]. It solves primarily for a simple specification of the linear
dynamical system, ∈ ×
, , which generates the wide-sense stationary random process
∈
. The solution of ̂ = is obtained by the minimum-mean squares error method which
is the optimization of the -norm ‖ − ‖
, where ∈
is the actual signal. For a class of
wide sense stationary random processes, the linear time invariant system matrix is a circulant-
Toeplitz, so the computation is fast with the fast Fourier transform. Other fast approaches, which
are based on the wavelet coefficients thresholding, introduce considerable artefacts [2]-[5].
Dohono's reconstruction by soft-thresholding of wavelet coefficients is at least as smooth as the
minimum mean squares solution, therefore it removes several of these artefacts. Current parallel
methods for signal restoration or filtration rely on splitting of a complex problem into partial
problems whose independent solutions lead to a list of feasible solutions, or one final solution.
The restoration problem is re-formulated as a set theoretic constrained optimization [7],[8]. The
solution space is usually a Hilbert space, where the original signal is described solely by a family
of constraints arising from a priori knowledge about the problem and from the observed data. We
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
2
also mention the parallel approaches such as block-iterative surrogate constraint splitting methods
for solving a quadratic minimization problem under convex constraints [9].
In this paper, we consider the transformation of the signal
of length = 2!
, 2, into a set
of splitting-signals ,$
%
;' = 0: /(2+) − 1, + = 2,
, - = 0: . This is a frequency (+) and
time (') representation of the signal, which is accomplished by the fast paired transform [11]-[17].
The linear filtration of the noisy signal, including the optimal (Wiener) filtration, is described.
Such filtration as the linear convolution can be reduced to filtration of splitting-signals in two
different ways. In the first way, the Fourier transform method of filtration is reduced to
calculation of the transform of each splitting-signal and filtration on the shifted grid of frequency-
points. In other words in the traditional Fourier method, the shifted DFTs over the splitting-
signals are calculated. All values of the filter function are reordered into subsets and used for
filtering the splitting-signals. These sub-filter functions or other such functions can also be used
for denoising the signal in the frequency domain with the points on the non shifted grid. In this
second way of filtration, one can save a large number of operations of complex multiplication and
design new linear filters. Such filters can be transformed into equivalent filters in the shifted grid,
which lead to the same results of filtration.
2. PAIRED REPRESENTATION OF THE SIGNAL
Given signal
and noise , we consider the linear signal-independent-noise model . =
+
, = 0: ( − 1). In paired representation, the signals
, and . are described by the
corresponding sets of short splitting-signals of lengths /2, /4, . . . ,2,1, and 1. For instance, the
signal
is transformed into the following set of splitting-signals:
/0
%
: →
2
3
3
3
4
3
3
3
5
67,$
%
; ' = 0: (/2 − 1)8
6,$
%
; ' = 0: (/4 − 1)8
69,9$
%
; ' = 0: (/8 − 1)8
… … …
0/9,
%
, 0/9,0/9
%
0/,
%
,
%
.
=
The 1-D -point discrete paired transform /0
%
is defined by the complete system of paired
functions [11][12]:
/,$
% () =
1; if = 'mod/+,
−1; if = (' + /(2+)) mod /+,
0; otherwise,
= t = 0: (N/(2p) − 1), (1)
where p = 2L
, k = 0: (r − 1) and /,
% () ≡ 1, = 0: ( − 1). The double numbering of the
paired functions /,$
% () refers to the frequency-point (p = 2L
) and time (+') which runs the
interval of time points 0,1,2, . . . , /2 − 1 with the step (“velocity”) equals +. The paired
transform is the frequency-time representation of the signal.
The components of the splitting-signal 6,
′
,
,
′
, ,
′
, ,O
′
, … , ,0/P
′
8 of the signal
are
calculated by
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
3
,$
%
= /,$
%
∘
= R /,$
%
()
0P7
S
, ' = 0: (/(2+) − 1).
The last one-point splitting-signal is ,
%
= ,
%
= + 7 + ⋯ + 0P + 0P7.
Example 1 (Case N=8) The matrix of the eight-point discrete paired transform (DPT) is defined
as
U/V
% W =
X
Y
Y
Y
Y
Y
Y
Y
Y
Y
Z
[/7,
%
[/7,7
%
[/7,
%
[/7,O
%
[/,
%
[/,
%
[/9,
%
[/,
%
]
^
^
^
^
^
^
^
^
^
_
=
X
Y
Y
Y
Y
Y
Y
Z
1 0 0 0 −1 0 0 0
0 1 0 0 0 −1 0 0
0 0 1 0 0 0 −1 0
0 0 0 1 0 0 0 −1
1 0 −1 0 1 0 −1 0
0 1 0 −1 0 1 0 −1
1 −1 1 −1 1 −1 1 −1
1 1 1 1 1 1 1 1]
^
^
^
^
^
^
_
The normalized coefficients 6√2, √2, √2, √2, 2,2,2√2, 2√28 of rows can be considered, to make
this transform unitary. The first four basis paired functions correspond to the frequency + = 1, the
next two functions correspond to frequency + = 2, and the last two functions correspond to the
frequencies + = 4 and 0, respectively. The process of composition of these functions from the
corresponding cosine waves defined in the interval U0,7W is illustrated in Figure 1.
Fig. 1. (a) Cosine waves and (b) discrete paired functions of the eight-point DPT
Let be the signal 1,2,2,4,5,3,1,3. The paired transform of this signal results in the following
four splitting-signals:
/0
%
: →
2
3
4
3
567,
%
, 7,7
%
, 7,
%
, 7,O
%
8,
6,
%
, ,
%
8,
9,
%
,
,
%
= = d
−4, −1,1,1,
3, −2,
−3,
21.
=
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
4
Splitting-signals carry the spectral information of the signal
in different and disjoint subsets of
frequency-points. In paired representation, the splitting of the -point DFT
e = R
f
0
0P7
S
, + = 0: ( − 1), f0 = exp(−h2i/),
by ( + 1) short DFTs is based on the following formula [16]:
e(j7) = ∑ l,$
%
f0/
$
m
0/P7
$S f0/()
$
, = 0: (/(2+) − 1). (2)
For the set of frequency-points + = 2,
; - = 0: ( − 1), the disjoint subsets
n
%
= (2 + 1)+ mod ; = 0: (/(2+) − 1)
together with n
%
= 0 divide the set of frequency-points o0 = 0, 1, 2, . . . , − 1. Therefore,
the splitting-signal generated by the frequency + = 2
is denoted by
pq
r = 6,
%
,
,
%
, ,
%
, ,O
%
, … , ,0/P
%
8.
2.1. The first and second splitting-signals
To describe the meaning of equation (2), we first consider the + = 1 case, when the first splitting-
signal carries the spectral information of the signal at all odd frequency-points,
ej7 = ∑ l7,$
%
f0
$
m
0/P7
$S f0/
$
, = 0: (/2 − 1). (3)
The /2-point DFT over the modified splitting-signal
spt
r = 67,
%
, 7,7
%
f7
, 7,
%
f
, 7,O
%
fO
, … , 7,0/P7
%
f0/P7
8
equals the -point DFT of the original signal at odd frequency-points. It should be noted, that the
right part of equation (3) is referred to as the /2-point shifted DFT (SDFT) on the set of
frequency-points + 1/2; = 0: (/2 − 1), which we denote by
ej7/
(7)
= ∑ 7,$
%
f
0/
$uj
t
v
w
0/P7
$S , +
7
=
7
,
O
,
x
, …
0P7
. (4)
Thus, the /2-point shifted DFT of the first splitting-signal coincides with the odd-indexed
DFT values of the original signal, ej7 = ej7/
(7)
, = 0: (/2 − 1).
For the p = 2 case, the splitting-signal is pv
r = 6,
%
, ,
%
, ,9
%
, ,y
%
, … , ,0/P
%
8 and
e(j7) = ∑ l,$
%
f0/
$
m
0/9P7
$S f0/9
$
, = 0: (/4 − 1). (5)
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
5
The /4-point DFT over the modified splitting-signal
spv
r = ,
%
, ,
%
f
, ,9
%
f9
, ,y
%
fy
, … ,
,0/P
%
f0/P
equals the -point DFT of the original signal at all frequency-points which are
odd-integer multiple of 2. The right part of equation (5) is referred to as the shifted /4-point
DFT on the set of frequency-points + 1/2; = 0: (/4 − 1), which is denoted by
ej7/
()
= ∑ ,$
%
f
0/9
$(j7/)
0/9P7
$S , = 0: (/4 − 1). (6)
In the general case + = 2,
where - , the -point DFT of the signal
at frequency-points of
n
%
is the /(2+)-point shifted DFT of the splitting-signal pq
r, i.e.,
e(j7) = ej7/
()
, = 0: /(2+) − 1.
Thus, in paired representation, when the signal is described by the set of splitting-signals, the
DFT of the signal
is described by a set of shifted DFTs of the splitting-signals. The shifts of
frequency-points are by 0.5.
2.2. Discrete Linear Filters
In this section, we consider the traditional method of DFT for convoluting the signals. Let z,
+ = 0: ( − 1), be the frequency characteristic of a given linear filter which describes the circular
convolution of the noisy signal . =
+ with the impulse response of the filter,
{
= | ⊛
.. We denote by ~ and e
the -point DFTs of the noisy and filtered signals, respectively.
Then e
= z~, for + = 0: ( − 1). The block-diagram of the linear filtration of the noisy
signal is given in Figure 2, where RDFT stands for the reversible DFT.
It follows directly from equation (2) that the process of filtration can be reduced to separate
filtration of splitting-signals, as shown in Figure 3. The /(2+)-point shifted DFTs of the
splitting-signals ,$
%
are multiplied by the corresponding filter functions z(j7), and then, the
reversible SDFTs are calculated to obtain the filtered splitting-signals.
Fig. 2. Block-diagram of the Fourier transform-based method of filtration of the noisy signal.
Fig. 3. Block-diagram of the Fourier transform-based method of filtration by splitting-signals.
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
6
The last two one-point splitting-signals .0/,
%
and .,
%
are not shown in the block diagram. The
processing of these splitting-signals is reduced to one multiplication for each. The convoluted
signal
{
is calculated as the inverse -point paired transform of the filtered splitting-signals.
This convolution can be obtained by calculating the -point reversible DFT over the transform
~, + = 0: ( − 1). Thus, the frequency characteristics are permuted onto ( + 1) subsets as
6z; + = 0: ( − 1)8 → z(j7) €‚0; = 0: /(2+) − 1; + = 1,2,4,8, . . . , /2, z. The
filter is considered as the set of ( + 1) sub-filters.
Example 2: Consider the signal
of length = 256, which is shown in Figure 4(a). The paired
transform of the signal, as the set of splitting-signals pt
r, pv
r, p„
r, p…
r, … . , ptv…
r , p†
r is shown
in part b. The vertical dashed lines separate the first seven splitting-signals.
Fig. 4. (a) The signal of length 256 and (b) the 256-point discrete paired transform.
Figure 5 shows the 256-point DFT of the signal in absolute mode in part a. In part b, the same
DFT is shown, but in the order that corresponds to the partition of the frequency-points {0,1,2, ...
, 255} by subsets n7
%
, n
%
, n9
%
, … , n7V
%
, and n
%
. The first 128 values corresponds to the 128-point
DFT of the modified splitting-signal 7,$
%
f$
which is also the 128-point SDFT of the splitting-
signal 7,$
%
. The next 64 values corresponds to the 64-point DFT of the modified splitting-signal
,$
%
f7V
$
, which is also the 64-point SDFT of the splitting-signal ,$
%
, and so on.
7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
7
Fig. 5. (a) DFT of the signal and (b) DFTs of the modified splitting-signals.
(The transforms are shown in absolute mode.)
We consider the smoothing of the signal by the circular convolution with the impulse response
| = 0,0, . . , 0,1,2,3, 5, 3,2,1,0,0, . . . , 0/17 with value | = 5. Figure 6 shows the frequency
characteristic z of | in absolute mode in part a. In part b, this characteristic is shown in the
same order as e
in part b of Figure 5. These characteristics are used to filter the corresponding
modified splitting-signals and they are separated in the graph by vertical lines.
Fig. 6. (a) DFT of the filter and (b) the reordered filter (both in absolute scale).
The result of convolution of the signal,
{
=
⊛ |, is shown in Figure 7 in part a, along with
nine splitting-signals which were filtered separately in b. Together these splitting-signals define
the filtered signal
{
in a. In other words, the 256-point inverse paired transform over the set of
these splitting-signals
{pt
r,
{pv
r,
{p„
r,
{p…
r, … . ,
{ptv…
r ,
{p†
r is the signal
{
. The following equation
describes the inverse paired transform [14][15],[17]:
{
= R
1
2,j7
{
‡,‡ €‚ 0/
%
!P7
,S
+
1
{,
%
, = 0: ( − 1). (7)
8. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
8
Fig. 7. (a) Filtered signal and (b) filtered splitting-signals.
All signals in part (a) and (b) are smooth. The last two one-point signal have not been changed,
because the filter was defined with ∑ | = 1. Thus the process of filtration of the original signal
by the discrete Fourier transform was reduced to separate processing of eight splitting-signals.
These signals were processed by the shifted DFTs.
3. NEW SIMPLIFIED SCHEME OF THE SIGNAL FILTRATION
It should be noted, that the calculation of the -point DFT by the DFTs of the splitting-signals
corresponds to the paired FFT with ˆ() = /2( − 3) + 2 operations of multiplication. The
same number of multiplications is required for the reversible DFT. If we propose the process of
filtration of each splitting-signal by the regular (not shifted) DFT, in other words, without
modifying the splitting-signal by the twiddle-coefficients f$, then many multiplications could be
saved. Indeed, let us consider two types of filtration of the first splitting-signal with the block-
diagrams shown in Figure 8.
Fig. 8. Block-diagrams of filtration of the first splitting-signal.
In the first diagram, the splitting-signal is processed in the traditional way, with the /2-point
shifted DFT, which requires multiplications by the factors f0
$
, ' = 0: (/2 − 1), and then, the
calculation of the /2-point DFT. In the second diagram, the same filter is used but the shifted
DFTs are substituted by the DFTs, and therefore, it allows for saving /2 operations of
multiplication by the twiddle factors. To be more accurate, this number is /2 − 2, because of
two trivial multiplications by f0
= 1 and f
0
0/9
= −h. The same number of multiplications is
saved when using the reversible /2-point DFT instead of the shifted DFT.
9. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
9
We now consider the complete block-diagram of the simplified signal filtration without shifted
DFTs, as shown in Figure 9.
Fig. 9. Block-diagram of the filtration by splitting-signals.
The filtration of the signal by this method saves the following number of operations of complex
multiplication:
∆ˆ() = 2U(/2 − 2) + (/4 − 2) + (/8 − 2) + . . . + (8 − 2)W
= 2U( − 8) − 2( − 3)W = 2 − 4 − 4.
For instance when = 256, such saving equals ∆ˆ(256) = 480. For the method of convolu-tion
with the block-diagram in Figure 3, the total number of required operations of multiplication
equals 2ˆ() + = ( − 2) + 4. For the = 256 case, this number is 1540, and the saving
in operations of multiplication is almost 32%.
3.1. Convolution of Splitting-Signals
To describe the process of filtration of the splitting-signals in the frequency-time domain, we
consider in detail this operation over the first-splitting signal. On the set of frequency-points n7
%
,
the application of the filter to the noisy signal . is described by
e
Šj7 = ~j7/
(7)
zj7/
(7)
, = 0 ∶ (/2 − 1), (8)
where we denote the shifted DFTs by ~j7/
(7)
= ~j7 and zj7/
(7)
= zj7.
Let
{
be the filtered signal and
{7,$
%
be its first splitting-signal, and let |7,$
%
be the first splitting-
signal of the impulse response of the filter z. Then, the following calculations hold:
{
7,$
%
f$
=
2
R ~j7/
(7)
zj7/
(7)
0/P7
$S
f0/
P$
=
2
R ŒR ŒR .7,
%
f
|7,Ž
%
fŽ
0/P7
S
0/P7
$tS
f
0/
$t
0/P7
S
f0/
P$
( = ('7 − ‘)’“ /2)
=
2
R ŒR .7,
%
f
|7,Ž
%
fŽ
0/P7
S
R f
0/
($tP$)
0/P7
S
0/P7
$tS
10. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
10
=
2
R ŒR .7,
%
f
|7,Ž
%
fŽ
0/P7
S
2
”('7 − ')
0/P7
$tS
= R .7,
%
f
|7,($P)€‚ 0/
%
f($P)€‚ 0/
.
0/P7
S
Thus, we obtain the following:
{7,$
%
= ∑ .7,
%
|7,($P)€‚ 0/
%
f
0
(P$)
f
0
($P)€‚ 0/
0/P7
S .
Since 0 ≤ ', ‘ /2 and (' − ‘)mod /2 = (' − ‘), if ' ≥ ‘, and (' − ‘) mod /2 =
(' − ‘) + /2, if ' ‘, we have the following:
f
0
(P$)
f
0
($P)€‚ 0/
= sign(' − ‘) = ™
1, š ' − ‘ ≥ 0,
−1, š ' − ‘ 0.
=
Therefore,
{7,$
%
= ∑ .7,
%
sign(' − ‘)|7,($P)€‚ 0/
%
0/P7
S , ' = 0: (/2 − 1). (9)
We remind here, that the function |7,$
%
as the function of ' has the period not /2, and the
function is “odd,” i.e., |7,P$
%
= −|7,0/P$
%
, ' = 1,2, . . . , (/2 − 1). Equation (9) can be written as
the linear (not circular) convolution calculated in /2 points:
{
7,$
%
= ∑ .7,
%
|7,($P)
%
0/P7
S , ' = 0: (/2 − 1). (10)
The linear convolution is calculated only in the original time interval 0,1, . . . , /2 − 1 and can
be calculated by the -point circular convolution. We call this operation the incomplete circular
linear convolution of length /2. Let |
›7,$
%
be the extended function
6|
›7,$; ' = 0: ( − 1)8 = 6|7,$
%
, '; ' = 0: (/2 − 1), −|7,0/P$
%
; ' = /2: ( − 1)8.
Let extended splitting-signal .
›7,$ of length be defined as
.
›7,$, '; ' = 0: ( − 1) = 6.7,$
%
; ' = 0: (/2 − 1), 0, 0, 0, . . . , 08.
It is not difficult to see, that the following holds:
{
7,$
%
= ∑ .
›7,$|
›7,($P)€‚ 0
0/P7
S , ' = 0: (/2 − 1). (11)
The linear convolution of the second and remaining splitting-signals with the filter is described
similarly. Thus, the cyclic convolution of length can be reduced to ( + 1) incomplete circular
convolutions, when representing the signal in form of splitting-signals. As an example, we
consider the signal of length 512 shown in part a of Figure 10. The paired transform as the set of
ten splitting-signals is shown in b.
11. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
11
Fig. 10. (a) Random signal and (b) its splitting-signals.
Figure 11 in part a shows the signal . =
+ with an additive random Gaussian noise
which is normally distributed as (0,17). The first splitting-signal .7,$
%
, '; ' = 0: 255 of the
noisy signal is shown in b, and the result of the optimal filtration of this splitting-signal in c. The
result of the optimal filtration of the entire noisy signal . by ten splitting-signals is shown in d.
The same result is achieved by using the Wiener filter over the noisy signal.
3.2. Parseval’s Theorem and Paired Transform
The normalization of the basic paired functions as /,
%
→ /,
%
/√ and /,$
%
→ /,$
%
/œ2+, when
' = 0: (/(2+) − 1), + = 1,2,4,8, … , /2, leads to the unitary property of the paired transform.
Therefore, according to Parseval’s’theorem, the difference of two signals
and
{
in metric
can be written as
l,
{m = Rl −
{
m
0P7
S
= R
1
2+
!P7
,S
R (,$
%
−
{,$
%
)
0/()P7
$S
+
1
(,
%
−
{,
%
)
= R
1
2+
!P7
,S
upq
r ,
{pq
r w +
1
up†
r,
{p†
rw
where we denote + = 2,
, and
=
upq
r ,
{pq
r w are the differences of splitting-signals of
and
{
in the same metric. The difference
l,
{m can be referred to as the error of estimation, or
filtration of the signal, and
as the errors of filtration of the splitting-signals pq
r .
12. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
12
Fig. 11. (a) The noisy signal of length 512, (b) the noisy first splitting-signal of length 256, (c) filtered
splitting-signal, and (d) the filtered signal.
It is clear that, the minimization of all errors
may give not the best restoration
{ of the signal.
The splitting-signal can be processed separately, but they are not independent. All together they
represent the signal . They carry the information of the signal, which is transformed to them
through the paired transform. On the other hand, the minimization of the error
l,
{m provides
good, if not the best minimum values for errors
for the splitting-signals. The optimal filter as
the set of coefficients, z, + = 0: ( − 1), after being reordered as z(j7); =
0: (/(2+) − 1), + = 2,
, - = 0: , may be modified or used directly, to define the set of
optimal, or close to optimal filters for splitting-signals. Our preliminary experimental results
show that the set of these filters can be used for filtration of splitting-signals, if we want to save
∆ˆ() = 2 − 4 − 4 operations of multiplication. We now consider the results of such
filtration for the Wiener filter. There is a small difference in the results of the filtration of the
signal ., by the Wiener filter over . and by the Wiener filters over splitting-signals.
3.3. Traditional Wiener filter (grid is shifted)
When denoising of the signal . =
+ is accomplished by the Wiener filter, the frequency
characteristic of this filter is defined as
z =
žŸ(+)
žŸ(+) + ž(+)
=
1
1 + ž/Ÿ(+)
, + = 0: ( − 1).
Here, žŸ(+) = 〈|e|〉 and ž(+) = 〈||〉 denote the power spectra of the original and noise
signals, respectively. The notation ž/Ÿ(+) is used for the noise-to-signal ratio, ž(+)/žŸ(+),
and · is used for the mathematical expectation of a random variable.
The Wiener filter results in a minimum error of signal reconstruction,
l,
{m = min, where
{ is
the inverse Fourier transform of z~. In paired representation, the Wiener filtration of the noisy
signal . is reduced to separate processing of splitting-signals. Each splitting-signal .pq
r,
+ ∈ 1,2,4,8, . . . , /2, 0, is modified by the cosine and sine waves and then the convolution with
the “impulse response” |,$
%
is calculated. Thus, in the frequency-time domain (+, +'), the
optimal filtration is achieved by amplifying the splitting-signal prior and after the convolution,
15. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
15
Fig. 14. (a) The random signal with Gaussian noise, and (b) the signal reconstruction.
Figure 15 shows another signal of length 256 in part a, along with the degraded signal in b, and
signal restored by processing separately the splitting-signals in c. The error of such filtration
equals 0.234 (and 0.239 when using the direct Wiener filtration).
In filtration, the Wiener filter is optimal which means the error
l,
{m is minimum. The set of
optimal filters for splitting-signals may not improve the result in the optimal filter over the
original signal. However such filtration will be effective, because it allows for saving many
operations of multiplication, 2 − 4 − 4.
Fig. 15. (a) The random signal, (b) signal with Gaussian noise, and (c) the signal reconstruction.
5. OPTIMAL FILTRATION BY SPLITTING-SIGNALS: GENERAL MODEL
In this section, analytical equations are described for equivalent filter functions over the splitting-
signals, when processing them in two models of filtration. Filter functions are called similar, or
equivalent if they lead to the same results of signal filtration
{
. We first describe the filtration of
the splitting-signal 6.7,$
%
; ' = 0: (/2 − 1)8 when this signal is modified and then filtered by the
optimal Wiener sub-filter z7, zO, zx, . . . , z0P7. We also consider a such sub-filter with
20. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
20
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Authors
Artyom M. Grigoryan received the MS degrees in mathematics from Yerevan State
University (YSU), Armenia, USSR, in 1978, in imaging science from Moscow Institute of
Physics and Technology, USSR, in 1980, and in electrical engineering from Texas AM
University, USA, in 1999, and Ph.D. degree in mathematics and physics from YUS, in
1990. In December 2000, he joined the Department of Electrical Engineering, University
of Texas at San Antonio, where he is currently an Associate Professor. SM of IEEE since
1998. He is the author of three books, three book-chapters, two patents, and many journal papers and
specializing in the theory and application of fast Fourier transforms, tensor and paired transforms, unitary
heap transforms, image enhancement, computerized tomography, processing biomedical images, and image
cryptography.
Sree P.K. Devieni is a PhD student in the University of Texas at San Antonio in Electrical Engineering
with a concentration in digital signal and image processing, transforms and wavelets.