Digital Signal Processing Lab Manual ECE studentsUR11EC098
This document describes a MATLAB program to perform operations on discrete-time signals. It discusses amplitude manipulation operations like amplification, attenuation, and amplitude reversal. Time manipulation operations covered include time shifting and time reversal. It also describes adding and multiplying two discrete signals. The program takes user input, performs the selected operations, and plots the output waveforms to verify results.
The document discusses performing a discrete wavelet transform (DWT) on a 1D signal using MATLAB. It loads a test signal, performs a 5-level DWT decomposition using the coif3 wavelet, then reconstructs the approximation and detail signals at each level. Plots of the original, approximation, and detail signals are generated.
This document provides an overview of 14 labs covering topics in digital signal processing using MATLAB. The labs progress from basic introductions to MATLAB and signals and systems concepts to more advanced topics like filters, the z-transform, the discrete Fourier transform, image processing, and signal processing toolboxes. Lab 1 focuses on introducing basic MATLAB operations and functions for defining variables, vectors, matrices, and m-files.
Digital signal Processing all matlab code with Lab report Alamgir Hossain
Digital signal processing(DSP) laboratory with matlab software....
Problem List :
1.To write a Matlab program to evaluate the impulse response of the system.
2.Computation of N point DFT of a given sequence and to plot magnitude and phase spectrum.
3.To Generate continuous time sinusoidal signal, discrete time cosine signal.
4.To find the DFT / IDFT of given signal.
5.Program for generation of Sine sequence.
6.Program for generation of Cosine sequence.
7. Program for the generation of UNIT impulse signal
8. Program for the generation of Exponential signal.
This document summarizes key information about wireless communication technologies including Wireless Local Loop (WLL), Wireless Local Area Network (WLAN), and Bluetooth.
WLL uses radio signals to connect subscribers to telephone networks as an alternative to copper wiring, reducing construction and operating costs. WLAN allows wireless connectivity between devices within a limited area like a home or office, providing installation flexibility and reduced costs compared to wired networks. Bluetooth operates in the unlicensed 2.4 GHz band using frequency-hopping spread spectrum, chopping up and transmitting data across multiple bands to enable short-range wireless communication between devices.
Fourier Transform : Its power and Limitations – Short Time Fourier Transform – The Gabor Transform - Discrete Time Fourier Transform and filter banks – Continuous Wavelet Transform – Wavelet Transform Ideal Case – Perfect Reconstruction Filter Banks and wavelets – Recursive multi-resolution decomposition – Haar Wavelet – Daubechies Wavelet.
Convolution and correlation are similar mathematical operations used to extract information from images. Convolution operates on two functions to produce a third, and is equivalent to multiplication in the frequency domain. It can be linear or circular. Linear convolution of signals x(n) and h(n) produces output y(n)=x(n)*h(n). Circular convolution uses the maximum length of the signals. Correlation provides a measure of similarity between two functions and can be auto correlation, comparing a function to its shifted self, or cross correlation, comparing two different functions. Both are used in applications like image processing, signal processing, and more.
Introduction to multiple signal classifier (music)Milkessa Negeri
This document provides an introduction to the MUSIC algorithm, which is used to estimate the frequency content of a signal or autocorrelation matrix using an eigenspace method. It assumes a signal consists of complex exponentials in noise. MUSIC is a high-resolution algorithm that uses the eigenvectors of the autocorrelation matrix to separate the signal and noise subspaces. The document also describes how MUSIC can be used for adaptive beamforming to enhance a desired signal while suppressing interference using an array of sensors. It compares MUSIC to the ESPRIT algorithm for direction of arrival estimation.
Digital Signal Processing Lab Manual ECE studentsUR11EC098
This document describes a MATLAB program to perform operations on discrete-time signals. It discusses amplitude manipulation operations like amplification, attenuation, and amplitude reversal. Time manipulation operations covered include time shifting and time reversal. It also describes adding and multiplying two discrete signals. The program takes user input, performs the selected operations, and plots the output waveforms to verify results.
The document discusses performing a discrete wavelet transform (DWT) on a 1D signal using MATLAB. It loads a test signal, performs a 5-level DWT decomposition using the coif3 wavelet, then reconstructs the approximation and detail signals at each level. Plots of the original, approximation, and detail signals are generated.
This document provides an overview of 14 labs covering topics in digital signal processing using MATLAB. The labs progress from basic introductions to MATLAB and signals and systems concepts to more advanced topics like filters, the z-transform, the discrete Fourier transform, image processing, and signal processing toolboxes. Lab 1 focuses on introducing basic MATLAB operations and functions for defining variables, vectors, matrices, and m-files.
Digital signal Processing all matlab code with Lab report Alamgir Hossain
Digital signal processing(DSP) laboratory with matlab software....
Problem List :
1.To write a Matlab program to evaluate the impulse response of the system.
2.Computation of N point DFT of a given sequence and to plot magnitude and phase spectrum.
3.To Generate continuous time sinusoidal signal, discrete time cosine signal.
4.To find the DFT / IDFT of given signal.
5.Program for generation of Sine sequence.
6.Program for generation of Cosine sequence.
7. Program for the generation of UNIT impulse signal
8. Program for the generation of Exponential signal.
This document summarizes key information about wireless communication technologies including Wireless Local Loop (WLL), Wireless Local Area Network (WLAN), and Bluetooth.
WLL uses radio signals to connect subscribers to telephone networks as an alternative to copper wiring, reducing construction and operating costs. WLAN allows wireless connectivity between devices within a limited area like a home or office, providing installation flexibility and reduced costs compared to wired networks. Bluetooth operates in the unlicensed 2.4 GHz band using frequency-hopping spread spectrum, chopping up and transmitting data across multiple bands to enable short-range wireless communication between devices.
Fourier Transform : Its power and Limitations – Short Time Fourier Transform – The Gabor Transform - Discrete Time Fourier Transform and filter banks – Continuous Wavelet Transform – Wavelet Transform Ideal Case – Perfect Reconstruction Filter Banks and wavelets – Recursive multi-resolution decomposition – Haar Wavelet – Daubechies Wavelet.
Convolution and correlation are similar mathematical operations used to extract information from images. Convolution operates on two functions to produce a third, and is equivalent to multiplication in the frequency domain. It can be linear or circular. Linear convolution of signals x(n) and h(n) produces output y(n)=x(n)*h(n). Circular convolution uses the maximum length of the signals. Correlation provides a measure of similarity between two functions and can be auto correlation, comparing a function to its shifted self, or cross correlation, comparing two different functions. Both are used in applications like image processing, signal processing, and more.
Introduction to multiple signal classifier (music)Milkessa Negeri
This document provides an introduction to the MUSIC algorithm, which is used to estimate the frequency content of a signal or autocorrelation matrix using an eigenspace method. It assumes a signal consists of complex exponentials in noise. MUSIC is a high-resolution algorithm that uses the eigenvectors of the autocorrelation matrix to separate the signal and noise subspaces. The document also describes how MUSIC can be used for adaptive beamforming to enhance a desired signal while suppressing interference using an array of sensors. It compares MUSIC to the ESPRIT algorithm for direction of arrival estimation.
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.
A seminar on INTRODUCTION TO MULTI-RESOLUTION AND WAVELET TRANSFORMमनीष राठौर
This document provides an introduction to multi-resolution analysis and wavelet transforms. It discusses that multi-resolution analysis analyzes signals at varying levels of detail or resolutions simultaneously. The Fourier transform has limitations for non-stationary signals as it does not provide time information. The short-term Fourier transform was developed to analyze non-stationary signals, but it has limitations in time-frequency resolution. Wavelet transforms were developed to analyze signals using variable time-frequency resolutions. Wavelet transforms have features like varying time-frequency resolutions and are suitable for analyzing non-stationary signals. They have applications in fields like signal compression, noise removal, and image processing.
This document defines key concepts in signal processing including signals, systems, and digital signal processing. It provides examples of signals that vary with time or other variables and carry information. Characteristics of signals like amplitude, frequency, and phase are described. Systems are defined as physical devices that operate on signals, with examples of filters. Signal processing involves passing signals through systems to perform operations like filtering. A block diagram shows the basic components of a digital signal processing system including analog to digital conversion, processing, and digital to analog conversion. Finally, advantages of digital over analog signal processing are listed such as programmability, accuracy, storage, and lower cost.
Using Mean Filter And Show How The Window Size Of The Filter Affects Filtering
The document discusses mean filtering and how the window size affects filtering. It defines mean filtering as replacing the center value in a window with the average of all values. A larger window size results in more smoothing as the average is taken over more points. The document provides examples of mean filtering a 3x3 window and pseudocode for a mean filter with a window size of 5. It also discusses edge effects, functions, sampling, filtering, noise addition, and signal observations at different points in the process.
The document discusses the Discrete Fourier Transform (DFT). It begins by explaining the limitations of the Discrete Time Fourier Transform (DTFT) and Discrete Fourier Series (DFS) from a numerical computation perspective. It then introduces the DFT as a numerically computable transform obtained by sampling the DTFT in the frequency domain. The DFT represents a periodic discrete-time signal using a sum of complex exponentials. It defines the DFT and inverse DFT equations. The document also discusses properties of the DFT such as linearity and time/frequency shifting. Finally, it notes that the Fast Fourier Transform (FFT) implements the DFT more efficiently by constraining the number of points to powers of two.
DSP_FOEHU - Lec 05 - Frequency-Domain Representation of Discrete Time SignalsAmr E. Mohamed
The document describes the process of analyzing a signal using the Fourier transform and synthesizing it back using the inverse Fourier transform. It involves taking the Fourier transform of the original signal to analyze it in the frequency domain, applying various operations, and then taking the inverse Fourier transform to synthesize the signal back in the time domain. Key steps include the Fourier transform, inverse Fourier transform, applying filters, and ensuring convergence in the mean square sense during synthesis.
The document discusses implementing convolution on an FPGA. It begins by introducing convolution and its applications in image processing. It then discusses the scope and technical approach of implementing discrete linear convolution on FPGA kits in order to perform convolution on images in real-time. The document outlines the structure of FPGAs, including configurable logic blocks and wiring tracks. It also discusses software requirements and provides an organization plan for subsequent chapters on linear convolution, FPGA technology, and a literature survey.
Александр Заричковый "Faster than real-time face detection"Fwdays
I will talk about object and face detection problems, evolution of different approaches to solving these problems and about the ideas behind each of these approaches. Also I will describe meta-architecture that achieve state of the art results on faces detection problem and works faster than real-time.
The document discusses two algorithms for object detection: HOG and SIFT.
HOG (Histogram of Oriented Gradients) focuses on the shape of an object by using the magnitude and direction of gradients to generate histograms and compute features. SIFT (Scale Invariant Feature Transform) describes local image areas by extracting invariant features to generate a set of key points for matching objects across different scales and rotations. Both algorithms can be used to detect objects by matching image features to trained models.
The document describes the design and implementation of a log periodic antenna by a group of students. It provides details on:
1) The history and purpose of log periodic antennas in being able to operate over a wide bandwidth of frequencies.
2) The design process for the antenna, including calculations of its impedance, minimum/maximum frequencies, boom length, gain, number of elements, and values for each element.
3) The development of a MATLAB program to calculate antenna parameters and structure dimensions to more accurately design the log periodic dipole antenna.
4) Potential applications of this type of wide bandwidth antenna in areas like TV, HF communications, and EMC measurements.
The document compares wavelet transforms and Fourier transforms. Wavelet transforms provide time-frequency localization while Fourier transforms only provide frequency localization. Wavelet transforms use small wave functions that are scaled and translated, allowing time-frequency localization. They also provide multiresolution analysis which is useful for applications like image processing. Wavelet transforms represent piecewise smooth signals like images and speech better than Fourier transforms as they require fewer coefficients around discontinuities. The document also discusses wavelet filter banks, discrete wavelet transforms, multiresolution analysis, and applications of wavelet transforms like image denoising.
The document discusses several algorithms used for medical image compression. It describes the JPEG algorithm which uses discrete cosine transformation and is widely used. It also covers region of interest compression which focuses on important areas. Finally, it examines embedded zerotree wavelet compression and proposes a new unit embedded zerotree wavelet algorithm to improve on existing methods by reducing the number of nodes checked in the wavelet tree structure.
satellite communication jntuh
Satellite Link Design: Basic Transmission Theory, System Noise Temperature, and G/T Ratio,
Design of Down Links, Up Link Design, Design Of Satellite Links For Specified C/N, System Design
Examples.
Multiple Access: Frequency Division Multiple Access (FDMA), Inter modulation, Calculation of C/N,
Time Division Multiple Access (TDMA), Frame Structure, Examples, Satellite Switched TDMA
Onboard Processing, DAMA, Code Division Multiple Access (CDMA), Spread Spectrum Transmission
and Reception.
presentation on digital signal processingsandhya jois
The document discusses digital signal processing (DSP). It defines key terms like digital, signal, and processing. It explains how analog signals are converted to digital form by sampling and quantization. It also describes common digital modulation schemes and compares DSP processors to microprocessors. Finally, it discusses digital filters and their types as well as applications of DSP in areas like audio processing, communications, and imaging.
1) Convolution represents a discrete-time (DT) or continuous-time (CT) linear time-invariant (LTI) system as the summation or integral of the input signal multiplied by the impulse response.
2) The impulse response completely characterizes an LTI system.
3) Convolution exploits the properties of time-invariance and linearity of LTI systems to represent the output of the system in terms of a convolution between the input and impulse response.
The document describes the Histogram of Oriented Gradients (HOG) feature descriptor technique. HOG counts occurrences of gradient orientation in localized portions of an image to represent a distribution of intensity fluctuations along different orientations. It works by first calculating gradient images, then calculating histograms of gradients in 8x8 cells, followed by block normalization to account for lighting variations before forming the final HOG feature vector.
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.
Wavelet Applications in Image Denoising Using MATLABajayhakkumar
The document discusses digital image processing and noise reduction techniques. It covers the following key points:
- Digital image processing uses computer algorithms to process digital images, with advantages over analog processing like a wider range of algorithms.
- Noise reduction is important as images can be contaminated by noise during acquisition, storage, or transmission, degrading quality. Common noise types include Gaussian, salt and pepper, and speckle noise.
- Filtering techniques for noise reduction include spatial filters, frequency domain filters, and wavelet domain techniques like thresholding, with the goal of removing noise while preserving useful image information.
The document discusses denoising signals using wavelet transforms. It begins with an overview of denoising and its goal of reconstructing a signal from a noisy one. It then compares denoising using wavelets to other methods like Fourier filtering and spline methods. The key advantages of wavelets are their ability to localize properties and concentrate a signal's energy. The document outlines the basic denoising process using wavelet transforms which involves decomposition, thresholding, and reconstruction. It also discusses different thresholding methods and commonly used thresholds like VisuShrink.
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.
A seminar on INTRODUCTION TO MULTI-RESOLUTION AND WAVELET TRANSFORMमनीष राठौर
This document provides an introduction to multi-resolution analysis and wavelet transforms. It discusses that multi-resolution analysis analyzes signals at varying levels of detail or resolutions simultaneously. The Fourier transform has limitations for non-stationary signals as it does not provide time information. The short-term Fourier transform was developed to analyze non-stationary signals, but it has limitations in time-frequency resolution. Wavelet transforms were developed to analyze signals using variable time-frequency resolutions. Wavelet transforms have features like varying time-frequency resolutions and are suitable for analyzing non-stationary signals. They have applications in fields like signal compression, noise removal, and image processing.
This document defines key concepts in signal processing including signals, systems, and digital signal processing. It provides examples of signals that vary with time or other variables and carry information. Characteristics of signals like amplitude, frequency, and phase are described. Systems are defined as physical devices that operate on signals, with examples of filters. Signal processing involves passing signals through systems to perform operations like filtering. A block diagram shows the basic components of a digital signal processing system including analog to digital conversion, processing, and digital to analog conversion. Finally, advantages of digital over analog signal processing are listed such as programmability, accuracy, storage, and lower cost.
Using Mean Filter And Show How The Window Size Of The Filter Affects Filtering
The document discusses mean filtering and how the window size affects filtering. It defines mean filtering as replacing the center value in a window with the average of all values. A larger window size results in more smoothing as the average is taken over more points. The document provides examples of mean filtering a 3x3 window and pseudocode for a mean filter with a window size of 5. It also discusses edge effects, functions, sampling, filtering, noise addition, and signal observations at different points in the process.
The document discusses the Discrete Fourier Transform (DFT). It begins by explaining the limitations of the Discrete Time Fourier Transform (DTFT) and Discrete Fourier Series (DFS) from a numerical computation perspective. It then introduces the DFT as a numerically computable transform obtained by sampling the DTFT in the frequency domain. The DFT represents a periodic discrete-time signal using a sum of complex exponentials. It defines the DFT and inverse DFT equations. The document also discusses properties of the DFT such as linearity and time/frequency shifting. Finally, it notes that the Fast Fourier Transform (FFT) implements the DFT more efficiently by constraining the number of points to powers of two.
DSP_FOEHU - Lec 05 - Frequency-Domain Representation of Discrete Time SignalsAmr E. Mohamed
The document describes the process of analyzing a signal using the Fourier transform and synthesizing it back using the inverse Fourier transform. It involves taking the Fourier transform of the original signal to analyze it in the frequency domain, applying various operations, and then taking the inverse Fourier transform to synthesize the signal back in the time domain. Key steps include the Fourier transform, inverse Fourier transform, applying filters, and ensuring convergence in the mean square sense during synthesis.
The document discusses implementing convolution on an FPGA. It begins by introducing convolution and its applications in image processing. It then discusses the scope and technical approach of implementing discrete linear convolution on FPGA kits in order to perform convolution on images in real-time. The document outlines the structure of FPGAs, including configurable logic blocks and wiring tracks. It also discusses software requirements and provides an organization plan for subsequent chapters on linear convolution, FPGA technology, and a literature survey.
Александр Заричковый "Faster than real-time face detection"Fwdays
I will talk about object and face detection problems, evolution of different approaches to solving these problems and about the ideas behind each of these approaches. Also I will describe meta-architecture that achieve state of the art results on faces detection problem and works faster than real-time.
The document discusses two algorithms for object detection: HOG and SIFT.
HOG (Histogram of Oriented Gradients) focuses on the shape of an object by using the magnitude and direction of gradients to generate histograms and compute features. SIFT (Scale Invariant Feature Transform) describes local image areas by extracting invariant features to generate a set of key points for matching objects across different scales and rotations. Both algorithms can be used to detect objects by matching image features to trained models.
The document describes the design and implementation of a log periodic antenna by a group of students. It provides details on:
1) The history and purpose of log periodic antennas in being able to operate over a wide bandwidth of frequencies.
2) The design process for the antenna, including calculations of its impedance, minimum/maximum frequencies, boom length, gain, number of elements, and values for each element.
3) The development of a MATLAB program to calculate antenna parameters and structure dimensions to more accurately design the log periodic dipole antenna.
4) Potential applications of this type of wide bandwidth antenna in areas like TV, HF communications, and EMC measurements.
The document compares wavelet transforms and Fourier transforms. Wavelet transforms provide time-frequency localization while Fourier transforms only provide frequency localization. Wavelet transforms use small wave functions that are scaled and translated, allowing time-frequency localization. They also provide multiresolution analysis which is useful for applications like image processing. Wavelet transforms represent piecewise smooth signals like images and speech better than Fourier transforms as they require fewer coefficients around discontinuities. The document also discusses wavelet filter banks, discrete wavelet transforms, multiresolution analysis, and applications of wavelet transforms like image denoising.
The document discusses several algorithms used for medical image compression. It describes the JPEG algorithm which uses discrete cosine transformation and is widely used. It also covers region of interest compression which focuses on important areas. Finally, it examines embedded zerotree wavelet compression and proposes a new unit embedded zerotree wavelet algorithm to improve on existing methods by reducing the number of nodes checked in the wavelet tree structure.
satellite communication jntuh
Satellite Link Design: Basic Transmission Theory, System Noise Temperature, and G/T Ratio,
Design of Down Links, Up Link Design, Design Of Satellite Links For Specified C/N, System Design
Examples.
Multiple Access: Frequency Division Multiple Access (FDMA), Inter modulation, Calculation of C/N,
Time Division Multiple Access (TDMA), Frame Structure, Examples, Satellite Switched TDMA
Onboard Processing, DAMA, Code Division Multiple Access (CDMA), Spread Spectrum Transmission
and Reception.
presentation on digital signal processingsandhya jois
The document discusses digital signal processing (DSP). It defines key terms like digital, signal, and processing. It explains how analog signals are converted to digital form by sampling and quantization. It also describes common digital modulation schemes and compares DSP processors to microprocessors. Finally, it discusses digital filters and their types as well as applications of DSP in areas like audio processing, communications, and imaging.
1) Convolution represents a discrete-time (DT) or continuous-time (CT) linear time-invariant (LTI) system as the summation or integral of the input signal multiplied by the impulse response.
2) The impulse response completely characterizes an LTI system.
3) Convolution exploits the properties of time-invariance and linearity of LTI systems to represent the output of the system in terms of a convolution between the input and impulse response.
The document describes the Histogram of Oriented Gradients (HOG) feature descriptor technique. HOG counts occurrences of gradient orientation in localized portions of an image to represent a distribution of intensity fluctuations along different orientations. It works by first calculating gradient images, then calculating histograms of gradients in 8x8 cells, followed by block normalization to account for lighting variations before forming the final HOG feature vector.
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.
Wavelet Applications in Image Denoising Using MATLABajayhakkumar
The document discusses digital image processing and noise reduction techniques. It covers the following key points:
- Digital image processing uses computer algorithms to process digital images, with advantages over analog processing like a wider range of algorithms.
- Noise reduction is important as images can be contaminated by noise during acquisition, storage, or transmission, degrading quality. Common noise types include Gaussian, salt and pepper, and speckle noise.
- Filtering techniques for noise reduction include spatial filters, frequency domain filters, and wavelet domain techniques like thresholding, with the goal of removing noise while preserving useful image information.
The document discusses denoising signals using wavelet transforms. It begins with an overview of denoising and its goal of reconstructing a signal from a noisy one. It then compares denoising using wavelets to other methods like Fourier filtering and spline methods. The key advantages of wavelets are their ability to localize properties and concentrate a signal's energy. The document outlines the basic denoising process using wavelet transforms which involves decomposition, thresholding, and reconstruction. It also discusses different thresholding methods and commonly used thresholds like VisuShrink.
Images may contain different types of noises. Removing noise from image is often the first step in image processing, and remains a challenging problem in spite of sophistication of recent research. This ppt presents an efficient image denoising scheme and their reconstruction based on Discrete Wavelet Transform (DWT) and Inverse Discrete Wavelet Transform (IDWT).
A Review on Image Denoising using Wavelet Transformijsrd.com
This document discusses image denoising using wavelet transforms. It begins with an introduction to wavelet transforms and their advantages over Fourier transforms for denoising non-stationary signals like images. It then describes the basic steps of image denoising using wavelets: decomposing the noisy image into wavelet coefficients, modifying the coefficients using thresholding, and reconstructing the denoised image. Thresholding techniques like hard and soft thresholding are explained. The document concludes that wavelet-based denoising is computationally efficient and can effectively remove noise from images.
1) The Bloody Mary cocktail originated in the 1920s at Harry's New York Bar in Paris. The bartender there experimented with vodka, which had recently been introduced by Russian émigrés, and mixed it with tomato juice to create the signature drink.
2) Champagne has evolved from a still, pale pink wine made from Pinot Noir grapes to the sparkling wine now associated with the region. While early producers saw bubbles as a fault and tried to eliminate them, the British developed a taste for the sparkling style in the 18th century.
3) The Romans first planted vineyards in Champagne and the region has been cultivated for wine since at least the 5th century. Hugh Cap
El documento habla sobre tres herramientas de búsqueda de información en internet: el metabuscador Kartoo, el motor de búsqueda Bing y el directorio temático InfoMine. Explica brevemente cómo funciona cada una y cómo presentan los resultados de una búsqueda.
Citizens Bank Park in Philadelphia is offering a new crabcake sandwich that is made without crab meat, using a wheat protein and soy substitute instead. The crab-free crabcake sandwich costs $9.25 and is available at the Planet Hoagie stand, along with other meat-free sandwiches and wrap options, as the stadium works to maintain its ranking as the most vegetarian-friendly baseball facility according to PETA. While the veggie versions may not outsell regular cheesesteaks and hoagies, the 100 orders per game make continuing to offer meat-free options worthwhile.
1) India has a population of over 1.1 billion people with high birth and fertility rates and growing at around 1.5% annually. Major ethnic groups include Indo-Aryans, Dravidians, and various religious communities.
2) A pilot study in rural Indian schools found high rates of parasites like ascariasis along with anemia and malnutrition among students. While the government distributed deworming medications, many traditional remedies were still preferred.
3) The conclusion calls for educating communities about disease complications to increase confidence in modern medicine alongside traditional methods and government initiatives.
This document discusses Canadian contributions to the journal Ecological Economics over the past 25 years. It begins with a brief history of ecological economics and the founding of the Canadian Society of Ecological Economics (CANSEE) in 1993. The document then analyzes 198 articles in Ecological Economics with at least one Canadian author, representing approximately 7 contributions per year or 3.82% of total articles. It categorizes the articles' topics and keywords to understand themes in Canadian ecological economic research. Provincial and institutional contributions are also examined. The analysis is limited by incomplete 2015 data and articles published outside Ecological Economics.
Wavelet packets provide an adaptive decomposition that overcomes limitations of the discrete wavelet transform (DWT). In wavelet packets, signal decomposition using high-pass and low-pass filters is applied recursively to both low-pass and high-pass outputs, allowing more flexible time-frequency analysis. This results in a redundant dictionary with increased flexibility but also higher computational costs. Pruning algorithms are used to select an optimal subset of bases for a given application based on cost functions related to properties like sparsity, entropy, or energy concentration.
El buen servidor público - Jesus Neira QuinteroManuel Bedoya D
Este documento es un libro titulado "El buen servidor público" en su tercera edición, escrito por Jesús Neira Quintero. El libro contiene 10 capítulos que cubren temas como las actitudes del servidor público, la cultura y el clima organizacional, la ética del servidor, cómo ser un buen gerente público, la responsabilidad de los servidores públicos y elementos básicos de derecho público. El objetivo del libro es proporcionar información y herramientas para que los servidores públicos mejoren
Ultra Low Power Wireless Sensors for E-HealthCare -Interactive RFIDajayhakkumar
This document discusses the design of ultra low power wireless sensors for e-healthcare using interactive RFID technology. It describes the requirements for wireless healthcare devices to be high performance, miniature, lightweight, flexible, reliable, secure, power efficient and intelligent but also low cost. It discusses using passive, active and semi-passive RFID systems with capacitive or inductive coupling in lower frequencies or radiative coupling in higher frequencies. The document outlines the challenges in designing such interactive RFID sensor systems and proposes solutions like using flexible electronic paper displays, UWB technology and flexible sensors. It provides examples of applications like using printed ECG electrodes and electronic paper displays on RFID tags for real-time wireless ECG monitoring.
This document proposes a CAD system for analyzing mammogram images to detect breast cancer. It discusses:
1) The need for mammogram analysis and microcalcification/mass detection due to the prevalence of breast cancer.
2) The CAD system which uses preprocessing, feature extraction including GLSDM, Gabor and wavelet transforms, and classifiers like ELM, SVM and Bayes for microcalcification detection, achieving up to 98% accuracy.
3) A comprehensive GUI tool is developed for abnormality detection, cancer characterization, risk analysis and tissue density classification achieving up to 100% accuracy for some classifiers and features. The tool also performs speckle noise denoising.
WAVELET THRESHOLDING APPROACH FOR IMAGE DENOISINGIJNSA Journal
This document presents a wavelet thresholding approach for image denoising. It proposes using a Bayesian technique to determine an adaptive threshold based on modeling wavelet coefficients with a generalized Gaussian distribution. The proposed threshold performs better than traditional methods like Donoho and Johnston's SureShrink. Experimental results on the Lena image show the proposed method significantly outperforms hard and soft thresholding in terms of peak signal-to-noise ratio, especially at higher noise levels. It concludes the adaptive thresholding approach effectively removes noise from images.
The document discusses several topics in digital signal processing including polyphase decomposition, discrete cosine transform (DCT), Gibbs phenomenon, and oversampled analog-to-digital converters (ADCs). Polyphase decomposition allows for more efficient implementation of decimation and interpolation filters. DCT is used for image compression and represents data in the frequency domain using cosine waves. Gibbs phenomenon causes ripples near discontinuities that cannot be fully removed. Oversampling ADCs sample at a higher rate than Nyquist to reduce noise and simplify anti-aliasing filters.
Image Denoising Using Wavelet TransformIJERA Editor
In this project, we have studied the importance of wavelet theory in image denoising over other traditional methods. We studied the importance of thresholding in wavelet theory and the two basic thresholding method i.e. hard and soft thresholding experimentally. We also studied why soft thresholding is preferred over hard thresholding, three types of soft thresholding (Bayes shrink, Sure shrink, Visu shrink) as well as advantages and disadvantage of each of them
For ease of analog or digital information transmission and reception, modulation is the foremost important technique. In the present project, we’ll discuss about different modulation scheme in digital mode done by operating a switch/ key by the digital data. As we know, by modifying basic three parameters of the carrier signal, three basic modulation schemes can be obtained; generation and detection of these three modulations are discussed and compared with respect to probability of error or bit error rate (BER).
1) The document discusses audio compression using Daubechie wavelets. It involves using optimal wavelet selection and quantizing wavelet coefficients along with A-law and U-law companding methods.
2) The key steps of wavelet-based audio compression are thresholding and quantizing wavelet coefficients, then encoding the data to remove redundancy and reduce the number of coefficients.
3) A psychoacoustic model is incorporated to determine inaudible quantization noise levels based on auditory masking principles. The masking thresholds are converted to constraints in the wavelet domain to guide coefficient quantization and selection of an optimal wavelet basis.
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Thresholding eqns for wavelet
1. Denoising by Wavelets
What is Denoising
Denoising refers to manipulation of
wavelet coefficients for noise reduction.
Coefficient values not exceeding a
carefully selected threshold level are
replaced by zero followed by an inverse
transform of modified coefficients to
recover denoised signal.
Denoising by thresholding of wavelet
coefficients is therefore a nonlinear (local)
operation
2. Noise Reduction by Wavelets and in
Fourier Domains
Comments
Denoising is a unique feature of signal
decomposition by wavelets
It is different from noise reduction as used in
spectral manipulation and filtering.
Denoising utilizes manipulation of wavelet
coefficients that are reflective of time/space
behavior of a given signal.
Denoising is an important step in signal
processing. It is used as one of the steps in
lossy data compression and numerous noise
reduction schemes in wavelet analysis.
3. Denoising by Wavelets
For denoising we use thresholding approach applied
on wavelet coefficients.
This is to be done by a judiciously chosen
thresholding levels.
Ideally each coefficients may need a unique
threshold level attributed to its noise content
In the absence of information about true signal, this
is not only feasible but not necessary since
coefficients are somewhat correlated both at inter
and intra decomposition levels ( secondary features
of wavelet transform).
4. True Signal Recovery
Thresholding modifies empirical
coefficients (coefficients belonging to the
measured noisy signal) in an attempt to
reconstruct a replica of the true signal.
Reconstruction of the signal is aimed to
achieve a ‘best’ estimate of the true
(noise-free) signal.
‘Best estimate’ is defined in accordance
with a particular criteria chosen for
threshold selection.
5. Thresholding
Mathematically, thresholding of the
coefficients can be described by a
transformation of the wavelet coefficients
Transform matrix is a diagonal matrix
with elements 0 or 1. Zero elements
forces the corresponding coefficient below
or equal to a given threshold to be set to
zero while others corresponding to one,
retains coefficients as unchanged.
∆=diag(δ1, δ2,….. δN) with δi={0,1}, i=1,…N.
6. Hard or Soft Thresholding.
Hard Thresholding. Only wavelet coefficients with
absolute values below or at the threshold level
are affected, they are replaced by zero and
others are kept unchanged.
Soft Thresholding. Coefficiens above threshold
level are also modified where they are reduced
by the threshold size.
Donoho refers to soft threshoding as ‘shrinkage’
since it can be proven that reduction in coef-
ficient amplitudes by soft thresholding, also
results in a reduction of the signal level thus a ‘
shrinkage’.
7. Hard and Soft Thresholding
Mathematically hard and soft
thresholding is described as
Hard threshold:
wm= w if |w|≥th,
wm= 0 if |w|<th
Soft threshold :
wm = sign(w)(|w|-th), |w|≥th,
wm=0 , |w|<th
8. Global and Local Thresholding
Thresholding can be done globally
or locally i.e.
single threshold level is applied across
all scales,
or it can be scale-dependent where
each scale is treated separately.
It can also be ‘zonal’ in which the given
function is divided into several
segments (zones) and at each segment
different threshold level is applied.
9. Additive Noise Model and
Nonparametric Estimation Problem
Additive Noise Model. Additive noise model is
superimposed on the data as follows.
f(t) = s(t) + n(t)
n(t) is a random variable assumed to be white
Gaussian N(0, σ). S(t) is a signal not
necessarily a R.V.
Original signal can be described by the given basis
function in which coefficients of expansion are
unknown se(t)=∑αi φi (t)
Se(t) is the estimate of the true signal s(t). Note the
estimate s^(t) is described by set of spanning
function φi(t), chosen to minimize the L2 error
function of the approximation
||s(t)-se (t)||2 .
As such denoising is considered as a non-
parametric estimation problem.
10. Properties of Wavelets Utilized in
Denoising
Sparse Representation.
Wavelet expansion of class of
functions that exhibit sufficient degree
of regularity and smoothness, results
in pattern of coefficient behavior that
can often be categorized into two
classes:
1) a few large amplitude coefficients and
2) large number of small coefficients. This
property allows compaction for compression
and efficient feature extraction.
11. Wavelet Properties and Denoising
Decorrelation. Wavelets are referred to as
decorrelators in reference to a property in which
wavelet expansion of a given signal results in
coefficients that often exhibit a lower degree of
correlation among the coefficients as
compared with that of the signal components.
Orthogonality. Intuitively, under a given
standard DWT of a signal, this can be explained by
the orthogonality of expansion and corresponding
bases functions.
12. i.i.d. assumption
Under certain assumptions,
coefficient in highest frequency
band, can be considered to be
statistically identically
independent of each other
13. Examples of Signal Compaction and
Decorrelation at Coefficient Domain
0 10 20 30
-10
0
10
Cycle#8
0 100 200 300
-5
0
5
MildKnock
0 100 200 300
-1
0
1
NoKnock
Signal at High Freq.Band #36
0 100 200 300
-5
0
5
HardKnock
0 10 20 30
-10
0
10
20
Cycle#1
0 10 20 30
-5
0
5
Cycle#6
Coeffs
15. Why Decorrelation by Wavelets
Coefficients carry signal information in
subspaces that are spanned by basis
functions of the given subspace.
Such bases can be orthogonal
( uncorrelated) to each other, therefore
coefficients tend to be uncorrelated to
each other
Segmentation of signal by wavelets
introduce decorrealtion at coefficient
domain
16. White Noise and Wavelet Expansion
No wavelet function can model noise
components of a given signa ( no match in
waveform for white noise).
White noise have spectral distribution which
spreads across all frequencies. There is no
match ( correlation) between a given wavelet
and white noise
As such an expansion of noise component of
the signal, results in small wavelet coefficients
that are distributed across all the details.
17. Search fro Noise in Small Coeffs
S(t) = x(t) +n(t)
An expansion of white noise
component of the signal, results
in small wavelet coefficients that
are distributed across all the
details.
We search for n(t) of white noise at
small coefficients in DWT that often
residing in details
18. White Noise and High Frequency
Details
At high frequency band d1, the number
of coefficients is largest under DWT or
other similar decomposition
architectures.
As such, a large portion of energy of
the noise components of a signal,
resides on the coefficients of high
frequency details d1
At high frequency band d1, are short
length basis functions and there is high
decorrelation at this level( white noise)
19. White Noise Model and Statistically i.i.d. Coeffs
Decorrelation property of the wavelet
transform at the coefficient level, can be
examined in terms of statistical property of
wavelet coefficients.
At one extreme end, coefficients may be
approximated as a realization of a stochastic
process characterized as a purely random
process with i.i.d (identically independently
distributed) random variable.
Under this assumption, every coefficient is
considered statistically independent of
the neighboring coefficients both at the
inter-scale (same sale) and intra-scale
(across the scales) level.
20. White Noise Model and Multiblock denoising
However, in practice, often there exist
certain degree of interdependence among
the coefficients, and we need to consider
correlated coefficients for noise
models( such as Markov models).
In other models used to estimate noise,
blocks of coefficients instead of single
coefficients, are used as statistically
independent
In Matlab, multi-block denoising at each
level is considered
21. Main Task
Main task in denoising by
wavelets:
Identification of underlying statistical
distribution of the coefficients attributed
to noise.
For signal, no structural assumption is
made since in general it is assumed to be
unknown. However, if we have additional
information on the signal, we can use
them and improve our estimation results
22. Main Task
Denoising problem is treated as an
statistical estimation problem.
The task is followed by the evaluation
of variance and STD of statistics of the
noise model that are used as metrics
for thresholding.
A’priori distributions may be imposed
on the coefficients using Bay’s
estimation approach after which
denoising is treated as a parametric
estimation problem
23. Alternative Models for Noise Reduction.
Basic Considerations
Additive Noise Model. Basic modeling
structure utilizes additive noise model as
stated earlier.
x(i)=s(i)+ σn(i) , i=1,2, … N
N is signal length, x(i) are the measurements,
s(i) is the true signal value(unknown) and n(i)
are the noise components(unknown)
n(i) is assumed to be white Gaussian with zero
mean N(0,1). Standard deviation is to be
estimated
24. Additive Noise Model and Linearity of
Wavelet Transform
Under an orthogonal decomposition and
additive noise model, linearity of wavelet
transform insures that the statistical
distribution of the measurements and
white noise remain unchanged in the
coefficient domain.
Under an orthogonal decomposition, each
coefficient is decomposed into component
attributed to the true signal values s(n)
and to signal noise component n(k) as
follows.
cj= uj+dj i=1,2, .. n
25. Orthogonal vs Biorthogonal
In vector form
C=U+D
C, U and D are vector representation of empirical
wavelet coefficients, true coefficient
values( unknown) and noise content of the
coefficients respectively.
Note ‘additive noise model’ at the coefficient level
while preserving statistical property of the signal
and noise at the coefficient as stated above, is valid
under orthogonal transformation where Parseval
relationship holds.
It is not valid under biorthogonal transform. Under
biorthogonal transform, white noise at the signal
level will exhibit itself as colored noise since the
coefficients here are no longer i.i.d but they are
correlated to each other.
26. Principle Considerations
1. Assumption of Zero Mean Gaussian.
Under additive noise model and
assumption of i.i.d. for the wavelet
coefficients, we consider zero mean
Gaussian distribution at the coefficient
domain.
Mean centering of data can always be
done to insure zero mean Gaussian
assumption as used above.
27. Main Considerations
Preservation of Smoothness.
It can be proved that under soft
thresholding, smoothness property of the
original signal remains unchanged with high
probability under variety of smoothness
measures ( such as Holden or Sobolov
smoothness measures).
Smoothness may be defined in terms of
integral of squared mth derivative of a
given function to be of finite value
This property and structural correlation of
wavelet coefficients at consecutive scales,
are used in wavelet-based zero-tree
compression algorithm
28. Main Considerations
Shrinkage. Under soft thres-
holding ( nonlinear operation at the
coefficient level), it can be shown
that
| xid |≤|xi| where xid is
denoised signal component i.e.
denoising results in reduction of all
the coefficients and shrinkage at
the signal level as well.
29. Denoising Problem
Denoising problem is mainly
estimation of STD and Threshold
Level
Basic problem in noise reduction
under Gaussian white noise, is
centered around the estimation of
standard deviation of the Gaussian
noise σ
It is then used to determine a
suitable threshold
30. Alternative Considerations.
White Noise Model-Global (Universal)
Thresholding.
Assume coefficients at the highest
frequency details gives a good estimate
of the noise content .
A white noise model is superimposed on
the coefficients at the highest frequency
detail level d1
An estimate of the standard deviation at
the d1 level is then used to arrive at a
suitable threshold level for coefficient
thresholding at all levels.
This approach is a global thresholding
which is applied to all detail coefficients
31. Level Dependent Thresholds
Nonwhite (Colored) Noise Model.
Under this model, still white noise model
is imposed on the coefficients of details,
however threshold levels are considered
to be level(scale) dependent.
Gaussian white noise model is imposed on
detail coefficients using standard
deviation and threshold level at each level
separately.
32. Comments on Estimation Problem, Near
Optimality under other Optimality Criteria
Wavelet denoising (WaveShrink) utilizes
a nonparametric function estimation
approach for noise thresholding.
It has been shown that statistically,
denoising is considered to be:
asymptotically near optimal over a wide
range of optimality criteria and for large
class of functions found in scientific and
engineering applications( see ref by
Donoho).
33. Inaccuracy of Assuming Gaussian
Distribution N(1,0), Result Evaluations
Assumption of Gaussian distribution at d1
level may not always be valid
Distribution of the coefficient at d1 often
exhibit a long tail as compared with
standard Gaussian(peaky distribution) This
can also be observed in the case of sparsely
distributed large amplitude coefficients or
outliers.
Under such condition, application of global
thresholding may be revised and results of
the thresholding be examined in light of
actual data analysis and performance of
denoising.
34. Inaccuracy of Assuming Gaussian Distribution
Fig.2 Peaky Gaussian-like pdf of the
coefficients with long tail ends
35. Signal Estimation and Threshold
Selection Rules
Use statistical estimation theory applied
on probability distribution of the wavelet
coefficients
Use criteria for estimation of statistical
parameters and selecting threshold levels
A loss function which is referred to as
‘risk function’ is defined first.
For Loss function we often use L2
norm of the error i.e. variance of
estimation error, i.e. difference between
the estimated value and actual unknown
value
36. Risk (Loss Function)
We use expected value of the error
as loss function since we are dealing
with noisy signal which is a random
variable and is therefore described
in term of expected value.
Minimization of risk function results
in an estimate of the variance of the
coefficient.
2
||)'(||)',( XXEXXR −=
37. Risk (Loss) Function
X is the actual (true) value of the signal to be
estimated ( or coefficients) and X’ is an
estimate of the signal X ( or coefficients ).
Since noise component is assumed to be zero
mean Gaussian, the difference is a measure
of an error based on the additive noise model
and given risk function.
It is a measure of the energy of the noise
i.e.∑[n(k)]2
Thus optimization procedure as defined above,
attempts to reduce the energy of the signal X by
an amount equal to the energy of the noise and
thus compensating for the noise in the sense of
L2 norms.
38. Minimization of the risk function at
coefficient level
Under an orthogonal decomposition, minimization of
risk function at the signal level, can equivalently be
defined at the coefficient level.
R(X^
,X)= E||X^
- X||2
=E||W-1
(C^
-C)||2
=E||(C^
-C)||2
C^ is the estimate of the true coefficient values. We
have used additive noise model and wavelet transform
in matrix form C=WX as described below.
X=S + σn,
C=WX, X=W-1
C
Accordingly, minimization of the risk function at the
coefficient level results equivalently in estimating the
true value of the signal.
39. Use of Minimax Rule
One ‘best’ estimate is obtained
using minimax rule indicated below:
Minmax R(X^,X)= inf sup R(X^,X)
Under minmax rule, worst case
condition is considered, i.e.
Sup R(X^,X)
Here our objective is to mimimize
the risk under worst case condition
(i.e. obtain Min Max R) .
40. Global/Universal Thresholding Rule
Under the assumption of i.i.d. for the wavelet coefficients
and Gaussian white noise, one can show that
Under soft thresholding, the actual risk is within
log(n) factor of the ideal risk where the error is
minimal (on the average).
This results in the following threshold value referred to as
Universal Thresholding which minimizes max risk as
defined above.
Th=σ√(2 log n), σ=MAD/.6745
MAD is ‘median absolute deviation’ of the coefficients
median({|d J−1,k |: k = 0, 1, . . . , 2^(J−1) −1})
Ref: Donoho D.L. ”Denoising by Soft thresholding”, IEEE
Trans on Information Theory, Vol 41,No.3 May 1995,pp
613-627
41. Universal Thresholding Rule
Underlying basis for above threshold rule
is based on the assumption of i.i.d for set
of random variables X1, . . . , Xn having
a distribution N(0, 1).
Under this assumption, we can say the
following for the probability of maximum
absolute value of the coefficients.
P{max |Xi|, 1≤i≤n> √ 2 logn}→ 0, as n →
∞
Note Xi refers to noise
42. Universal Thresholding Rule
Therefore, under universal thresholding
applied to wavelet coefficients, we can say
the following.
with high probability every sample in
the wavelet transform (i.e.coefficient)
in which the underlying function is
exactly zero will be estimated as zero
43. Universal Thresholding Rule in WP
Universal threshold estimation rule
when applied to wavelet packet is to
be adjusted to the length of
decomposition which is nlog(n).
Threshold is then
Th=σ√[2 log(nlog(n)].
44. Level Dependent Thresholding
In level dependent thresholding, thresholds
are rescaled at each level to arrive at a new
estimate corresponding to the standard
deviation of wavelet coefficients at that level.
We consider white noise model and Gaussian
distribution for the coefficient at each level.
This is referred to ‘mln’ [multilevel noise
model] in Matlab toolbox. Threshold level is
determined as follows.
Th(j,n) = σj√(2 log nj), σj =MADj /.6745
45. Stein Unbiased Risk Estimator( SURE)
A criteria referred to as Stein Unbiased Risk
Estimator abbreviated by SureSrink, utilizes
statistical estimation theory in which an unbiased
estimate of loss function is derived
Suppose X1, . . . , Xs are independent N(μi, 1), i =
1, . . . , s, random variables. The problem is to
estimate:
mean vector μ = (μ1, . , μs) with minimum L 2-risk.
Stein states that the L2-loss can be estimated
unbiasedly using any estimator μ that can be
written as
μ(X) = X + g(X),
where the function
g = (g1, . . . , gs) is weakly differentiable.
46. SURE Estimator
Under SURE criteria, following is considered as an
estimate of the loss function.
E||µ(x)- µe||^2 =E SURE(th:x)
where
SURE(th;x)=s-2#B{i:|Xi|≤ th}+ (min(|
xi|,th)^2
where µ(x) is a fixed estimate of the mean of the
coefficients and #B denotes the cardinality of a set B.
It can be shown that SURE(th;x) is an unbiased estimate
of the L2-risk, i.e.
µ|| µλ (X) - µ||^2 = µSURE(th; X).
Threshold level λ is based on minimum value of SURE loss
function which is defined as
Ths = arg min th Sure(th;x)
47. Other Thresholding Rules
Fixed Form thresholding is the same
as Universal Thresholding
Th=σ√(2 log n), σ=MAD/.6745
Minimax refers to finding the
minimum of the maximum mean
square error obtained for the worst
function in a given set
48. Rigorous SURE Denoising
Rigorous SURE (Stein’s Unbiased
Risk Estimate), a threshold-based
method with a threshold
where n is the number of samples
in the signal(i.e. coefficients)
49. Heuristic SURE
Heuristic SURE is a combination of
Fixed Form and Rigorous SURE
( for details refer to Matlab
Helpdesk)
50. Results of Denoising Application on
CDMA Signal
At SNR = 3 dB, MSE between the original signal
and the noisy signal is 0.99.
The following table shows MSE after denoising:
Wavelet Haar,Bior3.1,Db10,Coif5,
fixed form, white noise 0.55 0.64 0.46 0.46
RigSURE, white noise 0.36 0.41 0.27 0.27
HeurSURE,wh. Noise 0.42 0.41 0.27 0.28,
Minimax, white noise 0.46 0.46 0.34 0.33,
Minimax, nonwhite 0.53 1.09 0.44 0.32,
51. Observations on Denoising Applied on
CDMA Signals
It was found that soft thresholding gives better
performance ( in terms of SNR)than hard thresholding
in this project.
Since the noise model used in this project is WGN,
selecting the correct noise type (white noise) will also
give better results.
Db10 and Coif5 outperform Haar and Bior3.1 in
denoising because they have higher order of vanishing
moments.
At SNR equals –3 dB, Db10 and Coif5 with soft
thresholding and rigorous SURE threshold selection rule
give very good denoising performance. The MSE is
brought from 3.9 to approximately 0.7.
In general, Fixed Form and Heuristic SURE are more
aggressive in removing noise. Rigorous SURE and
Minimax are more conservative and they give better
results in this project because some details of the
CDMA signal lie in the noise range.
52. Denoising in MATLAB
In Matlab, command ‘wden’ is used for denoising:
Sig=wden(s,tptr,sorh,scal,n,wav)
for determining σ for noise thresholding where:
s=signal,
sorh= soft or hard thresholding ‘s’ ,‘h’
scal=1 original model( white noise with unscaled noise),
scal=’sln’ first estimate of the noise variance based on 1st
level details. This uses basic model.
Scal=’mln’, is for nonwhite noise, i.e scale dependent
noise thresholding.
[For further details, please refer to Matlab wavemenu
toolbox]
53. Artifacts at points of Singularity and
Stationary Wavelet Transform
Gaussian noise model for the coefficients does not fully
agree with peaky shape of the distribution at d1 level.
Gibbs type of oscillations and artifacts are also observed
at points of singularity, though not as much prominent as
Gibbs oscillation.
To correct such phenomena, stationary wavelet transform
is used and has been incorporated in Matlab toolbox.
In stationary wavelet transform, DWT is applied for all
circular shifts of the signal of length N and coefficients are
evaluated and threshold levels are determined.
An average of all the N denoised signals is used for the
final denoised signal
The only limitation here is that signal length must be a
factor of 2J .
Denoising using SWT often results in a conservative noise
reduction results as compared with standard soft
thresholding using ‘Fixed Form’ or RigSure.
58. Hidden Markov Model for Denoising
Please refer to class notes posted
on site
59. THE CLASSICAL APPROACH TO WAVELET THRESHOLDING
A wavelet based linear approach, extending simply spline smoothing estimation methods as described by
Wahba (1990), is the one suggested by Antoniadis (1996) and independentlyby Amato & Vuza (1997). Of
non-threshold type, this method is appropriate for estimating relatively regular functions. Assuming that
the smoothness index s of the function g to be
recovered is known, the resulting estimator is obtained by estimating the scaling coefficients
cj0k by their empirical counterparts ˆ cj0k and by estimating thewavelet coefficients djk via a linear
shrinkage
d˜jk =
dˆjk
1 + λ22js ,
where λ > 0 is a smoothing parameter. The parameter λ is chosen by cross-validation in Amato
& Vuza (1997), while the choice of λ in Antoniadis (1996) is based on risk minimization and
depends on a preliminary consistent estimator of the noise level σ. The above linear methods
are not designed to handle spatially inhomogeneous functions with low regularity. For such
functions one usually relies upon nonlinear thresholding or nonlinear shrinkage methods.
Donoho & Johnstone (1994, 1995, 1998) and Donoho, Johnstone, Kerkyacharian & Picard
(1995) proposed a nonlinear wavelet estimator of g based on reconstruction by keeping the
empirical scaling coefficients ˆ cj0k in (2) intact and from a more judicious selection of the
empirical wavelet coefficients dˆjk in (3).
60. They suggested the extraction of the significant wavelet
coefficients by thresholding in which wavelet coefficients are set to zero if their absolute value is
below a certain threshold level, λ ≥ 0, whose choice we discuss in more detail in Section 4.1.
Under this scheme we obtain thresholded wavelet coefficients using either the hard or soft
thresholding rule given respectively by
δH
λ (dˆjk) =
0 if|
dˆjk| ≤ λ
dˆjk if | dˆjk| > λ
(4)
and
δSλ
(dˆjk) =
0 if|dˆjk| ≤ λ
dˆjk − λ if dˆjk > λ
dˆjk + λ if dˆjk < −λ.
(5)
11
Thresholding allows the data itself to decide which wavelet coefficients are significant; hard
thresholding (a discontinuous function) is a ‘keep’ or ‘kill’ rule, while soft thresholding (a
continuous function) is a ‘shrink’ or ‘kill’ rule.