The main advantage of wavelet transforms are
1. Wavelet transforms has multiresolution properity
2. Better Spectral localization properity
What is Multiresolution properity:
Multiresolution properity means that wavelet transform used different Scales for the analysis of different frequency componants of any signal .
What is scale:
Scale is inversely propertional to the frequency of any signal
large scale is used for the analysis of small frequency componants presents in any signal Whereas Small scale is used for the analysis of high frequency componants of any signal
What is Spectral localisation
Spectral localization properity means that wavelet transform tells us that what frequency componants are present in any given signal and at time axis where these frequency componants are presents
Process of taking Wavelet transform of any signal
3.Frequency Domain Representation of Signals and SystemsINDIAN NAVY
This document provides an overview of frequency domain representation of signals and systems. It defines key concepts such as the Fourier transform, which converts a signal from the time domain to the frequency domain. The frequency spectrum shows the distribution of frequencies within a signal. Periodic signals can be represented using Fourier series, while aperiodic signals use the Fourier transform. Properties of the Fourier transform such as linearity, time shifting, and the convolution theorem are also covered.
fast-Fourier-transform-presentation and Fourier transform for wave
in
signal possessing for
physics and
geophysics
spectra analysis
periodic and non periodic wave
data sampling
The Nyquist frequency
The document discusses different classifications of signals:
1) Analog signals are continuous-time signals that can take any value in a range, like sine waves. Digital signals are discrete-time signals that can only take values in a finite set, like -1 or 1.
2) Signals can be real if they only take real values, or complex if they can also take imaginary values. Complex signals are useful for analyzing communication systems.
3) Deterministic signals have defined values at all times, like sine waves. Random signals have random values at different times, like noise or coin toss outcomes.
4) Even signals satisfy f(t)=f(-t) and odd signals satisfy f(t)=-f
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.
The document discusses the convergence of the Fourier transform. It provides an outline covering topics like the Fourier transform, Fourier series, the difference between them, and conditions for convergence like the Dirichlet condition. It also discusses Fourier analysis of discrete time signals and types of convergence like uniform and mean square convergence. Examples are given to illustrate concepts like Gibbs phenomenon where the Fourier approximation oscillates at points of discontinuity.
This document provides an overview of angle modulation techniques including frequency modulation (FM) and phase modulation (PM). It defines PM and FM mathematically. For PM, the phase deviation is a linear function of the baseband message signal. For FM, the instantaneous frequency deviation is a linear function of the message signal. The key advantages of FM and PM over amplitude modulation are constant envelope and better noise immunity. However, FM and PM require increased bandwidth compared to amplitude modulation. The document derives expressions for the pre-envelope and spectrum of an FM signal and discusses bandwidth requirements of FM.
The document discusses Fast Fourier Transform (FFT) analysis. It begins by explaining what Fourier Transform and Discrete Fourier Transform (DFT) are and how they convert signals from the time domain to the frequency domain. It then states that FFT is an efficient algorithm for performing DFT, allowing it to be done much faster on computers. The document proceeds to describe different types of FFT algorithms like Cooley-Tukey, Prime Factor, Bruun's, and Rader's algorithms. It concludes by discussing characteristics of FFT like approximation, accuracy, and complexity bounds, as well as applications and how FFT can be used to analyze vibration signals in the frequency domain.
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.
3.Frequency Domain Representation of Signals and SystemsINDIAN NAVY
This document provides an overview of frequency domain representation of signals and systems. It defines key concepts such as the Fourier transform, which converts a signal from the time domain to the frequency domain. The frequency spectrum shows the distribution of frequencies within a signal. Periodic signals can be represented using Fourier series, while aperiodic signals use the Fourier transform. Properties of the Fourier transform such as linearity, time shifting, and the convolution theorem are also covered.
fast-Fourier-transform-presentation and Fourier transform for wave
in
signal possessing for
physics and
geophysics
spectra analysis
periodic and non periodic wave
data sampling
The Nyquist frequency
The document discusses different classifications of signals:
1) Analog signals are continuous-time signals that can take any value in a range, like sine waves. Digital signals are discrete-time signals that can only take values in a finite set, like -1 or 1.
2) Signals can be real if they only take real values, or complex if they can also take imaginary values. Complex signals are useful for analyzing communication systems.
3) Deterministic signals have defined values at all times, like sine waves. Random signals have random values at different times, like noise or coin toss outcomes.
4) Even signals satisfy f(t)=f(-t) and odd signals satisfy f(t)=-f
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.
The document discusses the convergence of the Fourier transform. It provides an outline covering topics like the Fourier transform, Fourier series, the difference between them, and conditions for convergence like the Dirichlet condition. It also discusses Fourier analysis of discrete time signals and types of convergence like uniform and mean square convergence. Examples are given to illustrate concepts like Gibbs phenomenon where the Fourier approximation oscillates at points of discontinuity.
This document provides an overview of angle modulation techniques including frequency modulation (FM) and phase modulation (PM). It defines PM and FM mathematically. For PM, the phase deviation is a linear function of the baseband message signal. For FM, the instantaneous frequency deviation is a linear function of the message signal. The key advantages of FM and PM over amplitude modulation are constant envelope and better noise immunity. However, FM and PM require increased bandwidth compared to amplitude modulation. The document derives expressions for the pre-envelope and spectrum of an FM signal and discusses bandwidth requirements of FM.
The document discusses Fast Fourier Transform (FFT) analysis. It begins by explaining what Fourier Transform and Discrete Fourier Transform (DFT) are and how they convert signals from the time domain to the frequency domain. It then states that FFT is an efficient algorithm for performing DFT, allowing it to be done much faster on computers. The document proceeds to describe different types of FFT algorithms like Cooley-Tukey, Prime Factor, Bruun's, and Rader's algorithms. It concludes by discussing characteristics of FFT like approximation, accuracy, and complexity bounds, as well as applications and how FFT can be used to analyze vibration signals in the frequency domain.
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.
Performance analysis of wavelet based blind detection and hop time estimation...Saira Shahid
This document discusses a wavelet-based algorithm for blind detection and hop time estimation of frequency hopping signals in the HF band.
The algorithm first uses the temporal correlation function (TCF) of the received signal to extract phase information. Discrete wavelet transform (DWT) is then applied to the TCF to detect frequency transition points, which indicate the time of hopping between frequencies. Simulation results showing the detection performance at different signal-to-noise ratios are presented.
The key steps are: 1) Extracting the phase from the TCF of the received signal. 2) Applying DWT as a de-noising technique to detect the times of hopping between frequencies. 3) Presenting simulation
This document provides an overview of circuits and communication topics covered in an electrical engineering course. It discusses voltage sources, driving circuits, operational amplifier circuits, and communications concepts like matched filtering and receiver synchronization. The goal is to introduce practical circuit ideas and fundamental communication principles, with a focus on robustly detecting signals and data in the presence of noise. Worked examples are provided for repeating codes, on-off keying, and antipodal signalling transmission scenarios.
The document discusses Fourier analysis and signal processing. It introduces the continuous and discrete Fourier transforms which decompose a signal into weighted basis functions of cosine and sine. The continuous Fourier transform uses integrals while the discrete version uses sums, introducing aliasing effects. Convolution is also introduced, where a signal passed through a linear time-invariant system results in an output that is the convolution of the input signal and impulse response.
The document discusses the discrete Fourier transform (DFT) and its applications. It provides an overview of DFT and how it represents a signal in the frequency domain. It then describes the fast Fourier transform (FFT) algorithm, which efficiently computes the DFT. The document outlines algorithms to compute the inverse DFT and circular convolution using the DFT. It includes MATLAB code implementations of DFT, inverse DFT, FFT, and circular convolution. Graphs are shown comparing computation times of the algorithms.
This document contains a summary of key concepts from a chapter on Fourier transforms and their properties. It begins with an overview of the motivation for Fourier transforms as an extension of Fourier series to allow representation of aperiodic signals. It then provides examples of Fourier transforms for common functions like a rectangular pulse and exponential. The remainder summarizes important properties of Fourier transforms including: time-frequency duality, symmetry of direct and inverse transforms, scaling which relates time/bandwidth compression, time-shifting which causes phase change, and frequency-shifting which translates the spectrum.
Ch7 noise variation of different modulation scheme pg 63Prateek Omer
This document summarizes the noise performance of various modulation schemes. It begins by introducing a receiver model and defining figures of merit used to evaluate performance. It then analyzes the noise performance of coherent demodulation for DSB-SC and SSB modulation. The following key points are made:
1) Coherent detection of DSB-SC signals results in signal and noise being additive at both the input and output of the detector. The detector completely rejects the quadrature noise component.
2) For DSB-SC, the output SNR and reference SNR are equal, resulting in a figure of merit of 1.
3) Analysis of SSB modulation shows it achieves a 3 dB improvement in output SNR over
This document discusses concepts related to signals and systems. It begins by defining a signal as a time-varying quantity of information and a system as an entity that processes input signals to produce output signals. It then covers signal classification including continuous vs discrete time, analog vs digital, periodic vs aperiodic, deterministic vs random, and causal vs non-causal signals. Signal operations like time shifting, scaling, and inversion are described. Key concepts discussed in detail include signal size using energy and power, signal components and orthogonality, correlation as a measure of signal similarity, and trigonometric Fourier series. Worked examples are provided to illustrate various topics.
The discrete Fourier transform has many applications in science and engineering. For example, it is often used in digital signal processing applications such as voice recognition and image processing.
This document discusses model checking of time Petri nets (TPN). TPN extend ordinary Petri nets by associating time intervals with transitions, specifying minimum and maximum times for a transition to remain enabled before firing. The document outlines TPN semantics based on clocks or intervals, temporal logics like TCTL for specifying timed properties, and techniques for abstracting the generally infinite TPN state space into a finite representation to enable model checking of properties. Abstract states group markings by time variables and various abstractions aim to preserve linear or branching properties of the original state space.
Fourier Analysis Techniques has series and transforms. This slideshow gives a basic idea about the fourier series analysis for both trigonometric and exponential terms and gives an insight of odd, even and half wave symmetry, spectrum generation and composite signal
The document discusses sampling theory and analog-to-digital conversion. It begins by explaining that most real-world signals are analog but must be converted to digital for processing. There are three steps: sampling, quantization, and coding. Sampling converts a continuous-time signal to a discrete-time signal by taking samples at regular intervals. The sampling theorem states that the sampling frequency must be at least twice the highest frequency of the sampled signal to avoid aliasing. Finally, it provides an example showing how to calculate the minimum sampling rate, or Nyquist rate, given the highest frequency of a signal.
Ch2 probability and random variables pg 81Prateek Omer
This document discusses probability and random variables. It begins by defining key terms like random experiment, random event, sample space, mutually exclusive events, union and intersection of events, occurrence of an event, and complement of an event. It then discusses definitions of probability, including the relative frequency definition and classical definition. It also covers conditional probability, Bayes' theorem, and statistical independence. The key concepts of probability theory and random variables are introduced to enable analysis and characterization of random signals.
The document describes methodology for estimating the channel impulse response from acoustic signals transmitted during the SAVEX15 experiment. Stationary source experiments involved transmitting chirp and m-sequence signals from a fixed source location and receiving the signals on a vertical receiver array up to 5 km away. Matched filtering of the received signals with the transmitted source signals was used to estimate the time-varying channel impulse response, which characterizes how the underwater acoustic channel responds to any input signal.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
The document discusses sampling and the Hilbert transform. It begins by defining sampling as the reduction of a continuous-time (CT) signal to a discrete-time (DT) signal using a sampler. It describes the Nyquist sampling theorem which specifies the minimum sampling rate to reconstruct the original signal. It then discusses different types of sampling including impulse, natural, and flat top sampling. The document also covers aliasing, the Hilbert transform, and properties and examples of using the Hilbert transform including on bandpass signals and for system representation.
This document discusses principles of communication and representation of signals. It begins with an introduction to the communication process and challenges involved. Signals exist in the time and frequency domains, and Fourier analysis using the Fourier series and Fourier transform helps characterize signals in the frequency domain. Periodic signals can be represented by a Fourier series which decomposes the signal into a sum of complex exponentials at discrete frequencies that are integer multiples of the fundamental frequency. Examples are provided to illustrate calculation of Fourier coefficients and representation of periodic signals in the exponential and trigonometric forms of the Fourier series. Spectral plots from a spectrum analyzer are also presented for various waveforms.
The document discusses Fourier transforms of discrete signals. It covers topics like sampling, the discrete time Fourier transform (DTFT), the discrete Fourier transform (DFT), and fast Fourier transform (FFT) algorithms. The key points are:
- Continuous signals are converted to discrete signals through sampling at discrete time intervals. The sampling rate must be high enough to avoid aliasing.
- The DTFT analyzes the frequency spectrum of a discrete signal and is periodic in frequency space. The DFT provides the frequency spectrum of a finite discrete signal.
- FFT algorithms like the radix-2 algorithm improve the efficiency of computing DFTs and DFTs by decomposing the computation into smaller subproblems.
This document provides an overview of Fourier analysis techniques for communication engineering experiments. It introduces Fourier series as a way to expand periodic signals into a sum of complex exponentials. The Fourier series coefficients represent the contribution of each harmonic frequency. MATLAB will be used to implement Fourier analysis and observe its applications in communication systems. Students are expected to review basic MATLAB commands and complete pre-lab exercises on vector operations and plotting signals before conducting the experiment.
It is a basic ppt of pattern recognition using wavelates and contourlets.... I will describe the algo into next slide... Thank you... It is a good ppt you can learn the basic of this project
1. Fourier transforms can be used to analyze aperiodic signals by extending the period to infinity, turning the aperiodic signal into a periodic one. This allows the computation of Fourier coefficients using the continuous-time Fourier transform (CTFT).
2. The CTFT of an aperiodic signal results in a continuous function of frequency rather than discrete frequencies. Key examples are computed, such as the CTFT of an impulse function being 1 for all frequencies and the CTFT of a constant function being an impulse at zero frequency.
3. The CTFT represents the frequency content of a signal and is useful for analyzing aperiodic real-world signals. Examples demonstrate how the CTFT can be used to analyze signals like sinusoids
Performance analysis of wavelet based blind detection and hop time estimation...Saira Shahid
This document discusses a wavelet-based algorithm for blind detection and hop time estimation of frequency hopping signals in the HF band.
The algorithm first uses the temporal correlation function (TCF) of the received signal to extract phase information. Discrete wavelet transform (DWT) is then applied to the TCF to detect frequency transition points, which indicate the time of hopping between frequencies. Simulation results showing the detection performance at different signal-to-noise ratios are presented.
The key steps are: 1) Extracting the phase from the TCF of the received signal. 2) Applying DWT as a de-noising technique to detect the times of hopping between frequencies. 3) Presenting simulation
This document provides an overview of circuits and communication topics covered in an electrical engineering course. It discusses voltage sources, driving circuits, operational amplifier circuits, and communications concepts like matched filtering and receiver synchronization. The goal is to introduce practical circuit ideas and fundamental communication principles, with a focus on robustly detecting signals and data in the presence of noise. Worked examples are provided for repeating codes, on-off keying, and antipodal signalling transmission scenarios.
The document discusses Fourier analysis and signal processing. It introduces the continuous and discrete Fourier transforms which decompose a signal into weighted basis functions of cosine and sine. The continuous Fourier transform uses integrals while the discrete version uses sums, introducing aliasing effects. Convolution is also introduced, where a signal passed through a linear time-invariant system results in an output that is the convolution of the input signal and impulse response.
The document discusses the discrete Fourier transform (DFT) and its applications. It provides an overview of DFT and how it represents a signal in the frequency domain. It then describes the fast Fourier transform (FFT) algorithm, which efficiently computes the DFT. The document outlines algorithms to compute the inverse DFT and circular convolution using the DFT. It includes MATLAB code implementations of DFT, inverse DFT, FFT, and circular convolution. Graphs are shown comparing computation times of the algorithms.
This document contains a summary of key concepts from a chapter on Fourier transforms and their properties. It begins with an overview of the motivation for Fourier transforms as an extension of Fourier series to allow representation of aperiodic signals. It then provides examples of Fourier transforms for common functions like a rectangular pulse and exponential. The remainder summarizes important properties of Fourier transforms including: time-frequency duality, symmetry of direct and inverse transforms, scaling which relates time/bandwidth compression, time-shifting which causes phase change, and frequency-shifting which translates the spectrum.
Ch7 noise variation of different modulation scheme pg 63Prateek Omer
This document summarizes the noise performance of various modulation schemes. It begins by introducing a receiver model and defining figures of merit used to evaluate performance. It then analyzes the noise performance of coherent demodulation for DSB-SC and SSB modulation. The following key points are made:
1) Coherent detection of DSB-SC signals results in signal and noise being additive at both the input and output of the detector. The detector completely rejects the quadrature noise component.
2) For DSB-SC, the output SNR and reference SNR are equal, resulting in a figure of merit of 1.
3) Analysis of SSB modulation shows it achieves a 3 dB improvement in output SNR over
This document discusses concepts related to signals and systems. It begins by defining a signal as a time-varying quantity of information and a system as an entity that processes input signals to produce output signals. It then covers signal classification including continuous vs discrete time, analog vs digital, periodic vs aperiodic, deterministic vs random, and causal vs non-causal signals. Signal operations like time shifting, scaling, and inversion are described. Key concepts discussed in detail include signal size using energy and power, signal components and orthogonality, correlation as a measure of signal similarity, and trigonometric Fourier series. Worked examples are provided to illustrate various topics.
The discrete Fourier transform has many applications in science and engineering. For example, it is often used in digital signal processing applications such as voice recognition and image processing.
This document discusses model checking of time Petri nets (TPN). TPN extend ordinary Petri nets by associating time intervals with transitions, specifying minimum and maximum times for a transition to remain enabled before firing. The document outlines TPN semantics based on clocks or intervals, temporal logics like TCTL for specifying timed properties, and techniques for abstracting the generally infinite TPN state space into a finite representation to enable model checking of properties. Abstract states group markings by time variables and various abstractions aim to preserve linear or branching properties of the original state space.
Fourier Analysis Techniques has series and transforms. This slideshow gives a basic idea about the fourier series analysis for both trigonometric and exponential terms and gives an insight of odd, even and half wave symmetry, spectrum generation and composite signal
The document discusses sampling theory and analog-to-digital conversion. It begins by explaining that most real-world signals are analog but must be converted to digital for processing. There are three steps: sampling, quantization, and coding. Sampling converts a continuous-time signal to a discrete-time signal by taking samples at regular intervals. The sampling theorem states that the sampling frequency must be at least twice the highest frequency of the sampled signal to avoid aliasing. Finally, it provides an example showing how to calculate the minimum sampling rate, or Nyquist rate, given the highest frequency of a signal.
Ch2 probability and random variables pg 81Prateek Omer
This document discusses probability and random variables. It begins by defining key terms like random experiment, random event, sample space, mutually exclusive events, union and intersection of events, occurrence of an event, and complement of an event. It then discusses definitions of probability, including the relative frequency definition and classical definition. It also covers conditional probability, Bayes' theorem, and statistical independence. The key concepts of probability theory and random variables are introduced to enable analysis and characterization of random signals.
The document describes methodology for estimating the channel impulse response from acoustic signals transmitted during the SAVEX15 experiment. Stationary source experiments involved transmitting chirp and m-sequence signals from a fixed source location and receiving the signals on a vertical receiver array up to 5 km away. Matched filtering of the received signals with the transmitted source signals was used to estimate the time-varying channel impulse response, which characterizes how the underwater acoustic channel responds to any input signal.
DSP_2018_FOEHU - Lec 06 - FIR Filter DesignAmr E. Mohamed
This lecture discusses the design of finite impulse response (FIR) filters. It introduces the window method for FIR filter design, which involves truncating the ideal impulse response with a window function to obtain a causal FIR filter. Common window functions are presented such as rectangular, triangular, Hanning, Hamming, and Blackman windows. These windows trade off main lobe width and side lobe levels. The document provides an example design of a low-pass FIR filter using the Hamming window to meet given passband and stopband specifications.
The document discusses sampling and the Hilbert transform. It begins by defining sampling as the reduction of a continuous-time (CT) signal to a discrete-time (DT) signal using a sampler. It describes the Nyquist sampling theorem which specifies the minimum sampling rate to reconstruct the original signal. It then discusses different types of sampling including impulse, natural, and flat top sampling. The document also covers aliasing, the Hilbert transform, and properties and examples of using the Hilbert transform including on bandpass signals and for system representation.
This document discusses principles of communication and representation of signals. It begins with an introduction to the communication process and challenges involved. Signals exist in the time and frequency domains, and Fourier analysis using the Fourier series and Fourier transform helps characterize signals in the frequency domain. Periodic signals can be represented by a Fourier series which decomposes the signal into a sum of complex exponentials at discrete frequencies that are integer multiples of the fundamental frequency. Examples are provided to illustrate calculation of Fourier coefficients and representation of periodic signals in the exponential and trigonometric forms of the Fourier series. Spectral plots from a spectrum analyzer are also presented for various waveforms.
The document discusses Fourier transforms of discrete signals. It covers topics like sampling, the discrete time Fourier transform (DTFT), the discrete Fourier transform (DFT), and fast Fourier transform (FFT) algorithms. The key points are:
- Continuous signals are converted to discrete signals through sampling at discrete time intervals. The sampling rate must be high enough to avoid aliasing.
- The DTFT analyzes the frequency spectrum of a discrete signal and is periodic in frequency space. The DFT provides the frequency spectrum of a finite discrete signal.
- FFT algorithms like the radix-2 algorithm improve the efficiency of computing DFTs and DFTs by decomposing the computation into smaller subproblems.
This document provides an overview of Fourier analysis techniques for communication engineering experiments. It introduces Fourier series as a way to expand periodic signals into a sum of complex exponentials. The Fourier series coefficients represent the contribution of each harmonic frequency. MATLAB will be used to implement Fourier analysis and observe its applications in communication systems. Students are expected to review basic MATLAB commands and complete pre-lab exercises on vector operations and plotting signals before conducting the experiment.
It is a basic ppt of pattern recognition using wavelates and contourlets.... I will describe the algo into next slide... Thank you... It is a good ppt you can learn the basic of this project
1. Fourier transforms can be used to analyze aperiodic signals by extending the period to infinity, turning the aperiodic signal into a periodic one. This allows the computation of Fourier coefficients using the continuous-time Fourier transform (CTFT).
2. The CTFT of an aperiodic signal results in a continuous function of frequency rather than discrete frequencies. Key examples are computed, such as the CTFT of an impulse function being 1 for all frequencies and the CTFT of a constant function being an impulse at zero frequency.
3. The CTFT represents the frequency content of a signal and is useful for analyzing aperiodic real-world signals. Examples demonstrate how the CTFT can be used to analyze signals like sinusoids
Analog signals can have an infinite number of values within a range, while digital signals can have only a limited number of values. Periodic signals repeat a pattern over time periods, while aperiodic signals change constantly without a repeating pattern. The three attributes of analog signals are amplitude, frequency, and phase. A signal's bandwidth is the width of its frequency spectrum, calculated by subtracting the lowest frequency from the highest. According to Fourier analysis, any composite signal can be represented as a combination of simple sine waves with different frequencies, phases, and amplitudes.
This document discusses Fourier series and the frequency domain. It begins by defining the frequency domain as analyzing signals based on frequency rather than time. The frequency spectrum shows the amplitudes and phases of frequency components. Fourier series can be used to analyze periodic signals by expressing them as sums of sinusoids. There are different types of Fourier series including trigonometric, complex exponential, and cosine with phase. Fourier series satisfy Dirichlet conditions of being periodic, having a finite number of discontinuities and maxima/minima to be absolutely integrable. Examples are provided of applying Fourier series to periodic signals.
(1) An analog signal varies continuously over time while a digital signal has discrete values.
(2) A periodic signal repeats at regular intervals, while a non-periodic signal does not.
(3) A time-domain plot shows how a signal's amplitude changes over time, while a frequency-domain plot shows amplitude changes by frequency.
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.
This document provides an overview of key concepts in communications systems, including:
1) It describes the basic components of a communications system including the input/output transducers, transmitter, channel, and receiver.
2) It discusses different types of signals that can be transmitted through a channel including analog modulation techniques like AM, FM and PM as well as digital modulation.
3) It provides an overview of electromagnetic waves and the electromagnetic spectrum used for wireless communication.
This document discusses frequency domain representation of periodic signals. It defines spectrum as the measurable range of a physical property like frequency or wavelength. A signal's frequency domain representation plots amplitude and phase versus frequency, rather than versus time as in the time domain. The frequency domain reveals the frequencies and proportions of frequency components that make up the signal's shape. It can be obtained from the signal's Fourier series or Fourier transform. Sinusoids in continuous and discrete time are used as examples to demonstrate how their frequency domain representations graph amplitude versus frequency and phase versus frequency.
This document discusses the physical layer and media in networking. It covers:
- The physical layer is responsible for carrying information between nodes by converting data to signals and transmitting them across a medium.
- Data must be transformed into electromagnetic signals to be transmitted. Both analog and digital signals can be used for transmission.
- Periodic analog signals like sine waves are commonly used. They are defined by amplitude, frequency, and phase. Composite signals can be decomposed into sums of sine waves.
- The bandwidth of a signal is the range of its component frequencies. Frequency spectrum refers to the specific frequencies present in a signal.
Running Head Fourier Transform Time-Frequency Analysis. .docxcharisellington63520
This document provides an overview of time-frequency analysis and the Fourier transform. It discusses how the Fourier transform expresses a time function in terms of its frequency components. Time-frequency analysis looks at signals in both the time and frequency domains simultaneously using representations like the short-time Fourier transform. The document outlines applications of time-frequency analysis in fields like signal processing, optics, acoustics, and economic data analysis.
Ch3Data communication and networking by neha g. kuraleNeha Kurale
Data can exist in either analog or digital form. Analog data is continuous while digital data takes on discrete values. Both analog and digital signals can be periodic or non-periodic. Periodic signals can be decomposed into simpler sine waves using Fourier analysis. Non-periodic signals result in a combination of sine waves with continuous frequencies. The bandwidth of a signal is the difference between its highest and lowest frequencies.
Frequency modulation (FM) varies the carrier frequency within a small range based on the information signal. The peak frequency deviation (Df) is the farthest the FM signal can be from the original carrier frequency (fc). The modulation index (β) represents Df as a multiple of the maximum modulating frequency (fm), with Df = βfm. The FM spectrum has multiple sidebands separated by the modulating frequency on either side of the carrier frequency, with the extent of sidebands approximately (β + 1)fm above and below fc.
This document summarizes a lecture on data communications and networking. It discusses different types of signals including digital signals, which are discrete, and analog signals, which are continuous. Periodic signals repeat in a consistent pattern, while aperiodic signals do not. Simple analog signals like sine waves are characterized by amplitude, frequency, and phase. Composite signals contain multiple frequencies. Digital signals have a bit rate and bit interval. The document also covers network terminology, transmission impairments like attenuation and distortion, and noise.
The document introduces the limitations of Fourier transforms for analyzing non-stationary signals and discusses how wavelet transforms provide a solution. Fourier transforms can only provide frequency content over the entire signal duration and cannot localize this content in time. Wavelet transforms overcome this by analyzing the signal with translated and scaled versions of an analyzing wavelet, allowing time-frequency localization. This provides a time-frequency representation of non-stationary signals and indicates when different frequency components occur.
This document provides a tutorial on the Fast Fourier Transform (FFT). It begins by explaining that the FFT is a faster version of the Discrete Fourier Transform (DFT) that takes a discrete signal in the time domain and transforms it into the discrete frequency domain. It then reviews other transforms taught in previous courses and explains how the DFT relates to the Discrete-Time Fourier Transform (DTFT). The document provides MATLAB examples demonstrating how to use the FFT function to compute the DFT of signals and understand the results. It concludes by discussing how to properly analyze a signal's spectrum using the FFT with MATLAB.
Signals and Systems-Fourier Series and TransformPraveen430329
This document discusses analysis of continuous time signals. It begins by introducing Fourier series representation of periodic signals using trigonometric and exponential forms. It describes properties of Fourier series such as linearity, time shifting, and frequency scaling. It then introduces the Fourier transform which transforms signals from the time domain to the frequency domain. Common Fourier transform pairs are listed. The Laplace transform is also introduced which transforms signals from the time domain to the complex s-domain. Key properties of the Laplace transform include linearity, scaling, time shifting, and the initial and final value theorems. Conditions for the existence of the Laplace transform are also provided.
On Fractional Fourier Transform Moments Based On Ambiguity FunctionCSCJournals
The fractional Fourier transform can be considered as a rotated standard Fourier transform in general and its benefit in signal processing is growing to be known more. Noise removing is one application that fractional Fourier transform can do well if the signal dilation is perfectly known. In this paper, we have computed the first and second order of moments of fractional Fourier transform according to the ambiguity function exactly. In addition we have derived some relations between time and spectral moments with those obtained in fractional domain. We will prove that the first moment in fractional Fourier transform can also be considered as a rotated the time and frequency gravity in general. For more satisfaction, we choose five different types signals and obtain analytically their fractional Fourier transform and the first and second-order moments in time and frequency and fractional domains as well.
The document discusses Fast Fourier Transform (FFT) analysis. It begins by explaining what Fourier Transform and Discrete Fourier Transform (DFT) are and how they convert signals from the time domain to the frequency domain. It then states that FFT is an efficient algorithm for performing DFT, allowing it to be done much faster on computers. The document proceeds to describe different types of FFT algorithms like Cooley-Tukey, Prime Factor, Bruun's, and Rader's algorithms. It concludes by discussing characteristics of FFT like approximation, accuracy, and complexity bounds, as well as applications and how FFT can be used to analyze vibration signals in the frequency domain.
The document discusses the frequency domain and its key concepts. The frequency domain represents periodic functions as a sum of sines and cosines with different amplitudes and frequencies. Any periodic function can be characterized by its amplitude, period, frequency (the inverse of period), and phase. The frequency domain shows the frequency spectrum, or range of frequencies that make up a function. It indicates the amplitudes of different frequency components.
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1. Any signal in time domain is
concidered as raw signal. The Propose of all Transformation
techniques are to convert time domain signal in a form so that
desired information can be extracted from these signal’s and after
the application of certain transform the resultant signal is known as
processed signal.
The commonly used transformation techniques are
• Discreet Fourier Transform
• Short time Fourier transform
• Wavelet transform
1.DISCREAT FOURIER TRANSFORMS: The Discrete Fourier
transform is used for converting time domain signal into frequency
domain
In time domain representation of the signal there is a graph
between time and amplitude. In this time amplitude graph time is
taken at the x Axis as an independent variable Whereas amplitude
of the signal is taken at y axis .the time domain representation of
the signal gives an information about the signal that at which time
instants what is the amplitude of the signal or how amplitude of the
signal is varying with respect to time but it gives no information
about the Different frequency contents that are presents in the
signal.
In frequency domain representation of the signal there is a graph
between frequency and amplitude .In this frequency Amplitude
graph The frequency of the signal is taken at the x axis as
independend variable whereas the Amplitude of the signal is taken
at y Axis.The frequency domain representation of the signal tells
2. us that in a given signal what different frequency components are
present and what are there respective amplitudes.But again
frequency domain representation gives no idea that at which time
these frequency componants are presents. In some applications
frequency domain representation is more important. Or we can say
that frequency domain representation gives more information
about any signal (for example in any audio music signal).
Suppose
x= 100*sin(2*pi*2*t)+50*cos(2*pi*3*t)
is a time Domain representation of a given signal and
y=fft(x)
is frequency doamin representaion of signal x
0 1 2 3 4 5 6 7 8 9 10
-200
-100
0
100
200
TIME
AMPLITUDE
0 1 2 3 4 5 6 7 8 9 10
-1000
0
1000
2000
3000
FREQUENCY
AMPLITUDE
TIME DOMAINREPRESENTATIONOF SIGNAL
FREQUENCY DOMAINREPRESENTATIONOF SIGNAL
3. As it is clear from fig (1) that time domain representation of signal
gives no idea about the frequency componants of signal it simply
shows that how the amplitude of the signal is varying with respect
to time whereas frequency domain representation of the signal
shows the different frequency componants presents in a signal and
their respective amplitudes but it gives no idea that at which time
instants these frequency componants presents.
In the other word we can say in time domain representation of
signal the time resolution of signal is very high but its frequency
resolution is zero because it gives no idea about different
frequency componants presents in a signal where as in frequency
domain representation of the signal the frequency resolution of the
signal is very high but its time resolution is zero because it gives
no idea about time.
Both domain of signal analysis has its own utilities and has its own
importance and having its own sets of advantages and
disadvantages.
4. HOW FOURIER TRANSFORM CONVERT TIME DOMAIN
SIGNAL INTO FREQUENCY DOMAIN:
Suppose x(t) shows the time domain representation of
signal and X(f) shows the frequency domain signal
Then
Where
e2jpift
= Cos(2*pi*f*t)+J*Sin(2*pi*f*t) …..(3)
Equation (1) and equation (2) gives Fourier transform and inverse
Fourier transform of any signal the exponential term of equation
(1) can be expressed in terms of sine and cosine function shown by
equation(3)
with the help of equation (1) and equation (2) we can convert any
time domain signal into frequency domain signal and frequency
domain signal into time domain signal respectively.
As clear from equation (1) that in Fourier transform the signal is
integrated from – infinity to + infinity over time for each
frequency In the other word we can say that equation 1 take a
frequency for example f1 and search it from –infinity to + infinity
over time if it find the f1 frequency components it simply adds the
magnitude of all f1 frequency components.
Again take an another frequency for example f2 and search it from
– infinity to + infinity over time if it find the f2 frequency
5. components it simply adds the magnitude of all f2 frequency
components
Again repeat the same process with f3 ,f4, f5……. and so on
No matter in time axis where these frequency components exits
from – infinity to + infinity it will effect the result of integration
in the same way
For every frequency fourier transform check that wheather this
perticular frequency componant present or not present in time
from minus infinte to plus infine.And if present then how many
time this perticular frequency componants presents and what is
the amplitude of this perticular frequency componant and then
simply add that perticular frequency componant and calculate the
amplitude of any perticular frequency componant.
Again take a second frequency componant and check that in time
from minus infinite to plus infinite how many times this perticular
frequency componant exists and what is amplitude of this
perticular frequency componant and then simply adds them.In this
way the fourier transform calculated the amplitude of every
frequency componants presents in a given signal and draw a graph
between frequency and amplitude
Again there are certain disadvantage of frequency domain
representation of the signal first disadvantage is that it gives no
idea about time .The frequency transform of any signal simply tells
us that in any given signal what spectral components are present
and what are their respective amplitudes but it gives no idea that
in time axis where these frequency components exists
6. So again the D.F.T. prove its suitability for the signals which are
stationary in nature but this transform is not suitable for non
stationary signal
By stationary signal we simply means the signal in which the
frequency does not change with respect to time or we can say that
all frequency components exits for all the time
By non stationary signal we simply means the signal in which
frequency changes with respect to the time. or in which all the
frequency components does not exits for all the time interval .but
some frequencies are exits for some particular time interval
whereas some other frequency components exists for some other
time interval.
To understand the suitability of the DFT only for stationary signal
and not for non stationary signal take the following example
suppose there are two signals S1 and S2 the signal S1 has three
frequency components f1,f2 and f3 all the times and suppose the
signal S2 contains the same frequency components f1,f2 and f3 but
for the different -different time interval’s so we can say that these
two signals are completely different in nature. But in spite of it the
Fourier transform of these two signals will be the same because
these two signals have the same frequency components of course
one signal contains all the frequency components all the time
whereas second signal contains these frequency components at
different time intervals but as we know that Fourier transform has
nothing to do with the time. No matter where these frequency
components exits over time the matter is only that whether they
occur or not and what are their amplitudes.
7.
8. Again take an another example suppose there are two signals S3
and S4
Signal S3 contains frequencies f1 for time interval t0 to t1,frequency
f2 for time interval t1 to t2,and frequency f3 for time interval t2 to t3.
And signal S4 contain frequency f3 for time interval t0 to t1,
frequency f2 for t1 to t2 and frequency f1 for time interval t2 to t3
So we can say that these two signals are quie different though both
signals are having the same frequency componants in same amount
but the time instances where these frequency componants exists
are different so the overall characteristics of above these two
signals S3 and S4 will be different but inspite of this the Fourier
transform of these two signals will be the same because fourier
transform has nothing to do with time.
Fourier transform simply watch that what frequency componants
any signal has and what are their respective amplitudes
9.
10. 2 SHOT TIME FOURIER TRANSFORM:
The Short time Fourier Transform is a modified version of Fourier
transform. S.T.F.T. is nothing it is simply the Fourier transform of
any signal multiplied by a window function.
STFTX
(w)
(t, f) = ∫t [x(t). w*( t – t')].e-j2Πf t
dt …….6.3
The basic idea behind the STFT is that any non stationary signal
can be considered stationary for a short time interval. So we can
say that the STFT gives an idea about time frequency and
amplitude
But again the problem with STFT is that how to choose the size of
window (time interval of window) because if we choose
A small size window than it will give good time resolution but
poor frequency resolution i.e. it gives good information about the
time but its frequency information is poor.
Again if we choose large size window than it will provide us a
very good frequency information but the time information is poor
again if we choose the large size of window then the signal can not
be considered stationary(because the signal is stationary only for
the short time interval)
Again at the same time we can’t get good time and frequency
resolution either we get good time resolution or good frequency
resolution
The small window size is suitable for high frequencies whereas
Large window size is suitable for low frequencies
11. But the problem with STFT is that the window size remain same
for the all analysis we can’t choose different -different window
Size for the analysis of different frequency components .
As shown in fig that for all frequencies the size of window is same
Once window size is choosen then we can not change the size of
window .And any single window size can not suitable for different
different frequency componants
Again in S.T.F.T. it is very toufh task to choose the size of window
If we choose a narrow window size then it will provide a good
time resolution and bad frequency resolution
If we choose a large window size then it will give good frequency
resolution but poor time resolution ( though we can’t choose very
large otherwise it will against the concept of Short time fourier
transform)
12. Hence we can say that there is some resolution problem with
S.T.F.T. at the same time we can’t obtain both time as well as
frequency resolution means we can not know Exactly at which
perticular time instant which perticular frequency componants
Exists We can know only that in which time intervals(not time
instants) what frequency specturm occurs(not exact frequency)
Again if we want to increase the time resolution (by decresing
window size) then frequency resolution get decreased
And if we want to increased frequency resolution(by increasing
window size)then time resolution get decreased
So there is a trade off between the time resolution and frequency
resolution
13. WAVELET TRANSFORM:
By using wavelet Transform we can overcome the problem
with S.T.F.T. in wavelet transform we used different window
size for different frequency componants.Low scale(small
window size or small time scale) is used for higher frequencies
and higher scale (Large Window size or large time scale) is
used for low frequencies
The main advantage of wavelet transforms are
1. Wavelet transforms has multiresolution properity
2. Better Spectral localization properity
What is Multiresolution properity:
Mutiresolution properity means different frequency componants
presents in any signal are resolved at different scale(different time
scale or different window size)
Scale is inversly proportional to the frequency means small scale is
used for higher frquencies whereas lage scale is used for small
frequencies
Small scale (Small window Size or Small time scale)is used for the
analysis of higher frequency components.Means wavelet transform
provide higher time resolution for high frequency means if any signal
contain a very high frequency componant then with the help of wavelet
transform we can know that at which exact time interval (very small time
interval) these frequency componants exists
whereas the large scale (large window size or large time scale)is used for
the analysis of small frequency components.
Wavelet transform provide good time resolution for higher frequencies
whereas for small frequencies time resolution is not so good
But if we study real word signals then we find that genrally higher
frequencies are occurs only for very short time interval whereas small
frequency componants are presents for long time interval.So wavelet
transform prove its suitability for real time signals.
14. WHAT IS SCALE:
Scale is inversely propertional to the frequency of any signal
large scale is used for the analysis of small frequency componants
presents in any signal Whereas Small scale is used for the analysis of high
frequency componants of any signal
What is Spectral localisation
Spectral localization properity means that wavelet transform tells us that
what frequency componants are present in any given signal and at time
axis where these frequency componants are presents
The Wavelet transform has
multiresolution capability it means it resolve the different-different
15. frequency componants of the signal at different scales.Scale is
inversely propertional to the frequency of the signal it means the
high frequency signals are resolved at low scale whereas low
frequency signals are resolved at high scale because by choosing a
single scale we can not resolve the all frequency componants
present in any signal or in other word if we want to capture every
detail of the signal then we need to resolve the different frequency
componants present in signal at different scale.
To understand the concept of scale take the following example we
can draw the map of glope which shows only where Sea(water)
exist And where earth exists.If we can resolve more this map then
we will be able to see the boundary of each nation.If We can more
resolve this map then we will be able to see the boundaries of
states of each nation.If we can further increased resolution then we
will be able to see the map of major cities of countaries and If we
further increased resolution then we will be able to see the
different sector ,blocks or different area of the city. So as we are
increasing resolution then we are able to know each detail of map
or each detail of locations .Or in the other word we can say that by
choosing a single resolution level we can not capture the every
detail in the map for capturing the different detail of map we need
to resolve the map at different different levels this is called
multiresolution.
Again take an another example
suppose we are taking the photograph of city from the peak of any
mountain then the whole city will look like a dense population area
or look like that the group of several houses again if we zoom our
camera then we can focus any sector of city we can also watch the
roads,garden,electric cables etc.Again if we can further zoom our
camera then we can focus at any perticular home and can watch
that perticular home how many windows are there in any perticular
home or how many doors that perticular home has again we can
focus at the roof of any perticular home and even focus at the face
16. of any persion who is standing at the roof.we can also zoom at the
name-plate infront of any house and can read the name of
people.So by zomming our cemera at different different levels we
can able to capturing the different level of details By choosing a
single zooming level we cant capture every details of any picture
The same thing with wavelet transform in wavelet transform we
resolve each frequency componants at different different scales.
With the help of above equation we can understand that In the
process of taking wavelet Transform what exctly going on first we
choose a perticular frequency suppose f1 and choose a suitalbe
scale for it and search this frequency in time from minus infinity
to plus infinty and keeping record of both amplitude of frequency
as well as time duration where we found that perticular frequency
contents
Then increases the frequency to f2 and accordingly choose a
suitable scale (window size) and search this frequency from minus
infinity to plus infinity in time and keep track of both amplitude of
frequency contents and the time duration where we found this
frequency contents
In this way we collect all the information about any signal what
frequency contents it has what are the amplitude of these frequency
contents and in time where these frequency contents exists
It is just like to watch any signal with a microscope or lens with
different different megnification factor. So by choosing any
17. constant megnification factor we can not wacth all the frequency
componants present in any signal.we need to vary the
megnification factor of the lens of microscope as the frequency of
signal varies.
So we can draw a three dimentional plot between time frequency
and amplitude of the signal
Problem with fourier transform is that its
time resolution is zero means it gives no idea about the time.fourier
transform simply tells us that what different frequency componants
are present in a given signal but it gives no idea that at time axis
where these different frequency componants exists
S.T.F.T. solves the above problem of fourier transform but up to
certain extend S.T.F.T. also provide the time resolution but in
STFT the window size remain constant so the time resolution of
STFT is also constant
In case of wavelet transform the window size is
adjusting means we can analysis the same sagnal at different time
scale and according to the frequencies which are present in any
signal we can choose different different time scale for the analysis
of the different –different frequency componants
Scale is inversely proportional to the fequency so for the analysis
of low frequencies we choose the large time scale whereas for the
analysis of high frequency componants we choose low time scale
The small window size gives High time resolution but poor
frequency resolution
The large window size gives high frequency resolution but bad
time reolution
18. In wavelet transform we choose a small window size for high
frequency so it provide a very high time reolution for higer
frequency
again a large window size is choosen for low frequencies so it
provide good frequency resolution for low frequency
In practice all the practical signals has low frequency componants
for long time duration whereas has high frequency componants for
a very small time duration so wavelet transform is very suitable for
all real word signals.
The main problem with sort time fourier transform is that the size
of window function in STFT is same for the analysis of all spectral
componants of any signal.
So if we choose a small size window then it gives good time
approximation but poor frequency approximation whereas if we
choose a large window size then it gives good frequency
approximation but very poor time approximation again problem
with choosing a large window size is that for long window size a
signal can not be concidred statonary.again we can say that no
single window size is suitable for all the frequency componants
presents in any signal
Above Equation shows the wavelet transfor of any signal It is
clear from the above equation that we can change the scale by
varying the value of s .Now it is also clear from the above equation
that the different different scales are used for the analysis of
different different frequency componants as their suitabily for the
different different frequency componants it is just like to use
different different filters (high pass and low pass filter with
different cut off frequencies) for different different frequency
componants what exact we are doing in the process of taking
wavelet transform we passes the different frequency componants
with a filter of different cut off frequency.
Equation (1) gives us a theoritical approach about the wavelet
transform that for different time interval how we can change the
time scale or how we can change the time scale for the analysis of
19. different frequency componants.Now how we can converted this
theortical approach in to practice
Process of taking Wavelet transform of any signal
In the process of wavelet transform the original signal( S ) is first
decompose into Approximate Cofficients and Detailed Cofficients by
simply passing the signal through low pass filter and high pass filter
respectively.
The output of low pass filter is called Approximate[A1](Low freqency
componants) cofficient of the signal
The output of High pass filter is called Detailed [D1](High freqency
componants) Cofficients Of the signal.
This Approximate cofficient[A1] again passed through a low pass and
high pass filter
And again Decompose the signal into Approximate[A 2] and Detailed
Cofficients[D 2]
Further Approximate Componants[A2] can be decomposed into
Approximate cofficients[A3] And Detailed Cofficients[D3]
The number of Decomposition levels depends on the length of signal and
our requirements
S =A1+D1 [First level Wavelet Decomposition]…(a)
A1=A2+D2 [Second level Wavelet Decomposition]…(b)
A2=A3+D3 [Third level Wavelet Decomposition]…(c)
S=A3+D3+D2+D1 ………………………………….(1)
20. The Origional Signal S can be reconstruct with the help of
A3,D3,D2 and D1.
Fig(a) Wavelet Decomposition of signal
So it is clear that with the help of equation(1) and equation (a),(b),(c)
We can decompose any original signal sequences in to wavelet decomposition.
And with these wavelet decomposition again we can construst the origional
signal.
The number of sample in next decomposition level is half as
compaired to previous stage.Supose The original signal S has N samples then A1
and D1 will have N/2 Samples and A2 and D2 will have N/4 Samples.
Signal(S)
A-1D-1
D-2 A-2
A-3D-3
21. So wavelet transform is highly suitable for the analysis of local
behaviour of the signal such as spikes or discontinuties. Because at
the point of discontinuty the frequencies changes very fast only for
a very littile time so by choosing suitable time scale we can also
study or analysis these sudden changes.
Again an another advantage of wavelet transform over fourier
transform is that the fourier transform convert the signal into the
different sinusoids of different different frequencies
The shape of sine and cosine waves are predefined and
predectable.wheareas in wavelet transform we convert the signal
into the mother wavelets of different amplitude and scale the local
behavour of any signal can be discribed in better way by using
wavelets
Again with W.T. we have a freedom to choose the shape of
wavelets (mother wavelet).there are lot of standard wavelets
families (wavelet families contain different wavelets of different
orders) suitable for different applications.
Again with wavelet transform we have freedom to design our own
wavelet hence we can define our own wavelet by defining Two
functions
[1]Wavelet function:
[2]Scale function:
[1] Wavelet function: wavelet function capture the details(high
frequencies) present in any signal And the intrgation of wavelet
function should be zero or the mean value of wavelet function
should be zero
∫ Ψ(x).d(x)=0
[2] Scale function: Scale function capture the low frequencies
information(approximate ) presents in any signal.the intregation of
scale function should be one it means its average value is one.
∫Ǿ(x).d(x)=1
22. WAVELETS FAMILIES:
Some standard wavelet families are
Family name short name
Haar haar
Daubechies db
Symlets sym
Coiflets coif
BiorSplines bior
ReverseBior rbio
Meyer meyr
Dmeyer dmey
Gaussian gaus
Mexican_hat mexh
Morlet morl
Complex Gaussian cgau
Shannon shan
Frequency B-Spline fbsp
Complex Morlet cmor
Different wavelet of standard families
S.N. Standard Wavelet
family
Different wavelet
1 Haar Haar
2 Daubechies db1 db2 db3 db4 db5 db6 db7 db8
db9 db10
3 Symlets sym sym2 sym3 sym4 sym5 sym6 sym7
23. sym8
4 Coiflets coif1 coif2 coif3 coif4 coif5
5 BiorSplines bior1.1 bior1.3 bior1.5 bior2.2
bior2.4 bior2.6 bior2.8 bior3.1
bior3.3 bior3.5 bior3.7 bior3.9
bior4.4 bior5.5 bior6.8
6 ReverseBior rbio1.1 rbio1.3 rbio1.5 rbio2.2
rbio2.4 rbio2.6 rbio2.8 rbio3.1
rbio3.3 rbio3.5 rbio3.7 rbio3.9 rbio4.4
rbio5.5 rbio6.8
7 Meyer Meyr
8 DMeyer Dmey
9 Gaussian gaus1 gaus2 gaus3 gaus4
gaus5 gaus6 gaus7 gaus8
10 Mexican_hat Mexh
11 Morlet Morl
12 Complex Gaussian cgau1 cgau2 cgau3 cgau4
cgau5
13 Shannon shan1-1.5 shan1-1 shan1-0.5 shan1-0.1
shan2-3
14 Frequency B-
Spline
fbsp1-1-1.5 fbsp1-1-1 fbsp1-1-0.5 fbsp2-
1-1
fbsp2-1-0.5 fbsp2-1-0.1
15 Complex Morlet cmor1-1.5 cmor1-1 cmor1-0.5 cmor1-1
cmor1-0.5 cmor1-0.1
The choice of any perticular wavelet depend on that perticular
application and property of wavelet. The most common properties
of any wavelets are The support of scaling and wavelet
function(with compact support or without compact
support),Regularity,Orthoganality or Biorthogobnality,Number of
zero moment,Wavelet with FIR filter or without FIR filter etc
24. [1]the support of wavelet function Ψ(t) and scaling function Ǿ(x):-
This is the most important criteria that the wavelet function and
scaling function have compact support or not have compact
support.This properity of wavelet decides the localisation of
wavelet in time and frequency domain.the support of wavelet
function decides
the speed of convergence to 0 of wavelet function (Ψ(t) or Ψ(w))
when the time t or the frequency w goes to infinity.
[2] wavelet with F.I.R. filter or without F.I.R. filter:
[3] Symmetry of wavelets: wavelet is symmetric,near symmetric or
asymetric
[4] Orthogonality or biorthogonality
[5] Regularity of wavelet wich decide the smoothness of
reconstructing signal or image is a very important criteria.
[6] the number of vanishing moment (zero moment)for wavelet
function Ψ or scaling function Ǿ this is very useful for
compression purpose.
[7] the scaling function exist or does not exist
[8] an explict mathamatical expression available or not for scaling
function (if exist) and wavelet function
[9]Continue or discreate
25. Wavelet with filters Wavelet without filter
Classification of wavelet families
With compact
Support
Without compact
Support
Orthogonal Bi-Orthogonal Orthogonal
Real Complex
Db
Haar
Sym
Coif
Bior Meyr
Dmey
btlm
Gaus
Mexh
morl
Cgau
Shan
Fbsp
cmor