Wavelet transformIn mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. A function is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space of square integrable functions.
WaveletA wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second.
Discrete wavelet transformIn numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time). Haar wavelet The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum.
Filter bankIn signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, each one carrying a single frequency sub-band of the original signal. One application of a filter bank is a graphic equalizer, which can attenuate the components differently and recombine them into a modified version of the original signal.
Error correction codeIn computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code or error correcting code (ECC). The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors.
Signal processingSignal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, , potential fields, seismic signals, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. According to Alan V. Oppenheim and Ronald W.
Continuous wavelet transformIn mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. The continuous wavelet transform of a function at a scale (a>0) and translational value is expressed by the following integral where is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate.
Signal transmissionIn telecommunications, transmission is the process of sending or propagating an analog or digital signal via a medium that is wired, wireless, or fiber-optic. Transmission system technologies typically refer to physical layer protocol duties such as modulation, demodulation, line coding, equalization, error control, bit synchronization and multiplexing, but it may also involve higher-layer protocol duties, for example, digitizing an analog signal, and data compression.
Data compressionIn information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information.
Reliability (computer networking)In computer networking, a reliable protocol is a communication protocol that notifies the sender whether or not the delivery of data to intended recipients was successful. Reliability is a synonym for assurance, which is the term used by the ITU and ATM Forum. Reliable protocols typically incur more overhead than unreliable protocols, and as a result, function more slowly and with less scalability. This often is not an issue for unicast protocols, but it may become a problem for reliable multicast protocols.
Haar waveletIn mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar.
Noisy-channel coding theoremIn information theory, the noisy-channel coding theorem (sometimes Shannon's theorem or Shannon's limit), establishes that for any given degree of noise contamination of a communication channel, it is possible to communicate discrete data (digital information) nearly error-free up to a computable maximum rate through the channel. This result was presented by Claude Shannon in 1948 and was based in part on earlier work and ideas of Harry Nyquist and Ralph Hartley.
Digital signal processingDigital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.
Image compressionImage compression is a type of data compression applied to s, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior results compared with generic data compression methods which are used for other digital data. Image compression may be lossy or lossless. Lossless compression is preferred for archival purposes and often for medical imaging, technical drawings, clip art, or comics.
Lossless compressionLossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. Lossless compression is possible because most real-world data exhibits statistical redundancy. By contrast, lossy compression permits reconstruction only of an approximation of the original data, though usually with greatly improved compression rates (and therefore reduced media sizes).
Fourier transformIn physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.
Coding theoryCoding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science—for the purpose of designing efficient and reliable data transmission methods.
Transmission systemSee Transmission (mechanics) for a car's transmission system In telecommunications, a transmission system is a system that transmits a signal from one place to another. The signal can be an electrical, optical or radio signal. The goal of a transmission system is to transmit data accurately and efficiently from point A to point B over a distance, using a variety of technologies such as copper cable and fiber optic cables, satellite links, and wireless communication technologies.
Digital signal processorA digital signal processor (DSP) is a specialized microprocessor chip, with its architecture optimized for the operational needs of digital signal processing. DSPs are fabricated on MOS integrated circuit chips. They are widely used in audio signal processing, telecommunications, , radar, sonar and speech recognition systems, and in common consumer electronic devices such as mobile phones, disk drives and high-definition television (HDTV) products. The goal of a DSP is usually to measure, filter or compress continuous real-world analog signals.
Data communicationData communication or digital communications, including data transmission and data reception, is the transfer and reception of data in the form of a digital bitstream or a digitized analog signal transmitted over a point-to-point or point-to-multipoint communication channel. Examples of such channels are copper wires, optical fibers, wireless communication using radio spectrum, storage media and computer buses. The data are represented as an electromagnetic signal, such as an electrical voltage, radiowave, microwave, or infrared signal.
