Signal processing

Signal processing is a subfield of mathematics and electrical engineering that concerns the analysis and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, subjective quality, to emphasize or detect components of interest in a measured signal. According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. Analog signal processing is for signals that have not been digitized, as in legacy radio, telephone and television systems; this involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers and delay lines.

Non-linear circuits include compandors, voltage-controlled filters, voltage-controlled oscillators and phase-locked loops. Continuous-time signal processing is for signals; the methods of signal processing include time domain, frequency domain, complex frequency domain. This technology discusses the modeling of linear time-invariant continuous system, integral of the system's zero-state response, setting up system function and the continuous time filtering of deterministic signals Discrete-time signal processing is for sampled signals, defined only at discrete points in time, as such are quantized in time, but not in magnitude. Analog discrete-time signal processing is a technology based on electronic devices such as sample and hold circuits, analog time-division multiplexers, analog delay lines and analog feedback shift registers; this technology was a predecessor of digital signal processing, is still used in advanced processing of gigahertz signals. The concept of discrete-time signal processing refers to a theoretical discipline that establishes a mathematical basis for digital signal processing, without taking quantization error into consideration.

Digital signal processing is the processing of digitized discrete-time sampled signals. Processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued and addition. Other typical operations supported by the hardware are circular buffers and lookup tables. Examples of algorithms are the Fast Fourier transform, finite impulse response filter, Infinite impulse response filter, adaptive filters such as the Wiener and Kalman filters. Nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains. Nonlinear systems can produce complex behaviors including bifurcations, chaos and subharmonics which cannot be produced or analyzed using linear methods. Statistical signal processing is an approach which treats signals as stochastic processes, utilizing their statistical properties to perform signal processing tasks.

Statistical techniques are used in signal processing applications. For example, one can model the probability distribution of noise incurred when photographing an image, construct techniques based on this model to reduce the noise in the resulting image. Audio signal processing – for electrical signals representing sound, such as speech or music Speech signal processing – for processing and interpreting spoken words Image processing – in digital cameras and various imaging systems Video processing – for interpreting moving pictures Wireless communication – waveform generations, filtering, equalization Control systems Array processing – for processing signals from arrays of sensors Process control – a variety of signals are used, including the industry standard 4-20 mA current loop Seismology Financial signal processing – analyzing financial data using signal processing techniques for prediction purposes. Feature extraction, such as image understanding and speech recognition. Quality improvement, such as noise reduction, image enhancement, echo cancellation.

Including audio compression, image compression, video compression. Genomics, Genomic signal processing In communication systems, signal processing may occur at: OSI layer 1 in the seven layer OSI model, the Physical Layer. Filters – for example analog or digital Samplers and analog-to-digital converters for signal acquisition and reconstruction, which involves measuring a physical signal, storing or transferring it as digital signal, later rebuilding the original signal or an approximation thereof. Signal compressors Digital signal processors Differential equations Recurrence relation Transform theory Time-frequency analysis – for processing non-stationary signals Spectral estimation – for determining the spectral content of a

Mathematics

Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.

The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to