1.
Non-sinusoidal waveform
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Non-sinusoidal waveforms are waveforms that are not pure sine waves. They are usually derived from simple math functions, while a pure sine consists of a single frequency, non-sinusoidal waveforms can be described as containing multiple sine waves of different frequencies. These component sine waves will be whole number multiples of a fundamental or lowest frequency, the frequency and amplitude of each component can be found using a mathematical technique known as Fourier analysis. Non-sinusoidal waveforms are important in, for example, mathematics, music, examples of non-sinusoidal waveforms include square waves, rectangular waves, triangle waves, spiked waves, trapezoidal waves and sawtooth waves
2.
Saw
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A saw is a tool consisting of a tough blade, wire, or chain with a hard toothed edge. It is used to cut material, very often wood. The cut is made by placing the edge against the material and moving it forcefully forth. This force may be applied by hand, or powered by steam, water, an abrasive saw has a powered circular blade designed to cut through metal. Abrasive saw, A saw that cuts with a disc or band. Back, the edge opposite the toothed edge, fleam, The angle of the faces of the teeth relative to a line perpendicular to the face of the saw. Gullet, The valley between the points of the teeth, heel, The end closest to the handle. For example, a blade can cause excessive wobble, creating a wider-than-expected kerf. The kerf created by a blade can be changed by adjusting the set of its teeth with a tool called a saw tooth setter. Points per inch, The most common measurement of the frequency of teeth on a saw blade. It is taken by setting the tip of one tooth at the point on a ruler. There is always one point per inch than there are teeth per inch. Some saws do not have the number of teeth per inch throughout their entire length. Those with more teeth per inch at the toe are described as having incremental teeth, rake, The angle of the front face of the tooth relative to a line perpendicular to the length of the saw. In most modern serrated saws, the teeth are set, so that the kerf will be wider than the blade itself and this allows the blade to move through the cut easily without binding. The set may be different depending on the kind of cut the saw is intended to make, for example, a rip saw has a tooth set that is similar to the angle used on a chisel, so that it rips or tears the material apart. A flush-cutting saw has no set on one side, so that the saw can be flat on a surface. The set of the teeth can be adjusted with a tool called a saw set
3.
Rake angle
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Rake angle is a parameter used in various cutting and machining processes, describing the angle of the cutting face relative to the work. There are two angles, namely the back rake angle and side rake angle, both of which help to guide chip flow. There are three types of angles, positive, negative, and zero. Generally, positive rake angles, Make the tool more sharp and this reduces the strength of the tool, as the small included angle in the tip may cause it to chip away. Reduce cutting forces and power requirements, helps in the formation of continuous chips in ductile materials. Can help avoid the formation of a built-up edge, negative rake angles, by contrast, Make the tool more blunt, increasing the strength of the cutting edge. Can increase friction, resulting in higher temperatures, a zero rake angle is the easiest to manufacture, but has a larger crater wear when compared to positive rake angle as the chip slides over the rake face. Recommended rake angles can vary depending on the material being cut, tool material, depth of cut, cutting speed, machine and this table summarizes recommended rake angles for single-point turning on a lathe, rake angles for drilling, milling, or sawing are often different
4.
Triangle wave
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A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function, like a square wave, the triangle wave contains only odd harmonics, demonstrating odd symmetry. However, the higher harmonics roll off much faster than in a square wave
5.
Piecewise linear function
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In mathematics, a piecewise linear function is a real-valued function defined on the real numbers or a segment thereof, whose graph is composed of straight-line sections. It is a function, each of whose pieces is an affine function. Usually – but not always – the function is assumed to be continuous, in that case, since the graph of a linear function is a line, the graph of a piecewise linear function consists of line segments and rays. Other examples of linear functions include the absolute value function, the square wave, the sawtooth function. An approximation to a curve can be found by sampling the curve. An algorithm for computing the most significant points subject to a given error tolerance has been published, if partitions are already known, linear regression can be performed independently on these partitions. However, continuity is not preserved in that case, a stable algorithm with this case has been derived. If partitions are not known, the sum of squares can be used to choose optimal separation points. A variant of decision tree learning called model trees learns piecewise linear functions, the notion of a piecewise linear function makes sense in several different contexts. In each case, the function may be real-valued, or it may take values from a space, an affine space. In dimensions higher than one, it is common to require the domain of each piece to be a polygon or polytope and this guarantees that the graph of the function will be composed of polygonal or polytopal pieces. Important sub-classes of piecewise linear functions include the continuous linear functions. In general, for every n dimensional continuous piecewise linear function f, R n → R, there is a Π ∈ P such that, f = min Σ ∈ Π max ∈ Σ a → ⋅ x → + b. If f is convex as well as continuous, then there is a Σ ∈ P such that, splines generalize piecewise linear functions to higher-order polynomials, which are in turn contained in the category of piecewise-differentiable functions, PDIFF. Piecewise constant function Linear interpolation Spline interpolation Tropical geometry Polygonal chain Apps, P. Long, N. & Rees, journal of Public Economic Theory,16, 523-545
6.
Floor function
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In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, the language APL uses ⌊x, other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets, the ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard symbols, uses >. for ceiling. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\. x[, the fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. Unicode contains codepoints for these symbols at U+2308–U+230B, ⌈x⌉, ⌊x⌋, in the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉
7.
Frequency
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Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
8.
Phase (waves)
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Phase is the position of a point in time on a waveform cycle. A complete cycle is defined as the interval required for the waveform to return to its initial value. The graphic to the right shows how one cycle constitutes 360° of phase, the graphic also shows how phase is sometimes expressed in radians, where one radian of phase equals approximately 57. 3°. Phase can also be an expression of relative displacement between two corresponding features of two waveforms having the same frequency, in sinusoidal functions or in waves phase has two different, but closely related, meanings. One is the angle of a sinusoidal function at its origin and is sometimes called phase offset or phase difference. Another usage is the fraction of the cycle that has elapsed relative to the origin. Phase shift is any change that occurs in the phase of one quantity and this symbol, φ is sometimes referred to as a phase shift or phase offset because it represents a shift from zero phase. For infinitely long sinusoids, a change in φ is the same as a shift in time, if x is delayed by 14 of its cycle, it becomes, x = A ⋅ cos = A ⋅ cos whose phase is now φ − π2. It has been shifted by π2 radians, Phase difference is the difference, expressed in degrees or time, between two waves having the same frequency and referenced to the same point in time. Two oscillators that have the frequency and no phase difference are said to be in phase. Two oscillators that have the frequency and different phases have a phase difference. The amount by which such oscillators are out of phase with each other can be expressed in degrees from 0° to 360°, if the phase difference is 180 degrees, then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then interference will occur. It is common for waves of electromagnetic, acoustic or other energy to become superposed in their transmission medium, when that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes, time is sometimes used to express position within the cycle of an oscillation. A phase difference is analogous to two athletes running around a track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time, but the time difference between them is a constant - same for every pass since they are at the same speed and in the same direction. If they were at different speeds, the difference is undefined
9.
Harmonic
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A harmonic is any member of the harmonic series, the divergent infinite series. Every term of the series after the first is the mean of the neighboring terms. The phrase harmonic mean likewise derives from music, the term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves, a harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the frequency, the sum of harmonics is also periodic at that frequency. On strings, harmonics that are bowed have a glassy, pure tone, harmonics may also be called overtones, partials or upper partials. In some music contexts, the harmonic, overtone and partial are used fairly interchangeably. Most acoustic instruments emit complex tones containing many individual partials, rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials. Oscillators that produce harmonic partials behave somewhat like one-dimensional resonators, and are long and thin. Wind instruments whose air column is open at one end, such as trumpets and clarinets. However they only produce partials matching the odd harmonics, at least in theory, the reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials, antique singing bowls are known for producing multiple harmonic partials or multiphonics. An overtone is any partial higher than the lowest partial in a compound tone, the relative strengths and frequency relationships of the component partials determine the timbre of an instrument. This chart demonstrates how the three types of names are counted, In many musical instruments, it is possible to play the upper harmonics without the note being present. In a simple case this has the effect of making the note go up in pitch by an octave, in some cases it also changes the timbre of the note. This is part of the method of obtaining higher notes in wind instruments. The extended technique of playing multiphonics also produces harmonics, on string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch
10.
Fundamental frequency
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The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as f0, in other contexts, it is more common to abbreviate it as f1, the first harmonic. All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. The period of a waveform is the T for which the equation is true. This means that this equation and a definition of the values over any interval of length T is all that is required to describe the waveform completely. Every waveform may be described using any multiple of this period, there exists a smallest period over which the function may be described completely and this period is the fundamental period. The fundamental frequency is defined as its reciprocal, f 0 =1 T Since the period is measured in units of time, when the time units are seconds, the frequency is in s −1, also known as Hertz. For a tube of length L with one end closed and the end open the wavelength of the fundamental harmonic is 4 L. If the ends of the tube are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2 L. By the same method as above, the frequency is found to be f 0 = v 2 L. At 20 °C the speed of sound in air is 343 m/s and this speed is temperature dependent and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature. The fundamental may be created by vibration over the length of a string or air column. The fundamental is one of the harmonics, a harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The reason a fundamental is also considered a harmonic is because it is 1 times itself, the fundamental is the frequency at which the entire wave vibrates. Overtones are other components present at frequencies above the fundamental. All of the components that make up the total waveform, including the fundamental
11.
Stick-slip phenomenon
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The stick-slip phenomenon, also known as the slip-stick phenomenon or simply stick-slip, is the spontaneous jerking motion that can occur while two objects are sliding over each other. Below is a simple, heuristic description of stick-slip phenomena using classical mechanics that is relevant for engineering descriptions, however, in actuality, there is little consensus in academia regarding the actual physical description of stick-slip which follows the lack of understanding about friction phenomena in general. The stiffness of the spring, the load at the interface, the duration of time the interface has existed. A description using common phonons provides explanations for noise that generally accompanies stick-slip through surface acoustic waves, the use of complicated constitutive models that lead to discontinuous solutions end up requiring unnecessary mathematical effort and do not represent the true physical description of the system. However, such models are useful for low fidelity simulations. Engineering description Stick-slip can be described as alternating between sticking to each other and sliding over each other, with a corresponding change in the force of friction. Typically, the friction coefficient between two surfaces is larger than the kinetic friction coefficient. If an applied force is enough to overcome the static friction. The attached picture shows symbolically an example of stick-slip, V is a drive system, R is the elasticity in the system, and M is the load that is lying on the floor and is being pushed horizontally. The load starts sliding and the friction coefficient decreases from its value to its dynamic value. At this moment the spring can give power and accelerates M. During M’s movement, the force of the spring decreases, until it is insufficient to overcome the dynamic friction, from this point, M decelerates to a stop. The drive system however continues, and the spring is loaded again etc, examples of stick-slip can be heard from hydraulic cylinders, tractor wet brakes, honing machines etc. Special dopes can be added to the fluid or the cooling fluid to overcome or minimize the stick-slip effect. Stick-slip is also experienced in lathes, mill centres, and other machinery where something slides on a slideway, slideway oils typically list prevention of stick-slip as one of their features. Other examples of the phenomenon include the music that comes from bowed instruments, the noise of car brakes and tires. Stick-slip also has been observed in articular cartilage in mild loading and sliding conditions, another example of the stick-slip phenomenon occurs when musical notes are played with a glass harp by rubbing a wet finger along the rim of a crystal wine glass. One animal that produces sound using stick-slip friction is the spiny lobster which rubs its antennae over smooth surfaces on its head, another, more common example which produces sound using stick-slip friction is the grasshopper
12.
Additive synthesis
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Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together. The timbre of instruments can be considered in the light of Fourier theory to consist of multiple harmonic or inharmonic partials or overtones. Each partial is a wave of different frequency and amplitude that swells. Additive synthesis most directly generates sound by adding the output of multiple sine wave generators, alternative implementations may use pre-computed wavetables or the inverse Fast Fourier transform. These sinusoids are called harmonics, overtones, or generally, partials, in general, a Fourier series contains an infinite number of sinusoidal components, with no upper limit to the frequency of the sinusoidal functions and includes a DC component. Frequencies outside of the audible range can be omitted in additive synthesis. As a result, only a number of sinusoidal terms with frequencies that lie within the audible range are modeled in additive synthesis. A waveform or function is said to be periodic if y = y for all t, the bandwidth of r k should be significantly less than f 0. Additive synthesis can also produce sounds in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency. While many conventional musical instruments have harmonic partials, some have inharmonic partials, inharmonic additive synthesis can be described as y = ∑ k =1 K r k cos , where f k is the constant frequency of k th partial. In the general case, the frequency of a sinusoid is the derivative of the argument of the sine or cosine function. If this frequency is represented in hertz, rather than in angular frequency form and this is the case whether the partial is harmonic or inharmonic and whether its frequency is constant or time-varying. For example, F. Richard Moore listed additive synthesis as one of the four categories of sound synthesis alongside subtractive synthesis, nonlinear synthesis. In this broad sense, pipe organs, which also have pipes producing non-sinusoidal waveforms, summation of principal components and Walsh functions have also been classified as additive synthesis. Modern-day implementations of additive synthesis are mainly digital, Additive synthesis can be implemented using a bank of sinusoidal oscillators, one for each partial. Group additive synthesis is a method to group partials into harmonic groups, an inverse Fast Fourier transform can be used to efficiently synthesize frequencies that evenly divide the transform period or frame. It is possible to analyze the components of a recorded sound giving a sum of sinusoids representation. This representation can be re-synthesized using additive synthesis, one method of decomposing a sound into time varying sinusoidal partials is short-time Fourier transform -based McAulay-Quatieri Analysis
13.
Fourier series
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In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a set of simple oscillating functions, the discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1, Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem. The study of Fourier series is a branch of Fourier analysis, the Mémoire introduced Fourier analysis, specifically Fourier series. Through Fouriers research the fact was established that a function can be represented by a trigonometric series. The first announcement of this discovery was made by Fourier in 1807. The heat equation is a differential equation. These simple solutions are now sometimes called eigensolutions, Fouriers idea was to model a complicated heat source as a superposition of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series, from a modern point of view, Fouriers results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fouriers results with greater precision, in this section, s denotes a function of the real variable x, and s is integrable on an interval, for real numbers x0 and P. We will attempt to represent s in that interval as a sum, or series. Outside the interval, the series is periodic with period P and it follows that if s also has that property, the approximation is valid on the entire real line. We can begin with a summation, s N = A02 + ∑ n =1 N A n ⋅ sin . S N is a function with period P. The inverse relationships between the coefficients are, A n = a n 2 + b n 2 ϕ n = atan2 , when the coefficients are computed as follows, s N approximates s on, and the approximation improves as N → ∞. The infinite sum, s ∞, is called the Fourier series representation of s, both components of a complex-valued function are real-valued functions that can be represented by a Fourier series. This is the formula as before except cn and c−n are no longer complex conjugates. In particular, the Fourier series converges absolutely and uniformly to s whenever the derivative of s is square integrable, if a function is square-integrable on the interval, then the Fourier series converges to the function at almost every point
14.
Nyquist frequency
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The Nyquist frequency, named after electronic engineer Harry Nyquist, is half of the sampling rate of a discrete signal processing system. It is sometimes known as the frequency of a sampling system. An example of folding is depicted in Figure 1, where fs is the rate and 0.5 fs is the corresponding Nyquist frequency. The black dot plotted at 0.6 fs represents the amplitude, the other three dots indicate the frequencies and amplitudes of three other sinusoids that would produce the same set of samples as the actual sinusoid that was sampled. The symmetry about 0.5 fs is referred to as folding, the Nyquist frequency should not be confused with the Nyquist rate, which is the minimum sampling rate that satisfies the Nyquist sampling criterion for a given signal or family of signals. The Nyquist rate is twice the maximum component frequency of the function being sampled, for example, the Nyquist rate for the sinusoid at 0.6 fs is 1.2 fs, which means that at the fs rate, it is being undersampled. Thus, Nyquist rate is a property of a continuous-time signal, when the function domain is time, sample rates are usually expressed in samples/second, and the unit of Nyquist frequency is cycles/second. When the function domain is distance, as in a sampling system, the sample rate might be dots per inch. Referring again to Figure 1, undersampling of the sinusoid at 0.6 fs is what allows there to be a lower-frequency alias and that condition is usually described as aliasing. Samples of a pure 0.6 fs sinusoid would produce a 0.4 fs sinusoid instead. If the true frequency was 0.4 fs, there would still be aliases at 0.6,1.4,1.6, etc. but the reconstructed frequency would be correct. In a typical application of sampling, one first chooses the highest frequency to be preserved and recreated, based on the expected content, then one inserts an anti-aliasing filter ahead of the sampler. Its job is to attenuate the frequencies above that limit, finally, based on the characteristics of the filter, one chooses a sample-rate that will provide an acceptably small amount of aliasing. In applications where the sample-rate is pre-determined, the filter is based on the Nyquist frequency. For example, audio CDs have a rate of 44100 samples/sec. The Nyquist frequency is therefore 22050 Hz, the anti-aliasing filter must adequately suppress any higher frequencies but negligibly affect the frequencies within the human hearing range. A filter that preserves 0–20 kHz is more adequate for that. Early uses of the term Nyquist frequency, such as those cited above, are all consistent with the definition presented in this article, aliasing Kell factor Sampling frequency Superoscillation Signal
15.
Sampling (signal processing)
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In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a wave to a sequence of samples. A sample is a value or set of values at a point in time and/or space, a sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the value of the continuous signal at the desired points. Sampling can be done for functions varying in space, time, or any other dimension, then the sampled function is given by the sequence, s, for integer values of n. The sampling frequency or sampling rate, fs, is the number of samples obtained in one second. Reconstructing a continuous function from samples is done by interpolation algorithms, the Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated by the sample values. When the time interval between adjacent samples is a constant, the sequence of functions is called a Dirac comb. Mathematically, the modulated Dirac comb is equivalent to the product of the function with s. That purely mathematical abstraction is sometimes referred to as impulse sampling, most sampled signals are not simply stored and reconstructed. But the fidelity of a reconstruction is a customary measure of the effectiveness of sampling. That fidelity is reduced when s contains frequency components whose periodicity is smaller than 2 samples, the quantity ½ cycles/sample × fs samples/sec = fs/2 cycles/sec is known as the Nyquist frequency of the sampler. Therefore, s is usually the output of a lowpass filter, without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process. In practice, the signal is sampled using an analog-to-digital converter. This results in deviations from the theoretically perfect reconstruction, collectively referred to as distortion, various types of distortion can occur, including, Aliasing. Some amount of aliasing is inevitable because only theoretical, infinitely long, Aliasing can be made arbitrarily small by using a sufficiently large order of the anti-aliasing filter. Aperture error results from the fact that the sample is obtained as a time average within a sampling region, in a capacitor-based sample and hold circuit, aperture error is introduced because the capacitor cannot instantly change voltage thus requiring the sample to have non-zero width. Jitter or deviation from the precise sample timing intervals, noise, including thermal sensor noise, analog circuit noise, etc
16.
Fast Fourier transform
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A fast Fourier transform algorithm computes the discrete Fourier transform of a sequence, or its inverse. Fourier analysis converts a signal from its domain to a representation in the frequency domain. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from O, which if one simply applies the definition of DFT, to O. Fast Fourier transforms are used for many applications in engineering, science. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805, the DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields but computing it directly from the definition is too slow to be practical. The difference in speed can be enormous, especially for data sets where N may be in the thousands or millions. In practice, the time can be reduced by several orders of magnitude in such cases. The best-known FFT algorithms depend upon the factorization of N, but there are FFTs with O complexity for all N, even for prime N. Since the inverse DFT is the same as the DFT, but with the sign in the exponent. The development of fast algorithms for DFT can be traced to Gausss unpublished work in 1805 when he needed it to interpolate the orbit of asteroids Pallas and Juno from sample observations. His method was similar to the one published in 1965 by Cooley and Tukey. While Gausss work predated even Fouriers results in 1822, he did not analyze the computation time, between 1805 and 1965, some versions of FFT were published by other authors. Yates in 1932 published his version called interaction algorithm, which provided efficient computation of Hadamard, yates algorithm is still used in the field of statistical design and analysis of experiments. In 1942, Danielson and Lanczos published their version to compute DFT for x-ray crystallography, Cooley and Tukey published a more general version of FFT in 1965 that is applicable when N is composite and not necessarily a power of 2. To analyze the output of these sensors, a fast Fourier transform algorithm would be needed, garwin gave Tukeys idea to Cooley for implementation. Cooley and Tukey published the paper in a short six months
17.
Bandlimiting
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Bandlimiting is the limiting of a signals Fourier transform or power spectral density to zero above a certain finite frequency. A band-limited signal is one whose Fourier transform or power density has bounded support. The signal may be random or non-random. A bandlimited signal can be reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency in the bandlimited signal. This minimum sampling frequency is called the Nyquist rate and this result, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem. An example of a simple deterministic bandlimited signal is a sinusoid of the form x = sin . If this signal is sampled at a rate f s =1 T >2 f so that we have the samples x, for all integers n, similarly, sums of sinusoids with different frequencies and phases are also bandlimited to the highest of their frequencies. The signal whose Fourier transform is shown in the figure is also bandlimited, suppose x is a signal whose Fourier transform is X, the magnitude of which is shown in the figure. The highest frequency component in x is B, as a result, the Nyquist rate is R N =2 B or twice the highest frequency component in the signal, as shown in the figure. A bandlimited signal cannot be also timelimited, more precisely, a function and its Fourier transform cannot both have finite support. This fact can be proved by using complex analysis and properties of Fourier transform, proof, Assume that a signal f which has finite support in both domains exists. Lets sample it faster than the Nyquist frequency, and compute respective Fourier transform F T = F1, according to properties of DTFT, F2 = ∑ n = − ∞ + ∞ F1, where f x is the frequency used for discretization. If f is bandlimited, F1 is zero outside of an interval, so with large enough f x, F2 will be zero in some intervals too. According to DTFT definition, F2 is a sum of functions, and since f is time-limited. All trigonometric polynomials are holomorphic on a complex plane. But this contradicts our earlier finding that F2 has intervals full of zeros, thus the only time- and bandwidth-limited signal is a constant zero. All real-world signals are, by necessity, timelimited, which means that they cannot be bandlimited, nevertheless, the concept of a bandlimited signal is a useful idealization for theoretical and analytical purposes. Furthermore, it is possible to approximate a bandlimited signal to any level of accuracy desired
18.
Aliasing
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In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable when sampled. It also refers to the distortion or artifact that results when the signal reconstructed from samples is different from the continuous signal. Aliasing can occur in signals sampled in time, for digital audio. Aliasing can also occur in spatially sampled signals, for instance moiré patterns in digital images, aliasing in spatially sampled signals is called spatial aliasing. Aliasing is generally avoided by applying low pass filters- anti-aliasing filters to the signal before sampling. When a digital image is viewed, a reconstruction is performed by a display or printer device, and by the eyes and the brain. If the image data is processed in some ways during sampling or reconstruction, the image will differ from the original image. An example of aliasing is the moiré pattern observed in a poorly pixelized image of a brick wall. Spatial anti-aliasing techniques avoid such poor pixelizations, aliasing can be caused either by the sampling stage or the reconstruction stage, these may be distinguished by calling sampling aliasing prealiasing and reconstruction aliasing postaliasing. Temporal aliasing is a concern in the sampling of video. Music, for instance, may contain components that are inaudible to humans. If a piece of music is sampled at 32000 samples per second, to prevent this an anti-aliasing filter is used to remove components above the Nyquist frequency prior to sampling. In video or cinematography, temporal aliasing results from the frame rate. Aliasing has changed its apparent frequency of rotation, a reversal of direction can be described as a negative frequency. Like the video camera, most sampling schemes are periodic, that is, digital cameras provide a certain number of samples per degree or per radian, or samples per mm in the focal plane of the camera. Audio signals are sampled with a converter, which produces a constant number of samples per second. Some of the most dramatic and subtle examples of aliasing occur when the signal being sampled also has periodic content, actual signals have finite duration and their frequency content, as defined by the Fourier transform, has no upper bound. Some amount of aliasing always occurs when such functions are sampled, functions whose frequency content is bounded have infinite duration
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A440 (pitch standard)
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A440 or A4, which has a frequency of 440 Hz, is the musical note A above middle C and serves as a general tuning standard for musical pitch. Prior to the standardization on 440 Hz, many countries and organizations followed the French standard since the 1860s of 435 Hz, which had also been the Austrian governments 1885 recommendation. Johann Heinrich Scheibler recommended A440 as a standard in 1834 after inventing the tonometer to measure pitch, the American music industry reached an informal standard of 440 Hz in 1926, and some began using it in instrument manufacturing. In 1936 the American Standards Association recommended that the A above middle C be tuned to 440 Hz and this standard was taken up by the International Organization for Standardization in 1955 as ISO16. It is designated A4 in scientific pitch notation because it occurs in the octave that starts with the fourth C key on a standard 88-key piano keyboard, A440 is widely used as concert pitch in the United Kingdom and the United States. In continental Europe the frequency of A4 commonly varies between 440 Hz and 444 Hz, in the period instrument movement, a consensus has arisen around a modern baroque pitch of 415 Hz, baroque for some special church music at 466 Hz, and classical pitch at 430 Hz. A440 is often used as a reference in just intonation regardless of the fundamental note or key. The US time and frequency station WWV broadcasts a 440 Hz signal at two minutes past every hour, with WWVH broadcasting the same tone at the first minute past every hour and this was added in 1936 to aid orchestras in tuning their instruments. History of pitch standards in Western music Concert pitch Pitch Electronic tuner A
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Switched-mode power supply
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A switched-mode power supply is an electronic power supply that incorporates a switching regulator to convert electrical power efficiently. Like other power supplies, an SMPS transfers power from a DC or AC source, to DC loads, such as a computer, while converting voltage. Ideally, a power supply dissipates no power. Voltage regulation is achieved by varying the ratio of on-to-off time, in contrast, a linear power supply regulates the output voltage by continually dissipating power in the pass transistor. This higher power efficiency is an important advantage of a switched-mode power supply. Switched-mode power supplies may also be smaller and lighter than a linear supply due to the smaller transformer size. Switching regulators are used as replacements for linear regulators when higher efficiency and they are, however, more complicated, their switching currents can cause electrical noise problems if not carefully suppressed, and simple designs may have a poor power factor. 1836 Induction coils use switches to generate high voltages,1910 An inductive discharge ignition system invented by Charles F. Kettering and his company Dayton Engineering Laboratories Company goes into production for Cadillac. The Kettering ignition system is a version of a flyback boost converter. Variations of this system were used in all non-diesel internal combustion engines until the 1960s when it was displaced with capacitive discharge ignition systems. 1926 On 23 June, British inventor Philip Ray Coursey applies for a patent in his country and United States, the patent mentions high frequency welding and furnaces, among other uses. Ca 1936 Car radios used electromechanical vibrators to transform the 6 V battery supply to a suitable B+ voltage for the vacuum tubes,1959 Transistor oscillation and rectifying converter power supply system U. S. 1972 HP-35, Hewlett-Packards first pocket calculator, is introduced with transistor switching power supply for light-emitting diodes, clocks, timing, ROM,1973 Xerox uses switching power supplies in the Alto minicomputer 1977 Apple II is designed with a switching mode power supply. Rod Holt was brought in as engineer and there were several flaws in Apple II that were never publicized. One thing Holt has to his credit is that he created the power supply that allowed us to do a very lightweight computer. 1980 The HP8662A10 kHz –1.28 GHz synthesized signal generator went with a switched mode power supply, a linear regulator provides the desired output voltage by dissipating excess power in ohmic losses. Ideal switching elements have no resistance when closed and carry no current when open and this is because the inductor responds to changes in current by inducing its own voltage to counter the change in current, and this voltage adds to the source voltage while the switch is open. This boost converter acts like a transformer for DC signals
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Comparator
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In electronics, a comparator is a device that compares two voltages or currents and outputs a digital signal indicating which is larger. It has two input terminals V + and V − and one binary digital output V o. The output is ideally V o = {1, if V + > V −0 and they are commonly used in devices that measure and digitize analog signals, such as analog-to-digital converters, as well as relaxation oscillators. The differential voltages must stay within the limits specified by the manufacturer, early integrated comparators, like the LM111 family, and certain high-speed comparators like the LM119 family, require differential voltage ranges substantially lower than the power supply voltages. Rail-to-rail comparators allow any differential voltages within the power supply range.3 volts below the supply rail. Specific ultra-fast comparators, like the LMH7322, allow input signal to swing below the rail and above the positive rail. Differential input voltage of a modern rail-to-rail comparator is usually limited only by the swing of power supply. An operational amplifier has a well balanced difference input and a high gain. This parallels the characteristics of comparators and can be substituted in applications with low-performance requirements, in theory, a standard op-amp operating in open-loop configuration may be used as a low-performance comparator. When the non-inverting input is at a higher voltage than the inverting input, when the non-inverting input drops below the inverting input, the output saturates at the most negative voltage it can output. The op-amps output voltage is limited by the supply voltage, an op-amp operating in a linear mode with negative feedback, using a balanced, split-voltage power supply, has its transfer function typically written as, V o u t = A o. Hence, an op-amp typically has a recovery time from saturation. Almost all op-amps have an internal compensation capacitor which imposes slew rate limitations for high frequency signals, consequently, an op-amp makes a sloppy comparator with propagation delays that can be as long as tens of microseconds. Since op-amps do not have any internal hysteresis, an external hysteresis network is necessary for slow moving input signals. The quiescent current specification of an op-amp is valid only when the feedback is active, some op-amps show an increased quiescent current when the inputs are not equal. A comparator is designed to produce well limited output voltages that easily interface with digital logic, compatibility with digital logic must be verified while using an op-amp as a comparator. Some multiple-section op-amps may exhibit extreme channel-channel interaction when used as comparators, many op-amps have back to back diodes between their inputs. Op-amp inputs usually follow each other so this is fine, but comparator inputs are not usually the same
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Deflection (physics)
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Deflection, in physics, refers to the change in an objects acceleration as a consequence of contact with a surface or the influence of a field. An objects deflective efficiency can never equal or surpass 100%, for example, a mirror will never reflect exactly the same amount of light cast upon it. Also, on hitting the ground, a previously in free-fall will never bounce back up to the place where it first started to descend
