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Rustic Log Holder Custom Names Centerpiece 20 Inch 4 Candle Rustic Wedding Cabin Decor Table Piece Slab Ash Wood
Custom Name(s) hand wood burned into the 'debarked' area in the center makes for an eye catching heirloom centerpiece! Use on the head table at your wedding and later; display it on the mantel in your future home together!
This wild-looking piece of natures art is simply eye candy when lit up and displayed prominently in low light. The flickering light from the candles deepens the shadows in the natural ripples of the bark and enhances the bits of moss still clinging to the wood. One could get lost in a trance staring at all the natural details of the wood and still not catching all the fine delicacies that nature grew there over many years.
A wonderful way to beautify, light up and enjoy your cozy space!
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PLEASE SPECIFY your custom personalization desires in the "special notes" box at checkout! Four word limit.
Natural hardwood ash. Finished with a nontoxic eco-friendly laqcuer. Measures approx. 20" long.
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Regarding the nature of rustic wood: We do our best to use photos of typical pieces that will resemble the product you receive as closely as possible. Please note, however, that no two pieces will ever look identical. Your item may or may not contain certain features displayed such as knots, darkened heartwood, shrinkage cracks, etc. We will always try to use the wood's natural character to the item's artistic advantage. Please do not consider these irregularities to be imperfections unless they somehow drastically hinder the function of the piece.
Thanks for looking, and have a wonderful day!


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In rooms of patients with multidrug-resistant organisms (MDROs), the outside of the packages containing sterile items can become contaminated. Unused medical supplies are often thrown away to prevent the items from becoming pathways for transmission of drug-resistant microbes, and in the process this leads to increased healthcare costs. Researchers at Johns Hopkins Hospital found that hydrogen peroxide vapor (HPV) is an effective way to sanitize the outside of the packages of these sterile supplies.
In addition to protecting patients, use of HPV could provide financial and environmental benefits. The study is published in the May issue of Infection Control and Hospital Epidemiology, in a special topic issue focused on the role of the environment in infection prevention.
In a two-phase study, Johns Hopkins Hospital researchers assessed the rate of contamination of packages containing sterile supplies and the ability of HPV to disinfect the outside of these packages. In each phase, five pairs of supplies from 20 rooms of patients colonized or infected with various MDROs were selected. One of each pair was sampled without exposure to HPV and the other was sampled after decontamination using HPV on a metal rack in patients rooms.
In the initial study of packaged supplies taken from rooms of patients known to be colonized with vancomycin-resistant enterococci (VRE), 7 percent of 100 untreated packages were found to be contaminated with VRE, versus none of the 100 packaged supplies exposed to HPV. In the subsequent study that assessed all MDROs, one or more MDROs was grown from nine of the 100 supply items not exposed to HPV and from none of the packages exposed to HPV.
Using inventory lists and electronic medical records, researchers projected the annual cost of discarded supplies because of potentially contaminated packages from patient isolation rooms from the six units included in the study to be $387,055. The researchers see this as a conservative estimate since medical waste disposal costs were not included.
Our study results show that supplies in the rooms of patients in isolation precautions can become contaminated with multidrug-resistant organisms, which may present a risk to other patients if they are not discarded or disinfected when the patient is discharged, says infectious disease specialist and study senior investigator Trish Perl, MD, MSc. Hydrogen peroxide vapor looks to be a cost-effective and environmentally-friendly means to manage potentially contaminated packaged supplies.
Several study authors are employed by Bioquell, which provided HPV disinfection services without charge to Johns Hopkins Hospital for the duration of this study.
Reference: Jonathan A. Otter, Elaine Nowakowski, James A. G. Salkeld, Mike Duclos, Catherine L. Passaretti, Saber Yezli, Tracy Ross, Karen C. Carroll, Trish M. Perl. Saving Costs through the Decontamination of the Packaging of Unused Medical Supplies using Hydrogen Peroxide Vapor. Infection Control and Hospital Epidemiology 34:5 (May 2013).

subset: commoncrawl


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I, robot and gardener: MIT droids tend plants
A class of undergraduates at the Massachusetts Institute of Technology has created a set of robots that can water, harvest and pollinate cherry tomato plants.
These gardeners would have green thumbs, if they had thumbs.
A class of undergraduates at the Massachusetts Institute of Technology has created a set of robots that can water, harvest and pollinate cherry tomato plants.
The small, $3,000 robots, which move through the garden on a base similar to a Roomba vacuum, are networked to the plants. When the plants indicate they need water, the robots can sprinkle them from a water pump. When the plants have a ripe tomato, the machines use their arms to pluck the fruit.
Even though robots have made few inroads into agriculture, these robots' creators hope their technology eventually could be used by farmers to reduce the natural resources and the difficult labor needed to tend crops.
Last spring, Daniela Rus, a professor who runs the Distributed Robotics Lab at MIT, began a two-part course. In the first semester, the students learned the basics of creating and using robots. By the fall, the students were ready to have robots tackle a real-world problem. Rus and Nikolaus Correll, a postdoctoral assistant in Rus' lab, challenged the students to create a "distributed robotic garden" by the end of the semester.
The 12 students broke into groups, each tasked with solving a different problem, such as creating the mechanical arm needed to harvest the tomatoes or perfecting the network that let the plants and robots share information.
By the end of the fall term, the "garden" inside Rus' lab was green and growing.
Now there are four cherry tomato plants nestled into a plywood base covered in fake grass. Next to each pot is a gray docking station for the robots.
Each plant and robot is connected to a computer network. The plants, through sensors in their soil, can tell the network when they need water or fertilizer, while the robots use a camera to inventory the plants' fruit. The robots also are programmed with a rudimentary growth model of the cherry tomato plants, which tells them roughly when a tomato will be ripe enough to be picked.
But the students quickly encountered challenges, both robotic and biologic.
Huan Liu, a 21-year-old computer science major, said designing the robot to pick the delicate tomatoes was made more difficult because the fruit would grow in unreachable places, such as behind stems or above where the robot's arm could reach.
"The tomatoes, they come out of nowhere, or just in weird places," Liu said.
Robots have made factory assembly lines more efficient and are being developed for in-home purposes, such as serving as health care aides. Yet there hasn't been much use for robotics in agriculture, partly because of the challenge of getting machines to work in unpredictable environments.
There have been attempts to get robots to replace humans at farm tasks, from thinning apple trees to picking asparagus, but none of the machines "have sufficient capacity to compete with human beings," said Tony Grift, an associate professor in the Department of Agricultural and Biological Engineering at the University of Illinois.
Even when technology has proven to be useful in agriculture, such as on tractors equipped with satellite imagery of fields, it often is prohibitively expensive.
Rus and Correll hope to conquer those kinds of challenges and get robots to work in farms.
"Agriculture contributes a lot of damage to the land, the soil, the water and the environment," Rus said. "So if we can figure out a way of using robots and automation to deliver nutrients to plants - pesticides, fertilizers, water when it's needed, instead of sort of mass spreading them, then we hope we would have an impact on the environment."
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Lighting 1
How you will look in your photographs is greatly affected by your photographer's use of light, perspective, and optics. See below how they make a huge difference! Take a look at Sarah. Unlike most people, her face never changes. But see how very different she can appear. Just to show the big difference that your photographer will make in how you look in your wedding pictures and why you need real photographer who truly understands light and knows how to use it for maximum advantage at each moment. This is why Rabenko Photography will mean more to you later!Read More


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出典(authority):フリー百科事典『ウィキペディア(Wikipedia)』「2015/02/17 09:43:13」(JST)
|Fourier transforms
|Continuous Fourier transform
|Fourier series
|Discrete-time Fourier transform
|Discrete Fourier transform
The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time.
Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency,[note 1] so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as an electronic filter applied to a signal, can be expressed relatively simply as an operation on frequencies.[note 2] So significant simplification is often achieved by transforming time functions to the frequency domain, performing the desired operations, and transforming the result back to time. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to almost all areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa. The critical case is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion), which with appropriate normalizations goes to itself under the Fourier transform. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.[note 3] For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.[1] The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional space to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either space or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued.[2] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on ℝ or ℝn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = ℤ), the discrete Fourier transform (DFT, group = ℤ mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The Fast Fourier transform (FFT) is an algorithm for computing the DFT.
There are several common conventions for defining the Fourier transform of an integrable function (Kaiser 1994, p. 29), (Rahman 2011, p. 11). This article will use the following definition:
When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, is determined by via the inverse transform:
The statement that can be reconstructed from is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat (Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The functions and often are referred to as a Fourier integral pair or Fourier transform pair (Rahman 2011, p. 10).
For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.
The motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series, complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity (Taneja 2008, p. 192).
Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral. In many cases it is desirable to use Euler's formula, which states that e2πiθ = cos(2πθ) + i sin(2πθ), to write Fourier series in terms of the basic waves e2πiθ. This has the advantage of simplifying many of the formulas involved, and provides a formulation for Fourier series that more closely resembles the definition followed in this article. Re-writing sines and cosines as complex exponentials makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. These complex exponentials sometimes contain negative "frequencies". If θ is measured in seconds, then the waves e2πiθ and e−2πiθ both complete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is still closely related.
There is a close connection between the definition of Fourier series and the Fourier transform for functions f that are zero outside of an interval. For such a function, we can calculate its Fourier series on any interval that includes the points where f is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of f begins to look like the inverse Fourier transform. To explain this more precisely, suppose that T is large enough so that the interval [−T/2, T/2] contains the interval on which f is not identically zero. Then the n-th series coefficient cn is given by:
Comparing this to the definition of the Fourier transform, it follows that since f(x) is zero outside [−T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1/T, multiplied by the grid width 1/T.
Under appropriate conditions, the Fourier series of f will equal the function f. In other words, f can be written:
where the last sum is simply the first sum rewritten using the definitions ξn = n/T, and Δξ = (n + 1)/T − n/T = 1/T.
This second sum is a Riemann sum, and so by letting T → ∞ it will converge to the integral for the inverse Fourier transform given in the definition section. Under suitable conditions this argument may be made precise (Stein & Shakarchi 2003).
In the study of Fourier series the numbers cn could be thought of as the "amount" of the wave present in the Fourier series of f. Similarly, as seen above, the Fourier transform can be thought of as a function that measures how much of each individual frequency is present in our function f, and we can recombine these waves by using an integral (or "continuous sum") to reproduce the original function.
The following images provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The function depicted f(t) = cos(6πt) e−πt2 oscillates at 3 hertz (if t measures seconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse. Its general form is a Gaussian function). This function was specially chosen to have a real Fourier transform that can easily be plotted. The first image contains its graph. In order to calculate we must integrate e−2πi(3t)f(t). The second image shows the plot of the real and imaginary parts of this function. The real part of the integrand is almost always positive, because when f(t) is negative, the real part of e−2πi(3t) is negative as well. Because they oscillate at the same rate, when f(t) is positive, so is the real part of e−2πi(3t). The result is that when you integrate the real part of the integrand you get a relatively large number (in this case 0.5). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at , you see that both real and imaginary component of this function vary rapidly between positive and negative values, as plotted in the third image. Therefore in this case, the integrand oscillates fast enough so that the integral is very small and the value for the fourier transform for that frequency is nearly zero.
The general situation may be a bit more complicated than this, but this in spirit is how the Fourier transform measures how much of an individual frequency is present in a function f(t).
Original function showing oscillation 3 hertz.
Real and imaginary parts of integrand for Fourier transform at 3 hertz
Real and imaginary parts of integrand for Fourier transform at 5 hertz
Fourier transform with 3 and 5 hertz labeled.
Here we assume f(x), g(x) and h(x) are integrable functions, are Lebesgue-measurable on the real line, and satisfy:
We denote the Fourier transforms of these functions by , and respectively.
The Fourier transform has the following basic properties: (Pinsky 2002).
That is, the evaluation of the Fourier transform in the origin () equals the integral of f all over its domain.
Under suitable conditions on the function f, it can be recovered from its Fourier transform Indeed, denoting the Fourier transform operator by so then for suitable functions, applying the Fourier transform twice simply flips the function: , which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: In particular the Fourier transform is invertible (under suitable conditions).
More precisely, defining the parity operator that inverts time, :
These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equality almost everywhere?) and defining equality of operators – that is, defining the topology on the function space and operator space in question. These are not true for all functions, but are true under various conditions, which are the content of the various forms of the Fourier inversion theorem.
This four-fold periodicity of the Fourier transform is similar to a rotation of the plane by 90°, particularly as the two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the time–frequency domain (considering time as the x-axis and frequency as the y-axis), and the Fourier transform can be generalized to the fractional Fourier transform, which involves rotations by other angles. This can be further generalized to linear canonical transformations, which can be visualized as the action of the special linear group SL2(R) on the time–frequency plane, with the preserved symplectic form corresponding to the uncertainty principle, below. This approach is particularly studied in signal processing, under time–frequency analysis.
In mathematics, one often does not think of any units as being attached to the two variables and . But in physical applications, must have inverse units to the units of . For example, if is measured in seconds, should be in cycles per second for the formulas here to be valid. If the scale of is changed and is measured in units of seconds, then either must be in the so-called "angular frequency", or one must insert some constant scale factor into some of the formulas. If is measured in units of length, then must be in inverse length, e.g., wavenumbers. That is to say, there are two copies of the real line: one measured in one set of units, where ranges, and the other in inverse units to the units of , and which is the range of . So these are two distinct copies of the real line, and cannot be identified with each other. Therefore the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.
In general, must always be taken to be a linear form on the space of s, which is to say that the second real line is the dual space of the first real line. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalisations of the Fourier transform to general symmetry groups, including the case of Fourier series.
That there is no one preferred way (often, one says "no canonical way") to compare the two copies of the real line which are involved in the Fourier transform---fixing the units on one line does not force the scale of the units on the other line---is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants. If the units of are in seconds but the units of are in angular frequency, then the angular frequency variable is often denoted by one or another Greek letter, for example, is quite common. Thus (writing for the alternative definition and for the definition adopted in this article)
as before, but the corresponding alternative inversion formula would then have to be
To have something involving angular frequency but with greater symmetry between the Fourier transform and the inversion formula, one very often sees still another alternative definition of the Fourier transform, with a factor of , thus
and the corresponding inversion formula then has to be
Furthermore, there is no way to fix which square root of negative one will be meant by the symbol (it makes no sense to speak of "the positive square root" since only real numbers can be positive, similarly it makes no sense to say "rotation counter-clockwise", because until is chosen, there is no fixed way to draw the complex plane), and hence one occasionally sees the Fourier transform written with in the exponent instead of , and vice versa for the inversion formula, a convention that is equally valid as the one chosen in this article, which is the more usual one.
For example, in probability theory, the characteristic function of the probability density function of a random variable of continuous type is defined without a negative sign in the exponential, and since the units of are ignored, there is no either:
(In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat discontinuous distribution functions, i.e., measures which possess "atoms".)
From the higher point of view of group characters, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, on the notion of the Fourier transform of a function on an Abelian locally compact group.
The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties.
The Fourier transform, , of any integrable function f is uniformly continuous and (Katznelson 1976). By the Riemann–Lebesgue lemma (Stein & Weiss 1971),
However, need not be integrable. For example, the Fourier transform of the rectangular function, which is integrable, is the sinc function, which is not Lebesgue integrable, because its improper integrals behave analogously to the alternating harmonic series, in converging to a sum without being absolutely convergent.
It is not generally possible to write the inverse transform as a Lebesgue integral. However, when both f and are integrable, the inverse equality
holds almost everywhere. That is, the Fourier transform is injective on L1(R). (But if f is continuous, then equality holds for every x.)
Let f(x) and g(x) be integrable, and let and be their Fourier transforms. If f(x) and g(x) are also square-integrable, then we have Parseval's Formula (Rudin 1987, p. 187):
where the bar denotes complex conjugation.
The Plancherel theorem, which follows from the above, states that (Rudin 1987, p. 186)
Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on L2(R). On L1(R)∩L2(R), this extension agrees with original Fourier transform defined on L1(R), thus enlarging the domain of the Fourier transform to L1(R) + L2(R) (and consequently to Lp(R) for 1 ≤ p ≤ 2). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Liapounoff. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem.
See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.
The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. The Poisson summation formula says that for sufficiently regular functions ,
It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. The formula has applications in engineering, physics, and number theory. The frequency-domain dual of the standard Poisson summation formula is also called the discrete-time Fourier transform.
Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle. The fundamental solution of the heat equation on a circle is called a theta function. It is used in number theory to prove the transformation properties of theta functions, which turn out to be a type of modular form, and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula.
The Fourier transform translates between convolution and multiplication of functions. If f(x) and g(x) are integrable functions with Fourier transforms and respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms and (under other conventions for the definition of the Fourier transform a constant factor may appear).
This means that if:
where ∗ denotes the convolution operation, then:
In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system with input f(x) and output h(x), since substituting the unit impulse for f(x) yields h(x) = g(x). In this case, represents the frequency response of the system.
Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the Fourier transform of f(x) is given by the convolution of the respective Fourier transforms and .
In an analogous manner, it can be shown that if h(x) is the cross-correlation of f(x) and g(x):
then the Fourier transform of h(x) is:
As a special case, the autocorrelation of function f(x) is:
for which
One important choice of an orthonormal basis for L2(R) is given by the Hermite functions
where Hen(x) are the "probabilist's" Hermite polynomials, defined by
Under this convention for the Fourier transform, we have that
In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L2(R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only four different eigenvalues of the Fourier transform (±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L2(R) as a direct sum of four spaces H0, H1, H2, and H3 where the Fourier transform acts on Hek simply by multiplication by ik.
Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed. This approach to define the Fourier transform was first done by Norbert Wiener (Duoandikoetxea 2001). Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time-frequency analysis (Boashash 2003). In physics, this transform was introduced by Edward Condon (Condon 1937).
The integral for the Fourier transform
can be studied for complex values of its argument . Depending on the properties of , this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of , or something in between. [3]
The Paley--Wiener theorem says that is smooth (i.e., -times differentiable for all positive integers ) and compactly supported if and only if is a holomorphic function for which there exists a constant such that for any integer ,
for some constant . (In this case, is supported on .) This can be expressed by saying that is an entire function which is rapidly decreasing in (for fixed ) and of exponential growth in (uniformly in ). [4]
(If is not smooth, but only , the statement still holds provided .) [5] The space of such functions of a complex variable is called the Paley--Wiener space. This theorem has been generalised to semi-simple Lie groups. [6]
If is supported on the half-line , then is said to be "causal" because the impulse response function of a physically realisable filter must have this property, as no effect can precede its cause. Paley and Wiener showed that then extends to a holomorphic function on the complex lower half-plane which tends to zero as goes to infinity. [7] The converse is false and it is not known how to characterise the Fourier transform of a causal function. [8]
The Fourier transform is intimately related with the Laplace transform , which is also used for the solution of differential equations and the analysis of filters. Chatfield, indeed, has said that "... the Laplace and the Fourier transforms [of a causal function] are the same, provided that the real part of is zero." [9]
It may happen that a function for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the complex plane.
For example, if is of exponential growth, i.e.,
for some constants , then [10]
convergent for all , is the two-sided Laplace transform of .
The more usual version ("one-sided") of the Laplace transform is
If is also causal, then
Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case---the case of causal functions---but with the change of variable .
If has no poles for , then
by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis. [11]
Theorem: If for , and for some constants , then
for any .
This theorem implies the Mellin inversion formula for the Laplace transformation, [12]
for any , where is the Laplace transform of .
The hypotheses can be weakened, as in the results of Carleman and Hunt, to being , provided that is in the interior of a closed interval on which is continuous and of bounded variation, and provided that the integrals are taken in the sense of Cauchy principal values. [13]
versions of these inversion formulas are also available.[14]
The Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions. For an integrable function f(x), this article takes the definition:
where x and ξ are n-dimensional vectors, and x · ξ is the dot product of the vectors. The dot product is sometimes written as .
All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds. (Stein & Weiss 1971)
Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, its Fourier transform "stretches out" in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.
The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.
Suppose f(x) is an integrable and square-integrable function. Without loss of generality, assume that f(x) is normalized:
It follows from the Plancherel theorem that is also normalized.
The spread around x = 0 may be measured by the dispersion about zero (Pinsky 2002, p. 131) defined by
In probability terms, this is the second moment of |f(x)|2 about zero.
The Uncertainty principle states that, if f(x) is absolutely continuous and the functions x·f(x) and f′(x) are square integrable, then
The equality is attained only in the case (hence ) where σ > 0 is arbitrary and so that f is L2–normalized (Pinsky 2002). In other words, where f is a (normalized) Gaussian function with variance σ2, centered at zero, and its Fourier transform is a Gaussian function with variance σ−2.
In fact, this inequality implies that:
for any x0, ξ0 ∈ R (Stein & Shakarchi 2003, p. 158).
In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle (Stein & Shakarchi 2003, p. 158).
A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as:
where H(p) is the differential entropy of the probability density function p(x):
where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case.
Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function for which Fourier inversion holds good can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically[15]) by
This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions and can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):
Older literature refers to the two transform functions, the Fourier cosine transform, , and the Fourier sine transform, .
The function f can be recovered from the sine and cosine transform using
together with trigonometric identities. This is referred to as Fourier's integral formula.[16]
Let the set of homogeneous harmonic polynomials of degree k on Rn be denoted by Ak. The set Ak consists of the solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f(x) = e−π|x|2P(x) for some P(x) in Ak, then . Let the set Hk be the closure in L2(Rn) of linear combinations of functions of the form f(|x|)P(x) where P(x) is in Ak. The space L2(Rn) is then a direct sum of the spaces Hk and the Fourier transform maps each space Hk to itself and is possible to characterize the action of the Fourier transform on each space Hk (Stein & Weiss 1971). Let f(x) = f0(|x|)P(x) (with P(x) in Ak), then where
Here J(n + 2k − 2)/2 denotes the Bessel function of the first kind with order (n + 2k − 2)/2. When k = 0 this gives a useful formula for the Fourier transform of a radial function (Grafakos 2004). Note that this is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n+2 and n (Grafakos & Teschl 2013) allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.
In higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L2(Rn) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in Lp for 1 < p < 2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit sphere in Rn is of particular interest. In this case the Tomas-Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in Rn is a bounded operator on Lp provided 1 ≤ p ≤ (2n + 2) / (n + 3).
One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets ER indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R. For a given integrable function f, consider the function fR defined by:
Suppose in addition that f ∈ Lp(Rn). For n = 1 and 1 < p < ∞, if one takes ER = (−R, R), then fR converges to f in Lp as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for n > 1. In the case that ER is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball ER = {ξ : |ξ| < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in Lp(Rn). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2 (Duoandikoetxea 2001). In fact, when p ≠ 2, this shows that not only may fR fail to converge to f in Lp, but for some functions f ∈ Lp(Rn), fR is not even an element of Lp.
The definition of the Fourier transform by the integral formula
is valid for Lebesgue integrable functions f; that is, f ∈ L1(Rn).
The Fourier transform : L1(Rn) → L∞(Rn) is a bounded operator. This follows from the observation that
which shows that its operator norm is bounded by 1. Indeed it equals 1, which can be seen, for example, from the transform of the rect function. The image of L1 is a subset of the space C0(Rn) of continuous functions that tend to zero at infinity (the Riemann–Lebesgue lemma), although it is not the entire space. Indeed, there is no simple characterization of the image.
Since compactly supported smooth functions are integrable and dense in L2(Rn), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L2(Rn) by continuity arguments. The Fourier transform in L2(Rn) is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an L2 function f,
where the limit is taken in the L2 sense. Many of the properties of the Fourier transform in L1 carry over to L2, by a suitable limiting argument.
Furthermore : L2(Rn) → L2(Rn) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any f,g∈L2(Rn) we have
In particular, the image of L2(Rn) is itself under the Fourier transform.
The definition of the Fourier transform can be extended to functions in Lp(Rn) for 1 ≤ p ≤ 2 by decomposing such functions into a fat tail part in L2 plus a fat body part in L1. In each of these spaces, the Fourier transform of a function in Lp(Rn) is in Lq(Rn), where is the Hölder conjugate of p. by the Hausdorff–Young inequality. However, except for p = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in Lp for the range 2 < p < ∞ requires the study of distributions (Katznelson 1976). In fact, it can be shown that there are functions in Lp with p > 2 so that the Fourier transform is not defined as a function (Stein & Weiss 1971).
One might consider enlarging the domain of the Fourier transform from L1+L2 by considering generalized functions, or distributions. A distribution on Rn is a continuous linear functional on the space Cc(Rn) of compactly supported smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier transform on Cc(Rn) and pass to distributions by duality. The obstruction to do this is that the Fourier transform does not map Cc(Rn) to Cc(Rn). In fact the Fourier transform of an element in Cc(Rn) can not vanish on an open set; see the above discussion on the uncertainty principle. The right space here is the slightly larger space of Schwartz functions. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions (Stein & Weiss 1971). The tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support.
For the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let and be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula (Stein & Weiss 1971),
Every integrable function f defines (induces) a distribution Tf by the relation
So it makes sense to define Fourier transform of Tf by
for all Schwartz functions φ. Extending this to all tempered distributions T gives the general definition of the Fourier transform.
Distributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.
The Fourier transform of a finite Borel measure μ on Rn is given by (Pinsky 2002, p. 256):
This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures (Katznelson 1976). In the case that dμ = f(x)dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier-Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take eix·ξ instead of e−2πix·ξ (Pinsky 2002). In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants.
The Fourier transform may be used to give a characterization of measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle (Katznelson 1976).
Furthermore, the Dirac delta function is not a function but it is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).
The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an abelian group that is at the same time a locally compact Hausdorff topological space so that the group operation is continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called Haar measure. For a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its characters. With its natural group structure and the topology of pointwise convergence, the set of characters is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L1(G), its Fourier transform is defined by (Katznelson 1976):
The Riemann-Lebesgue lemma holds in this case; is a function vanishing at infinity on .
The Fourier transform is also a special case of Gelfand transform. In this particular context, it is closely related to the Pontryagin duality map defined above.
Given an abelian locally compact Hausdorff topological group G, as before we consider space L1(G), defined using a Haar measure. With convolution as multiplication, L1(G) is an abelian Banach algebra. It also has an involution * given by
Taking the completion with respect to the largest possibly C*-norm gives its enveloping C*-algebra, called the group C*-algebra C*(G) of G. (Any C*-norm on L1(G) is bounded by the L1 norm, therefore their supremum exists.)
Given any abelian C*-algebra A, the Gelfand transform gives an isomorphism between A and C0(A^), where A^ is the multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology. The map is simply given by
It turns out that the multiplicative linear functionals of C*(G), after suitable identification, are exactly the characters of G, and the Gelfand transform, when restricted to the dense subset L1(G) is the Fourier-Pontryagin transform.
The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators (Hewitt & Ross 1970, Chapter 8). The Fourier transform on compact groups is a major tool in representation theory (Knapp 2001) and non-commutative harmonic analysis.
Let G be a compact Hausdorff topological group. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation U(σ) on the Hilbert space Hσ of finite dimension dσ for each σ ∈ Σ. If μ is a finite Borel measure on G, then the Fourier–Stieltjes transform of μ is the operator on Hσ defined by
where is the complex-conjugate representation of U(σ) acting on Hσ. If μ is absolutely continuous with respect to the left-invariant probability measure λ on G, represented as
for some f ∈ L1(λ), one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ.
The mapping defines an isomorphism between the Banach space M(G) of finite Borel measures (see rca space) and a closed subspace of the Banach space C∞(Σ) consisting of all sequences E = (Eσ) indexed by Σ of (bounded) linear operators Eσ: Hσ → Hσ for which the norm
is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of C* algebras into a subspace of C∞(Σ). Multiplication on M(G) is given by convolution of measures and the involution * defined by
and C∞(Σ) has a natural C*-algebra structure as Hilbert space operators.
The Peter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if f ∈ L2(G), then
where the summation is understood as convergent in the L2 sense.
The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry.[citation needed] In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka–Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.
In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent.
As alternatives to the Fourier transform, in time-frequency analysis, one uses time-frequency transforms or time-frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform or fractional Fourier transform, or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform. (Boashash 2003).
Perhaps the most important use of the Fourier transformation is to solve partial differential equations. Many of the equations of the mathematical physics of the nineteenth century can be treated this way. Fourier studied the heat equation, which in one dimension and in dimensionless units is
The example we will give, a slightly more difficult one, is the wave equation in one dimension,
As usual, the problem is not to find a solution: there are infinitely many. The problem is that of the so-called "boundary problem": find a solution which satisfies the "boundary conditions"
Here, and are given functions. For the heat equation, only one boundary condition can be required (usually the first one). But for the wave equation, there are still infinitely many solutions which satisfy the first boundary condition. But when one imposes both conditions, there is only one possible solution.
It is easier to find the Fourier transform of the solution than to find the solution directly. This is because the Fourier transformation takes differentiation into multiplication by the variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function. After is determined, we can apply the inverse Fourier transformation to find .
Fourier's method is as follows. First, note that any function of the forms
satisfies the wave equation. These are called "the elementary solutions."
Second, note that therefore any integral
(for arbitrary , , , and ) satisfies the wave equation. (This integral is just a kind of continuous linear combination, and the equation is linear.)
Now this resembles the formula for the Fourier synthesis of a function. In fact, this is the real inverse Fourier transform of and in the variable .
The third step is to examine how to find the specific unknown coefficient functions and that will lead to 's satisfying the boundary conditions. We are interested in the values of these solutions at . So we will set . Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable ) of both sides and obtain
Similarly, taking the derivative of with respect to and then applying the Fourier sine and cosine transformations yields
These are four linear equations for the four unknowns and , in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found.
In summary, we chose a set of elementary solutions, parametrised by , of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter . But this integral was in the form of a Fourier integral. The next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functions and . But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative. The last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functions and in terms of the given boundary conditions and .
George Mackey has called this procedure "Fourier's algorithm." "Harmonic Analysis as the Exploitation of Symmetry---A Historical Survey." Rice University Studies Volume 64 (1978), Numbers 2 and 3, pp. 73–228. Reprinted in Bulletin of the American Mathematical Society (New Series) Volume 3 (1980), Number 1 (July—November), pp. 543–698.
In fact, one can perform further manipulations on the answer obtained, but those belong rather to the subject of partial differential equations than to the subject of the Fourier transformation itself.
From a higher point of view, Fourier's procedure can be reformulated more conceptually. Since there are two variables, we will use the Fourier transformation in both and rather than operate as Fourier did, who only transformed in the spatial variables. Note that must be considered in the sense of a distribution since is not going to be : as a wave, it will persist through time and thus is not a transient phenomenon. But it will be bounded and so its Fourier transform can be defined as a distribution. The operational properties of the Fourier transformation that are relevant to this equation are that it takes differentiation in to multiplication by and differentiation with respect to to multiplication by where is the frequency. Then the wave equation becomes an algebraic equation in :
This is equivalent to requiring unless . Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously will be solutions. Applying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier. But from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conic .
We may as well consider the distributions supported on the conic that are given by distributions of one variable on the line plus distributions on the line as follows: if is any test function,
where , and , are distributions of one variable.
Then Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put , which is clearly of polynomial growth):
Now, as before, applying the one-variable Fourier transformation in the variable to these functions of yields two equations in the two unknown distributions (which can be taken to be ordinary functions if the boundary conditions are or ).
From a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform. Closed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate. For practical calculations, other methods are often used.
The twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well.
The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in magnetic resonance imaging (MRI) and mass spectrometry.
The Fourier transform is useful in Quantum Mechanics in two different ways. To begin with, the basic conceptual structure of Quantum Mechanics postulates the existence of pairs of complementary variables, connected by the Heisenberg uncertainty principle. For example, in one dimension, the spatial variable of, say, a particle, can only be measured by the quantum mechanical "position operator" at the cost of losing information about the momentum of the particle. Therefore the physical state of the particle can either be described by a function, called "the wave function", of or by a function of but not by a function of both variables. The variable is called the conjugate variable to . In Classical Mechanics, the physical state of a particle (existing in one dimension, for simplicityof exposition) would be given by assigning definite values to both and simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a -axis and a -axis.
In contrast, Quantum Mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the -axis alone, but instead of considering only points, takes the set of all complex-valued "wave function" on this axis. Nevertheless, choosing the -axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle which is related to the first representation by the Fourier transformation
Physically realisable states are , and so by the Plancherel theorem, their Fourier transforms are also . (Note that since is in units of distance and is in units of momentum, the presence of Planck's constant in the exponent makes the exponent dimensionless, as it should be.)
Therefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum. Infinitely many different polarisations are possible, and all are equally valid. Being able to transform states from one representation to another is sometimes convenient.
The other use of the Fourier transform in both Quantum Mechanics and Quantum Field Theory is to solve the applicable wave equation. In non-relativistic Quantum Mechanics, Schroedinger's equation for a time-varying wave function in one-dimension, not subject to external forces, is
This is the same as the heat equation except for the presence of the imaginary unit . Fourier methods can be used to solve this equation.
In the presence of a potential, given by the potential energy function , the equation becomes
The "elementary solutions", as we referred to them above, are the so-called "stationary states" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of given its values for . Neither of these approaches is of much practical use in Quantum Mechanics. Boundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.
In relativistic Quantum Mechanics, Schroedinger's equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered. A simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein--Gordon--Schroedinger--Fock equation, this time in dimensionless units,
This is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods). This is of great use in Quantum Field Theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantised, a procedure known as "second quantisation". Fourier methods have been adapted to also deal with non-trivial interactions.
The Fourier transform is used for the spectral analysis of time-series. The subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself. Even if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time. The Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its auto-correlation function.
The auto-correlation function of a function is defined by
This function is a function of the time-lag elapsing between the values of to be correlated.
For most functions that occur in practice, is a bounded even function of the time-lag and for typical noisy signals it turns out to be uniformly continuous with a maximum at zero.
The auto-correlation function, more properly called the auto-covariance function unless it is normalised in some appropriate fashion, measures the strength of the correlation between the values of separated by a time-lag. This is a way of searching for the correlation of with its own past. It is useful even for other statistical tasks besides the analysis of signals. For example, if represents the temperature at time , one expects a strong correlation with the temperature at a time-lag of 24 hours.
It possesses a Fourier transform,
This Fourier transform is called the power spectral density function of . (Unless all periodic components are first filtered out from , this integral will diverge, but it is easy to filter out such periodicities.)
The power spectrum, as indicated by this density function , measures the amount of variance contributed to the data by the frequency . In electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal. This process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series (ANOVA).
Knowledge of which frequencies are "important" in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses. It can also be useful for the scientific analysis of the phenomena responsible for producing the data.
The power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out.
Spectral analysis is carried out for visual signals as well. The power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool.
Other common notations for include:
Denoting the Fourier transform by a capital letter corresponding to the letter of function being transformed (such as f(x) and F(ξ)) is especially common in the sciences and engineering. In electronics, the omega (ω) is often used instead of ξ due to its interpretation as angular frequency, sometimes it is written as F(jω), where j is the imaginary unit, to indicate its relationship with the Laplace transform, and sometimes it is written informally as F(2πf) in order to use ordinary frequency.
The interpretation of the complex function may be aided by expressing it in polar coordinate form
in terms of the two real functions A(ξ) and φ(ξ) where:
is the amplitude and
is the phase (see arg function).
Then the inverse transform can be written:
which is a recombination of all the frequency components of f(x). Each component is a complex sinusoid of the form e2πixξ whose amplitude is A(ξ) and whose initial phase angle (at x = 0) is φ(ξ).
The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted and is used to denote the Fourier transform of the function f. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function f) can be used to write instead of . Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as or as . Notice that in the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluated at ξ, not the other way around.
In mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted f(x). This means that a notation like formally can be interpreted as the Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed.
For example, is sometimes used to express that the Fourier transform of a rectangular function is a sinc function,
or is used to express the shift property of the Fourier transform.
Notice, that the last example is only correct under the assumption that the transformed function is a function of x, not of x0.
The Fourier transform can also be written in terms of angular frequency: ω = 2πξ whose units are radians per second.
The substitution ξ = ω/(2π) into the formulas above produces this convention:
Under this convention, the inverse transform becomes:
Unlike the convention followed in this article, when the Fourier transform is defined this way, it is no longer a unitary transformation on L2(Rn). There is also less symmetry between the formulas for the Fourier transform and its inverse.
Another convention is to split the factor of (2π)n evenly between the Fourier transform and its inverse, which leads to definitions:
Under this convention, the Fourier transform is again a unitary transformation on L2(Rn). It also restores the symmetry between the Fourier transform and its inverse.
Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.
|ordinary frequency ξ (hertz)
|angular frequency ω (rad/s)
As discussed above, the characteristic function of a random variable is the same as the Fourier–Stieltjes transform of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is defined .
As in the case of the "non-unitary angular frequency" convention above, there is no factor of 2π appearing in either of the integral, or in the exponential. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponential.
|It has been suggested that this section be split into a new article. (Discuss) Proposed since October 2014.
The following tables record some closed-form Fourier transforms. For functions f(x), g(x) and h(x) denote their Fourier transforms by , , and respectively. Only the three most common conventions are included. It may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.
The Fourier transforms in this table may be found in Erdélyi (1954) or Kammler (2000, appendix).
|Fourier transform
unitary, ordinary frequency
|Fourier transform
unitary, angular frequency
|Fourier transform
non-unitary, angular frequency
|Shift in time domain
|Shift in frequency domain, dual of 102
|Scaling in the time domain. If is large, then is concentrated around 0 and spreads out and flattens.
|Duality. Here needs to be calculated using the same method as Fourier transform column. Results from swapping "dummy" variables of and or or .
|This is the dual of 106
|The notation denotes the convolution of and — this rule is the convolution theorem
|This is the dual of 108
|For purely real
|Hermitian symmetry. indicates the complex conjugate.
|For a purely real even function
|, and are purely real even functions.
|For a purely real odd function
|, and are purely imaginary odd functions.
|Complex conjugation, generalization of 110
The Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdélyi 1954), or the appendix of (Kammler 2000).
|Fourier transform
unitary, ordinary frequency
|Fourier transform
unitary, angular frequency
|Fourier transform
non-unitary, angular frequency
|The rectangular pulse and the normalized sinc function, here defined as sinc(x) = sin(πx)/(πx)
|Dual of rule 201. The rectangular function is an ideal low-pass filter, and the sinc function is the non-causal impulse response of such a filter. The sinc function is defined here as sinc(x) = sin(πx)/(πx)
|The function tri(x) is the triangular function
|Dual of rule 203.
|The function u(x) is the Heaviside unit step function and a>0.
|This shows that, for the unitary Fourier transforms, the Gaussian function exp(−αx2) is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α)>0.
|For a>0. That is, the Fourier transform of a two-sided decaying exponential function is a Lorentzian function.
|Hyperbolic secant is its own Fourier transform
|is the Hermite's polynomial. If a = 1 then the Gauss-Hermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see Hermite polynomial. The formula reduces to 206 for n = 0.
The Fourier transforms in this table may be found in (Erdélyi 1954) or the appendix of (Kammler 2000).
|Fourier transform
unitary, ordinary frequency
|Fourier transform
unitary, angular frequency
|Fourier transform
non-unitary, angular frequency
|The distribution δ(ξ) denotes the Dirac delta function.
|Dual of rule 301.
|This follows from 103 and 301.
|This follows from rules 101 and 303 using Euler's formula:
|This follows from 101 and 303 using
|Here, n is a natural number and is the n-th distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all polynomials.
|Here sgn(ξ) is the sign function. Note that 1/x is not a distribution. It is necessary to use the Cauchy principal value when testing against Schwartz functions. This rule is useful in studying the Hilbert transform.
|1/xn is the homogeneous distribution defined by the distributional derivative
|This formula is valid for 0 > α > −1. For α > 0 some singular terms arise at the origin that can be found by differentiating 318. If Re α > −1, then is a locally integrable function, and so a tempered distribution. The function is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted for α ≠ −2, −4, ... (See homogeneous distribution.)
|Special case of 311.
|The dual of rule 309. This time the Fourier transforms need to be considered as Cauchy principal value.
|The function u(x) is the Heaviside unit step function; this follows from rules 101, 301, and 312.
|This function is known as the Dirac comb function. This result can be derived from 302 and 102, together with the fact that
as distributions.
|The function J0(x) is the zeroth order Bessel function of first kind.
|This is a generalization of 315. The function Jn(x) is the n-th order Bessel function of first kind. The function Tn(x) is the Chebyshev polynomial of the first kind.
|is the Euler–Mascheroni constant.
|This formula is valid for 1 > α > 0. Use differentiation to derive formula for higher exponents. u is the Heaviside function.
|Fourier transform
unitary, ordinary frequency
|Fourier transform
unitary, angular frequency
|Fourier transform
non-unitary, angular frequency
To 400: The variables ξx, ξy, ωx, ωy, νx and νy are real numbers. The integrals are taken over the entire plane.
To 401: Both functions are Gaussians, which may not have unit volume.
To 402: The function is defined by circ(r)=1 0≤r≤1, and is 0 otherwise. This is the Airy distribution, and is expressed using J1 (the order 1 Bessel function of the first kind). (Stein & Weiss 1971, Thm. IV.3.3)
|Fourier transform
unitary, ordinary frequency
|Fourier transform
unitary, angular frequency
|Fourier transform
non-unitary, angular frequency
To 501: The function χ[0, 1] is the indicator function of the interval [0, 1]. The function Γ(x) is the gamma function. The function Jn/2 + δ is a Bessel function of the first kind, with order n/2 + δ. Taking n = 2 and δ = 0 produces 402. (Stein & Weiss 1971, Thm. 4.15)
To 502: See Riesz potential. The formula also holds for all α ≠ −n, −n − 1, … by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See homogeneous distribution.
To 503: This is the formula for a multivariate normal distribution normalized to 1 with a mean of 0. Bold variables are vectors or matrices. Following the notation of the aforementioned page, and
To 504: Here . See (Stein & Weiss 1971, p. 6).
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|「フーリエ変換」「Fourier transform」
|「fast Fourier transformation」


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It's another new year, and our county's spiritual leaders are busy implementing goals to move their churches forward.
Pastor John E. Morse of the Pleasant Grove African Methodist Episcopal Church in Hinesville said the church's theme for 2014 is, "Follow up and follow through."
"My goals are to increase spiritual growth and increase direct community involvement," he said.
Mike Buckley, pastor of the Westside Baptist Church in Hinesville since September 2012, stated that his goal is to see improvement in people's relationship with God.
"I want to make sure people have the proper relationship with God. My job is to be a shepherd for the sheep," he said. "I have to ensure the sheep are healthy. When I stand before God, He is not going to be concerned about the numbers, but whether the sheep are healthy."
According to Pastor S. Dyann Robertson of the Thebes AME in Midway, her church plans to get out into the community.
"We plan to visit the nursing home, the elderly and the sick and shut-in more," she said. "Additionally, we intend to have more activities for the young people, and to go out and recapture former members."
Pastor Tom Gardner of A New Beginning Church in Ludowici said his church's directive focuses on the three powerful words, "I Am Persuaded." Once a person is persuaded, he said, there is no stopping them; persuasion takes us past believing.
"This year, we are persuaded that we can double our efforts to reach the nearly 17,000 people in Long County," he said. "Long County is our mission field. We would like to give to legitimate needs and outreach up to $40,000. Our goal is to better people's lives, i.e., their church life, work life and home life. All of this can be accomplished by bringing people into a relationship with Jesus Christ, with love and goodness."
Apostle Daisy Edwards of AGAPE Christian Fellowship of Hinesville said she plans to build upon her ministers' training classes.
"We are reaching out to new ministers to join us. We also plan to get our youth more involved in community outreach such as visiting the nursing home," she said. "I would also love to fellowship more with other churches in the community. God wants us to come together."
Dr. C. L. Anderson, who is pastor at Good Shepherd Missionary Baptist Church in Allenhurst, Greater Zachariah Baptist Church in Glennville and St. Luke Baptist Church, already has had a busy year.
"Prayerfully, we will bond more together as a church family, showing even more love and doing greater things in 2014," he said. "We began our 21-day Daniel Fast on Jan. 5 and our consecration services on Jan. 6–9. We are trusting God to lead and guide us so that we can become more healthy and spiritual so that our lights may shine just a little brighter in 2014."
"We are continuing our visits to the Coastal Manor Nursing Home in Ludowici and Glennville nursing home. Our combined churches' leadership training is growing and we are making plans for the leadership retreat at the conclusion of the training in May. I also intend to start a jailhouse ministry in 2014," Anderson added.
It's another new year, and our county's spiritual leaders are busy implementing goals to move their churches forward.


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While it's sad to see summer come to an end, we fortunately have fall fashion to look forward to. TODAY style editor and Bobbie.com's Bobbie Thomas helps us ease into autumn with tips on how to stay on trend while transitioning your wardrobe this season.
Go longer
As temperatures drop, so will hemlines. Fashion meets function, and there's no need to brave the winter with bare legs. Transition a summer maxi dress into fall with a chunky sweater and boots or try the new 'midi' length skirt for a little retro flair. Even outwear adds inches this season with longer trenches, knee-or-ankle length coats. (Midi Dress and Coat $36.59-137.21 asos.com; Nine West Abalene Pump $89, ; Bar III Skirt $59, Jones NY Cardi $94 macys.com)
Embrace color
After a long haze of gray, spicy hues, warm things up this autumn with shades of caramel, crimson, pumpkin, mustard and more. Red is red hot, and jewel tones reigned on the runways. Shop in your closet and combine unexpected colors, or invest in some new pieces to brighten your winter wardrobe. (Blouse $9.95, hm.com; Skirt, Bag $49.90-$89.90, zara.com; Kelsi Dagger Laurie Pumps $79.95, dsw.com)
Be bold
Step away from your basics and express yourself. Graphic prints, blocks of color, and even tribal accents add a little attitude. Don't be afraid to mix prints, and even consider pairing pieces that clash and contrast — they can actually balance each other if you stick with similar hues. And dare to wear strong accessories — whether it's a major cuff, a notice-me-now necklace, or drop-to-your-shoulder earrings. (Blouse $69.90 ; Skirt $76.00 ; Traisie pumps $89.95 )
Add texture
Indulge in fall's luxe landscape with rich fabrics and dimensional details. Layer different textiles such as lace and leather for an outfit with more visual interest. Add warmth with a velvet blazer, cozy up in a chunky knit sweater vest, or work in a little tweed. Fabulous faux fur is also everywhere from trim to handbags. (Studio M Dress $98, macys.com; Vest $69.95 nyandcompany.com; Capezio Legwear $18-$25 capezionewyork.com)
Glisten up
Metallics are an annual holiday favorite, but go beyond this year. Copper and rose gold are new shades of metal to melt over, but look no further than your beauty bag to shimmer and shine. From eyes to lips to fingertips, you can find a frosty shade that will work for you. (Jennifer Lopez for Kohls sequin top and pants $20-$120 kohls.com)


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, a direct to consumer company and leading designer and marketer of candles, accessories for the home, and health and wellness products, indicated that it has repurchased shares in the open market under its share repurchase authorization. During its fourth fiscal quarter, and subsequent to the finalization and disclosure of its new agreement with the ViSalus founders, the Company repurchased 411,336 shares, or approximately 2.5% of its 17 million shares outstanding. The company has 1.6 million shares remaining in its existing authorization. Chairman & CEO Robert B. Goergen noted that, "from time to time, the Company will repurchase shares when we believe it is in the best interest of our long-term shareholders based on our growth, profitability and cash flow outlook. We believe recent market conditions offered such an opportunity. We also intend to seek an increase in our existing authorization from Blyth's Board of Directors to continue our repurchase program in 2013 when market conditions warrant it." Blyth, Inc., headquartered in Greenwich, CT, USA, is a direct to consumer business focused on direct selling and direct marketing channels. We design and market home fragrance products and decorative accessories, as well as weight management products, nutritional supplements and energy drink mixes. These products are sold through Direct Selling from the home party plan method and network marketing. The Company also designs and markets household convenience items and personalized gifts through the catalog/internet channel. The Company manufactures most of its candles and sources nearly all of its other products. Its products are sold direct to the consumer under the PartyLite®, Two Sisters Gourmet® by PartyLite and ViSalus Sciences® brands and to consumers in the catalog/Internet channel under the As We Change®, Miles Kimball®, Exposures®, Walter Drake® and Easy Comforts brands. In Europe, Blyth's products are also sold under the PartyLite brand. Blyth, Inc. may be found on the Internet at www.blyth.com. This press release contains "forward-looking statements" within the meaning of the Private Securities Litigation Reform Act of 1995. Forward-looking statements include statements concerning plans, including plans to increase its share repurchase authorization and repurchases its shares at a price level it deems appropriate to enhancing shareholder value, objectives, goals, strategies, future events or performance and underlying assumptions and other statements that are other than statements of historical facts. Actual results could differ materially due to various factors, including the slowing of the United States or European economies or retail environments, the risk that we will be unable to maintain our historic growth rate, our ability to respond appropriately to changes in product demand, the risk that we will be unable to integrate the businesses that we acquire into our existing operations, the risks (including foreign currency fluctuations, economic and political instability, transportation delays, difficulty in maintaining quality control, trade and foreign tax laws and others) associated with international sales and foreign sourced products, risks associated with our ability to recruit new independent sales consultants, our dependence on key corporate management personnel, risks associated with the sourcing of raw materials for our products, competition in terms of price and new product introductions, risks associated with our information technology systems (including, susceptibility to outages due to fire, floods, power loss, telecommunications failures, computer viruses, break-ins and similar events) and other factors described in this press release and in the Company's most recently filed Annual Report on Form 10-K.
© 2023 Benzinga.com. Benzinga does not provide investment advice. All rights reserved.


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Only ten Km. from Trieste's airport, Aquileia is well connected by public transport both to Trieste and Venice, as well as a good stop-by on one's way to Austria, Slovenia and Croatia.
The sandy beaches of Grado are only 10 minutes by bus and in summer speed boats connect the resort to Trieste and Croatia.
The hostel has mostly got family rooms (for 2, 3 and 4 persons) with private adjoining bathroom.


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SB16-Ultra Subwoofer
SB16-Ultra Subwoofer
One Subwoofer to Rule Them All. Introducing the SB16-Ultra.
An unrelenting passion for awesome bass performance and engineering perfection guided every aspect of the SB16-Ultra subwoofer's design. Groundbreaking technology, rigorous design, and extreme testing in real world and laboratory environments allowed SVS to achieve massive output levels, extreme low-frequency extension, near-perfect frequency response accuracy, and pinpoint transient response. The culmination of all SVS design advancements, SB16-Ultra represents the greatest leap forward in performance and innovation since the inception of subwoofers.
Secure Checkout - 2048-Bit EV SSL Encryption
Delivery policy - Free Delivery on all orders over £75
Return policy - 14 Day right to cancel!
Masterful Engineering. 3 Game-Changing Subwoofer Innovations.
One of these groundbreaking innovations would've been enough to make the 16-Ultra Series an engineering tour-de-force, but the combination of 3 subwoofer firsts propels the SVS SB16-Ultra into the pantheon of all-time great audio products.
Massive New 16-inch SVS Ultra Driver with Unprecedented 8-inch Edge Wound Voice Coil.
When designing the biggest and most formidable driver ever used in an SVS subwoofer, it was of utmost importance to meet or exceed the same supreme standards of control and accuracy as every other subwoofer SVS manufactures. One huge advancement that allows the SB16-Ultra to handle the crushing force of a 16-inch driver is the 8-inch edge wound voice coil, the largest ever deployed in a consumer subwoofer. This innovation ensures ultra-low distortion, efficient heat dissipation, exceptional linearity and nearly zero power compression. By nature, larger drivers are more difficult to control, but the unprecedented voice coil design ensures accuracy in frequency response and speed in transients without ever sounding boomy.
Pushing the driver are four massive, dedicated toroidal ferrite magnets that unleash a staggering 78mm peak to peak excursion and the highest levels of motor force and palpable SPLs. The light and rigid 16-inch Ultra driver cone consists of a premium fiberglass resin composite with optimal stiffness to mass ratio to maximize movement of air and sonic energy without ever distorting or losing control.
Stunningly Powerful, New Sledge Amplifier with Fully Discrete MOSFET Output and 1,500 Watts Continuous, 5,000+ Watts Peak Power.
Conservatively rated at 1,500 watts continuous, 5,000+ watts peak power, the SB16-Ultra Sledge STA-1500D amplifier's 64 Amp 200-Volt MOSFETs are the sonic muscle behind arguably the most powerful and highest fidelity subwoofer amplifier ever developed. Discrete MOSFET output results in superior handling and tighter control of the immense amplifier power and represents a major upgrade over the integrated circuits found in nearly all other Class D subwoofers.
Effortless power must be harnessed to ensure refined behavior and pristine output at the most demanding drive levels. To achieve this, the SB16-Ultra subwoofer uses the audio industry's most advanced 50 Mhz Analog Devices Audio DSP with 56bit filtering to handle the nearly limitless power, only unleashing the perfect amount of low frequency energy at just the right time.
Acoustically Inert, Rigidly Braced Cabinet
A subwoofer capable of creating such extreme levels of sonic energy requires an equally robust cabinet design. The SB16-Ultra is built with a double thick MDF front baffle and rigid internal bracing to support the massive drive unit and to ensure an acoustically inert environment. The highly reinforced internal design eliminates any chance of coloration or resonance from the cabinet, even when the deepest low frequencies are played at reference sound pressure levels.
SVS Subwoofer Design Philosophy
Frequency Response / Acoustic Data:
- 16-460 Hz +/- 3 dB
- Roll-off slope specifically optimized to take maximum advantage of available room gain.
- 2-meter ground plane away from reflective structures (quasi-anechoic).
Subwoofer Specs:
- Black Oak and Piano Gloss Black finish options
- Protective non-resonant steel mesh grille
- Front-firing 16" driver
- Rear-mounted amplifier
- Screw-in rubber feet
- Robust Medium Density Fiberboard(MDF) cabinet construction
- Cabinet Dimensions: 20" (H) x 19.5" (W) x 20.1" (D) (without grille)
- Cabinet Dimensions (with grille): 20" (H) X 19.5" (W) X 22.9" (D)
- Shipped Dimensions: 24.9" H X 25.1" W X 30.3" D
- Weight (unboxed): 122 pounds
- Shipped Weight: 145.9 pounds
Driver Specs:
- SVS 16" Ultra-series driver
- Powder-coated cast aluminum basket
- Four high-grade ferrite magnets
- Finite Element Analysis (FEA)-optimized motor structure
- Dual shorting rings reduce gap induction and lower distortion
- Extended pole piece improves heat sink and thermal management properties
- 8" diameter, high-power edge wound voice coil
- 78mm peak to peak Xmech excursion (65mm Xmax)
- Upgraded Nomex spider for improved linearity and driver control at extreme drive levels
- Dual 24-strand copper and Nomex composite high-temperature leads provide a pristine electrical connection from the amplifier to the voice coil
- Premium glass fiber laminated dustcap and reinforced composite cone sub-structure ensure a light, ridged, and neutral radiating surface
- Low-creep rubber long-throw surround for excellent durability and longevity
Amplifier Specs:
- Sledge STA-1500D DSP amplifier
- 1,500 watts RMS continuous (5,000+ watts peak dynamic)
- Efficient and cool-running Class D topology
- High resolution 50Mhz Analog Devices Audio DSP with double precision 56bit filtering is most advanced subwoofer processor available for perfect audio reproduction
- Active PFC (Power Factor Correction) ensures the cleanest 1500W output regardless of your home mains supply
- XLR and RCA stereo input/output with ultra-wide input voltage for consumer and professional audio applications
- Convenient front-mounted display with subwoofer controls and 8 digit LED display
- Groundbreaking Subwoofer Control and Bass Management Smartphone App for Apple and Android Devices
- IR Remote Control
- Three user adjustable parametric EQs
- Continuously variable volume/gain control
- Continuously variable (0-180 degrees) phase control
- Continuously variable (30-200 Hz) 6, 12, 18, or 24 dB/octave low pass filter frequency with disable/LFE setting
- Main power rocker switch
- Auto-On / On / Trigger Controls
- 3-12V trigger with 1/8" (3.5 mm) TRS mini-jack input
- Green standby mode with > 0.5 watts consumption
- Input impedance 22kohm (RCA)/ 22kohm (XLR).
- RoHS compliant, lead-free construction and world-wide safety certifications
- Detachable power cord