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Magnetics
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Contents
Articles
Saturation (magnetic) 1
Intermodulation 2
Volterra series 7
References
Article Sources and Contributors 11
Image Sources, Licenses and Contributors 12
Article LicensesLicense 13
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Saturation (magnetic) 1
Saturation (magnetic)
Magnetization curves of 9 ferromagnetic
materials, showing saturation. 1.Sheet steel,
2.Silicon steel, 3.Cast steel, 4.Tungsten steel,
5.Magnet steel, 6.Cast iron, 7.Nickel, 8.Cobalt,
9.Magnetite
Seen in some magnetic materials, saturation is the state reached when
an increase in applied external magnetic field H cannot increase the
magnetization of the material further, so the total magnetic flux density
B levels off. It is a characteristic particularly of ferromagnetic
materials, such as iron, nickel, cobalt and their alloys.
Description
Saturation is most clearly seen in the magnetization curve (also called
BH curve or hysteresis curve) of a substance, as a bending to the right
of the curve (see graph at right). As the H field increases, the B field
approaches a maximum value asymptotically, the saturation level for
the substance. Technically, above saturation, the B field continues
increasing, but at the paramagnetic rate, which is 3 orders of magnitude
smaller than the ferromagnetic rate seen below saturation.
The relation between the magnetizing fieldH and the magnetic fieldB
can also be expressed as the magnetic permeability: or
the relative permeability , where is the vacuum
permeability. The permeability of ferromagnetic materials is not constant, but depends onH. In saturable materials
the relative permeability increases with H to a maximum, then as it approaches saturation inverts and decreases
toward one.
Different materials have different saturation levels. For example, high permeability iron alloys used in transformers
reach magnetic saturation at 1.6 - 2.2 teslas (T), whereas ferrites saturate at 0.2 - 0.5 T. Some amorphous alloys
saturate at 1.2-1.3 T. Mu metal saturates at around 0.8 T.
Due to saturation, the magnetic permeability fof
a ferromagnetic substance reaches a maximum
and then declines
Explanation
Ferromagnetic materials (like iron) are composed of microscopic
regions called magnetic domains, that act like tiny permanent magnets
that can change their direction of magnetization. Before an external
magnetic field is applied to the material, the domains are oriented in
random directions. Their tiny magnetic fields are oriented in random
directions and cancel each other out, so the material has no significant
magnetic field. When an external magnetizing fieldH is applied to the
material, it penetrates the material and aligns the domains, causing
their tiny magnetic fields to turn and align parallel to the external field,
adding together to create a large magnetic field B which extends out
from the material. This is called magnetization. The stronger the
external magnetic fieldH, the more the domains align yielding a higher magnetic flux densityB. Saturation occurs
when practically all the domains are lined up, so further increases inH can't increase B beyond the increment that
would be caused in a nonmagnetic material, in other words, cannot cause further alignment of the domains.
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Saturation (magnetic) 2
Effects and uses
Saturation limits the maximum magnetic fields achievable in ferromagnetic-core electromagnets and transformers to
around 2 T, which puts a limit on the minimum size of their cores. This is one reason why high power motors,
generators, and utility transformers are physically large; because they must have large magnetic cores.
In electronic circuits, transformers and inductors with ferromagnetic cores operate nonlinearly when the current
through them is large enough to drive their core materials into saturation. This means that their inductance and other
properties vary with changes in drive current. In linear circuits this is usually considered an unwanted departure from
ideal behavior. When AC signals are applied, this nonlinearity can cause the generation of harmonics and
intermodulation distortion. To prevent this, the level of signals applied to iron core inductors must be limited so they
don't saturate. To lower its effects, an air gap is created in some kinds of transformer cores. The saturation current,
the current through the winding required to saturate the magnetic core, is given by manufacturers in the
specifications for many inductors and transformers.
On the other hand, saturation is exploited in some electronic devices. Saturation is employed to limit current in
saturable-core transformers, used in arc welding, and ferroresonant transformers which serve as voltage regulators.
When the primary current exceeds a certain value, the core is pushed into its saturation region, limiting further
increases in secondary current. In a more sophisticated application, saturable core inductors and magnetic amplifiers
use a DC current through a separate winding to control an inductor's impedance. Varying the current in the control
winding moves the operating point up and down in the saturation curve, controlling the AC current through the
inductor. These are used in variable fluorescent light ballasts, and power control systems.
References
Intermodulation
A frequency spectrum plot showing intermodulation between two injected
signals at 270 and 275 MHz (the large spikes). Visible intermodulation products
are seen as small spurs at 280 MHz and 265 MHz.
Intermodulation or intermodulation
distortion (IMD) is the amplitude modulation
of signals containing two or more different
frequencies in a system with nonlinearities.
The intermodulation between each frequency
component will form additional signals at
frequencies that are not just at harmonic
frequencies (integer multiples) of either, but
also at the sum and difference frequencies of
the original frequencies and at multiples of
those sum and difference frequencies.
Intermodulation is caused by non-linear
behaviour of the signal processing being used.
The theoretical outcome of these
non-linearities can be calculated by generating
a Volterra series of the characteristic, while the
usual approximation of those non-linearities is
obtained by generating a Taylor series.
Intermodulation is rarely desirable in radio or audio processing, as it creates unwanted spurious emissions, often in
the form of sidebands. For radio transmissions this increases the occupied bandwidth, leading to adjacent channel
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Intermodulation 3
interference, which can reduce audio clarity or increase spectrum usage. It should not be confused with harmonic
distortion (which has common musical applications), nor with intentional modulation (such as a frequency mixer in
superheterodyne receivers) where signals to be modulated are presented to an intentional nonlinear element
(multiplied) (see non-linear mixers such as mixer diodes and even single-transistor oscillator-mixer circuits). In
audio, the intermodulation products are nonharmonically related to the input frequencies and therefore "off-key"
with respect to the common Western musical scale.
Causes of intermodulation
A linear system cannot produce intermodulation. If the input of a linear time-invariant system is a signal of a single
frequency, then the output is a signal of the same frequency; only the amplitude and phase can differ from the input
signal. However, non-linear systems generate harmonics, meaning that if the input of a non-linear system is a signal
of a single frequency, then the output is a signal which includes a number of integer multiples of the input
frequency; (i.e. some of ).
Intermodulation occurs when the input to a non-linear system is composed of two or more frequencies. Consider an
input signal that contains three frequency components at , , and ; which may be expressed as
where the and are the amplitudes and phases of the three components, respectively.
We obtain our output signal, , by passing our input through a non-linear function:
will contain the three frequencies of the input signal, , , and (which are known as thefundamental
frequencies), as well as a number of linear combinations of the fundamental frequencies, each of the form
where , , and are arbitrary integers which can assume positive or negative values. These are the
intermodulation products (or IMPs).
In general, each of these frequency components will have a different amplitude and phase, which depends on the
specific non-linear function being used, and also on the amplitudes and phases of the original input components.
More generally, given an input signal containing an arbitrary number of frequency components
, the output signal will contain a number of frequency components, each of which may be described by
where the coefficients are arbitrary integer values.
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Intermodulation 4
Intermodulation order
Distribution of third-order intermodulations: in blue the position of the fundamental
carriers, in red the position of dominant IMPs, in green the position of specific IMPs.
The order of a given
intermodulation product is the sum of
the absolute values of the coefficients,
For example, in our original example
above, third-order intermodulation
products (IMPs) occur where
:
In many radio and audio applications,
odd-order IMPs are of most interest, as
they fall within the vicinity of the
original frequency components, and
may therefore interfere with the
desired behaviour.
Passive Intermodulation
(PIM)
As explained in a previous section, intermodulation can only occur in non-linear systems. Non-linear systems are
generally composed of active components, meaning that the components must be biased with an external power
source which is not the input signal (i.e. the active components must be "turned on"). Passive intermodulation (PIM),
however, occurs in passive devices (which may include cables, antennas etc.) that are subjected to two or more high
power tones. The PIM product is the result of the two (or more) high power tones mixing at device nonlinearities
such as junctions of dissimilar metals, metal-oxide junctions and even loose connectors. The higher the signal
amplitudes, the more pronounced the effect of the nonlinearities, and the more prominent the intermodulation that
occurs - even though upon initial inspection, the system would appear to be linear and unable to generate
intermodulation.
Sources of PIM
Ferromagnetic materials are the most common materials to avoid and include ferrites, nickel, (including nickel
plating) and steels (including some stainless steels.) These materials exhibit hysteresis when exposed to reversingmagnetic fields resulting in PIM generation.
PIM can also be generated in components with manufacturing or workmanship defects, such as cold or cracked
solder joints or poorly made mechanical contacts. If these defects are exposed to high RF currents, PIM can be
generated. As a result, RF equipment manufacturers perform factory PIM tests on components, to eliminate PIM
caused by these design and manufacturing defects.
In the field, PIM can be caused by components that were damaged in transit to the cell site, installation workmanship
issues and by external PIM sources. Some of these include:
Contaminated surfaces or contacts due to dirt, dust, moisture or oxidation.
Loose mechanical junctions due to inadequate torque, poor alignment or poorly prepared contact surfaces. Loose mechanical junctions caused during transportation, shock or vibration.
Metal flakes or shavings inside RF connections.
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Intermodulation 5
Inconsistent metal-to-metal contact between RF connector surfaces caused by any of the following:
Trapped dielectric materials (adhesives, foam, etc.), cracks or distortions at the end of the outer conductor of coaxial
cables, often caused by overtightening the back nut during installation, solid inner conductors distorted in the
preparation process, hollow inner conductors excessively enlarged or made oval during the preparation process.
PIM can also occur in connectors, or when conductors made of two galvanically unmatched metals come in
contact with each other. Nearby metallic objects in the direct beam and side lobes of the transmit antenna including rusty bolts, roof
flashing, vent pipes, guy wires, etc.
PIM Testing
IEC 62037 is the international standard for PIM testing and gives specific details as to PIM measurement setups. The
standard specifies the use of two +43 dBm (20W) tones for the test signals for PIM testing. This power level has
been used by RF equipment manufacturers for more than a decade to establish PASS / FAIL specifications for RF
components.
Intermodulation in electronic circuits
Intermodulation is caused by nonlinearity or parameter limitations in an amplifier system. This nonlinearity can be
characterized in many ways, including the slew rate, crossover distortion, reduced transistor current gain, or
saturation of collector-emitter junctions near clipping. Slew-induced distortion (SID) can produce intermodulation
distortion (IMD) when the first signal is slewing (changing voltage) at the limit of the amplifier's power bandwidth
product. This induces an effective reduction in gain, partially amplitude-modulating the second signal. If SID only
occurs for a portion of the signal, it is called "transient" intermodulation distortion.[1]
This usually occurs due to soft
clipping of the signal peaks.[2]
Intermodulation in audio applications
Any system that produces harmonic distortion also produces intermodulation distortion. The difference between the
two types of distortion is the signal used to test it, not the device. Only sine waves produce harmonic distortion;
music signals always produce intermodulation.
Audio engineers usually strive to avoid intermodulation, as for anything other than extremely simple input
waveforms, it introduces frequency components that are not harmonically related, which tends to sound unmusical
and unpleasant. However, certain audio effects rely on amplitude modulation; these include tremolo and ring
modulation. One way to generate such effects is through deliberate intermodulation in a non-linear device, but may
also be achieved without intermodulation by an analog multiplier.Wikipedia:Disputed statement Transient
intermodulation distortion, or TIM, occurs in amplifiers that employ negative feedback when signal delays make theamplifier incapable of correcting distortion when exposed to fast, transient signals.
Harmonic distortion occurs when non-linearity (in an amplifier or loudspeaker, for instance) only creates new
frequencies that are harmonically related to the original sine wave signal. Intermodulation distortion occurs when a
different typeWikipedia:Disputed statement of non-linearity can create new frequencies that are not harmonically
related to the original signal. All audio devices give rise to distortion to some extent; harmonic distortion and
intermodulation distortion tests highlight different aspects of imperfections, and one type of distortion may be
inaudibly low while the other is significantly high for some equipment under certain conditions.Wikipedia:Disputed
statement[citation needed]
In analog recording, wow and flutter are forms of intermodulation distortion caused by speed variations in the
medium (usually tape). When the flutter rate is above a certain point, typically about 20Hz, the modulation products
impressed into the musical signal no longer present as an audibly obvious flutter, yet continue to interfere with the
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Intermodulation 6
signal as extraneous frequency modulation, and the resulting sideband products manifest as distortion. This
distortion results in a thicker, grainier texture due to the excess non-musical sum and difference components riding
above and below the harmonic content of the material.
Measurement
Intermodulation distortion in audio is usually specified as the Root Mean Square (RMS) value of the various
sum-and-difference signals as a percentage of the original signal's RMS voltage, although it may be specified in
terms of individual component strengths, in decibels, as is common with RF work. Audio IMD standard tests include
SMPTE standard RP120-1994 where two signals (at 60 Hz and 7 kHz, with 4:1 amplitude ratios) are used for the
test; many other standards (such as DIN, CCIF) use other frequencies and amplitude ratios. Opinion varies over the
ideal ratio of test frequencies (e.g. 3:4,[3]
or almost -but not exactly - 3:1 for example).
After feeding the equipment under test with low distortion input sinewaves, the output distortion can be measured by
using an electronic filter to remove the original frequencies, or spectral analysis may be made using Fourier
Transformations in software or a dedicated spectrum analyser, or when determining intermodulation effects in
communications equipment, may be made using the receiver under test itself.
Using a modern network analyzer with two internal RF sources and sensitive RF detectors simplifies the
measurement setup and also provides a sensitivity level comparable to spectrum analyzers. Furthermore, a calibrated
VNA setup also removes mismatch errors from measurements which otherwise would be present in spectrum
analyzer measurements. Meanwhile error-corrected IM measurement systems are available. These system support
frequency converting vector-measurements of S-parameters.[4]
The user can locate IM-sources and perform a vector
or time-domain fitting or modelling of the IM-signals and components.
External links
Lloyd Butler (1997). "Intermodulation Performance and Measurement of Intermodulation Components"[5]
.
VK5BR. "Amateur Radio," August 1997. Retrieved 30 January 2012.
References
[1] Rane Pro Audio Reference for IM (http://www.rane.com/par-i.html#IM)
[2] http://waltjung.org/PDFs/SID_TIM_TAA77_P1.pdf Slewing Induced Distortion in Audio Amplifiers, Part 1 by Walter Jung in The Audio
Amateur Issue 1/1977
[3] http://www.leonaudio.com. au/3-4.ratio.distortion.measurement.pdf Graeme John Cohen: 3-4 Ratio; A method of measuring distortion
products
[4] Thalayasingam, K. and Heuermann, H.,Novel Vector Non-Linear Measurement System for Intermodulation Measurements, European
Microwave Conference, Rom, Italy, IEEE, 2009, Available online (http://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&
queryText=thalayasingam&x=32&y=12=no)
[5] http://users.tpg.com.au/ldbutler/Intermodulation.htm
This article incorporates public domain material from the General Services Administration document "Federal
Standard 1037C" (http://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm) (in support of MIL-STD-188).
http://en.wikipedia.org/w/index.php?title=MIL-STD-188http://www.its.bldrdoc.gov/fs-1037/fs-1037c.htmhttp://en.wikipedia.org/w/index.php?title=General_Services_Administrationhttp://en.wikipedia.org/w/index.php?title=Copyright_status_of_work_by_the_U.S._governmenthttp://en.wikipedia.org/w/index.php?title=File:PD-icon.svghttp://users.tpg.com.au/ldbutler/Intermodulation.htmhttp://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=thalayasingam&x=32&y=12=nohttp://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=thalayasingam&x=32&y=12=nohttp://www.leonaudio.com.au/3-4.ratio.distortion.measurement.pdfhttp://waltjung.org/PDFs/SID_TIM_TAA77_P1.pdfhttp://www.rane.com/par-i.html#IMhttp://users.tpg.com.au/ldbutler/Intermodulation.htmhttp://en.wikipedia.org/w/index.php?title=Network_analyzer_%28electrical%29http://en.wikipedia.org/w/index.php?title=Spectrum_analyserhttp://en.wikipedia.org/w/index.php?title=Fast_Fourier_transformhttp://en.wikipedia.org/w/index.php?title=Fast_Fourier_transformhttp://en.wikipedia.org/w/index.php?title=Electronic_filterhttp://en.wikipedia.org/w/index.php?title=Audio_system_measurementshttp://en.wikipedia.org/w/index.php?title=Radio_frequencyhttp://en.wikipedia.org/w/index.php?title=Decibelhttp://en.wikipedia.org/w/index.php?title=Root_Mean_Squarehttp://en.wikipedia.org/w/index.php?title=Frequency_modulation -
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Volterra series 7
Volterra series
The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series
in its ability to capture 'memory' effects. The Taylor series can be used to approximate the response of a nonlinear
system to a given input if the output of this system depends strictly on the input at that particular time. In the
Volterra series the output of the nonlinear system depends on the input to the system at all other times. This provides
the ability to capture the 'memory' effect of devices such as capacitors and inductors.
It has been applied in the fields of medicine (biomedical engineering) and biology, especially neuroscience. It is also
used in electrical engineering to model intermodulation distortion in many devices including power amplifiers and
frequency mixers. Its main advantage lies in its generality: it can represent a wide range of systems. It is therefore
sometimes referred to as a non-parametric model.
In mathematics, a Volterra series denotes a functional expansion of a dynamic, nonlinear, time-invariant functional.
Volterra series are frequently used in system identification. The Volterra series, which is used to prove the Volterra
theorem, is a series of infinite sum of multidimensional convolutional integrals.
History
Volterra series is a modernized version of the theory of analytic functionals due to the Italian mathematician Vito
Volterra in work dating from 1887.[1]
Norbert Wiener became interested in this theory in the 1920s from contact
with Volterra's student Paul Lvy. He applied his theory of the Brownian motion to the integration of Volterra
analytic functionals. The use of Volterra series for system analysis originated from a restricted 1942 wartime
report[2]
of Wiener, then professor of mathematics at MIT. It used the series to make an approximate analysis of the
effect of radar noise in a nonlinear receiver circuit. The report became public after the war.[3]
As a general method of
analysis of nonlinear systems, Volterra series came into use after about 1957 as the result of a series of reports, at
first privately circulated, from MIT and elsewhere.[4]
The name Volterra series came into use a few years later.
Mathematical theory
The theory of Volterra series can be viewed from two different perspectives: either one considers an operator
mapping between two real (or complex) function spaces or a functional mapping from a real (or complex) function
space into the real (or complex) numbers. The latter, functional perspective is in more frequent use, due to the
assumed time-invariance of the system.
Continuous time
A continuous time-invariant system withx(t) as input andy(t) as output can be expanded in Volterra series as:
where and .
, are called n-th order Volterra kernel which can be regarded as a higher-order impulse
response of the system.
IfN is finite, the series operator is said truncated.
If a,b andN are finite, the series operator is called doubly-finite Volterra series.
http://en.wikipedia.org/w/index.php?title=Integral_kernelhttp://en.wikipedia.org/w/index.php?title=Massachusetts_Institute_of_Technologyhttp://en.wikipedia.org/w/index.php?title=Brownian_motionhttp://en.wikipedia.org/w/index.php?title=Paul_L%C3%A9vy_%28mathematician%29http://en.wikipedia.org/w/index.php?title=Norbert_Wienerhttp://en.wikipedia.org/w/index.php?title=Vito_Volterrahttp://en.wikipedia.org/w/index.php?title=Vito_Volterrahttp://en.wikipedia.org/w/index.php?title=System_identificationhttp://en.wikipedia.org/w/index.php?title=Functional_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Nonlinearhttp://en.wikipedia.org/w/index.php?title=Mathematicshttp://en.wikipedia.org/w/index.php?title=Non-parametrichttp://en.wikipedia.org/w/index.php?title=Frequency_mixerhttp://en.wikipedia.org/w/index.php?title=Taylor_series -
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Volterra series 8
Sometimes the n-th order term is divided by n!, a convention which is convenient when considering the combination
of Volterra systems by placing one after the other ('cascading').
The causality condition: Since in any physically realizable system the output can only depend on previous values of
the input, the kernels will be zero if any of the variables are negative. The
integrals may then be written over the half range from zero to infinity. So if the operator is causal, .
Frchet's approximation theorem: The use of the Volterra series to represent a time-invariant functional relation isoften justified by appealing to a theorem due to Frchet. This theorem states that such a system can be approximated
uniformly and to an arbitrary degree of precision by a sufficiently high finite order Volterra series. The input set over
which this approximation holds must be compact. This is usually taken to be the set of equicontinuous, uniformly
bounded functions which is compact by the ArzelAscoli theorem. In many physical situations this assumption
about the input set is a reasonable one. The theorem however gives no indication as to how many terms are needed
for a good approximation which is the important question in applications.
Discrete time
where and .
, are called Volterra kernels.
IfP is finite, the series operator is said truncated.
If a,b andP are finite the series operator is called doubly-finite Volterra series.
If the operator is causal.
We can always consider, without loss of the generality, the kernel as symmetrical. In fact, for the
commutativity of the multiplication it is always possible to symmetrize it without changing .
So for a causal system with symmetrical kernels we can write
Methods to estimate the kernel coefficients
Estimating the Volterra coefficients individually is complicated since the basis functionals of the Volterra series are
correlated. This leads to the problem of simultaneously solving a set of integral-equations for the coefficients. Hence,
estimation of Volterra coefficients is generally performed by estimating the coefficients of an orthogonalized series,
e.g. the Wiener series, and then recomputing the coefficients of the original Volterra series. The Volterra series main
appeal over the orthogonalized series lies in its intuitive, canonical structure, i.e. all interactions of the input have
one fixed degree. The orthogonalized basis functionals will generally be quite complicated.
An important aspect, with respect to which the following methods differ is whether the orthogonalization of the basis
functionals is to be performed over the idealized specification of the input signal (e.g. gaussian, white noise) or over
the actual realization of the input (i.e. the pseudo-random, bounded, almost-white version of gaussian white noise, or
any other stimulus). The latter methods, despite their lack of mathematical elegance, have been shown to be more
flexible (as arbitrary inputs can be easily accommodated) and precise (due to the effect that the idealized version of
the input signal is not always realizable).
http://en.wikipedia.org/w/index.php?title=Wiener_serieshttp://en.wikipedia.org/w/index.php?title=Arzel%C3%A0%E2%80%93Ascoli_theoremhttp://en.wikipedia.org/w/index.php?title=Uniformly_boundedhttp://en.wikipedia.org/w/index.php?title=Uniformly_boundedhttp://en.wikipedia.org/w/index.php?title=Equicontinuoushttp://en.wikipedia.org/w/index.php?title=Compact_spacehttp://en.wikipedia.org/w/index.php?title=Maurice_Ren%C3%A9_Fr%C3%A9chet -
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Volterra series 9
Crosscorrelation method
This method, developed by Lee & Schetzen, orthogonalizes with respect to the actual mathematical description of
the signal, i.e. the projection onto the new basis functionals is based on the knowledge of the moments of the random
signal.
To allow identification orthogonalization, Volterra series must be rearranged in terms of orthogonal
non-homogeneous G operators (Wiener series):
The G operators can be defined by the following
whenever is arbitrary omogeneous Volterra,x(n) is a Stationary white noise with zero mean and variance
A.
Recalling that every Volterra functional is orthogonal to all Wiener functional of greater order, and considering the
following Volterra functional
we can write
Ifx is SWN, and by letting , we have:
So if we exclude the diagonal elements, , it is
If we want to consider the diagonal points, the solution proposed by Lee and Schetzen is:
Efficient formulas and references for diagonal kernel point estimation can be found in
[5]
and .
[6]
Exact orthogonal algorithm
This method and its more efficient version (Fast Orthogonal Algorithm) were invented by Korenberg. In this method
the orthogonalization is performed empirically over the actual input. It has been shown to perform more precisely
than the Crosscorrelation method. Another advantage is that arbitrary inputs can be used for the orthogonalization
and that fewer data-points suffice to reach a desired level of accuracy. Also, estimation can be performed
incrementally until some criterion is fulfilled.
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Volterra series 10
Linear regression
Linear regression is a standard tool from linear analysis. Hence, one of its main advantages is the widespread
existence of standard tools for solving linear regressions efficiently. It has some educational value, since it highlights
the basic property of Volterra series: linear combination of non-linear basis-functionals. For estimation the order of
the original should be known, since the volterra basis-functionals are not orthogonal and estimation can thus not be
performed incrementally.
Kernel method
This method was invented by Franz & Schlkopf and is based on statistical learning theory. Consequently, this
approach is also based on minimizing the empirical error (often called empirical risk minimization). Franz and
Schlkopf proposed that the kernel method could essentially replace the Volterra series representation, although
noting that the latter is more intuitive.
Differential sampling
This method was developed by van Hemmen and coworkers and utilizes Dirac delta functions to sample the Volterra
coefficients.
References
[1][1] Vito Volterra. Theory of Functionals and of Integrals and Integro-Differential Equations. New York: Dover Publications, 1959.
[2] Wiener N:Response of a nonlinear device to noise. Radiation Lab MIT 1942, restricted. report V-16, no 129 (112 pp). Declassified Jul 1946,
Published as rep. no. PB-1-58087, U.S. Dept. Commerce. URL: http://www.dtic.mil/dtic/tr/fulltext/u2/a800212. pdf
[3] Ikehara S:A method of Wiener in a nonlinear circuit. MIT Dec 10 1951, tech. rep. no 217, Res. Lab. Electron.
[4][4] Early MIT reports by Brilliant, Zames, George, Hause, Chesler can be found on dspace.mit.edu.
[5] M. Pirani, S. Orcioni, and C. Turchetti, ``Diagonal kernel point estimation of n-th order discrete Volterra-Wiener systems,EURASIP Journal
on Applied Signal Processing, vol. 2004, no. 12, pp. 1807--1816, Sept. 2004.
[6] S. Orcioni, M. Pirani, and C. Turchetti, ``Advances in Lee-Schetzen method for Volterra filter identification,Multidimensional Systems and
Signal Processing, vol. 16, no. 3, pp. 265--284, 2005.
Further reading
Barrett J.F:Bibliography of Volterra series, Hermite functional expansions, and related subjects. Dept. Electr.
Engrg, Univ.Tech. Eindhoven, NL 1977, T-H report 77-E-71. (Chronological listing of early papers to 1977)
URL: http://alexandria.tue.nl/extra1/erap/publichtml/7704263.pdf
Bussgang, J.J.; Ehrman, L.; Graham, J.W:Analysis of nonlinear systems with multiple inputs,Proc. IEEE, vol.62,
no.8, pp. 10881119, Aug. 1974
Giannakis G.B & Serpendin E:A bibliography on nonlinear system identification. Signal Processing, 81 2001
533580. (Alphabetic listing to 2001) www.elsevier.nl/locate/sigpro Korenberg M.J. Hunter I.W: The Identification of Nonlinear Biological Systems: Volterra Kernel Approaches,
Annals Biomedical Engineering (1996), Volume 24, Number 2.
Kuo Y L:Frequency-domain analysis of weakly nonlinear networks, IEEE Trans. Circuits & Systems,
vol.CS-11(4) Aug 1977; vol.CS-11(5) Oct 1977 26.
Rugh W J:Nonlinear System Theory: The VolterraWiener Approach. Baltimore 1981 (Johns Hopkins Univ
Press) Many online versions, e.g. www.ece.jhu.edu/~rugh/volterra/book.pdf
Schetzen M: The Volterra and Wiener Theories of Nonlinear Systems, New York: Wiley, 1980.
http://en.wikipedia.org/w/index.php?title=Georgios_B._Giannakishttp://alexandria.tue.nl/extra1/erap/publichtml/7704263.pdfhttp://www.dtic.mil/dtic/tr/fulltext/u2/a800212.pdfhttp://en.wikipedia.org/w/index.php?title=Dirac_delta_functionhttp://en.wikipedia.org/w/index.php?title=Statistical_learning_theory -
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Article Sources and ContributorsSaturation (magnetic) Source: http://en.wikipedia.org/w/index.php?oldid=593004054 Contributors: A13ean, ARTE, Avoided, BZegarski, BenFrantzDale, Chetvorno, Dcshank, Francisco
Quiumento, Gene Nygaard, JPFen, Jim.henderson, KoenDelaere, Mnmngb, Red Act, Rjwilmsi, RockMagnetist, Rtdrury, Shaddack, StAnselm, Steve Quinn, Teaktl17, The Anome, TomyDuby,
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GirasoleDE, JFB80, Jitse Niesen, Jlataire, Konstable, Kuru, Mdd, Memming, Michael Hardy, Myasuda, Oleg Alexandrov, Omphaloscope, Orcioni, Rjwilmsi, Squids and Chips, Sawomir Biay,
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License
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