formulasearchengine - User contributions [en]

Graph partition

2014-11-10T20:32:55Z

173.13.169.18: /* Problem complexity */

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Backfitting algorithm

2014-08-26T14:13:02Z

173.13.0.38: /* Algorithm */

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Superelliptic curve

2014-02-03T00:05:55Z

173.13.181.233: /* Ramification */ Minor fix

[[File:Matrix Signal-Flow Graph.png|thumb|A multi-input, multi-output system represented as a noncommutative matrix signal-flow graph.]]

In [[automata theory]] and [[control theory]], branches of [[mathematics]], [[theoretical computer science]] and [[systems engineering]], a '''noncommutative signal-flow graph''' is a tool for modeling{{sfn|Lorens|1964}} interconnected systems and state machines by mapping the edges of a [[directed graph]] to a [[ring (mathematics)|ring]] or [[semiring]].

A single edge '''weight''' might represent an array of [[impulse response]]s of a complex system (see figure to the right),{{sfn|Riegle|Lin|1972}} or a character from an [[Alphabet (computer science)|alphabet]] picked off the [[Finite state transducer|input tape]] of a finite automaton,{{sfn|Brzozowski|McCluskey|1963}} while the graph might represent the flow of information or state transitions.

As diverse as these applications are, they share much of the same underlying theory.{{sfn|Book|Even|Greibach|Ott|1971}}{{sfn|Pliam|Lee|1995}}

==Definition==
[[File:Signal-Flow Graph Fragment.png|thumb|Signal-flow graph fragment.]]

Consider ''n'' equations involving ''n''+1 variables {''x''0, ''x''1,...,''x''n}.

:<math>x_i = \sum_{j=0}^n a_{ij}x_j, \;\;\; 1\leq i \leq n,</math>

with ''a''ij elements in a ring or semiring ''R''. The free variable ''x''0 corresponds to a source vertex ''v''0, thus having no defining equation. Each equation corresponds to a fragment of a [[directed graph]] ''G''=(''V'',''E'') as show in the figure.

The edge weights define a function ''f'' from ''E'' to ''R''. Finally fix an output vertex ''vm''. A signal-flow graph is the collection of this data ''S'' = (''G''=(''V'',''E''), ''v0'',''vm'' <math>\in</math> ''V'', ''f'' : ''E'' → ''R''). The equations may not have a solution, but when they do,

:<math>x_m = T x_0,</math>

with ''T'' an element of ''R'' called the '''gain'''.

==Successive Elimination==
{{expand section|date=May 2012}}

==Return Loop Method==
There exist several{{sfn|Riegle|Lin|1972}} noncommutative generalizations of [[Mason's rule]]. The most common is the '''return loop method''' (sometimes called the '''forward return loop method (FRL)''', having a dual '''backward return loop method (BRL)'''). The first rigorous proof is attributed to Riegle,{{sfn|Riegle|Lin|1972}} so it is sometimes called '''Riegle's rule'''.{{sfn|Andaloussi|Chalh|Sueur|2006|pp=2962}}

As with Mason's rule, these gain expressions combine terms in a graph-theoretic manner (loop-gains, path products, etc). They are known to hold over an arbitrary noncommutative ring and over the semiring of regular expressions.{{sfn|Pliam|Lee|1995}}

===Formal Description===
The method starts by enumerating all paths from input to output, indexed by ''j'' <math>\in</math> ''J''. We use the following definitions:

* The ''j''-th '''path product''' is (by abuse of notation) a tuple of ''kj'' edge weights along it:

::::<math>p_j = (w^{(j)}_{k_j},\ldots, w^{(j)}_2, w^{(j)}_1).</math>

* To '''split''' a vertex ''v'' is to replace it with a source and sink respecting the original incidence and weights (this is the inverse of the graph morphism taking source and sink to ''v'').
* The '''loop gain''' of a vertex ''v'' w.r.t. a subgraph ''H'' is the gain from source to sink of the signal-flow graph split at ''v'' after removing all vertices not in ''H''.
* Each path defines an ordering of vertices along it. The along path ''j'', the ''i''-th '''FRL (BRL) node factor''' is (1-''Si(j)'')−1 where ''Si(j)'' is the loop gain of the ''i''-th vertex along the ''j''-th w.r.t. the subgraph obtained by removing ''v''0 and all vertices ahead of (behind) it.

The contribution of the ''j''-th path to the gain is the product along the path, alternating between the path product weights
and the node factors:

:<math>T_j = \prod_{i=k_j}^1 (1-S^{(j)}_i)^{-1} w^{(j)}_i,</math>

so the total gain is

:<math>T = \sum_{j\in J} T_j.</math>

===An Example===
[[File:FRL BRL Example.png|thumb|A noncommutative signal-flow graph from ''x'' to ''z'']]

Consider the signal-flow graph shown. From ''x'' to ''z'', there are two path products: (''d'') and (''e,a''). Along (''d''), the FRL and BRL contributions coincide as both share same loop gain (whose split reappears in the upper right of the table below):

:<math>f+e(1-b)^{-1}c,</math>

Multiplying its node factor and path weight, its gain contribution is

:<math>T_d = \left[1 - f - e(1-b)^{-1}c \right]^{-1}d.</math>

Along path (''e,a''), FRL and BRL differ slightly, each having distinct splits of vertices ''y'' and ''z'' as shown in the following table.

:[[File:Return Loop Split Table.png|540px]]

Adding to ''Td'', the alternating product of node factors and path weights, we obtain two gain expressions:

:<math>T^{(FRL)} = \left[1 - f - e(1-b)^{-1}c \right]^{-1}d + \left[1 - f - e(1-b)^{-1}c \right]^{-1}e(1-b)^{-1}a,</math>

and

:<math>T^{(BRL)} = \left[1 - f - e(1-b)^{-1}c \right]^{-1}d + (1-f)^{-1}e\left[1 - b - c(1-f)^e\right]^{-1}a,</math>

These values are easily seen to be the same using identities (''ab'')−1 = ''b''−1''a''−1 and ''a''(1-''ba'')−1=(1-''ab'')−1''a''.

==Applications==

===Matrix Signal-Flow Graphs===
Consider equations

:<math>y_i=\sum_{j=1}^2 a_{ij} x_j + \sum_{j=1}^2 b_{ij}y_j</math>

and

:<math>z_i=\sum_{j=1}^2 c_{ij} y_j,</math>

This system could be modeled as scalar signal-flow graph with multiple inputs and outputs. But, the variables naturally fall into layers, which can be collected into vectors
''x''=(''x''1,''x''2)''t''
''y''=(''y''1,''y''2)''t'' and
''z''=(''z''1,''z''2)''t''.
This results in much simpler '''matrix signal-flow graph''' as shown in the figure at the top of the article.

Applying the forward return loop method is trivial as there's a single path product (''C'',''A'') with a single loop-gain ''B'' at ''y''. Thus as a matrix, this system has a very compact representation of its input-output map

:<math>T = C(1-B)^{-1}A.</math>

===Finite Automata===
[[File:Automaton as Signal-Flow Graph.png|thumb|Representation of a finite automaton as a (noncommutative) signal flow graph over a semiring.]]

An important kind of noncommutative signal-flow graph is a finite state [[automaton]] over an alphabet <math>\Sigma</math>.{{sfn|Brzozowski|McCluskey|1963}}{{sfn|Book|Even|Greibach|Ott|1971}}

Serial connections correspond to the concatenation of words, which can be extended to subsets of the [[free monoid]] <math>\Sigma^*</math>. For ''A'', ''B'' <math>\subseteq\Sigma^*</math>

:<math>A \cdot B = \{ab \mid a\in A, b\in B\}.</math>

Parallel connections correspond to [[set union]], which in this context is often written ''A''+''B''.

Finally, self-loops naturally correspond to the [[Kleene closure]]

:<math>A^* = \{\lambda\} + A + AA + AAA + \cdots,</math>

where <math>\lambda</math> is the [[empty word]]. The similarity to the infinite geometric series

:<math>(1-x)^{-1} = 1 + x + x^2 + x^3 \cdots,</math>

is more than superficial, as expressions of this form serve as 'inversion' in this [[semiring]].{{sfn|Kuich|Salomaa|1985}}

In this way, the subsets of <math>\Sigma^*</math> built of from finitely many of these three operations can be identified with the [[semiring]] of [[regular expressions]]. Similarly, finite graphs whose edges are weighted by subsets of <math>\Sigma^*</math> can be identified with finite automata, though generally that theory starts with [[Singleton (mathematics)|singleton]] sets as in the figure.

This automaton is deterministic so we can unambiguously enumerate paths via words. Using the return loop method, path contributions are:

* path ''ab'', has node factors (''c''*, <math>\lambda</math>), yielding gain contribution

::<math>ac^*b,</math>

* path ''ada'', has node factors (''c''*, ''c''*, <math>\lambda</math>), yielding gain contribution

::<math>ac^*dc^*a,</math>

* path ''ba'', has node factors (''c''*, <math>\lambda</math>), yielding gain contribution

::<math>bc^*a.</math>

Thus the [[Formal language|language]] accepted by this automaton (the gain of its signal-flow graph) is the sum of these terms

:<math>L = ac^*b+ac^*dc^*a+bc^*a.</math>

==Historical Notes==
{{expand section|date=May 2012}}

==See also==
*[[Signal-flow graph]]
*[[Mason's rule]]
*[[Finite automata]]
*[[Regular expressions]]

==Notes==
{{reflist}}

== References ==
*{{cite conference
| ref=harv
| title=Infinite zero of linear time varying bond-graph models: Graphical rules
| last1=Andaloussi
| first1=C.
| last2=Chalh
| first2=Z.
| last3=Sueur
| first3=C.
| booktitle=Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications
| pages=2962–2967
| year=2006
| url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4777109
}}
*{{cite journal
| ref=harv
| last1=Book
| first1=Ronald
| last2= Even
| first2=Shimon
| last3= Greibach
| first3=Sheila
| last4= Ott
| first4=Gene
| title=Ambiguity in graphs and expressions
| journal=IEEE Transactions on Computers
| volume=100
| number=2
| pages=149–153
| year=1971
| publisher=IEEE
| url=http://www.diku.dk/hjemmesider/ansatte/henglein/papers/book1971.pdf
}}
*{{cite journal
| ref=harv
| last1=Brzozowski
| first1=J.A.
| last2=McCluskey Jr.
| first2= E.J.
| title=Signal flow graph techniques for sequential circuit state diagrams
| journal=IEEE Transactions on Electronic Computers
| publisher=IEEE
| number=2
| pages=67–76
| year=1963
| url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4037802
}}
*{{cite journal
| ref=harv
| title=A new normal-form theorem for context-free phrase structure grammars
| last=Greibach
| first=Sheila
| journal=Journal of the ACM
| volume=12
| number=1
| pages=42–52
| year=1965
| publisher=ACM
| url=http://dl.acm.org/citation.cfm?id=321254
}}
*{{cite book
| ref=harv
| last1=Kuich
| first1=Werner
| last2=Salomaa
| first2=Arto
| title=Semirings, automata and languages
| year=1985
| publisher=Springer-Verlag
}}
*{{cite book
| ref=harv
| last=Lorens
| first=Charles S.
| title=Flowgraphs: For the Modeling and Analysis of Linear Systems
| year=1964
| publisher=McGraw-Hill
}}
*{{cite journal
| ref=harv
| last1=Pliam
| first1=John
| last2=Lee
| first2=E. Bruce
| title=On the global properties of interconnected systems
| journal=IEEE Transactions on Circuits and Systems I: Fund. Theory and Apps
| volume=42
| number=12
| pages=1013–1017
| year=1995
| publisher=IEEE
| url=http://www.atbash.com/node/8
}}
*{{cite journal
| ref=harv
| last1=Riegle
| first1=Daryle
| last2=Lin
| first2=P.M.
| title=Matrix signal flow graphs and an optimum topological method for evaluating their gains
| journal=IEEE Transactions on Circuit Theory
| volume=19
| number=5
| pages=427–435
| year=1972
| publisher=IEEE
| url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1083542
}}

[[Category:Control theory]]
[[Category:Automata theory]]

Configuration state function

2013-11-12T15:08:35Z

173.13.109.174: /* A genealogical algorithm for CSF construction */

The '''Wiener–Khinchin theorem''' (also known as the '''Wiener–Khintchine theorem''' and sometimes as the '''Wiener–Khinchin–Einstein theorem''' or the '''Khinchin–Kolmogorov theorem''') states that the [[autocorrelation]] function of a [[wide-sense-stationary random process]] has a spectral decomposition given by the [[power spectrum]] of that process.<ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | pages = 94–95}}</ref><ref>{{cite book | title = Time Series | author = Norbert Wiener | publisher = M.I.T. Press, Cambridge, Massachusetts | year = 1964 | page = 42}}</ref><ref>Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, ''The New Palgrave: A Dictionary of Economics. Time Series and Statistics'', Macmillan, London, 1990, p. 271.</ref><ref>{{cite book | title = Echo Signal Processing | author = Dennis Ward Ricker | publisher = Springer | year = 2003 | isbn = 1-4020-7395-X | url = http://books.google.com/books?id=NF2Tmty9nugC&pg=PA23&dq=%22power+spectral+density%22+%22energy+spectral+density%22&lr=&as_brr=3&ei=HZMvSPSWFZyStwPWsfyBAw&sig=1ZZcHwxXkErvNXtAHv21ijTXoP8#PPA23,M1 }}</ref><ref>{{cite book | title = Digital and Analog Communications Systems | author = Leon W. Couch II | edition = sixth | publisher = Prentice Hall, New Jersey | year = 2001 | isbn=0-13-522583-3| pages = 406–409}}</ref><ref>{{cite book | title = Wireless Technologies: Circuits, Systems, and Devices | author = Krzysztof Iniewski | publisher = CRC Press | year = 2007 | isbn = 0-8493-7996-2 | url = http://books.google.com/books?id=JJXrpazX9FkC&pg=PA390&dq=Wiener-Khinchin-Einstein&ei=1SxlSPGhB4jgsQPr5b3lDw&sig=ACfU3U2Phnk-zwJi57XrvNmdfosyg55FVA }}</ref><ref>{{cite book | title = Statistical Optics | author = Joseph W. Goodman | publisher = Wiley-Interscience | year = 1985 | isbn=0-471-01502-4}}</ref>

==History==
[[Norbert Wiener]] first published this [[theorem]] in 1930;<ref>{{cite journal|last=Wiener|first=Norbert|title=Generalized Harmonic Analysis|journal=Acta Mathematica|year=1930|volume=55|pages=117–258}}</ref> [[Aleksandr Khinchin]] independently<ref>{{cite book|last=Nahin|first=Paul J.|title=Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills|year=2011|publisher=Princeton University Press|isbn=9780691150376|pages=225|url=http://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA225}}</ref> discovered the result and published it in 1934.<ref>{{cite journal|last=Khintchine|first=A.|title=Korrelationstheorie der stationären stochastischen Prozesse |journal=[[Mathematische Annalen]]|year=1934 |volume=109 |issue=1 |pages=604–615 |doi=10.1007/BF01449156 }}</ref> [[Albert Einstein]] had probably anticipated the idea in a brief two-page memo in 1914.<ref>{{cite book|title=The Legacy of Norbert Wiener: A Centennial Symposium (Proceedings of Symposia in Pure Mathematics)|page=95|first1=David|last1=Jerison|first2=Isadore Manuel|last2=Singer|first3=Daniel W.|last3=Stroock|publisher=American Mathematical Society|year=1997|isbn=0-8218-0415-4}}</ref>

==The case of a continuous time process==
For continuous time, the Wiener—Khinchin theorem <ref>Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, ''The New Palgrave: A Dictionary of Economics. Time Series and Statistics'', Macmillan, London, 1990, p. 271.</ref><ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | pages = 94–95}}</ref> says that if <math> x </math> is a wide-sense stationary process such that its [[autocorrelation function]] (sometimes called [[autocovariance]]) defined in terms of statistical [[expected value]] E,
<math>r_{xx}(\tau) = \operatorname{E}\big[\, x(t)x^*(t-\tau) \, \big] \ </math>
exists and is finite at every lag <math> \tau </math>, then there exists a monotone function <math> F(f) </math> in the frequency domain <math> -\infty < f < \infty </math> such that

<math>
r_{xx} (\tau) = \int_{-\infty}^\infty e^{2\pi i\tau f} dF(f)
</math>

where the integral is a [[Stieltjes integral]]. This is a kind of spectral decomposition of the auto-correlation function. F is called the power spectral distribution function, and is a statistical distribution function. It is sometimes called the integrated spectrum.

(The asterisk denotes complex conjugate, and of course it can be omitted if the random process is real-valued.)

Note that the Fourier transform of <math>x(t)\,</math> does not exist in general, because stationary random functions are not generally either [[square-integrable function|square integrable]] or absolutely integrable. Nor is <math> r_{xx} </math> assumed to be absolutely integrable, so it need not have a Fourier transform, either.

But if <math> F(f) </math> is absolutely continuous, for example if the process is purely indeterministic, then one can define the power [[spectral density]] of <math>x(t)\,</math> by taking the derivative of <math> F </math>, putting <math> S_{xx}(f) = F'(f) </math> almost everywhere,<ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | page = 96}}</ref> and the theorem simplifies to

<math>
r_{xx} (\tau) = \int_{-\infty}^\infty S_{xx}(f) e^{2\pi i\tau f} df.
</math>

If now one assumes that r and S satisfy the necessary conditions for Fourier inversion to be valid, the Wiener—Khinchin theorem takes the simple form of saying that r and S are a Fourier transform pair, and

<math>
S_{xx}(f) = \int_{-\infty}^\infty r_{xx} (\tau) e^{-2\pi if\tau} d\tau.
</math>

==The case of a discrete time process==
For the discrete-time case, the power spectral density of the function with discrete values <math>x[n]\,</math> is
:<math> S_{xx}(f)=\sum_{k=-\infty}^\infty r_{xx}[k]e^{-i(2\pi f) k} </math>,

where

:<math>r_{xx}[k] = \operatorname{E}\big[ \, x[n] x^*[n-k] \, \big] \ </math>

is the discrete autocorrelation function of <math>x[n]\,</math>, provided this is absolutely integrable. Being a sampled and discrete-time sequence, the spectral density is periodic in the frequency domain.

==Application==
The theorem is useful for analyzing [[LTI system theory|linear time-invariant systems]], LTI systems, when the inputs and outputs are not square integrable, so their Fourier transforms do not exist. A corollary is that the Fourier transform of the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of the autocorrelation function of the input of the system times the squared magnitude of the Fourier transform of the system impulse response.<ref>
{{cite book
| title = Random signals and noise: a mathematical introduction
| author = Shlomo Engelberg
| publisher = CRC Press
| year = 2007
| isbn = 978-0-8493-7554-5
| page = 130
| url = http://books.google.com/books?id=Zl51JGnoww4C&pg=PA130
}}</ref>
This works even when the Fourier transforms of the input and output signals do not exist because these signals are not square integrable, so the system inputs and outputs cannot be directly related by the Fourier transform of the impulse response.

Since the Fourier transform of the autocorrelation function of a signal is the power spectrum of the signal, this corollary is equivalent to saying that the power spectrum of the output is equal to the power spectrum of the input times the power [[transfer function]].

This corollary is used in the parametric method for power spectrum estimation.

==Discrepancies in terminology==

In many textbooks and in much of the technical literature it is tacitly assumed that Fourier inversion of the [[autocorrelation]] function and the power spectral density is valid, and the Wiener—Khinchin theorem is stated, very simply, as if it said that the Fourier transform of the autocorrelation function was equal to the power [[spectral density]], ignoring all questions of convergence.<ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | page = 98}}</ref> (Einstein is an example.)
But the theorem (as stated here), was applied by [[Norbert Wiener]] and [[Aleksandr Khinchin]] to the sample functions (signals) of [[wide-sense-stationary random process]]es, signals whose Fourier transforms do not exist.
The whole point of Wiener's contribution was to make sense of the spectral decomposition of the autocorrelation function of a sample function of a [[wide-sense-stationary random process]] even when the integrals for the Fourier transform and Fourier inversion do not make sense.

Some authors refer to R as the autocovariance function. They then proceed to normalise it, by dividing by R(0), to obtain what they refer to as the autocorrelation function.

==References==
{{Reflist}}

==Further reading==
*{{cite book |last=Brockwell |first=Peter A. |last2=Davis |first2=Richard J. |title=Introduction to Times Series and Forecasting |edition=Second |publisher=Springer-Verlag |location=New York |year=2002 |isbn=038721657X }}
*{{cite book |last=Chatfield |first=C. |title=The Analysis of Time Series—An Introduction |edition=Fourth |publisher=Chapman and Hall |location=London |year=1989 |isbn=0412318202 }}
*{{cite book |last=Fuller |first=Wayne |title=Introduction to Statistical Time Series |series=Wiley Series in Probability and Statistics |edition=Second |publisher=Wiley |location=New York |year=1996 |isbn=0471552399 }}
*{{cite paper |last=Wiener |first=Norbert |title=Extrapolation, Interpolation, and Smoothing of Stationary Time Series |publisher=Technology Press and Johns Hopkins Univ. Press |location=Cambridge, Massachusetts |year=1949 }} (a classified document written for the Dept. of War in 1943).
*{{cite book |last=Yaglom |first=A. M. |title=An Introduction to the Theory of Stationary Random Functions |publisher=Prentice-Hall |location=Englewood Cliffs, New Jersey |year=1962 }}

{{DEFAULTSORT:Wiener-Khinchin theorem}}
[[Category:Fourier analysis]]
[[Category:Signal processing]]
[[Category:Probability theorems]]

Fieller's theorem

2012-06-13T21:09:28Z

173.13.62.33: /* Further reading */

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