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In [[mathematics]], '''Welch bounds''' are a family of [[inequality (mathematics)|inequalities]] pertinent to the problem of evenly spreading a set of unit [[vector space|vectors]] in a [[vector space]]. The bounds are important tools in the design and analysis of certain methods in [[telecommunication]] engineering, particularly in [[coding theory]]. The bounds were originally published in a 1974 paper by L. R. Welch. | |||
==Mathematical statement== | |||
If <math>\{x_1,\ldots,x_m\}</math> are unit vectors in <math>\mathbb{C}^n</math>, define <math>c_\max = \max_{i\neq j} |\langle x_i, x_j \rangle|</math>, where <math>\langle\cdot,\cdot\rangle</math> is the usual [[inner product]] on <math>\mathbb{C}^n</math>. Then the following inequalities hold for <math>k=1,2,\dots</math>: | |||
: <math>(c_\max)^{2k} \geq \frac{1}{m-1} \left[ \frac{m}{\binom{n+k-1}{k}}-1 \right]</math> | |||
==Applicability== | |||
If <math>m\leq n</math>, then the vectors <math>\{x_i\}</math> can form an [[orthonormal set]] in <math>\mathbb{C}^n</math>. In this case, <math>c_\max=0</math> and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if <math>m>n</math>. This will be assumed throughout the remainder of this article. | |||
==Proof for ''k'' = 1== | |||
The "first Welch bound," corresponding to <math>k=1</math>, is by far the most commonly used in applications. Its proof proceeds in two steps, each of which depends on a more basic mathematical inequality. The first step invokes the [[Cauchy-Schwarz inequality]] and begins by considering the <math>m\times m</math> [[Gram matrix]] <math>G</math> of the vectors <math>\{x_i\}</math>; i.e., | |||
: <math>G=\left[ \begin{array}{ccc} \langle x_1, x_1 \rangle & \cdots & \langle x_1, x_m \rangle \\ \vdots & \ddots & \vdots \\ \langle x_m, x_1 \rangle & \cdots & \langle x_m, x_m \rangle \end{array}\right]</math> | |||
The [[trace (linear algebra)|trace]] of <math>G</math> is equal to the sum of its eigenvalues. Because the [[rank (linear algebra)|rank]] of <math>G</math> is at most <math>n</math>, and it is a [[positive-semidefinite matrix|positive semidefinite]] matrix, <math>G</math> has at most <math>n</math> positive [[eigenvalue]]s with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of <math>G</math> as <math>\lambda_1,\ldots,\lambda_r</math> with <math>r\leq n</math> and applying the Cauchy-Schwarz inequality to the inner product of an <math>r</math>-vector of ones with a vector whose components are these eigenvalues yields | |||
: <math>(\mathrm{Tr}\;G)^2 = \left( \sum_{i=1}^r \lambda_i \right)^2 \leq r \sum_{i=1}^r \lambda_i^2 \leq n \sum_{i=1}^m \lambda_i^2 </math> | |||
The square of the [[Frobenius norm]] (Hilbert–Schmidt norm) of <math>G</math> satisfies | |||
: <math> ||G||^2 = \sum_{i=1}^{m} \sum_{j=1}^m |\langle x_i , x_j \rangle|^2 = \sum_{i=1}^m \lambda_i^2</math> | |||
Taking this together with the preceding inequality gives | |||
: <math>\sum_{i=1}^m \sum_{j=1}^m |\langle x_i , x_j \rangle|^2\geq \frac{(\mathrm{Tr}\;G)^2}{n}</math> | |||
Because each <math>x_i</math> has unit length, the elements on the main diagonal of <math>G</math> are ones, and hence its trace is <math>\mathrm{Tr}\;G = m</math>. So, | |||
: <math>\sum_{i=1}^{m} \sum_{j=1}^m |\langle x_i , x_j \rangle|^2 = m+\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m^2}{n}</math> | |||
or | |||
: <math>\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m(m-n)}{n}</math> | |||
The second part of the proof uses an inequality encompassing the simple observation that the average of a set of non-negative numbers can be no greater than the largest number in the set. In mathematical notation, if <math>a_{\ell}\geq 0</math> for <math>\ell=1,\ldots, L</math>, then | |||
: <math>\frac{1}{L}\sum_{\ell=1}^L a_{\ell} \leq \max a_{\ell}</math> | |||
The previous expression has <math>m(m-1)</math> non-negative terms in the sum,the largest of which is <math>c_\max^2</math>. So, | |||
: <math>(c_\max)^2\geq \frac{1}{m(m-1)}\sum_{i\neq j} |\langle x_i , x_j \rangle|^2\geq\frac{m-n}{n(m-1)}</math> | |||
or | |||
: <math>(c_\max)^2\geq \frac{m-n}{n(m-1)}</math> | |||
which is precisely the inequality given by Welch in the case that <math>k=1</math> | |||
==Achieving Welch bound equality== | |||
In certain telecommunications applications, it is desirable to construct sets of vectors that meet the Welch bounds with equality. Several techniques have been introduced to obtain so-called '''Welch Bound Equality''' (WBE) sets of vectors for the ''k'' = 1 bound. | |||
The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when <math>k=1</math>. The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix <math>G</math> are equal, which happens precisely when the vectors <math>\{x_1,\ldots,x_m\}</math> constitute a [[tight frame]] for <math>\mathbb{C}^n</math>. | |||
The other inequality in the proof is satisfied with equality if and only if <math>|\langle x_i, x_j \rangle|</math> is the same for every choice of <math>i\neq j</math>. In this case, the vectors are [[equiangular lines|equiangular]]. So this Welch bound is met with equality if and only if the set of vectors <math>\{x_i\}</math> is an equiangular tight frame in <math>\mathbb{C}^n</math>. | |||
==References== | |||
{{refbegin}} | |||
*{{cite journal |first1=S. |last1=Datta |first2=S.D. |last2=Howard |first3=D. |last3=Cochran |title=Geometry of the Welch Bounds |journal=Linear Algebra and its Applications |volume=437 |issue=10 |pages=2455–70 |year=2012 |doi=10.1016/j.laa.2012.05.036 |url=http://www.sciencedirect.com/science/article/pii/S0024379512004405 |arxiv=0909.0206v1}} | |||
*{{cite journal |first=L.R. |last=Welch |title=Lower Bounds on the Maximum Cross Correlation of Signals |journal=IEEE Trans. on Info. Theory |volume=20 |issue=3 |pages=397–9 |date=May 1974 |doi=10.1109/TIT.1974.1055219 |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1055219}} | |||
{{refend}} | |||
[[Category:Inequalities]] |
Revision as of 20:14, 27 January 2014
In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory. The bounds were originally published in a 1974 paper by L. R. Welch.
Mathematical statement
If are unit vectors in , define , where is the usual inner product on . Then the following inequalities hold for :
Applicability
If , then the vectors can form an orthonormal set in . In this case, and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if . This will be assumed throughout the remainder of this article.
Proof for k = 1
The "first Welch bound," corresponding to , is by far the most commonly used in applications. Its proof proceeds in two steps, each of which depends on a more basic mathematical inequality. The first step invokes the Cauchy-Schwarz inequality and begins by considering the Gram matrix of the vectors ; i.e.,
The trace of is equal to the sum of its eigenvalues. Because the rank of is at most , and it is a positive semidefinite matrix, has at most positive eigenvalues with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of as with and applying the Cauchy-Schwarz inequality to the inner product of an -vector of ones with a vector whose components are these eigenvalues yields
The square of the Frobenius norm (Hilbert–Schmidt norm) of satisfies
Taking this together with the preceding inequality gives
Because each has unit length, the elements on the main diagonal of are ones, and hence its trace is . So,
or
The second part of the proof uses an inequality encompassing the simple observation that the average of a set of non-negative numbers can be no greater than the largest number in the set. In mathematical notation, if for , then
The previous expression has non-negative terms in the sum,the largest of which is . So,
or
which is precisely the inequality given by Welch in the case that
Achieving Welch bound equality
In certain telecommunications applications, it is desirable to construct sets of vectors that meet the Welch bounds with equality. Several techniques have been introduced to obtain so-called Welch Bound Equality (WBE) sets of vectors for the k = 1 bound.
The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when . The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix are equal, which happens precisely when the vectors constitute a tight frame for .
The other inequality in the proof is satisfied with equality if and only if is the same for every choice of . In this case, the vectors are equiangular. So this Welch bound is met with equality if and only if the set of vectors is an equiangular tight frame in .
References
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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