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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.
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| ==Mathematical statement==
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| 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>:
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| : <math>(c_\max)^{2k} \geq \frac{1}{m-1} \left[ \frac{m}{\binom{n+k-1}{k}}-1 \right]</math>
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| ==Applicability==
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| 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.
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| ==Proof for ''k'' = 1==
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| 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.,
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| : <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> | |
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| 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
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| : <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>
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| The square of the [[Frobenius norm]] (Hilbert–Schmidt norm) of <math>G</math> satisfies
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| : <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>
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| Taking this together with the preceding inequality gives
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| : <math>\sum_{i=1}^m \sum_{j=1}^m |\langle x_i , x_j \rangle|^2\geq \frac{(\mathrm{Tr}\;G)^2}{n}</math> | |
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| 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,
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| : <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>
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| or
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| : <math>\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m(m-n)}{n}</math>
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| 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
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| : <math>\frac{1}{L}\sum_{\ell=1}^L a_{\ell} \leq \max a_{\ell}</math>
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| 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,
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| : <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>
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| or
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| : <math>(c_\max)^2\geq \frac{m-n}{n(m-1)}</math> | |
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| which is precisely the inequality given by Welch in the case that <math>k=1</math>
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| ==Achieving Welch bound equality==
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| 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.
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| 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>.
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| 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>.
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| ==References==
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| {{refbegin}}
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| *{{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}}
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| *{{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}}
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| {{refend}}
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| [[Category:Inequalities]]
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