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 formula_1 are unit vectors in formula_2, define formula_3, where formula_4 is the usual inner product on formula_2. Then the following inequalities hold for formula_6:formula_7Welch bounds are also sometimes stated in terms of the averaged squared overlap between the set of vectors. In this case, one has the inequalityformula_8
Applicability.
If formula_9, then the vectors formula_10 can form an orthonormal set in formula_2. In this case, formula_12 and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if formula_13. This will be assumed throughout the remainder of this article.
Proof for "k" = 1.
The "first Welch bound," corresponding to formula_14, 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 formula_15 Gram matrix formula_16 of the vectors formula_10; i.e.,
 formula_18
The trace of formula_16 is equal to the sum of its eigenvalues. Because the rank of formula_16 is at most formula_21, and it is a positive semidefinite matrix, formula_16 has at most formula_21 positive eigenvalues with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of formula_16 as formula_25 with formula_26 and applying the Cauchy-Schwarz inequality to the inner product of an formula_27-vector of ones with a vector whose components are these eigenvalues yields
 formula_28
The square of the Frobenius norm (Hilbert–Schmidt norm) of formula_16 satisfies
 formula_30
Taking this together with the preceding inequality gives
 formula_31
Because each formula_32 has unit length, the elements on the main diagonal of formula_16 are ones, and hence its trace is formula_34. So,
 formula_35
or
 formula_36
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 formula_37 for formula_38, then
 formula_39
The previous expression has formula_40 non-negative terms in the sum, the largest of which is formula_41. So,
 formula_42
or 
 formula_43
which is precisely the inequality given by Welch in the case that formula_14.
Achieving the Welch bounds.
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 formula_14 bound.
The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when formula_14. 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 formula_16 are equal, which happens precisely when the vectors formula_1 constitute a tight frame for formula_2.
The other inequality in the proof is satisfied with equality if and only if formula_50 is the same for every choice of formula_51. In this case, the vectors are equiangular. So this Welch bound is met with equality if and only if the set of vectors formula_10 is an equiangular tight frame in formula_2.
Similarly, the Welch bounds stated in terms of average squared overlap, are saturated for all formula_54 if and only if the set of vectors is a formula_55-design in the complex projective space formula_56.
References.
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