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 bounds are also sometimes stated in terms of the averaged squared overlap between the set of vectors.
can form an orthonormal set in
This will be assumed throughout the remainder of this article.
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
is equal to the sum of its eigenvalues.
, and it is a positive semidefinite matrix,
Writing the non-zero eigenvalues of
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
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.
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
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
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
constitute a tight frame for
The other inequality in the proof is satisfied with equality if and only if
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
Similarly, the Welch bounds stated in terms of average squared overlap, are saturated for all
-design in the complex projective space