Low coherence unit norm tight frames
Equiangular tight frames (ETFs) have found significant applications in signal processing and coding theory due to their robustness to noise and transmission losses. ETFs are characterized by the fact that the coherence between any two distinct vectors is equal to the Welch bound. This guarantees that the maximum coherence between pairs of vectors is minimized. Despite their usefulness and widespread applications, ETFs of a given size are only guaranteed to exist in or if . This leads to the problem of finding approximations of ETFs of vectors in or where To be more precise, one wishes to construct a unit norm tight frame (UNTF) such that the maximum coherence between distinct vectors of this frame is as close to the Welch bound as possible. In this paper low coherence UNTFs in are constructed by adding a strategically chosen set of vectors called an optimal set to an existing ETF of vectors. In order to do so, combinatorial objects called block designs are used. Estimates are provided on the maximum coherence between distinct vectors of this low coherence UNTF. It is shown that for certain block designs, the constructed UNTF attains the smallest possible maximum coherence between pairs of vectors among all UNTFs containing the starting ETF of vectors. This is particularly desirable if there does not exist a set of the same size for which the Welch bound is attained.
Key words and phrases:Block designs, coherence, equiangular frames, tight frames, Welch bound
1991 Mathematics Subject Classification:42C15; 94Axx
1.1. Background and Motivation
The maximum coherence between pairs in a set of unit vectors in satisfies the following inequality due to Welch :
The quantity appearing on the right side of (1.1) is known as the Welch bound. Sets of unit vectors attaining the lower bound in (1.1) are mathematical objects called equiangular tight frames (ETFs) [2, 3]. Such sets arise in many different areas as in communications, quantum information processing, and coding theory [1, 4, 5, 2, 6, 7, 8, 9, 10, 11]. Consequently, the problem of constructing ETFs and determining conditions under which they exist has gained substantial attention [2, 12, 13, 14, 15, 16, 17, 18, 19]. The Gram matrix of an ETF has two distinct eigenvalues: zero and with multiplicities and respectively . Conditions on eigenvalues for the existence of an ETF have been discussed in [2, 12, 14, 15, 16], among others. A graph theoretic approach to constructing ETFs has been studied in . A correspondence discovered by Fickus et al.  uses Steiner systems to directly construct the frame vectors of certain ETFs, bypassing the common technique of constructing a suitable Gram matrix. This approach lets one construct highly redundant sparse ETFs. However, in the real case, this approach can give rise to ETFs only if a real Hadamard matrix of a certain size exists.
Despite the desirability and importance of ETFs, these cannot exist for many pairs When the Hilbert space is the maximum number of equiangular lines is bounded by and for the bound is [20, 21]. Even when these restrictions hold, ETFs are very hard to construct and do not exist for many pairs . This leads to generalizations and approximations of ETFs. For a real ETF, the off-diagonal entries of the Gram matrix are either or , where is the Welch bound. In other words, the off-diagonal entries of the Gram matrix all have modulus equal to Generalizing this notion, a unit norm tight frame (UNTF) whose associated Gram matrix has off-diagonal entries with distinct moduli is called a -angle tight frame . These objects have also been explored in  under the name -angular frames. Note that for a set of unit vectors, the diagonal entries of the corresponding Gram matrix will equal . Under this definition, ETFs are viewed as -angle tight frames. -angle tight frames, or biangular tight frames, are discussed in [23, 24] and combinatorial constructions are provided. It is to be noted that often in the literature, a unit norm tight frame is called a two-distance tight frame [25, 26] if the off-diagonal entries of the associated Gram matrix take on either of two values and In that case, real ETFs are thought of as two-distance tight frames instead of -angle tight frames, as done here. Non-equiangular two-distance tight frames are examples of the biangular tight frames mentioned above. Explicit constructions of -angle tight frames can be found in . Besides generalizing the notion of an ETF, -angle tight frames prove to be important due to their connection to graphs and association schemes as discussed in .
The goal here is to come up with objects that can be considered approximations of ETFs. In particular, one seeks UNTFs with low coherence among the vectors. Section 2 contains the main contribution of the work presented here. In Section 2, a deterministic way of constructing low coherence UNTFs is given that has the added benefit of being applicable when an equiangular set of lines, and in particular an ETF, of a certain size is known not to exist. The idea is to start from an ETF of vectors in and determine an optimal set of vectors that can be added to this ETF such that the resulting union is a UNTF in for which the maximum coherence between distinct vectors can be minimized even if there does not exist a set of the same size for which the Welch bound is attained. This UNTF can be considered as an approximation of an ETF of the same size in . The exact meaning of “optimal” in this context is given in Definition 2.1. This approach relies on the fact that an ETF of vectors in or always exists [2, 14, 22, 27] which can be viewed as the vertices of a regular simplex centered at the origin. Combinatorial objects called block designs are used to determine the optimal sets to be added to the starting ETF. The main results of Section 2 are Theorem 2.9 and Theorem 2.18. Section 3 provides some concluding remarks.
1.2. Notation and Preliminaries
In a finite dimensional Hilbert space like or a spanning set is called a frame.111In an infinite dimensional space, the notion of a frame is far more subtle [28, 29] and will not be needed here. Given a set of vectors in or let be the matrix whose columns are the vectors will be called the synthesis operator of . For a tight frame the matrix is a multiple of the identity. That is, equals , where is the identity, and is called the frame bound. A tight frame for which each vector is unit norm is called a unit norm tight frame (UNTF). The matrix is the Gram matrix of the set and its non-zero eigenvalues are the same as the eigenvalues of The entry of the Gram matrix is the inner product . An equiangular tight frame (ETF) is a set in a -dimensional Hilbert space satisfying :
, i.e., the set is a tight frame.
for i.e., the set is unit norm.
where is the Welch bound.
Throughout, will be either or .222The results can be easily generalized to any -dimensional Hilbert space since would be isomorphic to or A frame of vectors in (respectively, ) will be referred to as an real (respectively, complex) frame. When is not specified, the frame will be called an frame. Let be the eigenvalues of the corresponding Gram matrix . The frame potential  of a set of unit vectors is the quantity given by
The following result on the frame potential is found in .
Theorem 1.1 (Theorem 6.2, ).
Let with and let be a set of unit norm vectors in or . Then is bounded below by with equality if and only if the frame is a unit norm tight frame (an orthonormal basis in the case ).
2. Constructing low coherence UNTFs with block designs
2.1. Optimal sets to add to ETFs
As already mentioned in Section 1.1, ETFs are useful due to their minimal maximum coherence between distinct vectors. Unfortunately, even though ETFs always exist, an ETF may not always exist if . Since ETFs already minimize the maximum coherence among pairs of its vectors, a natural approach to approximating an ETF, when , is to add an optimal set of vectors to a ETF in such a way that the resulting set is still a UNTF with maximum coherence among its distinct vectors as close to the Welch bound as possible. By definition, a ETF is a UNTF. Since the union of two UNTFs is another UNTF for the same space, the optimal set to be added will be taken to be a UNTF to ensure that the resulting set is a UNTF. This is accomplished below in Theorem 2.9 and Theorem 2.18. In the absence of an ETF, the low coherence UNTFs provided by Theorem 2.9 and Theorem 2.18 can be thought of as approximations of a corresponding ETF, if it were to exist.
From (1.1), for a ETF, the modulus of the inner product of any two distinct vectors is . In what follows, will denote a ETF where for . For any , this ETF can be constructed based on an explicit construction given in .
Let be a UNTF. Let be a ETF where for . is said to be optimal with respect to if, among all possible UNTFs , minimizes the maximum coherence between distinct vectors of .
Given a set of unit vectors containing the real ETF of Definition 2.1, Theorem 2.2 below gives a lower bound for the maximum coherence between distinct vectors of this set. Using Theorem 2.2, one can determine a set that is optimal with respect to in the sense of Definition 2.1. The bound in (2.1) is similar to the orthoplex bound .
Let denote an ETF satisfying for . Let be a unit vector. Then
Since is a unit vector and is a UNTF,
The sum must be bounded above by , which gives
Simplifying this gives
Note that Theorem 2.2 only relies on the fact that is a UNTF. Hence the result still holds if the given ETF is replaced with an arbitrary UNTF, although this fact will not be required in this paper. It is worthwhile to note here that if is odd then Theorem 2.4 shows that the inequality in Theorem 2.2 is sharp for the ETF given in the statement of this theorem.
If is a UNTF such that the maximum coherence between distinct vectors of is , then Theorem 2.2 implies that is optimal with respect to . It is possible to obtain such UNTFs using the construction given in Theorem 2.4 below. This is then further developed in Theorem 2.9, Theorem 2.14, and Theorem 2.18.
Let , set . Let denote the ETF satisfying for . Let denote a subset of of size , and define the vector by
First, note that
Hence , and it follows that
Now consider the case . If is odd then and If is even, then and
Since , if is even. ∎
Theorem 2.4 shows how to construct vectors whose maximum coherence with vectors of the given ETF either meets or comes very close to meeting the optimal bound of Theorem 2.2. In order to construct UNTFs that are optimal with respect to the ETF satisfying for , one can then consider UNTFs of the form , where by the proof of Theorem 2.4,
and is a collection of subsets of of size . The problem now is to determine how to select the subsets so that the vectors satisfy the following constraints:
the maximum coherence of is ,
the set is a UNTF.
Note that the constraint in (ii) is equivalent to being a UNTF since the union of two UNTFs is another UNTF. To see how to choose subsets that minimize the maximum coherence, the following computations will be useful. Recall that, for , . Also, due to Theorems 2.4 and 2.2, . It remains to determine . Due to (2.2),
For let . Then out of the terms in the double summation, there will be terms that equal and terms that equal . Therefore, for
Thus if is the closest integer to , then the above inner product is minimized. The next example shows this approach in action.
Example 2.5 (An optimal UNTF to add to a Etf).
Let be an ETF satisfying for . Such an ETF can be constructed from the vertices of a regular tetrahedron inscribed within the unit sphere. Set . Then the goal is to find a collection of subsets of of size such that the intersection of any two members has element. One such collection is given by .
Now define by
It can be checked that forms an orthonormal basis for This implies that is a UNTF. By Theorem 2.4, the maximum coherence between distinct vectors of the set is which is the lower bound given in Theorem 2.2. By (1.1), the corresponding Welch bound for a collection of unit vectors in is whereas . However, there does not exist a ETF. In fact, the largest ETF in is a ETF, and so no collection of unit vectors in has maximum coherence . Further, by Theorem 2.2, the maximum coherence between distinct vectors of the set is the smallest possible among all UNTFs containing the starting ETF In the absence of a ETF, the set can be thought of as a low coherence UNTF of vectors in that is an approximation of the hypothetical ETF.
2.2. Using block designs to determine optimal sets to add to ETFs
Let denote a set containing points and suppose there is a collection of subsets (“blocks”) of where each block has size . If for any there are precisely blocks in containing , and for any distinct there are precisely blocks containing , then is said to be a block design on , or more simply a block design.
Example 2.5 indirectly makes use of symmetric designs and Hadamard designs , which are particular examples of block designs 333Particular block designs known as Steiner systems have been used to construct equiangular tight frames .. This is explained below.
Let denote a block design. is a symmetric design if . is a Hadamard design if it is a symmetric design and the parameters satisfy and or and .
A or Hadamard design exists if and only if there exists a corresponding real Hadamard matrix . Hadamard designs, like all symmetric designs, satisfy the following important property.
Lemma 2.8 ().
Let denote a symmetric design. Then any two distinct blocks in have precisely elements in common.
Hadamard designs, specifically Hadamard designs, can now be used to extend the construction given in Example 2.5 to other dimensions besides .
Let for some and suppose that there exists a Hadamard design on the set . Let denote an ETF satisfying for . Define for and for each construct the vector by
Then is a UNTF, and the maximum coherence between distinct vectors of this UNTF is .
Example 2.10 (An optimal UNTF to add to an Etf).
Let be an ETF satisfying for . Then Theorem 2.9 shows that a Hadamard design on the set can be used to find an optimal UNTF (specifically, an orthonormal basis) to add to this ETF. One such design can be found in  and is given by
The next step is to add to each block to create the sets , and use these to construct the orthonormal basis :
The resulting UNTF then has maximum coherence between distinct vectors given by .
Theorem 2.9 uses Hadamard designs to construct low coherence UNTFs that meet the lower bound of Theorem 2.2. However, as mentioned previously these designs are equivalent to the existence of a real Hadamard matrix of the right size. It is therefore desirable to use more general block designs to construct low coherence UNTFs. This will require the following definition.
Let denote a finite set and let denote a block design on . The matrix for , , where
is called the incidence matrix of .
The following fundamental result on block designs will be necessary, and can be found in .
Lemma 2.12 ().
Let denote a finite set and let denote a block design on . Let denote the number of blocks in containing a given element of .
Let denote the incidence matrix of . Then
where denotes the matrix whose entries are all .
The following inequality will also be required and can be found in .
Theorem 2.13 (Fisher’s Inequality ).
If is a block design on then must contain at least blocks.
Let denote an ETF satisfying for , and suppose that is a block design on for some . Define for by
Then is a UNTF.
Let denote the synthesis operator of , let denote the corresponding Gram matrix, and let be the incidence matrix of the block design . Let denote the synthesis operator of and the corresponding Gram matrix. Then using the proof of Theorem 2.4, one can write
By Lemma 2.12 (ii),
Since is the zero matrix,
where the last equality follows from the fact that the frame potential of the original ETF is . By Lemma 2.12 (i),
Therefore is a UNTF by Theorem 1.1. ∎
As (2.3) shows, the coherence between two distinct vectors and obtained from this construction is related to the size of the intersection of the blocks and that determine and .
Let denote a block design on a set . An integer is said to be an intersection number of if there are blocks and such that .
Bounds on the possible intersection numbers of a block design are required in order to obtain UNTFs that are optimal in the sense of Definition 2.1. The bound below was originally given in  but the form used here is from .
Theorem 2.16 ().
Let denote a block design on a set , and let denote the number of blocks containing a given element of . Define and as follows:
Let and denote distinct blocks of . Then
Finally, the previous results may now be used to obtain a UNTF that is optimal or nearly optimal with respect to the ETF . Let In cases where an ETF does not exist, the low coherence UNTF can be considered an approximation. This is presented in Theorem 2.18.
Let denote an ETF satisfying for . Let denote a block design on . Let be given by
Furthermore, suppose and is a block design with . Then is a UNTF satisfying the following.
If is odd then the maximum coherence between distinct vectors of is bounded above by
if is even then the maximum coherence between distinct vectors of is bounded above by
First, recall (2.3):
Hence the intersection number of that is farthest from will give . Let . It will be shown that , which will imply that is closer to than . The definitions of and , as well as the relation which is obtained from Lemma 2.12, give
Thus gives the largest possible value in (2.3) and so
Using the expression for from Theorem 2.16 as well as the fact that ,
As shown in the proof of Theorem 2.14,
after substituting the expression for from Lemma 2.12. Therefore,
By Theorem 2.14, is a UNTF. Since is also a UNTF, is a UNTF as well. The rest of the proof involves finding the bounds of parts (i) and (ii). Now let and suppose that . If is odd, then and . Thus where , and it follows that
Similarly, if is even then and . Thus