Abstract

We present a new method to construct unit norm tight frames by applying altered Hadamard matrices. Also we determine an elementary construction which can be used to produce a unit norm frame with prescribed spectrum of frame operator.

1. Introduction and Preliminaries

Frames were first introduced in 1952 by Duffin and Schaeffer [1] in the context of nonharmonic Fourier series. They are system of functions in Hilbert spaces that provide numerically stable methods for finding overcomplete decompositions of vectors and such are useful tools in various signal processing applications, data compression, wireless communications, and so on [2–4]. Frames in finite dimensional Hilbert spaces have become of interests for many of researches [5, 6]. One of the important subjects in this area is the way for constructing such frames. Some methods of construction finite tight frames are stated by researchers [7–9].

Let be a positive integer. A sequence of vectors in Hilbert space is said to be a frame (see also [10]) for if there exist constants and such that and The numbers and are called frame bounds and they are not unique. The frame is said to be tight (or -tight) if . In this case, is said to be the frame constant and it is a Parsval frame, if . When the index set is a finite set, the frame will be called finite. A normalized frame or unit norm frame is the one in which elements have the norm one.

If only the right side of the inequalities (1) holds, then is called a Bessel sequence and if is a normalized Bessel sequence in , then is finite sequence [5]. According to this, our frame will be the form whenever is some positive integer. Also we will replace by . To each Bessel sequence corresponds an operator called analysis operator which is well-defined and bounded operator. Its adjoint is the operator called the synthesis operator. If is a frame with frame bounds and , then the operator is called the frame operator of the frame . It is a positive, self-adjoint, bounded, and hence invertible operator with the inverse . Benedetto and Fickus in [5] have proved that if is a normalized -tight frame for a -dimensional Hilbert space , then .

In this paper, we address the question of how to efficiently construct unit norm tight frame. After reviewing known results about the Spectral Tetris Construction and introducing the Spectral Hadamard-Tetris Construction in Section 2, our main result in Section 3 presents a way to construct unit norm tight frames that we call Hadamard-Tetris Construction. In Section 4, we introduce another version of Hadamard-Tetris which can be used to construct unit norm frames with different eigenvalues for the frame operator. We give necessary condition for this version of Hadamard-Tetris to produce the desired frames.

2. Special Hadamard-Tetris Construction

In [11] by using the Schur-Horn Theorem, the authors have shown how to construct every possible frame in which frame operator has a given spectrum and in which vectors are of given prescribed norms, but their method is complex. In [12], the authors have presented another construction of unit norm frames which is called “Spectral Tetris Construction” (STC). Spectral Tetris is a flexible and elementary method to construct unit norm frames with a given frame operator, having all of its eigenvalues greater than or equal to two. But that way is limited to the case of frames with elements for in which . However, they extended the existing construction to unit norm tight frame of redundancy less than two [13], but those frames are in not in .

In this paper, we implement an algorithm which is named Hadamard-Tetris for the construction of unit norm tight frames. Hadamard-Tetris constructs synthesis matrix with unit norm columns in which rows are pairwise orthogonal and square sum to . By this construction, the frame operator is . The main advantage here is given an elementary and easily implementable algorithm for constructing frames.

In our way, the assumption on the spectrum of the unit norm tight frame in STC can be dropped and construct unit norm tight frame for with elements, where and are positive integers and satisfy the following condition.

If we can decompose as summation   , that is, , which there exist Hadamard matrices of sizes ’s and they satisfy the condition then there exists a unit norm tight frame for with sparsity at most .

A frame constructed via the Hadamard-Tetris construction is called Hadamard-Tetris frame.

Aside from the fact that Hadamard-Tetris frames are easy to construct, their major advantage for applications is the sparsity of their synthesis matrices. This sparsity is dependent on the decomposition of .

In our construction for any positive integers and   , we put and if we could decompose as summation of power of two, then we give a unit normalized tight frame for with elements. Hence has minimum value. If is the binary representation of , where ’s are 0 or 1, then .

In special case, when , we can construct synthesis matrix of size which has ’s as its columns as in Algorithm 1.

Theorem 1. Let be a positive integer number. Then the sparsest synthesis matrix of the -element Hadamard-Tetris unit norm tight frame for which consists of blocks of size 2 can be constructed by HTC(I) and its sparsity is .

Proof. Let . By the construction of HTC(I), in lines (3) and (4), ’s are normalized. We should show that the square norm of each row of synthesis matrix is . It is clear that the square norm of the first row of the matrix is and, in the last row, since , the square norm of the last row is
Also, for , the square norm of th row is In this construction, synthesis matrix of frame consists of blocks of size , so the sparsity of the matrix is .

In [14], Casazza et al. proved that the algorithm STC can be performed to generate a unit norm tight frame of vectors in if and only if or is of the form , for some positive integer . Whereas we exhibit an algorithm that utilizes matrices of size 2 such as used in the construction of STC. For example, for and , the matrix is synthesis matrix of .

Example 2. If and , then is synthesis of 6-elements for .

3. Hadamard-Tetris Frame

In this section we provide a new method for constructing finite normalized tight frames (FNTFs). In brief, we want to construct synthesis matrix which has(i)columns of unit norm;(ii)orthogonal rows, meaning that the frame operator is diagonal;(iii)rows of constant norm, meaning that is a constant multiple of the identity matrix.

The Hadamard-Tetris Construction (HTC) is capable of constructing unit norm tight frames, with the number of elements that decompose as   . It constructs the synthesis matrices of such frames by successively filling of size of blocks. In this construction, we use altered Hadamard matrices.

The importance of Hadamard matrices to our construction stems from the fact that they have orthogonal rows and columns and that all entries have the same modulus. In the course of the construction, we will have to alter the row norms of the Hadamard matrices by multiplying rows with appropriate constants. While this will destroy the pairwise orthogonality of the columns, it will preserve the pairwise orthogonality of the rows, which is the crucial feature for our construction to work.

Definition 3. Given positive integers and , we denote a matrix by or call it a block, if it is derived from a Hadamard matrix of size by multiplying the entries of the th row of Hadamard matrix by for and the entries of the first and the last row of Hadamard matrix by and and if it has normalized columns. We call and the first and the last correction factor of , respectively.

Note that the row norm of a block equals , except possibly for the first and the last rows. We want to present an algorithm following the lines of Example 4; that is, we want to compose the desired synthesis matrix which consists of square blocks in which first and last rows successively overlap. If we use square matrix as building blocks of the synthesis matrix, then, due to the overlapping, we have .

It is perhaps most instructive to first look at the example of the construction that we are going to introduce in this section.

Example 4. We construct a 6-elements unit norm tight frame in . We can start filling the desired synthesis matrix with an altered Hadamard matrix in the upper left corner. The alteration we make is to multiply the entries of the first row by in size to make the first row have the desired norm . We multiply the second and the third rows of Hadamard matrix by to make the norm of those equal to .
To get normalized column, we multiply the forth row of the Hadamard matrix by . At this point, we have constructed the first three rows and the first four columns of the desired synthesis matrix:
Note that, so far, we have constructed a matrix in which first four rows are orthogonal, no matter how we keep filling the first four rows. The fourth row at this point has norm , while we need to make it have norm .
We can insert altered Hadamard matrix in the same fashion as above. To do this, we would multiply its first row by the factor in size to have the forth row of the synthesis matrix square sum to and its second row by the factor to get The latter matrix is the synthesis matrix of the desired frame, since its columns are normalized and its rows are pairwise orthogonal and square sum to .

The next theorem is the main theorem of this section.

Theorem 5 (main theorem). Let and    be positive integers and . Furthermore, assume that can be decomposed as such that, for each , and there exist Hadamard matrices of sizes ’s. Then HTC(II) gives a unit normalized tight frame for with elements.

In special case, when , then the condition on ’s is confirmed automatically. Indeed, if is selected such that , then .

Corollary 6. If such that there exist Hadamard matrices of sizes and , then the synthesis matrix of the elements for can be constructed via HTC(II).

To prove Theorem 5, we need to collect some information about the first and the last correction factors of ’s.

In construction of synthesis matrix of size , we use blocks of sizes ’s , where .

As was mentioned, each block is that derived from the Hadamard matrix of size . If the factor that multiplies the entries of the th row of Hadamard matrix of size is shown by , where , then, for each block of size , the correction factor is

The last row of a block and the first row of the following block appear in the same row of the synthesis matrix and while the last correction factor is chosen to ensure normalized columns, the following first correction factor is chosen to guarantee that the rows of the synthesis matrix have square norm .

Hence, we have the following relationships for and , where .

Lemma 7. If and are the first and the last correction factors, respectively, then

Proof. The correction factors of second row to th row of are
Since norm of each column of must be one, we have and so on
Also the last correction factor of and the first correction factor of must be chosen such that the square norm of that row is . Hence and so on

By combining formulas (15) in Lemma 7, we get the following recursive relations: for .

Lemma 8. If and are the first and the last correction factors of , respectively, then

Proof. We prove these by induction on . The first steps of the induction are trivially true. Assume that the identities are true for . For the case , the following identities hold by using formulas (21) and the induction hypothesis:

Now we are ready to prove the main Theorem 5.

Proof. We check that synthesis matrix constructed by HTC(II) has row norm equal and columns are normalized. First, we show that each row of synthesis matrix, which is constructed via HTC(II), has square norm . It is enough to show that the last row of together with the first row of makes square norm . In other words,
By Lemma 8, we have
Now, we show each column of synthesis matrix that is constructed with HTC(II) has norm 1; that is, By Lemma 8, we have
We have to ensure that the last correction factor of the final block inserted into the synthesis matrix equals (in other words, the next first correction factor would be zero, but we have arrived at this point where the algorithm terminates).
In Algorithm 2, line (9), when and , we have and in line (4), for , we have .

Note that if we do a HTC(II) as in Example 4 with altered Hadamard matrices of sizes , then the sparsity of the synthesis matrix is .

The next lemma states that if we construct synthesis matrix with blocks of sizes , each condition on must be established for and vice versa.

Lemma 9. Let and be positive integers with , , and such that ’s are satisfied in inequalities: Then

Proof. Since , so In the other hand, by replacing with in the inequality (30), we have and so on Hence, we have which give inequalities (31).

Following Lemma 9, if is a synthesis matrix of size , then , where , is also having the same situation.

Since Hadamard matrices of the size power of 2 exist, so we use altered Hadamard matrices of size , where for . However, it is provided that there exists a Hadamard matrix of size , with . In this case, the algorithm HTC builds the synthesis matrix of the tight frame as in Example 4 by inserting , where    and .

4. Spectral Hadamard Tetris Construction

Another case of HTC is the case of unit norm but not necessarily tight frames. Such frames are known to exist, provided that the eigenvalues of the frame operator sum up to the number of frame vectors [15]. In this section, we give a version of Hadamard-Tetris Construction which uses altered Hadamard matrices to construct unit norm frames with a given spectrum of frame operator. Our method cannot construct all such frames. We give necessary condition under which this construction works. To construct unit norm frame with prescribed spectrum, we are looking for a matrix with columns square summing up to 1 and rows square summing up to . In Algorithm 3, we present a modified Hadamard-Tetris algorithm HTC, allowing the construction of such frames.