Chapter IV: A Unified Theory of Union-of-Subspaces Representations of
4.1 Introduction
The Nested Periodic Matrices and Dictionaries of the previous chapter generalize Ramanujan subsapces to a much bigger family of subspaces. However, as mentioned in Chapter 1, there have been a few more subspace models for periodic signals that have been used for period estimation in the past [12]–[14], [25], [29]. Most of these other works use the column-extended versions of Identity matrices to span period-P signals. This results in a dimesnionPsubspace for each periodP, unlike the Nested Periodic Dictionaries which have a dimensionφ(P)subspace for each periodP. As we will explain in this chapter, this additional redundancy in the other techniques can cause some fundamental problems.
Inspite of many such works that use subspace models for periodicity, there is no unified theory in the literature that studies and compares all such models under one framework. As such, the connection between the elegant Exactly Periodic Subspaces theory of Muresan and Parks [14], the novel periodicity transforms of Sethares and Staley [13], the dictionary approach of Nakashizuka [25], the theory of intrinsic periodic functions of Pei and Lu [30], and the nested periodic dictionaries [23], has not been studied in the literature. Thus, all these above methods remain mostly as isolated pieces of work. Furthermore, there are a number of unanswered questions in the context of dictionaries constrcuted using such subspaces. For example, (a) what is the provably minimum required dictionary size for the periodicity problem?
(b) What are the required dimensions of the various subspaces representing hidden periods in the dictionary? (c) What is aminimal setof conditions on the dictionary so that it yields a unique representation for a periodic signal? (d) If a dictionary is based on single-frequency exponentials (Fourier atoms) then how should these frequencies be spaced on the unit circle? If the intuitively appealing uniform-grid is not the best, then what is the best grid to use? Many such questions remain unanswered so far.
Chapter Scope and Outline
The purpose of this chapter is therefore two-fold. First, we unify many of the above methods based onnested periodic matrices and subspaces. For this, a brief (but important) review of the relevant past methods is first given in Sec. 4.2. The con- nection between the methods is then described. For example, we show in Sec. 4.3 that theexactly periodic subspaces(EPS) of Muresan and Parks [14] are precisely the Ramanujan subspaces defined in [18] (Theorem 4.3.1). The complex theoreti- cal framework of [14] which arrives at these spaces through an orthogonalization approach can therefore be replaced by the direct methods of [9], [18] and [23]. In the EPS theory, the dimensions of the periodic subspaces are not properly explained or specified. We show, based on the connection with Ramanujan spaces, that this dimension is precisely the Euler totient function φ(P), where Pis the period asso- ciated with the subspace. (In a later section, we will show that the Euler totient is even more fundamental than this, please see below.) Third, we also show that the intrinsic integer periodic functions (IIPF), defined from a very different point of view in [30], are in fact identical to the Ramanujan space approach (Theorem 4.3.2).
The second purpose of this chapter is to go beyond this unification, and place the dictionary approaches [23], [25] on a firm theoretical footing. This gives rise to a number of theorems which answer several basic questions about the dictionary approaches, not addressed in any of the earlier papers including [23]. For example, what is the theoretically minimum number of atoms required in any type of dictio- nary, in order to represent periods 1 ≤ P ≤ Pmax? For each period P, what should be the minimum dimension of the subspace of atoms representing the Pth period itself? The answers are found in Theorems 4.6.1 and 4.6.2 (Sec. 4.6). In particular, the answer to the second question is precisely the Euler totient φ(P). We will also see, rather surprisingly, that a larger-than-minimal dictionary creates difficulties in the process of uniquely identifying periods even in the absence of noise (Sec. 4.6).
Next, what is the set of properties of the subspaces in a dictionary, which allows a signal to be decomposed into periodic components in auniqueway? We will answer this in Sec. 4.4, and show that among all the past approaches, there are very few methods which allow such a unique solution, and this set includes Ramanujan-space based methods. We answer this in Sec. 4.4. These results, presented as Theo- rems 4.4.1, 4.4.2 and 4.4.3, are much stronger than earlier results in the sense that, an absolutely minimum set of conditions are imposed on the dictionary subspaces (the LIPS conditions, Sec. 4.4) which make all the previously imposed conditions
Figure 4.1: The set of frequencies needed to span all periodic signals whose periods lie in the range 1 ≤ P ≤ 8. See Sec. 4.5.
[23] such as the Euler structure, the Nested Periodic Property, and so forth, come out as natural consequences of this!
When a signal is expressed as a linear combination of periodic subspace signals then it is sometimes possible to uniquely identify the period of the original signal from a mere knowledge of the indices (= periods) of the participating subspaces.
Namely, the period is exactly the LCM of these participating indices. This is a very important practical property, when we have to estimate the period from the dictionary representation. This LCM property is true for the Nested Periodic Dictionaries in Chapter 3 but not for the methods in [12], [13], [25]. What then is the fundamental theoretical condition under which this LCM property holds? Is there a broader class of methods than the Nested Periodic Subspaces methods, with the LCM property?
This question is addressed in Sec. 4.6 (Theorem 4.6.3).
Fourier dictionaries and frames are popularly used in a number of signal processing applications such as DOA estimation [31], [32] and [33]. These dictionaries are usually chosen such that the frequency grid is uniformly sampled. In order to avoid the inaccuracies in representation caused by the grid, it is customary to increase the number of atoms in the dictionary to decrease the grid size.1 We will show that for the case of period estimation, the correct representation departs from this conventional approach in two ways. First, increasing the size of the dictionary is detrimental to the period estimation problem (Sec. 4.6, Sec. 4.7), and second, the best frequency grids are necessarilynonuniform, and patterned after the so-called Farey seriesof number theory [24] (please see Fig. 4.1). While such a dictionary was first reported in [26], the fact that this is theonlydictionary that works if the atoms are Fourier exponentials is new and is proved for the first time in [37], as elaborated in Sec. 4.5.
1There also exist gridless methods, which take a different approach, e.g., MUSIC [15], or more recent methods [34] and [35], [36].
Finally in Sec. 4.7 we provide some numerical examples to demonstrate the effect of redundant versus minimal dictionaries in the representation of periodic signals in the presence of noise. We also demonstrate how denoised versions of noisy periodic signals can be reconstructed from such representations.