Chapter IV: A Unified Theory of Union-of-Subspaces Representations of
4.4 Fundamental Properties of Subspaces That Admit Unique Periodic
The basic idea behind using NPSs (including EPS/Ramanujan subspaces) for pe- riod estimation is as follows: A given periodic signal can be decomposed into its components along a set of linearly independent subspaces, revealing its period (the parallelogram completion idea of Fig. 4.4). These components turn out to be the projections in the case of Ramanujan subspaces and the EPSs, due to their orthog- onality. The parallelogram completion idea of Fig. 4.4 would not work when using theVP’s since every periodic signal can be expressed in an infinite number of ways as a sum of its components along the VP’s. This is due to the linear dependence among the variousVP’s3 A natural question then is as follows: Beyond the NPSs, are there more general subspaces that have this linear independence property? An analysis of this question reveals some surprising results. We start with the following definition:
Definition 4.4.1. Linearly Independent Periodic Subspaces:LetG= {G1,G2, . . . ,GPmax} be a set of subspaces such that the following hold:
1. Linear Independence: Gis a linearly independent set of subspaces. That is, if for eachi, Ri is a linearly independent set of signals in Gi, then∪iRi is a linearly independent set of signals too.
3An easy way of seeing this linear dependence is using the fact thatVP⊂ VN P, ∀N>1.
♦
Notice that, unlike in the definitions of EPSs, Ramanujan Subspaces or the Nested Periodic Subspaces, we have imposed very few conditions while defining LIPS here.
For example,we did not even insist that Gi must consist of periodisignals. While the linear independence condition in the above definition is motivated by the need for unique periodic decompositions, the direct sum condition ensures that only thoseTi, withi|P, are involved in spanning a periodPsignal. The latter makes it convenient to be able to estimate the period based on the indices of the subspaces along which the signal has large components. It is easy to verify from the discussion in Sec. 4.2, that the EPSs (equivalently Ramanujan Subspaces) and all other Nested Periodic Subspaces are specific examples of subspaces that satisfy the conditions of being LIPS.
We will now derive some fundamental properties of subspaces that satisfy the LIPS conditions.
Linear Independence Necessitates Euler Structure First, we prove the following theorem:
Theorem 4.4.1. Origin of the Euler-structure: LetBPbe a basis forVPsuch that for every divisord ofP, there exists a subset ofBP that is a basis forVd. ThenBP
must have exactlyφ(d)vectors with period d, for every divisordofP. ♦ Proof. Letdbe a divisor ofP. A basis forVdmust consist of signals with periods that are divisors ofd, since each basis vector must itself belong toVd. The rank of the set of all signals inBPthat have proper divisors ofdas periods must be≤ Í
di|d di<d φ(di) (see Appendix 4.10). So forBP to contain a basis for thed dimensional spaceVd, there must be at least d −Í
di|d
di<d φ(di) = φ(d)[24] linearly independent signals in BP with period d. But the maximum number of vectors in BP must be P, and Í
d|Pφ(d)= P[24]. So for each divisord ofP, there are exactlyφ(d)vectors with
perioddinBP. 5 5 5
In [23], it was shown that the Euler-structure of the NPMs leads to their nested basis property (Sec. II-E). But the above theorem shows that the converse is also
true. Namely, if A is a P× P matrix satisfying the nested basis property, then Theorem 4.4.1 states thatAmust have exactlyφ(d)columns with perioddfor every d|P, making it an NPM.
We will now show that every set of LIPS must necessarily have the Euler-structure.
Theorem 4.4.2. The Euler-structure in LIPS:LetG = {G1,G2, . . . ,GPmax} be a set of LIPS. Then for eachi,Gihas to be aφ(i)dimensional subspace consisting of
periodisignals only. ♦
Proof. For everyi, letRi denote a basis forGi. It follows from the direct sum and the linear independence conditions in Definition 4.4.1 that∪d|iRd is a basis forVi. We will first prove that Gi consists of period i signals only. It follows from the direct sum condition of LIPS that Gi ⊆ Vi, and so Gi must consist of signals that are periodic with periodior a proper divisor ofi. To show thatGi consists of only periodi signals, (and not signals with periods that are proper divisors ofi) we use a proof by contradiction. Suppose there was a signaly(n)with period j inGisuch that j < i and j|i. In that case,∪dj|jRdj must be able to span y(n). But y(n)can also be spanned byRi, which is a basis ofGi. This violates the linear independence condition in the definition of LIPS. Hence,Gimust consist of periodisignals only.
To show that the dimension ofGi is φ(i), notice that if d|i, then∪dj|dRdj, which is a subset of∪d|iRd, is a basis for Vd. So ∪d|iRd is a basis forVi that satisfies the conditions of Theorem 4.4.1. And sinceRimust consist of periodi signals only, it follows thatRi must consist of exactlyφ(i)periodi signals. Hence, the dimension
ofGimust beφ(i). 5 5 5
The above Theorem shows that the Euler-structure is necessary when one tries to design subspaces that offer unique decompositions for periodic signals. The Nested Periodic Subspaces, by definition, have their dimensions given by the Euler totient function (Definition 4.2.3). It follows from the properties of Nested Periodic Matrices that that this leads to unique decompositions of periodic signals (see Theorem 4.2.1). However, Theorem 4.4.2 gives us a converse result. Namely, it shows that if one wants unique periodic decompositions, then the dimensions of the subspaces must necessarily be given by the Euler totient function.
This leads us to suspect whether the Nested Periodic Subspaces (Definition 4.2.3) are theonlyexamples of subspaces that offer unique periodic decomposition. This is indeed true, as shown by the following theorem:
Proof. For every i, let Ri denote a basis for Gi. Let L = lcm{1,2, . . . ,Pmax}.
And let HPmax+1,HPmax+2, . . . ,HL represent bases for the Ramanujan Subspaces SPmax+1,SPmax+2, . . . ,SL respectively. Now, consider the set:
B= {R1,R2, . . . ,RPmax,HPmax+1,HPmax+2, . . . ,HL} (4.11) If we form a matrixB whose columns are the signals inB, truncated to their first L elements, then we can show that such a matrix is a Nested Periodic Matrix. The three conditions of an NPM from Sec. 3.1 are met as follows:
• Conditions 1 and 2: From Theorem 4.4.2, we know that eachRicorresponds to an L×φ(i)matrix consisting of periodisignals. And by construction, the Ramanujan Subspaces have theφ−structure.
• Condition 3: {R1,R2, . . . ,RPmax} is a linearly independent set because of the linear independence condition in Definition 4.4.1 of LIPS. And the following set:
{HPmax+1,HPmax+2, . . . ,HL} (4.12)
is also a linearly independent set because of the orthogonality of the Ramanu- jan Subspaces. Moreover, from Theorem 4.3.1, {HPmax+1,HPmax+2, . . . ,HL} is orthogonal to{R1,R2, . . . ,RPmax}. Hence,Bin (4.11) has full rankL.
So, we have proved thatGis a set of NPS according to Definition 4.2.3. 5 5 5 This Theorem shows that, although NPSs were proposed in [23] as basic generaliza- tions of the Ramanujan Subspaces, especially inheriting theirφ−structure in an ad hoc fashion, they arise as the only subspaces that offer unique decompositions of pe- riodic signals. This result will be used in Sec. 4.6 to prove some important properties of dictionaries that span periodic signals (Theorem 4.6.3 and Corollary 4.6.1).
Importance of the correct choice of subspaces: An example
Let us now illustrate the concept of unique periodic decomposition using an ex- ample. Consider a period 8 signal x(n), whose one period is given by the vector [1, 2, 3, 3, 3, 3, 2, 1]. In Fig. 4.5(a) and (b), we have shown two possible decompositions of this signal along the set of subspaces{V1,V2, . . . ,V8}:
x(n)=
8
Õ
i=1
xi(n) (4.13)
where, for eachi, xi(n)indicates the component alongVi. Due to the lack of linear independence among these subspaces, we can obtain an infinite number of such decompositions of the same signal. Notice that the LCM property does not apply to the decomposition in Fig. 4.5(a). The LCM property, if blindly applied, estimates the period to belcm(2,3,6,8) = 24, which is incorrect. This is because theVP’s are not linearly independent.
Next, let us decompose the same signal along subspaces that offer unique periodic decompositions. We used the Ramanujan subspaces{S1,S2, . . . ,S8}in Fig. 4.5(c) and the Natural Basis subspaces{T1,T2, . . . ,T8} in Fig. 4.5(d). The Natural Basis subspaces are an example of non-orthogonal NPSs, proposed in [23]. The decom- positions shown in Fig. 4.5 are the only possible decompositions ofx(n)along these subspaces. In Fig. 4.5(c), the subspaces with non-zero components areS4 andS8, while in Fig. 4.5(d), they areT2,T4andS8. Notice that the LCM property correctly predicts the period of the signal as 8 in each case. In these plots, we have ignored the period 1 components, since it is just DC.