Concluding Remarks

Chapter VII: Minimum Datalength for Integer Period Estimation

7.10 Concluding Remarks

This chapter derived fundamental bounds on the minimum necessary and suffi- cient data-length needed to estimate the integer period and the hidden periods of a sequence x(n). The notion of hidden periods in a mixture was carefully formal- ized, including a discussion on their uniqueness and identifiability. While most of the results in this chapter are algorithm-independent, the datalength requirements for a particular case of the recently proposed dictionary based period estimators of [23] was also presented. We also briefly analyzed the case where the discrete signal’s period might not exactly be an integer. Finally, the question of whether non-contiguous sampling helps to reduce the minimum number of samples required for period estimation was investigated using a simple case.

A possible extension of this chapter would be to multi-dimensional periodic signals.

The extension of Caratheodary’s results to the multi-dimensional setting in [112]

provides useful initial insights. We would also like to theoretically study the accuracy of period estimation and its dependence on the number of samples in the presence of noise, under a rigorous statistical framework. Finally, from a practical perspective, all the algorithms used in the sufficiency proofs in this chapter are computationally intensive. This was not an issue in this chapter, since these algorithms were only used as proof techniques. However, an interesting question is as follows: Can there be algorithms that are both computationally efficient, as well as mathematically guaranteed to work with the theoretical minimum number of samples? This is an

Lemma 7.11.1. LetAbe aN×N matrix with distinct, Vandermonde columns. Let B be the(N −1) × N matrix obtained by removing the last row of A. Then, any non-zero vectorzin the Null space ofBsatisfieszi ,0∀i. ♦ Proof. Assume the contrary, and thus letzbe a vector inNU L L(B), with say itsi^{t h} entry equal to 0. LetB⁰be the(N−1) × (N−1)matrix obtained by deleting thei^{t h} column ofB, and letz⁰be the(N−1) ×1 vector obtained by deleting thei^{t h} entry of z. Since Bz=0, it follows that B⁰z⁰=0. However, notice that B⁰ is a square Vandermonde matrix with distinct columns, and hence is full rank. Soz⁰must be0,

which then implies thatz=0. 5 5 5

Lemma 7.11.2. LetFbe an M-set. LetG ⊆ D.S.(F), such thatF⊆ G. LetHbe the following set:

H= {g ∈G:Mg<G∀M > 1} (7.60)

Then,H=F. ♦

Proof. Let f ∈ F. This implies that f ∈ G, since F ⊆ G. Suppose f < H, then, M f ∈Gfor someM > 1. But sinceG ⊆ D.S.(F), M f|l for somel ∈F. This then implies that f|l. However, we have obtained a contradiction, sinceF is an M-set.

Hence, f ∈H. This proves that

F ⊆ H (7.61)

Now, supposeq ∈Gsuch that q <F. Since G ⊆ D.S.(F), K q = pfor somep ∈ F andK > 1. But sinceF ⊆ G, this implies thatK q= pfor somep ∈GandK > 1.

Soq<H. This proves thatH ⊆ F. Combining it with (7.61), we can conclude that

H=F. 5 5 5

Lemma 7.11.3. For any two positive integersPiandPj, the following holds true:

Pi+Pj − (Pi,Pj)= Õ

d∈D.S.({P_i,P_j})

φ(d) (7.62)

♦

Proof. For any positive integerP, the following is true (from [24]):

Õ

d|P

φ(d)= P (7.63)

Expanding each term in the L.H.S. of (7.62) using (7.63) gives us:

Pi+Pj − (Pi,Pj)= Õ

d_i|P_i

φ(di)+ Õ

d_j|P_j

φ(dj) − Õ

d_k|G

φ(dk) (7.64)

where G = (Pi,Pj). Now, if dk|G, then dk|Pi and dk|Pj. So D.S.({P₁}) ∩ D.S.({P₂}) = D.S.({G}). Using the principle of inclusion and exclusion from set theory on the setsD.S.({P₁}),D.S.({P₂}),D.S.({P₁})∩D.S.({P₂})andD.S.({Pi,Pj}), it is now easy to see that the R.H.S. of (7.64) is equal to the R.H.S. of (7.62). 5 5 5

ARBITRARILY SHAPED PERIODS IN MULTI-DIMENSIONAL PERIODICITY

Traditionally, the above notion of periodicity has been generalized for multidimen- sional signals in the following way [113], [114]: A signalx(n),n ∈Z^D is said to be periodic if there exists a non-singular integer matrixPsuch that:

x(n+Pr)= x(n) ∀n,r ∈Z^D (8.1) Analogous to the one dimensional case, such aPcan be called a repetition matrix of x(n). For such anx(n), the parallelepiped whose edges are represented by the column vectors of P is known as a repetition region, since this parallelepiped when tiled periodically along the directions represented by the columns ofP, generates x(n).

Moreover, the absolute value of the determinant of Pgives the number of integer points inside the repetition region. If P is a repetition matrix with a determinant that has the smallest absolute value among all possible repetition matrices for x(n), then the parallelepiped whose edges are given by the columns of such aPis known as a period ofx(n). In this case,Pis said to be a periodicity matrix ofx(n). In other words, a period is defined as the smallest parallelepiped that is a repetition region for the signal. Fig. 8.1(a) shows an example of a two dimensional periodic signal with period represented by the following matrix:

P=

"

2 1 0 2

#

(8.2)

For simplicity, we will indicate periodic signals such as the one in Fig. 8.1(a) by plots similar to Fig. 8.1(b). (The horizontal direction in Fig. 8.1(a) and (b) represents the first coordinate.)

The question we want to address in this chapter is whether the above notion of periodicity captures all possible multidimensional periodic patterns. For example, according to the above definitions, the period and repetition regions are paral- lelepipeds. But there are many examples of signals with a much more diverse and rich set of geometries for the period. For example, the Dutch artist M. C. Escher had made several paintings consisting of figures that tile the two dimensional continuous

Figure 8.1: Part (a) - A two dimensional periodic signal according to the definition in (8.1), whose period is represented by the matrix in (8.2). The grid of dots is the setZ². The numbers shown indicate the value of the signal at those integer points.

Part (b) - A convenient way to indicate repetition regions for such periodic signals.

space plane. His tiles include many intricate shapes such as reptiles, fishes, men etc.

[115]. The website in the1, for example, investigates some of the techniques behind constructing Escher’s figures. By nature, all these paintings are continuous space signals. But taking inspiration from them, we constructed the discrete space exam- ples of Fig. 8.2 and Fig. 8.3. Such non-parallelepiped shaped tiles occur frequently in the real world, for example in honeycombs, scales of animals such as fish, in art, textile design, architecture and computer graphics. Such tilings are also useful in crystallography [116], [117].

Can we have a formal notion of periodicity with repetition regions and periods having such general shapes? Do each of these signals also have parallelepiped shaped periods? If so, how are these parallelepiped periods related to the ones shown here? Are there signals similar to Fig. 8.2 and Fig. 8.3 whose smallest repeating unit is not a parallelepiped?

To address these questions, we will formulate a new definition of periodicity in the following subsection, that is directly motivated from patterns such as in Fig. 8.2 and Fig. 8.3. We will then derive and discuss some properties of signals that satisfy

1J. Britton, Escher in the classroom [Online]. Available: http://britton.

disted.camosun.bc.ca/jbescher.htm

Figure 8.2: A two dimensional periodic signal with a non-parallelogram repetition region (shaded in black) tiling the plane.

Figure 8.3: Another example of a two dimensional periodic signal with a non- parallelogram repetition region (shaded in black) tiling the plane.

this new definition. These properties show how such signals relate to the above mentioned parallelepiped formulation (Eq. 8.1).

Multi-Dimensional Periodicity - An Alternate Definition

Consider the periodic signals in Fig. 8.2 and Fig. 8.3. The period (black area) is some shape that when repeated along two directions tiles the plane. When we look at these figures, it is quite obvious to our intuition that there is a periodically repeating pattern on the plane. This perception of periodicity can be transcribed into words as follows: “copies of a shape, obtained by translating it along fixed directions, tile the plane and the signal values on that shape are replicated on its copies”. We want to model the same into a mathematical formulation. Hence the following definition:

Definition 8.0.1. A signalx(n), is said to be periodic if there exists a non-empty set S⊂ Z^D and a matrixP∈Z^D×N, such that if we define the sets:

Si =S+Pi (8.3)

then,

1. ∪_iSi= Z^D

2. Si∩Sj = φforSi ,Sj

3. x(n+pr)= x(n)∀n∈Sandr ∈Z^N.

Here,S+Piindicates the set obtained by adding the vectorPito every element of S. As will be clear from the following discussion, for a signal x(n) satisfying the above definition, the set{S,P}is not unique. Some remarks on Definition 8.0.1:

1. The setS: In the above definition,Srepresents the set of points that form the shape that tiles the plane. For example, in Fig. 8.2 and Fig. 8.3,Scan be the shape shaded in black. We will call it a repetition region of the signal. Notice that a group of those shapes taken together also represents a valid repetition region that can tile the plane. So a signal satisfying Definition 8.0.1 will not have a unique repetition region. Also, note that unlike in the parallelepiped based periodicity framework in the beginning of this section, we are not explicitly specifying any shape for the repetition region.

matrix is not unique. This is because, the set of all integer vectors spanned by the columns ofPis the same as that spanned by the columns ofPUwhereU is any integer unimodular matrix [113]. So ifPis a translation matrix forS, thenPUwill also be a valid translation matrix forS.

3. The size of P: The motivating idea behind Definition 8.0.1 involves a shape being translated and copied along fixed directions. But along how many directions do we need to translate? By taking aD×N translation matrix, we are essentially choosingN such directions. In Section 8.1, we will point out that if Phas rank R < D, then the repetition region must be of infinite size along D− R dimensions. For simplicity, we will not consider such signals here. On the other hand, we will show that when P has rank D, one may assumeN = Dwithout loss of generality. So in the following discussions, we will assumePto be a full rankD×Dmatrix.

4. The three conditions: The first condition in the definition just says that the setSand its shifted copies must cover the entire space. The second condition requires that different shifted versions ofSmust not overlap with each other.

Finally, the third condition requires that the value of the signal x(n) on the points in S must replicate themselves on all the shifted copies of S. These conditions were directly motivated from signals such as those in Fig. 8.2 and Fig. 8.3.

Notice that a repetition region S might itself not be the period. For instance, in Fig. 8.2 and Fig. 8.3, a group of the indicated shapes taken together as one big shape also represent a valid repetition region. We will now formally define what we mean by a period of a periodic signal. To do so, we note that in the parallelepiped based definitions for repetition regions and periods discussed earlier, the period was defined as any parallelepiped that is a repetition region, and has the smallest number of points among all such repeating parallelepipeds. We will generalize it to more general shapes in the following definition:

Definition 8.0.2. For a signal that is periodic according to Definition 1, a repetition region is said to be a period if for any other repetition region of the signal,

Properties Arising From The New Definition

We will now prove some properties of signals that are periodic according to Def- inition 8.0.1 of the previous subsection. In the process, we will relate them to the conventional parallelepiped based formulation mentioned after (8.1). First we prove:

Theorem 8.0.1. If x(n)is periodic according to Definition 8.0.1, thenx(n+Pr) = x(n)∀n,r∈Z^D.

Note the difference between Theorem 8.0.1 and condition 3 of Definition 8.0.1.

Theorem 8.0.1 talks about all integer vectorsn∈ Z^D whereas condition 3 required only thatn ∈S.

Proof. Consider any n ∈ Z^D. From conditions 1 and 2 in Definition 8.0.1, there must exist a unique i ∈ Z^D such that n ∈ Si. In that case, there must exist a point m in S such that n=m+Pi because of (8.3). So x(n+Pr) = x(m+Pi+Pr) = x(m+P(i+r)) = x(m), where the last equality follows from condition 3 in Definition 8.0.1. Since m ∈ S, condition 3 in Definition 1 also implies that x(m)= x(m+Pi)= x(n). Hence, x(n+Pr)= x(n). 5 5 5 The above theorem is important for the following reason. Let us say we have a set Swhich was used in verifying Definition 8.0.1 to determine that a particular signal x(n)is periodic. Irrespective of the shape ofS, Theorem 8.0.1 tells us that we can always have a parallelepiped repetition region for x(n). This is because, it is well known in the literature ([113], [114] etc.) that for any signal x(n)that satisfies:

x(n+Pr)= x(n)∀n,r∈Z^D (8.4) a parallelepipedP ⊂ Z^D that has the columns of Pas its edge vectors, when tiled periodically along the directions represented by the columns ofP, generates x(n). It can easily be shown that such a parallelepiped satisfies all the conditions of Definition 8.0.1 too, and is hence a valid repetition region according to our new definitions for periodicity. It’s translation matrix is in fact P. Consider the signal in Fig. 8.2 for instance. LetSbe the repetition region shaded in black, and letPbe its translation matrix. If we denote the parallelogram that has its edges along the columns ofPbyP, then Fig. 8.4 shows the same signal as havingP(shaded part) as a repetition region.

Theorem 8.0.2. x n

regionSand a translation matrixP. LetPbe another repetition region forx(n)with the same translation matrix. Then, |S|= |P|= |det(P)| .

Proof. LetSandPrepresent two possible repetition regions for x(n)that have the translation matrix P. To every point in P, we can associate a point in S in the following way. Letmbe a point inP. Then, there exists a unique pointninSsuch thatm= n+Pvfor somev ∈Z^D. This is because:

Existence ofn: Follows from Condition 1 in Definition 8.0.1.

Uniqueness ofn: If there were two pointsnandn⁰inSsuch thatm= n+Pv=n⁰+Pv⁰, thenn⁰=n+P(v−v⁰). Letv−v⁰=i. Then, from Definition 8.0.1, sincen ∈ S, n⁰ must belong to Si. But S ∩Si is the null set unless i= 0 by Condition 2 of Definition 8.0.1. This then implies that n⁰=n+Pi= n. Hence, uniqueness has been proved.

From the above arguments, we can define a mapping that relates every point in P to a point inS. We now claim that this mapping is one-one. That is, two points in Pcannot be associated to the same point inS. We can prove this by contradiction.

Suppose there were two points P1and p2 inPsuch that p1+Pv1=p2+Pv2= s, for somes ∈ S. Then, p1= p2+P(v2−v1). Since p1 and p2 are distinct points, v1, v2. But this leads to a contradiction, since Pitself is a repetition region that satisfies Definition 8.0.1, and using an argument similar to the uniqueness of n argument above,p1 =p2+P(v2−v1)cannot be possible forv1,v2.

So we have proved that there exists a one-one mapping fromPtoS. This means that the number of points inPmust be less than or equal to the number of points inS. Interchanging P and S in the proof above, we can similarly construct a one-one mapping fromStoP. And hence, the number of points inSmust actually be equal to the number of points inP.

Finally, as we remarked earlier, the parallelepiped whose edges are represented by the columns of Palso satisfies Definition 8.0.1 as a repetition region with the same translation matrixP. The number of points in such a parallelepiped is given by |det(P)| [113], [114]. So from the above result, the number of points in any repetition region with translation matrixPis|det(P)|. 5 5 5

Theorem 8.0.1 tells us that any signal x(n) that is periodic according to Defini- tion 8.0.1, having possibly arbitrary shaped periods such as in Fig. 8.2 or Fig. 8.3, will also always have a parallelepiped shaped repetition region. Theorem 8.0.2 shows that the number of points in that parallelepiped is in fact equal to the num- ber of points in the “irregular” repetition region S. This means, for example, that the parallelograms shown in Fig. 8.4 have the same number of points as the region shaded in black in Fig. 8.2. Moreover, combining Theorem 8.0.1 and Theorem 8.0.2 we can arrive at an important conclusion that the smallest repetition region forx(n) can always be assumed to be a parallelepiped:

Theorem 8.0.3. Let x(n) be periodic according to Definition 8.0.1. Let S be an arbitrarily shaped period of x(n) according to Definition 8.0.2. Then, there will always exist a parallelepiped that is also a period ofx(n), having the same number of points asS.

Proof. Let P be the parallelepiped whose edges are represented by the columns of the translation matrix of S. As discussed earlier, from Theorem 8.0.1, P is a repetition region forx(n). And from Theorem 8.0.2,PandShave the same number of points. However, no repetition region can have smaller number of points thanS by Definition 8.0.2. And hence no other repetition region can have smaller number of points thanP. SoPis a period of according to Definition 8.0.2. 5 5 5

8.1 A Note on Translational Matrices

We had allowed the number of columns in the translation matrixPin Definition 8.0.1 to be any natural numberN. But we restricted our subsequent analysis toPbeing a full rank square matrix. We will now discuss what happens when has rank< D, or has rank= Dbut more thanDcolumns.

First, consider the case when the rank ofPis less than D. If a shapeSwere to tile the entire spaceZ^D using translations along the columns of such aP, then such a shapeSmust have infinite extent along the D−rank(P)dimensional subspace that is orthogonal to the space spanned by the columns ofP. For simplicity, we did not discuss such signals in this work.

On the other hand, ifPhas full rank but has N > D, then it can be shown that the set of all copies ofSobtained by translations along the columns of Pwould be the same set as that obtained by translating along the D directions represented by the

Figure 8.4: A parallelepiped repetition region always exists for any signal following Definition 8.0.1. Indicating this for the signal of Fig. 8.2.

columns of the greatest left divisor ofP. We will explain this in more detail in the following.

A left divisor ofPis defined as a matrixL ∈ Z^D×D that satisfiesP=LRfor some R ∈ Z^D×N [113]. A left divisor ofP, call itL0, that satisfies the property that any left divisor ofP is a left divisor ofL0, is known as a greatest left divisor (gld) of P. It can be shown that a gld ofPexists when has rankD. Notice that gld’s are not unique, since ifL0is a gld, so is L0Ufor any integer unitary matrixU. (For more details on gld’s, see [118]–[120]).

LetL0represent a gld ofP. We claimed above that the set {Sⁱ : i ∈ Z^D} obtained by using Pas a translation matrix is the same as that obtained by using L) as the translation matrix. This follows from the following theorem that essentially states that the lattice generated byPis the same as the lattice generated byL0.

Theorem 8.1.1. LetP∈ Z^D×N have rank D, and letL0be its gld. Then, there will existz∈Z^D andi ∈Z^N such thatz=Piiff there exits aj ∈Z^D such thatz=L0j.

Proof. Letz=Pi. SinceL0is a left divisor ofP, there must exist an integer matrix R such that P= L0R. Substituting this into z=Pi gives z= L0Ri=L0j where j=Ri.