A UNIFIED THEORETICAL HARMONIC ANALYSIS APPROACH TO THE CYCLIC WAVELET TRANSFORM (CWT) FOR PERIODIC SIGNALS OF PRIME DIMENSIONS

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1 Sahand Communications in Mathematical Analysis (SCMA Vol. 1 No. 2 (2014, A UNIFIED THEORETICAL HARMONIC ANALYSIS APPROACH TO THE CYCLIC WAVELET TRANSFORM (CWT FOR PERIODIC SIGNALS OF PRIME DIMENSIONS ARASH GHAANI FARASHAHI 1 AND MOZHGAN MOHAMMAD-POUR 2 Abstract. The article introduces cyclic dilation groups and finite affine groups for prime integers, and as an application of this theory it presents a unified group theoretical approach for the cyclic wavelet transform (CWT of prime dimensional periodic signals. 1. Introduction In signal processing, time-frequency analysis comprises those techniques that analyze a signal in both the time and frequency domains simultaneously, commonly called time-frequency methods or representations (see [6, 16, 25]. The mathematical motivation for this analysis is that signals and their transform representation are often tightly connected, and they can be understood better by analyzing them jointly, as a two-dimensional object, rather than separately. Commonly used coherent (structured methods and techniques in time-frequency analysis are Gabor analysis (see [10, 11, 14, 17] and wavelet analysis (see [1, 2, 21, 22, 32] as methods of windowed transforms. The theory of Gabor analysis is based on the modulations and translations of a given window signal (atom. The phase space (time-frequency plane has a unified group structure (see [8, 9, 17, 24], which implies concrete discretizations (see [11, 12]. The theory of finite dimensional Gabor analysis from abstract harmonic analysis point of view, is originally well studied in [11, 12, 24, 28, 29] Mathematics Subject Classification. 42C13, 15A60. Key words and phrases. Cyclic wavelet transform (CWT, Finite affine group, Prime integer, Digital signal processing. Received: 15 June 2014, Accepted: 5 November Corresponding author. 1

2 2 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR Wavelet analysis is a time-frequency method in signal processing which is based on the continuous affine group (0, R as the group of dilations and translation (see [7, 21, 31, 32, 33]. Standard discretization and quantization of the continuous wavelet transform use dyadic dilations and integer translates of the window single (wavelet, see [3, 4, 35]. Signal processing of periodic signals is the basis of digital signal processing. Classical methods for wavelet analysis of periodic signals or signals of finite size rely on embedding the vector space of finite signals in the Hilbert space of all complex valued sequences with finite energy (l 2 -norm, see [5, 30, 34]. Traditional wavelet techniques for periodic signals are not on finite dimensional analogous to the continuous setting as is the case in Gabor analysis (see [5, 24, 30]. In this article, first the dilation group U p and the finite affine group W p for the cyclic group Z p is introduced, where p is a positive prime integer. If y C p is a window signal, we define the cyclic wavelet transform (CWT W y as a transform defined on C p with complex values which are indexed in the finite affine group W p. These techniques imply a unified group theoretical based time-frequency (dilation and translation representations for signals in C p. The article presents conditions in such away that the transform W y will be isometric. It is shown that CWT satisfies various types of inversion formulas, as well. The matrix representation of CWT will be presented, which can be helpful for any programming related to the CWT computational algorithms. 2. Preliminaries and Notations Throughout this article the standard and traditional harmonic analysis modeling for the linear vector space of all periodic signals should be used. A given one dimensional (1D finite discrete data or signal x, i.e. a signal of a given length N N denoted by x = [x(0,..., x(n 1], which is a function defined on the set {0,..., N 1} Z. This type of modeling for indexing of finite signals persists us to consider finite signals as functions defined on the group of unit roots of order N, or equivalently as periodic discrete functions (sequences x : Z C with x(n + kn = x(n for all 0 n N 1, and k Z. With above mathematical modeling of one dimensional (1D finite discrete signals, the notation C N precisely stands for the complex linear vector space of all finite signals of size N. To make the formulas in this article more readable, the finite additive (Abelian group of all integers between zero and some non-zero integer number N by Z N or N shoud be denoted. Roughly speaking, Z N contains the set of all equivalence classes of remainders of all integer numbers module N. The set Z N = {0, 1,..., N 1} or with a simple notation Z N = {0, 1,..., N 1} is a finite cyclic group with

3 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 3 respect to the addition module N with the identity element 0 and the additive inverse N l for the element l Z N. The set of all bijective (i.e. injective and surjective homomorphisms on Z N (i.e. isomorphisms on Z N is denoted by Aut(Z N which is a group with respect to composition of automorphisms. For a matrix X M N M (C, the notation : used to denote all elements indexed by a variable. As an example, if 0 n N then X(n, : is the n-th row of X. This notation coincides with the notations in MATLAB and FORTRAN programming. For a matrix X M N M (C, the Frobenius norm is defined by (2.1 X Fr = ( N 1 M 1 n=0 m=0 The inner product of x, y C N is x, y = N 1 X(n, m 2 1/2 x(ky(k, which induces the l 2 norm or the Frobenius norm x 2 2 = x 2 Fr = x, x for x C N. Let l, l Z N. The translation operator T l : C N C N is T l x(k = x(k l for x C N and l, k Z N. The modulation operator M l : C N C N is given by M l x(k = e 2πilk/N x(k for x C N and l, k Z N. The translation and modulation operators on the Hilbert space C N are unitary operators in the Frobenius norm. For l, l Z N we have (T l = (T l 1 = T N l and (M l = (M l 1 = M N l. The circular convolution of x, y C N is defined by x y(k = 1 N 1 x(ly(k l for k Z N. N In terms of the translation operators we have x y(k = 1 N 1 x(lt l y(k N for k Z N. The circular involution or circular adjoint of x C N is given by x (l = x( l = x(n l. The complex linear space C N equipped with the l 1 -norm, the circular convolution, and involution is a Banach.

4 4 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR -algebra (see [13, 34]. The unitary discrete Fourier transform (DFT of a 1D discrete signal x C N is defined by x(l = 1 N 1 x(kw l (k, N for all l Z N where for all l, k Z N we have w l (k = e 2πilk/N. The set {w l : l Z N } is precisely the additive group of all pure frequencies (characters ẐN (i.e homomorphisms or characters into the circle group T on the additive group Z N. More precisely, the map l w l is a group isomorphism between Z N and ẐN (see [13]. Therefore, w l+l = w l w l and w l = w N l for all l, l Z N. Thus DFT of a 1D discrete signal x C N at the frequency l Z N is (2.2 x(l = F N (x(l = 1 N 1 x(kw l (k N = 1 N 1 x(ke 2πilk/N. N The discrete Fourier transform (DFT is a unitary transform in the Frobenius norm, i.e. for all x C N satisfies the Parseval formula F N (x Fr = x Fr, which equivalently implies (see [13, 34] N 1 x(l 2 = N 1 x(k 2. The Polarization identity implies the Plancherel formula x, y C N = x, ŷ C N for x, y C N, which equivalently implies N 1 x(lŷ(l = N 1 x(ky(k. The unitary DFT (2.2 satisfies (see [13, 19, 20, 34]; T l x = M l x, Ml x = T N l x, x = x, x y = x.ŷ,

5 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 5 for x, y C N and l Z N. The inversion discrete Fourier formula (IDFT for a 1D discrete signal x C N is (2.3 N 1 x(l = x(kw l (k N 1 = x(ke 2πilk/N, 0 l N Prime Dimensional Cyclic Wavelet Transform (CWT This section contains two parts. We start by an introduction of cyclic dilation operators and finite affine groups. The article studies basic and interesting properties of cyclic dilation operators. In second part theory of cyclic wavelet transform (CWT for periodic signals of prime dimension is presented. Throughout this section the study assumes that p is a positive prime integer Dilation operators and finite affine groups. The set (3.1 U p := Aut(Z p = Z p {0} = {1,..., p 1}, is a finite multiplicative Abelian group (see [18, 27] of order p 1 with respect to the multiplication module p with the identity element 1 ([18, 27]. The multiplicative inverse for m U p (i.e. an element m p U p with mm p p mp m p 1 is m p which satisfies m p m + np = 1 for some n Z, which can be done by Bezout lemma ([27]. For prime integer p and m U p, the cyclic dilation operator D m : C p C p is defined by D m x(k := x( m p k p for x = [x(0,..., x(p 1] C p, where m p is the multiplicative inverse of m in U p. The notation. p stands for the remainder module p. Throughout the article, for simplicity mk is used instead of mk p. Thus the dilation operator will be D m x(k = x(m p k for x C p. The following results give us generic properties of dilation operators. First we express algebraic properties of dilation operators. Proposition 3.1. Let p be a positive prime integer and m U p. Then, (i The cyclic dilation operator D m : C p C p is an -homomorphism. (ii For l Z p we have D m T l = T ml D m.

6 6 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR Proof. (i Let x, y C p and m U p. Then for all k Z p we have D m (x y(k = x y(m p k Replacing l with m p l we get = x(ly(m p k l. x(ly(m p k l = x(m p ly(m p k m p l = x(m p ly(m p (k l = x(m p ld m y(k l = D m x(ld m y(k l = (D m x (D m y(k, which implies D m (x y = (D m x (D m y. As well as, we can write (D m x (l = D m x(p l = x(m p (p l = x(m p p m p l = x(p m p l = x (m p l = D m x (l, which implies (D m x = D m x. (ii The proof is straightforward. Next proposition states analytic and geometric properties of dilation operators. Proposition 3.2. Let p be a positive prime integer. Then, (i Dilation operators are unitary operators in the Frobenius norm. (ii For m U p we have (D m = (D m 1 = D mp. (iii For m U p and x C p we have D m x = D mp x.

7 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 7 Proof. (i Let m U p and x C p. Then Replacing k with mk we get D m x 2 Fr = D m x(k 2 = x(m p k 2. x(m p k 2 = x(k 2 = x 2 Fr. (ii The proof is straightforward. (iii Let m U p and x C p. For l Z p we have Replacing k with mk we achieve D m x(l = 1 D m x(ke 2πilk/p p = 1 x(m p ke 2πilk/p. p 1 x(m p ke 2πilk/p = 1 p p = D mp x(l. x(ke 2πilmk/p For m U p, let m : Z p Z p be given by (3.2 m(l := ml for 0 l p 1. The map m is a bijection (i.e. surjective and injective for m U p. If m U p and 0 l, l p 1, then we have m(l + l = ml + ml, m(p l = mp ml p p ml, which implies that m is a bijection homomorphism (i.e. automorphism of the additive group Z p and so we get m Aut(Z p for m U p is obtained.

8 8 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR The homomorphism m m induces the semi-direct product group W p := U p Z p, which has the underlying set {(m, l : m U p, l Z p }, and it is equipped with the following group operations; (3.3 (m, l (n, k := (mn, l + mk for (m, l, (n, k W p, (3.4 (m, l 1 := (m p, m p.(p l for (m, l W p, where m p is the multiplicative inverse of m in U p (i.e m p.m and m.m p are 1 module p. Then W p is a finite non-abelian group of order (p 1 p. We call W p as the finite affine group on p integers or the wavelet group associated to the integer p or simply we call it as the p-wavelet group. Let L(W p be the complex linear space of all complex valued functions on the finite group W p. The linear space L(W p is precisely M ( p (C (i.e. the complex linear space of all (p 1 p-matrices X with complex entries. Proposition 3.3. Let p be a positive prime integer and W p = U p Z p. Then, W p is a non-abelian group of order p(p 1 which contains Z p as a normal Abelian subgroup and U p as a non-normal Abelian subgroup. Proof. The proof is straightforward Cyclic Wavelet Transform. Let y C p be a given window signal. The cyclic wavelet transform (CWT of a signal x C p with respect to the window signal y (y-wavelet transform is defined by (3.5 W y x(m, l := x(ky(m p (k l for (m, l W p. The map W y : C p M ( p (C given by x W y x is linear. From (3.5 we can see that (3.6 W y x(m, l = x(k[t l D m y](k = x, T l D m y, for (m, l W p. Using (3.6 we also have (3.7 W y x(m, l = x, T l D m y = T p l x, D m y, for (m, l W p. Remark 3.4. For a window signal y C p, we define

9 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 9 D y := D 1 y D 2 y... D y, T x := [T x, T p 2 x,..., T 0 x], where for 1 l p and 1 m p 1 we have D m y = [y(0, y(m p,..., y(m p (p 1], T p l x = [x(l p, x(1 + l p,..., x(l 1] T. Then for 1 m p 1, 1 l p and 1 k p we have Now for (m, l W p we can write T x (k, l = (T x kl = T p l x(k 1, D y (m, k = (D y mk = D m y(k 1. W y x(m, l = T p l x, D m y = T p l x(kd m y(k = T p l x(kd m y(k = = p T p l x(k 1D m y(k 1 k=1 p (D y mk (T x kl k=1 = (D y m.(t x l = (D y.t x ml, where (D y m = D y (m, : (resp. (T x l = T x (:, l is the m-th row of the matrix D y (resp. l-th column of the matrix T x and. is the standard product of matrices. Thus we achieve W y x = D y.t x. The following proposition gives us a Fourier representation and a circular convolution representation for the wavelet transform defined in (3.5. Proposition 3.5. Let x, y C p. For (m, l W p we have

10 10 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR (i W y x(m, l = pf p ( x. D m y(p l. (ii W y x(m, l = x D m y (l. Proof. Let (m, l W p. (i Using the Plancherel formula we have (ii Using (i we can write W y x(m, l = x(kt l D m y(k = x(l T l D m y(l = x(lm l D m y(l = x(l D m y(lw l (l = ( x. D m y (lw p l (l = pf p ( x. D m y(p l. W y x(m, l = x(l D m y(lw l (l = = x(l (D m y (lw l (l x(l (D m y (lw l (l = x D m y (lw l (l = x D m y (l. Theorem 3.6. Let y C p be a window signal and x C p. Then, (3.8 ( ( ( W y x 2 Fr = p (p 1 ŷ(0 2 x(0 2 + x(l 2 ŷ(m 2.

11 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 11 Proof. Let m U p. Using Proposition (3.5 we have W y x(m, l 2 = p F p ( x. D m y(p l Therefore we achieve Now we can write W y x(m, l 2 = p ( x(l 2 = p F p ( x. D m y(l = p ( x. D m y(l = p x(l. D m y(l = p ŷ(ml 2 = p x(l. D m y(l 2 x(l 2 D m y(l x(l 2 D m y(l 2 ( = p x(l D m y(l 2 ( = p x(l 2 D mp ŷ(l 2 ( = p x(l 2 ŷ(ml 2. ( = x(0 2 ŷ(0 2 ( + x(l 2 ŷ(ml 2.

12 12 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR Replacing m with ml p we have which implies ( x(l 2 ŷ(ml 2 = ŷ(ml 2 ŷ(m 2, = (p 1 ŷ(0 2 x(0 2 ( + x(l 2 ŷ(ml 2 = (p 1 ŷ(0 2 x(0 2 + ( ( x(l 2 ŷ(m 2. Thus we achieve (3.8. Now the following interesting result is proven, which shows when the windowed transform (3.5 is an isometric transform. Corollary 3.7. Let y C p be a non-zero window signal with ŷ(0 0. The cyclic wavelet transform W y : C p M ( p (C is an isometric transform (up to a normalization if and only if the window signal y satisfies the following p-admissibility condition; (3.9 y Fr = p ŷ(0. Proof. If the condition (3.9 holds, then we get Thus ŷ(l 2 = p ŷ(0 2. (p 1 ŷ(0 2 = ŷ(m 2.

13 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 13 Using (3.8 we achieve ( W 2 Fr = p (p 1 ŷ(0 2 x(0 2 + ( ( x(l 2 ( = p ((p 1 ŷ(0 2 x(0 2 + x(l 2 ( = p(p 1 ŷ(0 2 x(l 2 = p(p 1 ŷ(0 2 x 2 Fr = p(p 1 ŷ(0 2 x 2 Fr. Conversely, let for some positive real number α we have Thus (3.10 Using (3.8 we have (3.11 α/p W y x 2 Fr = α x 2 Fr, for x C p. ( W y x 2 Fr = α = α ( x(k 2 ( x(l 2. x(l 2 = (p 1 ŷ(0 2 x(0 2 + ( ŷ(m 2 ( ŷ(m 2 x(l 2. Splitting the left side of (3.11 we get ( α/p x(0 2 + α/p x(l = (p 1 ŷ(0 2 x(0 2 + ( ( ŷ(m 2 x(l 2, for x C p. Let x C p be such that x(0 = 1 and x(l = 0 for 1 l p 1. Then (3.12 α/p = (p 1 ŷ(0 2.

14 14 A. GHAANI FARASHAHI AND M. MOHAMMAD-POUR Similarly, if we assume x C p with x(0 = 0 and x(l = 1 for 1 l p 1. Then ( (p 1α/p = (p 1 ŷ(m 2, which implies (3.13 α/p = The equations (3.12 and (3.13 imply which is equivalent to (3.9. (p 1 ŷ(0 2 = ŷ(m 2. ŷ(m 2, A non-zero window signal y C p is called a p-wavelet if and only if y satisfies (3.14 y Fr = p ŷ(0. It is evident to see that each p-wavelet satisfies ŷ(0 0. From (3.9 we deduce that a window signal y C p is a p-wavelet if and only if 2 (3.15 y(k = y(k 2. If y is a p-wavelet, we call α y := (p 1 as the p-wavelet constant. y(k 2 = p(p 1 ŷ(0 2 = p ŷ(m 2, Corollary 3.8. Let y C p be a p-wavelet. For x, z C p we have (3.16 W y x(m, lw y z(m, l = α y x(lz(l. The following results state an inversion formula for the windowed transform given in (3.5. Corollary 3.9. Let y C p be a p-wavelet. For x C p we have the following reconstruction formula; (3.17 x(k = 1 α y W y x(m, lt l D m y(k, for 0 k p 1.

15 A UNIFIED GROUP THEORETICAL APPROACH TO CWT 15 Corollary Let y C p be a p-wavelet and X M ( p (C be a y-wavelet matrix (i.e. any matrix in the range of the y-wavelet transform. The signal x C p given by (3.18 x(k = 1 α y has the y-wavelet transform X. X(m, lt l D m y(k for 0 k p 1, Remark For a positive prime integer p, we define { } (3.19 B p := x C p : x(0 = x(k = 0. The set B p is a complex linear subspace of C p with dimension p 1. Using (3.8, for a window signal y C p and x B p we have ( (3.20 W y x 2 Fr = p ŷ(m 2 x 2 Fr. For a window signal y C p, we define (3.21 β y := p ŷ(m 2 = p ( y 2 Fr ŷ(0 2 = p y 2 Fr y(k 2. Then for any window signal y C p with β y 0 and x, z B p we have (3.22 W y x(m, lw yz (m, l = β y x(lz(l. If y C p is a window signal with β y 0, then each x B p satisfies the following reconstruction formula; (3.23 x(k = 1 β y W y x(m, lt l D m y(k for 0 k p 1. Remark It should be mentioned that when p is a prime integer F := Z p is a finite field, see [23]. In this case F = F {0} = Z p {0} = U p, and hence the finite affine group W p is precisely the group F F. Thus Corollaries 3.7 and 3.9 coincides with the direct consequences of results in [26]. The advantage of our approach which is deeply originated from

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