arXiv:1703.05012v1 [math.FA] 15 Mar 2017
A CONSTRUCTIVE APPROACH TO THE FINITE WAVELET FRAMES OVER PRIME FIELDS ASGHAR RAHIMI AND NILOUFAR SEDDIGHI Abstract. In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory. Keywords: Finite wavelet frames, finite wavelet group, prime fields. MSC(2010): Primary 42C15, 42C40, 65T60; Secondary 30E05, 30E10.
1. Introduction The mathematical theory of frames in Hilbert spaces was introduced in [10], and has been studied in depth for finite dimensions in [5, 7]. Finite frames have been applied as means to interpret periodic signals and privileged in areas ranging from digital signal processing to image analysis, filter banks, big data and compressed sensing, see [7, 28, 29] and references therein. The representations of a function/signal in time-frequency (resp. time-scale) domain are obtained through analyzing the signal with respect to an over complete system whose elements are localized in time and frequency (resp. scale) [3, 4, 19, 20]. In the framework of wave packet analysis [8], finite wavelet systems are particular classes of finite wave packet systems which have been introduced recently in [15, 16, 17]. The analytic structure of finite wavelet frames over prime fields (finite Abelian groups of prime orders) has been studied in [14, 21]. The notion of a wavelet transform over a prime field was introduced in [12] and extended for finite fields in [11, 23]. This notion is an example of the abstract generalization of wavelet transforms using harmonic analysis, see [1, 2, 6, 9, 13, 28] and classical references therein. The current paper consists of a constructive approach to the abstract aspects of nature of finite wavelet systems over prime fields. The motivation of this paper is to establish an alternative constructive formulation for the wavelet coefficients of finite wavelet frames over prime fields. This article is organized as follows. Section 2 introduces some notations as well as a brief review of Fourier transform on finite cyclic groups, periodic signal processing, and finite frames. Then in section 3 we present a constructive technique for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and theoretical group theory. In addition, we shall also give a constructive 1
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characterization for frame conditions of finite wavelet systems over prime fields using matrix analysis terminology. 2. Preliminaries This section is devoted to present a brief review of notations, basics, and preliminaries of Fourier analysis and computational harmonic analysis over finite cyclic groups, for any details we refer the readers to [25] and classical references therein. It should be mentioned that in this article p is a positive prime integer. We also employ notations of the author of the references [14, 15, 16]. For a finite group G, the complex vector space CG = {x : G → C} is a |G|-dimensional vector space with complex vector entries indexed by elements P in the finite group G. The inner product of x, y ∈ CG is defined by hx, yi = g∈G x(g)y(g), and the induced norm is the k.k2 -norm of x. For CZp , where Zp denotes the cyclic group of p elements {0, . . . , p − 1}, we write Cp . The notation kxk0 = |{k ∈ Zp : x(k) 6= 0}| counts non-zero entries in x ∈ Cp . The translation operator Tk : Cp → Cp is Tk x(s) = x(s − k) for x ∈ Cp and k, s ∈ Zp . The modulation operator Mℓ : Cp → Cp is given by Mℓ x(s) = e−2πiℓs/p x(s) for x ∈ Cp and ℓ, s ∈ Zp . The translation and modulation operators on the Hilbert space Cp are unitary operators in the k.k2 -norm. For ℓ, k ∈ Zp we have (Tk )∗ = (Tk )−1 = Tp−k and (Mℓ )∗ = (Mℓ )−1 = Mp−ℓ . The unitary discrete Fourier Transform (DFT) of a 1D P b(ℓ) = √1p p−1 discrete signal x ∈ Cp is defined by x k=0 x(k)wℓ (k), for all ℓ ∈ Zp , where for all ℓ, k ∈ Zp we have wℓ (k) = e2πiℓk/p . Thus, DFT of x ∈ Cp at ℓ ∈ Zp is (2.1)
p−1
p−1
k=0
k=0
1 X 1 X b(ℓ) = Fp (x)(ℓ) = √ x x(k)wℓ (k) = √ x(k)e−2πiℓk/p . p p
The DFT is a unitary transform in k.k2 -norm, i.e. for all x ∈ Cp satisfies the Parseval formula kFp (x)k2 = kxk2 . The Polarization identity implies bi for x, y ∈ Cp . The unitary DFT (2.1) the Plancherel formula hx, yi = hb x, y d b, M b, for x ∈ Cp and k, ℓ ∈ Zp . Also the satisfies Td k x = Mk x ℓ x = Tp−ℓ x inverse discrete Fourier formula (IDFT) for a 1D discrete signal x ∈ Cp is given by p−1
p−1
ℓ=0
ℓ=0
1 X 1 X b(ℓ)wℓ (k) = √ b(ℓ)e2πiℓk/p , 0 ≤ k ≤ p − 1. x(k) = √ x x p p
A finite sequence A = {yj : 0 ≤ j ≤ M − 1} ⊂ Cp is called a frame (or finite frame) for Cp , if there exists a positive constant 0 < A < ∞ such that [7] M −1 X |hx, yj i|2 , (x ∈ Cp ). Akxk22 ≤ j=0
FINITE WAVELET FRAMES OVER PRIME FIELDS
3
PM −1 −1 The synthesis operator F : CM → Cp is defined by F {cj }M j=0 = j=0 cj yj M −1 M ∗ p M for all {cj }j=0 ∈ C . The adjoint operator F : C → C is defined by −1 p F ∗ x = {hx, yj i}M j=0 for all x ∈ C . If A = {yj : 0 ≤ j ≤ M − 1} is a frame p ∗ for C , by composing F and F , we get the positive and invertible frame operator S : Cp → Cp given by ∗
x 7→ Sx = F F x =
M −1 X j=0
hx, yj iyj ,
(x ∈ Cp ),
and the set A spans the complex Hilbert space Cp which implies M ≥ p, where M = |A|. Each finite spanning set in Cp is a finite frame for Cp . The ratio between M and p is called the redundancy of the finite frame A (i.e. redA = M/p). If A is a finite frame for Cp , each x ∈ Cp satisfies x=
M −1 X j=0
hx, S −1 yj iyj =
M −1 X j=0
hx, yj iS −1 yj .
3. Construction of Wavelet Frames over Prime Fields In this section, we briefly state structure and basic properties of cyclic dilation operators, see [12, 14, 18, 23]. Then we present an overview for the notion and structure of finite wavelet groups over prime fields [24, 26]. 3.1. Structure of Finite Wavelet Group over Prime Fields. The set Up := Zp − {0} = {1, ..., p − 1},
is a finite multiplicative Abelian group of order p − 1 with respect to the multiplication module p with the identity element 1. The multiplicative inverse for m ∈ Up is mp which satisfies mp m + np = 1 for some n ∈ Z, see [22, 27]. For m ∈ Up , the cyclic dilation operator is defined by Dm : Cp → Cp by Dm x(k) := x(mp k)
for x ∈ Cp and k ∈ Zp , where mp is the multiplicative inverse of m in Up . In the following propositions, we state some basic properties of cyclic dilations. Proposition 3.1. Let p be a positive prime integer. Then (i) For (m, k) ∈ Up × Zp , we have Dm Tk = Tmk Dm . (ii) For m, m′ ∈ Up , we have Dmm′ = Dm Dm′ . (iii) For (m, k), (m′ , k′ ) ∈ Up × Zp , we have Tk+mk′ Dmm′ = Tk Dm Tk′ Dm′ . (iv) For (m, ℓ) ∈ Up × Zp , we have Dm Mℓ = Mmp ℓ Dm . The next result states some properties of cyclic dilations. Proposition 3.2. Let p be a positive prime integer and m ∈ Up . Then (i) The dilation operator Dm : Cp → Cp is a ∗-homomorphism.
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A. RAHIMI AND N. SEDDIGHI
(ii) The dilation operator Dm : Cp → Cp is unitary in k.k2 -norm and satisfies (Dm )∗ = (Dm )−1 = Dmp . [ b. (iii) For x ∈ Cp , we have D m x = Dmp x The underlying set
Up × Zp = {(m, k) : m ∈ Up , k ∈ Zp } , equipped with the following group operations (m, k) ⋉ (m′ , k′ ) := (mm′ , k + mk′ ), (m, k)−1 := (mp , mp .(p − k)),
is a finite non-Abelian group of order p(p − 1) denoted by Wp and it is called as finite affine group on p integers or the finite wavelet group associated to the integer p or simply as p-wavelet group. Next proposition states basic properties of the finite wavelet group Wp . Proposition 3.3. Let p > 2 be a positive prime integer. Then Wp is a nonAbelian group of order p(p − 1) which contains a copy of Zp as a normal Abelian subgroup and a copy of Up as a non-normal Abelian subgroup. 3.2. Wavelet Frames over Prime Fields. A finite wavelet system for the complex Hilbert space Cp is a family or system of the form W(y, ∆) := {σ(m, k)y = Tk Dm y : (m, k) ∈ ∆ ⊆ Wp } , for some window signal y ∈ Cp and a subset ∆ of Wp . If ∆ = Wp , we put W(y) := W(y, Wp ) and it is called a full finite wavelet system over Zp . A finite wavelet system which spans Cp is a frame and is referred to as a finite wavelet frame over the prime field Zp . If y ∈ Cp is a window signal then for x ∈ Cp , we have hx, σ(m, k)yi = hx, Tk Dm yi = hTp−k x, Dm yi,
((m, k) ∈ Wp ).
The following proposition states a formulation for wavelet coefficients via Fourier transform. Proposition 3.4. Let x, y ∈ Cp and (m, k) ∈ Wp . Then, hx, σ(m, k)yi = Proof. See Proposition 4.1 of [14].
√ [ pFp (b x.D m y)(p − k).
In [14] using an analytic approach the author has presented a concrete formulation for the k.k2 -norm of wavelet coefficients, the formula of which is just stated hereby.
FINITE WAVELET FRAMES OVER PRIME FIELDS
5
Theorem 3.5. Let p be a positive prime integer, M a divisor of p − 1 and M be a multiplicative subgroup of Up of order M . Let y ∈ Cp be a window signal and x ∈ Cp . Then, X X |hx, σ(m, k)yi|2 m∈M k∈Zp
= p M |b y(0)|2 |b x(0)|2 +
X
m∈M
where
γℓ (y, M) :=
X
|b y(m)|2
m∈M
!
|b y(mℓ)|2 ,
X
ℓ∈M
|b x(ℓ)|2
!
+
X
ℓ∈Up −M
γℓ (y, M)|b x(ℓ)|2 ,
(ℓ ∈ Up − M).
Proof. See Theorem 4.2 of [14].
Remark 3.6. The formulation presented in Theorem 3.5 is an analytic formulation of wavelet coefficients associated to the subgroup M. In details, that formulation originated from an analytic approach which was based on direct computations of cyclic dilations in the subgroup M. In the following theorem, we present a constructive formulation for the k.k2 -norm of wavelet coefficients. At the first, we need to prove some results. Proposition 3.7. Let p be a positive prime integer, M a divisor of p − 1 and M a multiplicative subgroup of Up of order M . Let ǫ be a generator of the cyclic group Up and a := p−1 M the index of M in Up . Then (i) For 0 ≤ r, s ≤ a − 1, we have ǫr M = ǫs M iff r = s. (ii) Up /M = {ǫt M : 0 ≤ t ≤ a − 1}. Proof. (i) First, suppose ǫr M = ǫs M for some 0 ≤ r, s ≤ a − 1. We will show r = s. For see this, assume to the contrary that r 6= s. Without loss of generality, suppose s > r. Thus, there exists 1 ≤ r ′ ≤ a − 1 < a,
(3.1)
′
′
such that s = r + r ′ , and we get ǫr M = ǫr+r M = ǫr ǫr M. Since ǫr is an p ′ ′ invertible element of Up , so M = ǫr M and by this ǫr ∈ M. Also, ǫp−1 ≡ p−1
p
(ǫr ) r′ ≡ 1. Hence the order of ǫr in Up can be at most p−1 r ′ . Since ǫ ′ p−1 r is a generator of Up , the order of ǫ is r′ . In addition, by (3.1) we get r′ M < p−1 r ′ ≤ p − 1, therefore ǫ can not be in M which is a contradiction. The inverse implication is straightforward. (ii) According to (i), {ǫt M : 0 ≤ t ≤ a − 1} is a set of disjoint cosets of M in Up and the cardinality of this set is equal to the index number of M in Up , so the equality in (ii) holds. In particular, this set of cosets creates a partition to Up . ′
′
Next we present a constructive formula for k.k2 of wavelet coefficients.
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Theorem 3.8. Let p be a positive prime integer, M a divisor of p − 1 and M a multiplicative subgroup of Up of order M . Let ǫ be a generator of the p cyclic group Up and a := p−1 M the index of M in Up . Let y ∈ C be a window p signal and x ∈ C . Then, p−1 XX
m∈M k=0
|hx, Tk Dm yi|2
X a−1 X X 2 2 2 2 |b y(w)| = p M |b x(0)| |b y(0)| + , |b x(ℓ)| t=0
w∈Ht
ℓ∈Ht
where Ht := ǫt M for all 0 ≤ t ≤ a − 1. Proof. Let y ∈ Cp be a window function, x ∈ Cp and m ∈ M. Then using Proposition 3.4 and Plancherel formula we have p−1 X k=0
|hx, Tk Dm yi|2 = p
p−1 X ℓ=0
y(mℓ)|2 . |b x(ℓ)|2 .|b
Hence we achieve p−1 XX
m∈M k=0
|hx, Tk Dm yi|2 =
X
p
ℓ=0
m∈M
=p
p−1 X
p−1 X
X
ℓ=0 m∈M
y(mℓ)|2 |b x(ℓ)|2 .|b y(mℓ)|2 . |b x(ℓ)|2 .|b
Then we have (3.2) X X X X p−1 p−1 X 2 2 2 2 2 2 |b x(ℓ)| |b y(mℓ)| = |b x(0)| |b y(0)| + |b x(ℓ)| |b y(mℓ)| . ℓ=0
m∈M
m∈M
ℓ=1
m∈M
Now by (3.2) and Proposition 3.7, we get p−1 XX
m∈M ℓ=0
|b x(ℓ)|2 |b y(mℓ)|2 = M |b x(0)|2 |b y(0)|2 +
a−1 X X
t=0 ℓ∈Ht
p
|b x(ℓ)|2
X
m∈M
|b y(mℓ)|2 .
Replacing mℓ with w we get, w ≡ mℓ ∈ M.Ht = M.ǫt M = ǫt M = Ht . For each ℓ ∈ Ht , the mapping m → mℓ defines a bijection of M onto Ht , and
FINITE WAVELET FRAMES OVER PRIME FIELDS
7
thus we deduce that p−1 XX
m∈M ℓ=0
|b x(ℓ)|2 |b y(mℓ)|2 = M |b x(0)|2 |b y(0)|2 +
a−1 X X t=0
ℓ∈Ht
2
|b x(ℓ)|
X
w∈Ht
2
|b y(w)|
.
Therefore p−1 XX
m∈M k=0
|hx, Tk Dm yi|2
X a−1 X X = p M |b x(0)|2 |b y(0)|2 + |b y(w)|2 |b x(ℓ)|2 . t=0
w∈Ht
ℓ∈Ht
The following result is an immediate consequence of Theorem 3.8. Corollary 3.9. Let p be a positive prime integer. Let y ∈ Cp be a window signal and x ∈ Cp . Then p−1 X X
m∈Up k=0
|hx, Tk Dm yi|2
2
2
= p (p − 1)|b x(0)| |b y(0)| +
X
ℓ∈Up
|b x(ℓ)|
2
X
w∈Up
|b y(w)|
2
.
Applying Theorem 3.8, we can present the following constructive characterization of a given window signal y ∈ Cp and a subgroup of the finite wavelet group Wp to guarantee that generated wavelet system is a frame for Cp . Theorem 3.10. Let p be a positive prime integer, ǫ a generator of Up , M a divisor of p − 1, M a multiplicative subgroup of Up of order M and a := p−1 M the index of M in Up . Let ∆M := M × Zp and y ∈ Cp be a non-zero window signal. The finite wavelet system W(y, ∆M ) is a frame for Cp if and only if the following conditions hold b (0) 6= 0 (i) y b(ǫt mt ) 6= 0. (ii) For each t ∈ {0, ..., a − 1}, there exists mt ∈ M such that y
Proof. Let y be a non-zero window signal which satisfies conditions (i), (ii). Then by definition of Ht we get ! ! X X |b y(w)|2 6= 0. ϑ := min min |b y(mǫt )|2 = t∈{0,...,a−1}
m∈M
t∈{0,...,a−1}
w∈Ht
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A. RAHIMI AND N. SEDDIGHI
Now, let 0 < A < ∞ be given by X p−1 2 A := min M y(k) , pϑ . k=0
Using Theorem 3.8 for x ∈ Cp , we have p−1 XX
m∈M k=0
2
2
2
|hx, Tk Dm yi| = pM |b x(0)| |b y(0)| + p
a−1 X X t=0
≥ pM |b x(0)|2 |b y(0)|2 + pϑ 2
≥ A|b x(0)| + A = A|b x(0)|2 + A =
Akb xk22
=
t=0
a−1 X X t=0
ℓ∈Ht
p−1 X
|b x(ℓ)|2
ℓ=1 Akxk22 .
ℓ∈Ht
a−1 X
|b x(ℓ)|
X
ℓ∈Ht 2
|b x(ℓ)|
2
X
|b x(ℓ)|2
w∈Ht
|b y(w)|
which implies that the finite wavelet system W(y, ∆M ) is a frame for Cp . For the inverse implication, consider y ∈ Cp be a non-zero window signal such that the finite wavelet system W(y, ∆M ) is a frame for Cp . Thus, there exists A1 > 0 such that p−1 XX
m∈M k=0
|hx, Tk Dm yi|2 ≥ A1 kxk22 ,
(x ∈ Cp ).
Then by Theorem 3.8 we have (3.3)
M |b y(0)|2 |b x(0)|2 +
a−1 X X t=0
ℓ∈Ht
|b x(ℓ)|2
X
w∈Ht
|b y(w)|2
2
≥ A2 kxk22
b′ (0) 6= 0 and for all x ∈ Cp , where A2 = Ap1 . Now let x′ ∈ Cp with x b′ (ℓ) = 0, for all ℓ ∈ {1, ..., p − 1}. Thus, by (3.3) we get y b (0) 6= 0. Next, x consider xt ∈ Cp be a non-zero vector in which for a fixed but arbitrary t ∈ {0, ..., a − 1}, bt (ℓ) = 0, ∀ℓ 6∈ Ht . x P y(w)|2 should be non-zero. Therefore Hence (3.4) assures that w∈Ht |b yb(w) 6= 0 for some w ∈ Ht , which indicates that there exists mt ∈ M b(mt ǫt ) 6= 0. such that y (3.4)
The following result shows that for a large class of non-zero window signals the finite wavelet system W(y) is a frame for Cp with redundancy p − 1.
FINITE WAVELET FRAMES OVER PRIME FIELDS
9
Corollary 3.11. Let p be a positive prime integer and y ∈ Cp be a non-zero window signal. The finite wavelet system W(y) constitutes a frame for Cp b (0) 6= 0 and kb with the redundancy p − 1 if and only if y yk0 ≥ 2.
The following corollary presents a characterization for finite wavelet systems over prime fields using matrix analysis language.
Corollary 3.12. Let p be a positive prime integer, ǫ a generator of Up , M a divisor of p − 1, M a multiplicative subgroup of Up of order M and a := p−1 M be the index of M in Up . Let ∆M := M × Zp and y ∈ Cp be a non-zero window signal. The finite wavelet system W(y, ∆M ) is a frame for Cp if b (0) 6= 0 and the matrix Y(M, y) of size a × M given by and only if y b (1) b (ǫa ) b (ǫ(M −1)a ) y y ··· y 1 y b (ǫa+1 ) · · · y b (ǫ(M −1)a+1 ) y b(ǫ ) Y(M, y) := . .. .. .. .. . . . b (ǫa−1 ) y b(ǫ2a−1 ) · · · y
b(ǫM a−1 ) y
a×M
is a matrix such that each row has at least a non-zero entry. Proof. Since ǫa is a generator of M and |M| = M , so (3.5)
M = {(ǫa )0 , (ǫa )1 , ..., (ǫa )M −1 } = {1, ǫa , ..., ǫ(M −1)a }.
By (3.5) and definition of Ht , we have Ht = {ǫt , ǫa .ǫt , ..., ǫ(M −1)a .ǫt } = {ǫt , ǫa+t , ..., ǫ(M −1)a+t } b for all t ∈ {0, ..., a − 1}. The entries of t-th row of Y(M, y) are entries of y on Ht . Hence by Theorem 3.10 the result holds. We can also deduce the following tight frame condition for finite wavelet systems generated by subgroups of Wp .
Proposition 3.13. Let p be a positive prime integer, M a divisor of p − 1 and M a multiplicative subgroup of Up of order M . Let ∆M := M × Zp and y ∈ Cp be a non-zero window signal. The finite wavelet system W(y, ∆M ) b(0) 6= 0 and is a tight frame for Cp if and only if y p−1 2 ! X X 2 |b y(w)| M y(k) = p k=0
w∈Ht
for all t ∈ {0, ..., a − 1}. In this case
αy := pM |b y(0)|2 = p
X
m∈M
|b y(m)|2
is the frame bound. Proof. It can be proven by a similar argument used in Theorem 3.10.
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A. RAHIMI AND N. SEDDIGHI
Acknowledgement Some of the results are obtained during the second author’s appointment from the NuHAG group at the University of Vienna. We would like to thank Prof. Hans. G. Feichtinger for his valuable comments and the group for their hospitality.
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[email protected] Niloufar Seddighi, Faculty Mathematics, University of Maragheh, Maragheh, Iran. E-mail address: stu
[email protected]
