A Constructive Inversion Framework for Twisted Convolution

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1 A Constructive Inversion Framework for Twiste Convolution Yonina C. Elar, Ewa Matusiak, Tobias Werther June 30, 2006 Subject Classification: 44A35, 15A30, 42C15 Key Wors: Twiste convolution, Wiener s Lemma, Gabor frame, Invertibility of operators Abstract In this paper we evelop constructive invertibility conitions for the twiste convolution. Our approach is base on splitting the twiste convolution with rational parameters into a finite number of weighte convolutions, which can be interprete as another twiste convolution on a finite cyclic group. In analogy with the twiste convolution of finite iscrete signals, we erive an antihomomorphism between the sequence space an a suitable matrix Dept. of Electrical Engineering, Technion Israel Institute of Technology, Haifa, Israel. Tel.: , Fax.: , yonina@ee.technion.ac.il Faculty of Math., University of Vienna, 1090 Vienna, Austria. Tel.: , Fax.: , ewa.matusiak@univie.ac.at, tobias.werther@univie.ac.at. 1

2 algebra which preserves the algebraic structure. In this way, the problem reuces to the analysis of finite matrices whose entries are sequences supporte on corresponing cosets. The invertibility conition then follows from Cramer s rule an Wiener s lemma for this special class of matrices. The problem results from a well known approach of stuying the invertibility properties of the Gabor frame operator in the rational case. The presente approach gives further insights into Gabor frames. In particular, it can be applie for both the continuous (on R ) an the finite iscrete setting. In the latter case, we obtain algorithmic schemes for irectly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature. 1 Introuction Twiste convolution arises naturally in the context of time frequency operators, more specifically in the treatment of Gabor frames [1, 8]. The stuy of inversion schemes of twiste convolution has, therefore, a major impact on the analysis of Gabor frames. Our metho is originate by the Janssen representation of Gabor frame operators [11] an simplifies the approach given in [12]. A ifferent, however, equivalent metho for stuying Gabor frame operators is the well known Zibulski-Zeevi representation [14] base on a generalize Zak-transform. In contrast to the stanar convolution, the twiste convolution is not commutative. This is oppose to the possibility of applying powerful tools from harmonic analysis, such as Wiener s Lemma, in orer to stuy twiste convolution operators. Recently, in [12], the authors escribe 2

3 an new approach to classify the invertibility of l 1 -sequences with respect to the twiste convolution for rational parameters. In this manuscript we exten the iea of [12] in the sense that we take a ifferent approach which allows far better insights into the problem. Specifically, we only eal with sequences an show explicitly how efficient inversion schemes can be erive by rather simple (though sophisticate) manipulations of the twiste convolution. The essential iea is to split up the twiste convolution into a finite number of sums that can be incorporate into a special matrix algebra. In this matrix algebra we then prove a special type of Wiener Lemma which is the most challenging part from a mathematical perspective. The paper is organize as follows. The first section briefly outlines the basic efinition of the twiste convolution. In this section we further iscuss the example of twiste convolution on the finite group Z p Z p. This example serves the purpose to motivate the introuction of the matrix algebra that appears in Section 3 where we prove Wiener s Lemma for a special subalgebra. Section 4 links the twiste convolution to timefrequency operators. More specifically, it shows how the results shown in Section 3 can be use in the context of Gabor frames. In the last section, we give a short outline of the application of the presente approach for inverting frame-like Gabor operators. 3

4 2 Twiste Convolution Let p an q be integers an relatively prime. We efine the twiste convolution for sequences a, b l 1 (Z 2 ) by (a b) m,n = k,l Z a k,l b m k,n l ω (m k) l (1) where ω = e 2πiq/p an enotes the inner prouct in R. Although the twiste convolution epens on p, q we o not specify this epenence because p, q will always be given an fixe beforehan. In Section 4 we show how the twiste convolution is relate to a class of operators with a special time-frequency representation. In contrast to the conventional convolution with symbol, in which ω = 1, the twiste convolution is not commutative, an turns l 1 (Z 2 ) into a non-commutative algebra with the elta-sequence δ as its unit element. We tackle the problem to stuy the invertibility of twiste convolution operators. Non-commutativity is the main subtle point in this problem. In fact, the question when the mapping C b : a l 1 a b l 1 for some b l 1 is invertible an how we can compute the inverse is more ifficult than for a commutative setting. In particular, Wiener s Lemma which eals with the problem that if, for some b l 1, C b is invertible on l 2 then the inverse is generate from an element again in l 1, has to be proven separately. An abstract an more general proof of Wiener s Lemma for twiste convolution is given in [10]. Herein, we focus on a constructive 4

5 metho for stuying the invertibility of the twiste convolution with the rational parameter q/p. In the following subsections we stuy the twiste convolution in a finite setting an raw analogies for approaching the problem of invertibility of C b in the general case. 2.1 Twiste convolution on Z p Z p In what follows we escribe the twiste convolution on the finite group F = Z p Z p. The stanar (commutative) convolution of two elements f, g C p p is efine by (f g) m,n = p 1 k,l=0 f k,l g m k,n l, where operations on inices is performe moulo p. In analogy to the infinite case, we efine the twiste convolution f g of two elements f, g C p p by (f g) m,n = p 1 k,l=0 f k,l g m k,n l ω (m k)l with ω = e 2πiq/p. For a fixe g, the twiste convolution can be seen as a linear mapping C g : f f g whose matrix G is block circulant with p blocks, i.e., G = C(G 0, G p 1,..., G 1 ) = Each block has entries of the form G 0 G p 1 G 1 G 1 G 0 G 2... G p 1 G p 2 G 0. (G j ) kl = ω jl g j,k l. 5

6 Note that for the regular convolution each block is itself circulant. For the invertibility of block circulant matrices we apply a well known result from Fourier analysis. Lemma 2.1. [3] The matrix G = C(G 0, G p 1,..., G 1 ) is invertible if an only if every Ĝs = p 1 r=0 e 2πisr/p G r, s = 0,..., p 1, is invertible. In this case where H r = 1 p G 1 = C(H 0, H p 1,..., H 1 ) p 1 s=0 e2πisr/p (Ĝs) 1. By analyzing Ĝs, we see that all blocks are unitary equivalent, in the sense that T r Ĝ s T r = Ĝs qr, where T r enotes the unitary matrix with entries 1 if p r = l k, (T r ) kl = 0 else. Since p an q are relatively prime, we obtain all blocks by such a unitary transformation. This implies that showing that if Ĝ0 is invertible, then all Ĝs are invertible for s = 1,..., p 1. In other wors, the p p matrix Ĝ 0 contains all the information about the invertibility of C g. An easy computation shows that the entries of Ĝ0 are given by p 1 (Ĝ0) n,l = ω nl g k,n l. (2) k=0 We will later see that this observation motivates the matrix algebra that we introuce to stuy the invertibility of the twiste convolution. 6

7 Now, also all Ĝ 1 s satisfy the same unitary equivalence. It follows that we can rea from Ĝ 1 0 the element g 1 which inverts the twiste convolution f f g, i.e., g 1 g = g g 1 = δ. The twiste convolution on Z p Z p serves as analogy for moelling the twiste convolution for the continuous an the finite imensional case. 3 Main results Our aim is to fin a way to escribe those sequences that have an inverse in (l 1 (Z 2 ), ). To this en we ivie the twiste convolution into a finite sum of weighte normal convolutions of sequences that have isjoint support. We efine such a sequence a r,s by a k,l if (k, l) p (r, s), (a r,s ) k,l = 0 else, (3) where r, s Zp. Obviously, a r,s is supporte on the coset (r +pz ) (s+ pz ) an a = r,s Zp ar,s. For a sequence a having a coset support only for one inex, e.g., on Z (s + pz ), we simply write a,s. We write p for enoting the equivalence of integers moulo p. The iea of splitting a sequence into a sum of sequences supporte on cosets has first been introuce by K. Gröchenig an W. Kozek in [9]. Lemma 3.1. Let a, b, c be in l 1 (Z 2 ). (a) For r, s, u, v Z p, a r,s b u,v is a sequence supporte on the coset (u + r + pz ) (v + s + pz ). 7

8 (b) If c = c,0 is invertible in (l 1, ), then c 1 is also supporte on Z pz. Proof. Let a r,s, b u,v be sequences in l 1 (Z 2 ) an k, l Zp. Then (a r,s b u,v ) k+pz,l+pz = (a r,s ) m,n (b u,v ) k+pz m,l+pz n m,n Z = (a r,s ) t,w (b u,v ) k+pz t,l+pz w. m,n Zp (t,w) p (m,n) Since a r,s is nonzero only for (t, w) p (r, s), an b u,v for (k t, l w) p (u, v), we obtain that (k, l) has to be equivalent to (u + r, v + s) moulo p for a r,s b u,v to be nonzero. To show (b), let c = c,0 be invertible an e be its inverse. Then δ = c e = c,0 ( s Z p e,s) = s Z p c,0 e,s, where, by previous calculations, c,0 e,s is a sequence supporte on Z (s + pz ) for each s Z p. Since δ = s Z p δ,s, an elements of the sum have isjoint supports, c,0 e,s = δ,s. But since δ,s = 0 for s 0 an δ,0 = δ, we conclue that δ s = 0 c,0 e,s = 0 s 0 an therefore e = e,0. 8

9 With Definition (3), we obtain for u, v Zp (a b) u+pz,v+pz = a k,l b u+pz k,v+pz lω (u k) l k,l Z = a k,l b u+pz k,v+pz lω (u r) s r,s Zp (k,l) p(r,s) = r,s Z p k,l Z (a r,s ) k,l (b u r,v s ) u+pz k,v+pz lω (u r) s = (a r,s b u r,v s ) u+pz,v+pz ω(u r) s. r,s Z p In a more compact notation we have (a b) u,v = a r,s b u r,v s ω (u r) s. (4) r,s Z p We observe now that the upper inices in (4) behave like a twiste convolution in Zp Zp. What changes is that we have sequences as elements an stanar convolution instea of multiplication. Motivate by the block circulant structure of the twiste convolution on Z p Z p as escribe in the previous section, we introuce a new matrix algebra which is isomorphic to (l 1 (Z 2 ), ). Before we o so, we fix an orering of the elements from Zp. Let N = p an I = {1,..., N}. Then, to each i I we assign an element k i from Zp an set k 1 = (0,..., 0). We will often write 0 instea of k 1. Let (M, ) be an algebra of p p -matrices whose entries are l 1 - sequences an multiplication of two elements A, B M is given by (A B) i,j = l I A i,l B l,j i, j I. The ientity element I is a matrix with δ sequences on the iagonal. 9

10 Theorem 3.2. Let { M 0 = A M A i,j = ω m k j a m,k i k j, a l 1 an i, j I m Zp Then M 0 is a subalgebra of M. Proof. Define a mapping φ: (l 1, ) (M, ) by (φ(a)) i,j = m Z p }. ω m k j a m,k i k j. (5) Then φ is linear, (φ(δ)) i,j = m Z p ωm k j δ m,k i k j = δ if i = j an zero otherwise. So φ(δ) = I. We emphasize that the mapping φ has been motivate by the matrix Ĝ0 escribe in the previous section. For i, j I, ( φ(a b) ) i,j = m Z p = m Zp = ω m k j (a b) m,k i k j ω m k j l,s Z p ω (m l) s a l,s b m l,k i k j s ω m k j+s) ω l s a l,s b m l,k i k j s = = m,l,s Z p m,l,s Z p ω m s ω l (s k j) a l,s k j ω l k j ω s (m l) a l,s k j b m l,k i s b m l,k i s m,l,s Z p = ( ω l k j a ) l,s k j ( ω s (m l) b ) m l,k i s s Zp l Zp m Zp = ( ω k n m b m,k i k n ) ( ω l k j a l,k n k j ) n I m Z p l Z p = n I φ(b) i,n φ(a) n,j = ( φ(b) φ(a) ) i,j. Therefore φ is an anti-homomorphism, that is, φ(a b) = φ(b) φ(a). 10

11 Hence M 0 is an algebra, being an image of an anti-homomorphism. Before stating the main theorem, we explore properties of elements of M 0. For i, j I an a matrix A M 0 we efine a new matrix A(j, i) obtaine from A by substituting the jth row of A with a vector of zeros having δ on the ith position, an the ith column with a column of zeros having δ on the jth position. Lemma 3.3. Let A M 0. Then (a) et(a) is a sequence supporte on Z pz. (b) et(a(1, i)) is a sequence supporte on Z (k i + pz ) for i I. Proof. Let S N be the group of permutations of the set I. Then et(a) = σ S N ( 1) σ = σ S N ( 1) σ N A σ(i),i = i=1 m 1,...,m N Z p Since σ is a permutation of I, σ S N ( 1) σ N ( i=1 m i Z p ω k i m i a m i,k σ(i) k i ) ω N i=1 m i k i a m 1,k σ(1) k 1 a m N,k σ(n) k N }{{}. G m1,...,m N (k σ(1) k 1 ) + (k σ(2) k 2 ) + + (k σ(n) k N ) = 0. Therefore, by Lemma 3.1, G m1,...,m N is a sequence supporte on the coset ( i I m i + pz ) pz. Since i I m i runs over all Zp, we see that et(a) is supporte on the coset Z pz, i.e., et(a) = et(a),0. In orer to compute the support of et(a(1, i)) for i I, let S N 1 enote the group of permutations of {2,..., N}. Then for i = 1,..., N, et(a(1, i)) = 11

12 = ( 1) i+1 ( 1) σ A σ(2),1 A σ(i),i 1 A σ(i+1),i+1 A σ(n),n σ S N 1 = ( 1) i+1 ω m 2 k 1 + +m i k i 1 +m i+1 k i+1 + +m N k N σ S N 1 ( 1) σ m 2,...,m N Z p a m 2,k σ(2) k 1 a m i,k σ(i) k i 1 a m i+1,k σ(i+1) k i+1 a m N,k σ(n) k N }{{}. G m2,...,m N Since σ is a permutation of {2,..., N}, (k σ(2) k 1 ) + + (k σ(i) k i 1 ) + (k σ(i+1) k i+1 ) + + (k σ(n) k N ) = (k σ(2) + + k σ(n) ) (k 1 + k k N ) + k i = (k σ(2) + + k σ(n) ) (k k N ) + k i = k i. Therefore, by Lemma 3.1, G m2,...,m N is supporte on ( N i=2 m i + pz ) (k i + pz ), an since each m i runs over all Z p, et(a(1, i)) is supporte on Z (k i + pz ). That is, et(a(1, i)) = et(a(1, i)),k i. Now we are in the position to state an prove the main result Theorem 3.4. [Wiener s Lemma for M 0 ] Let A M 0. If A is invertible in M, then B = A 1 M 0. Proof. Since A M 0 is invertible, et(a) is an invertible sequence in (l 1, ), an there exists a matrix B M such that A B = I. By Lemma 3.3, et(a) = et(a),0 an by Lemma 3.1 its inverse, e = et(a) 1, is also supporte on the same coset, hence e = e,0. By Cramer s rule the inverse of A is given by B i,j = et(a(j, i)) e. We see that by Lemma 3.3 (b), B i,1 is a sequence supporte on Z (k i + pz ). Let b be a sequence efine by b = B 1,1 + B 2, B N,1. 12

13 Then B i,1 = j I bk j,k i. Define a new matrix, enote by B, as B i,j = m Z p ω m k j b m,k i k j. Then B M 0 an we will show that B = B, that is, B is the inverse of A. Since B is the inverse of A, I i,1 = ( A B ) i,1 = j I A i,j B j,1 = j I ω m k j a m,k i k j B j,1 = j I m Z p ω m k j a m,k i k j ( b n,k j ) = j I m Z p ω m k j a m,k i k j n Z p b n,k j = m Z p = m Z p where G(m, k i ) = j I m Z p j I n Zp n Z p ω (m n) k j a m n,k i k j b n,k j G(m, k i ), supporte on (m + pz) (k i + pz). n Z p ω(m n) k j a m n,k i k j b n,k j is a sequence Therefore, G(k 1, k 1 ) = δ an G(m, k i ) = 0 for m k 1 an i 1. Using the above ientity we will show that A B = I, an by the uniqueness of the inverse we will 13

14 conclue that B = B: ( ) A B = A i,j i,s B s,j = s I = ( ) ( ω m k s a m,k i k s s I ω n k j b n,k s k j ) = m Z p = m Z p = m Z p = m Z p = m Z p m Z p n Z p ω m k s ω n k j a m,k i k s b n,k s k j s I n Zp ω (ks+k j) m ω n k j a m,k i k j k s b n,ks s I n Zp ω (k s+k j ) (m n) ω n k j a m n,(k i k j ) k s b n,k s s I n Zp ω m k j Hence, A B = I. ω (m n) k s a m n,(k i k j ) k s b n,k s s I n Zp ω m k j G(m, k i k j ) = δ k i k j = k 1 i = j; 0 k i k j k 1 i j; Theorem 3.4 provies the key result to stuy invertibility of twiste convolution. Inee, for a given sequence a in l 1 we look at the corresponing matrix A = φ(a) as efine in (5). If A is invertible in (M, ), which can be checke showing that the eterminant is invertible in (l 1, ), then its inverse A 1 is of the form φ(b) for another element b in l 1. This element b, in turn, provies the inverse of a in (l 1, ). The approach is constructive in the sense that algebraic methos such as Cramer s Rule can be applie to fin the inverse of A. Then, the sequence b can simply be rea from the entries of A 1 accoring to the mapping φ. In particular for small p an this metho leas to fast inversion schemes for the twiste convolution operator. In the last section we will show explicitly how this works in the case of = 1. 14

15 4 Twiste convolution an Gabor analysis Central objects in time frequency analysis are moulation an translation operators. Although most of the upcoming notation can be given in the more general setting of locally compact Abelian groups we restrict ourselves to R in orer to simplify the reaability of this article. For x, ω R we efine the translation operator an the moulation operator on L 2 (R ) by T x f( ) = f( x), M ω f( ) = e 2πiω f( ), respectively. Many technical etails in time-frequency analysis are linke to the commutation law of the translation an moulation operator, namely, M ω T x = e 2πix ω T x M ω. (6) The time-frequency shift for x, ω R is enote by π(x, ω) = T x M ω. It follows from (6) that π(x 1, ω 1 )π(x 2, ω 2 ) = e 2πix 2 ω 1 π(x 1 + x 2, ω 1 + ω 2 ). (7) This shows that time-frequency shifts almost allow a group structure. Incorporating the aitional phase factor into a more extene group law leas to the so-calle Heissenberg group. For more etails about this topic, the reaer is referre to [7]. Gabor analysis eals with the problem of ecomposing an reconstructing signals accoring to a special basis system which consists of 15

16 regular time-frequency shifts of a single so-calle winow function [5, 6]. Let Λ be a time-frequency lattice, i.e., a iscrete subgroup of the timefrequency plane R 2, an let g be in L 2 (R ). Then we efine a Gabor system G(g, Λ) by G(g, Λ) = { } π(λ)g λ Λ. We associate with this Gabor system the positive operator S : f L 2 Sf = λ Λ f, π(λ)g π(λ)g. If the operator S is boune an invertible on L 2 (R ), then G(g, Λ) is calle a frame an S the associate frame operator, cf. [1]. Many stuies in Gabor analysis are evote to the frame operator [8]. In what follows we will escribe the so-calle Janssen representation of such operators. To this en we nee the notion of the ajoint lattice, i.e., Λ = { } λ R 2 π(λ)π(λ ) = π(λ )π(λ), λ Λ. In [2, 4, 11] it is shown that the frame operator S satisfies Janssen representation, S = g, π(λ )g π(λ ). (8) λ Λ At this point, the question arises if we can euce the invertibility of the operator S from the Janssen coefficients ( g, π(λ )g ). It is known from frame theory that if S is invertible, then its inverse is of the same type, that is, it also has a Janssen representation. In orer to better unerstan the main ingreients of this problem we transfer the moel to an operator algebra. To this en we restrict our 16

17 iscussion to so-calle separable lattices of the form Λ = αz βz for some fixe positive numbers α an β. An easy computation base on (7) shows that Λ = β 1 Z α 1 Z. We efine the operator algebra A as in [10] by A = { S = } a k,l π(β 1 k, α 1 l) a = (a k,l ) l 1 (Z 2 ). k,l Z The restriction to l 1 -sequences guarantees absolute convergence of the sum of time-frequency shifts. Let κ be the mapping κ : a l 1 κ(a) = Then, as alreay observe in [11], we have k,l Z a k,l π(β 1 k, α 1 l) A. κ(a)κ(b) = κ(a b) an κ(δ) = I where δ an I enote the Dirac sequence an the ientity operator, respectively. Both represent the unit element of the corresponing algebra. It follows that κ is an algebra homomorphism from (l 1 (Z 2 ), ) to A, an invertibility of an element in A can be transferre to the invertibility of the associate l 1 -sequence with respect to the twiste convolution. It is important to observe, that all the results go through also for weighte l 1 -spaces. These facts are use to esign ual Gabor winows of a special type, cf. [10]. 17

18 In the following section we give an example of how this approach can be explicitly use in Gabor analysis of one-imensional signals. Remark. The above results, with the help of metaplectic operators, carry over to the more general class of lattices, calle symplectic lattices. A lattice Λ s R 2 is calle symplectic, if one can write Λ s = DΛ where Λ is a separable lattice an D GL 2 (R). To every D GL 2 (R), there correspons a unitary operator µ(d), calle metaplectic, acting on L 2 (R ). One can show that a Gabor system on a symplectic lattice is unitary equivalent to a Gabor system on a separable lattice uner µ(d), an S Λ s g = µ(d) 1 S Λ µ(d)gµ(d). Hence, to analyze the invertibility of a frame operator S associate to the winow function g L 2 (R ) an symplectic lattice Λ s, it suffices to analyze a frame operator associate to µ(d)g an a separable lattice Λ. For more etails see [8]. 5 Application to one-imensional signals In this section, we briefly escribe how the presente inversion scheme applies to Gabor frame operators in a one-imensional setting. A more etaile iscussion also for finite imensional signals is escribe in [13]. Assume = 1. Let a be in l 1 (Z 2 ) an α, β be constants such that αβ = p/q with p, q relative prime. Set κ(a) = k,l Z a k,l π(β 1 k, α 1 l). In orer to verify if κ(a) is invertible on L 2 (R) we simply look at the 18

19 coefficient sequence a an check whether a is invertible in (l 1 (Z 2 ), ). To this en, we apply the above results an switch to the matrix A whose entries are efine by A i,j = p 1 m=0 ω mj a m,i j, with ω = e 2πiq/p. Next, we nee to show that the matrix A is invertible in (M, ). For example, we can calculate the eterminate which is a sequence in l 1 an show that it is invertible in (l 1, ). Assume that the eterminant of A is invertible. We enote its inverse by e. By Cramer s Rule, we compute the first column of the inverse matrix B of A as B k,0 = et A(0, k) e, for k = 0,..., p 1. Then p 1 b = k=0 B k,0 provies the inverse sequence of a which, in turns, gives κ(a) 1 = κ(b). Note that for p = 1, the twiste convolution turn into normal convolution an we can simply apply the stanar Fourier inversion scheme of sequences in (l 1 (Z 2 ), ) since in this case the matrix A reuces to the sequence a. Acknowlegements The authors gratefully acknowlege support from the Ollenorff Minerva Center an from the European Union s Human Potential Programme, uner the contract HPRN-CT (HASSIP). We woul also like 19

20 to thank Karlheinz Gröchenig an Yehoshua Y. Zeevi for many fruitful iscussions. References [1] O. Christensen. An Introuction to Frames an Riesz Bases. Birkhäuser, [2] I. Daubechies, H. J. Lanau, an Z. Lanau. Gabor time-frequency lattices an the Wexler-Raz Ientity. J. Four. Anal. an Appl., 1(4): , [3] P. J. Davis. Circulant Matrices. New York, AMS Chelsea Publishing, 2n eition, [4] H. G. Feichtinger an W. Kozek. Quantization of TF lattice invariant operators on elementary LCA groups. In H.G. Feichtinger an T. Strohmer, eitors, Gabor Analysis an Algorithms: Theory an Applications, pages Birkhäuser, Boston, [5] Hans G. Feichtinger an Thomas Strohmer. Gabor analysis an algorithms. Theory an applications. Boston MA: Birkhäuser, [6] Hans G. Feichtinger an Thomas Strohmer. Avances in Gabor analysis. Boston MA: Birkhäuser, [7] G. B. Follan. Harmonic analysis in phase space. Annals of Mathematics Stuies, 122. Princeton, NJ: Princeton University Press,

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