Uncertainty Principles for the Offset Linear Canonical Transform

Uncertain t y Principles for the Offset Linear Canonical T ransform Haiye Huo ∗ Department of Mathemat ics, Sc ho ol of Science, N a nc hang Univ ersity , Nanc hang 33003 1, Jia ngxi, China A bstr act . The offset linear canonical transform (OLCT) pro vides a more general framew ork for a n um b er of w ell kno wn linear integral transforms in signal pro cessing and o ptics, suc h as F ourier transform, fractional F ourier transform, linear canonical transform. In this pap er, to c haracterize sim ultaneous lo calization of a signal and its OLCT, w e extend some differen t uncertain t y principles (UPs), including Nazarov’s UP , Hardy’s UP , Beurling’s UP , log arithmic UP and entropic UP , whic h hav e already b een w ell studied in the F ourier transform do ma in ov er the last few decades, t o the OLCT domain in a broader sense. Keywor ds. Offset linear canonical transform; Uncertain ty principle; Logarithmic uncertain t y estimate; En tropic inequality ; Lo calization 1 In tro d uction Uncertain t y princip le (UP) pla ys an imp o rtan t ro le in quan tum m ec hanics [9] and signal pro cessing [2]. In quantum mec hanics, UP was first pro p osed b y the German ph ysicist W. Heisen b erg in 1927 [9]. It basically sa ys that the more precisely the p osition of a particle is determined, the less precisely its momen tum can b e known, and vice v ersa. F rom the p erspective of signal pro cessing, UP can b e describ ed a s follo ws: “One cannot sharply lo calize a signal in b oth the time domain and frequency domain sim ultaneously” (see [2, 5] for more details). By using differen t notat ions of essen tial supp ort , there ar e man y differen t kinds of UPs asso ciated with the F ourier transform, like Heisen b erg’s UP [7, 9], Nazarov’s UP [11, 13], Hardy’s UP [7 , 8], Beurling’s UP [10 ], logarithmic UP [1], en tropic UP [4], and so on. It is w ell kno wn that the offset linear canonical transform (OLCT) [14, 16, 19 , 22, 24] is a generalized v ersion of F ourier transform and has wide applications in signal pro cessing and o ptics. Note that UP cannot b e av oided and o wns its sp ecific form for eac h time-frequency represen tation. Therefore, it is necessary to extend the aforemen tioned UPs to the OLCT domain. In 200 7, A. Stern extended Heisen b erg’s ∗ Email: hyh uo@ncu.edu.cn 1 UP from the F ourier transform domain to the OLCT domain [16]. It states that a nonzero function and its OLCT cannot b ot h b e sharply lo calized. T o the b est of our kno wledge, there are no other results published ab out UP s asso ciated with the O LCT. He nce, in this pap er, the o ther fiv e UPs f or the F ourier transform are extended to the OLCT domain, i.e., Nazaro v’s UP , Hardy’s UP , Beurling’s UP , logarithmic UP , and entropic UP . The rest o f the pap er is org anized as follo ws. In Section 2, w e recall the notations of t he O L CT and the g eneralized P arsev al formula for the OLCT. In Section 3, w e exten d the corres p onding results of Nazarov’s UP , Hardy’s UP , Beurling’s UP , logarithmic UP , a nd en tropic UP , to the OLCT domain, respective ly . In Section 4, w e conclude the pap er. 2 Preliminaries In this section, let us review the definition of the OLCT and its generalized P arsev al form ula. F or a g iv en function f ( t ) ∈ L 2 ( R ), the definition of its OLCT [12] with pa r a meter A =  a b τ c d η  is O A f ( u ) = O A [ f ( t )]( u ) = ( R + ∞ −∞ f ( t ) K A ( t, u )d t, b 6 = 0 , √ de j cd 2 ( u − τ ) 2 + j u η f ( d ( u − τ )) , b = 0 , (1) where K A ( t, u ) = 1 √ j 2 π b e j a 2 b t 2 − j 1 b t ( u − τ ) − j 1 b u ( dτ − bη )+ j d 2 b ( u 2 + τ 2 ) , parameters a, b, c, d, τ , η ∈ R , and ad − bc = 1. F rom the definition of the OLCT, when b = 0, the OLCT reduces to a c hirp mul- tiplication op erator. Hence, without loss of generalit y , w e assume b > 0 t hro ughout the pap er. By (1), one can easily c hec k that the OLCT includes man y w ell-kno wn linear transforms a s special cases. F or instance, let A =  0 1 0 − 1 0 0  , the OLCT reduces to the F ourier transform [5]; let A =  cos α sin α 0 − sin α cos α 0  , the OLCT reduces to the fractional F ourier transform [17]; let A =  a b 0 c d 0  , the OLCT reduces to the linear canonical transform [6, 15, 1 8, 20], etc. Next, w e introduce one of imp o rtan t prop erties for the OLCT, i.e., its generalize d P arsev al form ula [3 ], a s follo ws: Z R f ( t ) g ( t )d t = Z R O A f ( u ) O A g ( u )d u, (2) 2 where ¯ · denotes the complex conjuga te. This equation will be used in the fo llowing sections. 3 Uncertain t y Prin ciples in the OLCT Domain Recall that there are man y differen t forms of UPs in the F ourier t ransform domain, suc h as Heisen b erg’s UP , Nazaro v’s UP , Hardy’s UP , Beurling’s UP , logarithmic UP , en tropic UP , and so on, in terms of differen t notations of “lo calization”. As far as w e kno w, in 2009, G. Xu et al [21] exte nded the logarithmic UP and en tropic UP to the linear canonical transform domain. Recen tly , Q. Zhang [23] extended the other t hr ee UPs: Nazaro v’s UP , Hardy’s UP , Beurling’s UP to the linear canonical transform domain. Considering t ha t the OLCT is a generalized v ersion of the F ourier transform or the linear canonical transform, it is natur a l and intere sting to study the sim ultaneous localizatio n of a function and its OL CT b y further extending the aforemen tioned UPs to the OLCT domain. So far, there exists only one researc h w ork on Heis en b erg’s UP in the OLCT domain. Therefore, in this sec tion, w e in v estigate the o t her fiv e differen t forms of UPs asso ciated with the function f and its OLCT O A f , i.e., Nazaro v’s UP , Ha rdy’s UP , Beurling’s UP , logarithmic UP , a nd en tropic UP , for t he OLCT. 3.1 Nazaro v’s UP As for Heisen b erg’s UP , its lo calization is measured b y smallness of disp ersions. By considering a no ther criterion of lo calization, i.e., smallness of supp ort, Nazarov ’s UP w as first pro p osed b y F.L . Nazaro v in 1993 [13]. It argues what happ ens if a nonzero function and its F ourier transform are only small o ut side a compact set? Let us recall t he concept o f Nazaro v’s UP for t he F ourier transform [11, 13] as fo llo ws. Prop osition 3.1 ([11, 1 3]) . L et f ∈ L 2 ( R ) , and T , Ω b e two subsets of R with finite me asur e. Then , ther e exists a c onstant C > 0 , such that Z R | f ( t ) | 2 d t ≤ C e C | T || Ω |  Z R \ T | f ( t ) | 2 d t + Z R \ Ω |F f ( u ) | 2 d u  , (3) wher e F is the F ourier tr ansform define d by F f ( u ) = 1 √ 2 π Z + ∞ −∞ f ( t ) e − j u t d t, and | T | is denote d a s the L eb esgue me asur e of T . Motiv at ed b y Prop osition 3.1, we next extend the Nazarov’s UP to the OLCT domain. 3 Theorem 3.2. L et f ∈ L 2 ( R ) , and T , Ω b e two subsets of R with finite me asur e. Then, ther e exists a c onstant C > 0 , such that Z R | f ( t ) | 2 d t ≤ C e C | T || Ω |  Z R \ T | f ( t ) | 2 d t + Z R \ (Ω b ) | O A f ( u ) | 2 d u  . (4) Pr o of. By the definition of the OLCT (1), w e can rewrite the OLCT as follow s O A f ( u ) = 1 √ j b e − j 1 b u ( dτ − bη )+ j d 2 b ( u 2 + τ 2 ) G ( u ) , (5) where G ( u ) , 1 √ 2 π Z + ∞ −∞ f ( t ) e j a 2 b t 2 − j 1 b t ( u − τ ) d t. (6) Th us, | O A f ( u ) | = 1 √ b | G ( u ) | . (7) Let g ( t ) = f ( t ) e j a 2 b t 2 + j 1 b tτ , then G ( ub ) is the F ourier transform of g ( t ), and | f ( t ) | = | g ( t ) | . Since f ∈ L 2 ( R ), then g ∈ L 2 ( R ). Applying Prop osition 3.1 with the function g ( t ) and its F ourier t ransform G ( ub ), w e get Z R | g ( t ) | 2 d t ≤ C e C | T || Ω |  Z R \ T | g ( t ) | 2 d t + Z R \ Ω | G ( ub ) | 2 d u  . (8) Substituting (7 ) into (8), w e obtain Z R | f ( t ) | 2 d t = Z R | g ( t ) | 2 d t ≤ C e C | T || Ω |  Z R \ T | f ( t ) | 2 d t + Z R \ Ω | G ( ub ) | 2 d u  = C e C | T || Ω |  Z R \ T | f ( t ) | 2 d t + 1 b Z R \ (Ω b ) | G ( u ) | 2 d u  = C e C | T || Ω |  Z R \ T | f ( t ) | 2 d t + 1 b Z R \ (Ω b )  √ b | O A f ( u ) |  2 d u  = C e C | T || Ω |  Z R \ T | f ( t ) | 2 d t + Z R \ (Ω b ) | O A f ( u ) | 2 d u  , whic h completes the pro of. Theorem 3.2 is a quantitativ e v ersion of UP , t he constan t C on the righ t-hand side of (4) is hard to determine exactly . By Theorem 3.2, w e kno w that it is not p ossible for a nonzero function f and its OLCT O A f to b o t h b e supported on sets of finite Lebsgue measure. 4 3.2 Hardy’s UP Hardy’s UP w as first in tro duced by G.H. Hardy in 1933 [8]. Its lo calization is measured by fast decrease of a function and its F ourier transform. Hardy’s UP basically say s that it is imp o ssible for a nonzero function and its F ourier transform to dec rease v ery rapidly simu ltaneously . Let us review Hardy’s UP in the F ourier transform domain [8, 1 0] as follow s. Prop osition 3.3 ([10], Theorem 5.2.1) . If a function f ∈ L 2 ( R ) is such that | f ( t ) | = O ( e − π αt 2 ) and |F f ( u ) | = O  e − u 2 / (4 πα )  for some p ositive c on stant α > 0 , then f ( t ) = C e − π αt 2 (9) for some C ∈ C . Based on Prop osition 3.3, w e deriv e the cor r esp onding Hardy’s UP for the OLCT. Theorem 3.4. If a function f ∈ L 2 ( R ) is such that | f ( t ) | = O ( e − π αt 2 ) and | O A f ( u ) | = O  e − u 2 / (4 παb 2 )  for some p ositive c on stant α > 0 , then f ( t ) = C e −  π α + j a 2 b  t 2 − j 1 b tτ (10) for some C ∈ C . Her e τ is a p a r ameter of A . Pr o of. Let g ( t ) = f ( t ) e j a 2 b t 2 + j 1 b tτ . Since | f ( t ) | = O ( e − π αt 2 ), w e hav e | g ( t ) | = | f ( t ) | = O ( e − π αt 2 ). Let the OLCT O A f ( u ) b e rewritten as (5), and G ( u ) b e giv en b y (6). Th us, | G ( u ) | = √ b | O A f ( u ) | = O  e − u 2 / (4 παb 2 )  . Therefore, w e ha v e | G ( ub ) | = O  e − ( ub ) 2 / (4 παb 2 )  = O  e − u 2 / (4 πα )  . 5 Since G ( ub ) is the F our ier transform of g ( t ), it follo ws from Propo sition 3.3 that g ( t ) = C e − π αt 2 for some C ∈ C . Hence, w e obtain f ( t ) = g ( t ) e − j a 2 b t 2 − j 1 b tτ = C e −  π α + j a 2 b  t 2 − j 1 b tτ . This completes the proo f . It follows from T heorem 3.4 that it is imp ossible for a nonzero function f a nd its OLCT O A f b oth t o dec rease v ery rapidly . 3.3 Beurling’s UP Beurling’s UP is a v aria n t of Hardy’s UP . It implies the w eak form of Hardy’s UP immediately . Let us revisit Beurling’s UP in the F ourier transfor m domain [10] a s follo ws. Prop osition 3.5 ([10]) . L et f ∈ L 1 ( R ) and F f ∈ L 1 ( R ) . If Z R 2 | f ( t ) F f ( u ) | e | tu | d t d u < ∞ , (11) then f = 0 . Next, w e fo rm ulate the Beurling’s UP in the OLCT domain. Theorem 3.6. L et f ∈ L 1 ( R ) and O A f ∈ L 1 ( R ) . I f Z R 2 | f ( t ) O A f ( u ) | e | tu/b | d t d u < ∞ , (12) then f = 0 . Pr o of. Let g ( t ) = f ( t ) e j a 2 b t 2 + j 1 b tτ , the OLCT O A f ( u ) b e rewritten in the fo rm of (5), a nd G ( u ) b e defined b y (6). Since f ( t ) ∈ L 1 ( R ) and O A f ( u ) ∈ L 1 ( R ), we get g ( t ) ∈ L 1 ( R ) and G ( u ) ∈ L 1 ( R ). Thus , G ( ub ) ∈ L 1 ( R ). By (12) , w e obtain Z R 2 | g ( t ) G ( ub ) | e | tu | d t d u = 1 b Z R 2 | g ( t ) G ( u ) | e | tu/b | d t d u = 1 √ b Z R 2 | f ( t ) O A f ( u ) | e | tu/b | d t d u < ∞ . Hence, it fo llo ws from Prop o sition 3.5 that g = 0. Therefore, we ha v e f = 0 . F rom Theorem 3.6, we kno w that it is not p ossible for a nonzero f unction f a nd its OLCT O A f to decrease v ery rapidly sim ultaneously . 6 3.4 Logarithmic UP Logarithmic UP w as first in tro duced b y W. Bec kner in 1995 [1]. Its lo calizatio n is measured in terms of en trop y . It is derive d b y using Pitt’s inequalit y . First, let us recall Pitt’s inequality as follows . Lemma 3.7 ([1]) . F or f ∈ S ( R ) and 0 ≤ λ < 1 , we h a ve Z R | u | − λ |F f ( u ) | 2 d u ≤ Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) Z R | t | λ | f ( t ) | 2 d t, (13) wher e S ( R ) denotes the Schwartz cla ss, an d Γ( · ) is the Gamma function. In order to obtain logarithmic UP a sso ciated with the OL CT, w e deriv e the corresp onding g eneralized Pitt’s inequalit y f or the OLCT as fo llows. Theorem 3.8. F or f ∈ S ( R ) and 0 ≤ λ < 1 , we have b λ Z R | u | − λ | O A f ( u ) | 2 d u ≤ Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) Z R | t | λ | f ( t ) | 2 d t. (14) Pr o of. Let g ( t ) = f ( t ) e j a 2 b t 2 + j 1 b tτ , the OLCT O A f ( u ) b e rewritten as (5), and G ( u ) b e denoted as (6). Since G ( ub ) is the F o urier t r ansform of g ( t ), b y applying Lemma 3.7, w e obta in Z R | u | − λ | G ( ub ) | 2 d u ≤ Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) Z R | t | λ | g ( t ) | 2 d t. (15) Let u ′ = ub in (1 5), w e hav e 1 b Z R    u ′ b    − λ | G ( u ′ ) | 2 d u ′ ≤ Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) Z R | t | λ | g ( t ) | 2 d t. (16) Substituting | g ( t ) | = | f ( t ) | and G ( u ) = √ b | O A f ( u ) | in to (16), we g et 1 b Z R    u b    − λ | √ bO A f ( u ) | 2 d u ≤ Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) Z R | t | λ | f ( t ) | 2 d t, (17) i.e., b λ Z R | u | − λ | O A f ( u ) | 2 d u ≤ Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) Z R | t | λ | f ( t ) | 2 d t, whic h completes the pro of. Based on the generalized Pitt’s inequalit y for the OLCT prop osed in Theo- rem 3.8, we inv estigate the logarithmic UP asso ciated with the OLCT. 7 Theorem 3.9. L et f ∈ S ( R ) , and k f k 2 = 1 , then Z R | f ( t ) | 2 ln | t | d t + Z R | O A f ( u ) | 2 ln | u | d u ≥ ln b + Γ ′ (1 / 4) Γ(1 / 4) . (18) Pr o of. Let M λ , Γ 2 ( 1 − λ 4 ) Γ 2 ( 1+ λ 4 ) , and S ( λ ) , b λ Z R | u | − λ | O A f ( u ) | 2 d u − M λ Z R | t | λ | f ( t ) | 2 d t. T aking the deriv ativ e of S ( λ ) ab out the v ariable λ , w e ha v e S ′ ( λ ) = b λ ln b Z R | u | − λ | O A f ( u ) | 2 d u − b λ Z R | u | − λ ln | u || O A f ( u ) | 2 d u − M λ Z R | t | λ ln | t || f ( t ) | 2 d t − ( M λ ) ′ Z R | t | λ | f ( t ) | 2 d t, where ( M λ ) ′ =  − 1 2 Γ  1 − λ 4  Γ ′  1 − λ 4  Γ 2  1 + λ 4  − 1 2 Γ  1 + λ 4  Γ ′  1 + λ 4  × Γ 2  1 − λ 4   / Γ 4  1 + λ 4  . By Theorem 3.8, w e kno w S ( λ ) ≤ 0 for 0 ≤ λ < 1 . Since S (0 ) = 0, w e get S ′ (0+) ≤ 0 , that is, Z R ln | u || O A f ( u ) | 2 d u + Z R ln | t || f ( t ) | 2 d t ≥ ln b Z R | O A f ( u ) | 2 d u + Γ ′ (1 / 4) Γ(1 / 4) Z R | f ( t ) | 2 d t. (19) By the generalized P arsev al f orm ula for the OLCT (2), w e get k O A f k 2 = k f k 2 = 1 . (20) Substituting (2 0 ) in to (19), w e hav e Z R | f ( t ) | 2 ln | t | d t + Z R | O A f ( u ) | 2 ln | u | d u ≥ ln b + Γ ′ (1 / 4) Γ(1 / 4) . This completes the proo f . Applying Jassen’s inequalit y to (18), it is easily to sho w t ha t lo garithmic UP prop osed in Theorem 3.9 implies Heisen b erg’s UP derived in [16]. 8 3.5 En tropic UP En tropic UP is a fundamen tal to ol in infor ma t io n theory , ph ysical quantum, and harmonic a na lysis. Its lo calization is measured in terms of Shannon en tropy . Let ρ b e a probability densit y function on R . The Shannon en trop y of ρ is denoted as E ( ρ ) = − Z R ρ ( t ) ln ρ ( t )d t. (21) In what follows, w e revisit entropic UP asso ciated with the F o urier transform [4]. Prop osition 3.10 ([4]) . L et f ∈ L 2 ( R ) , and k f k 2 = 1 , then E ( | f | 2 ) + E ( |F f | 2 ) ≥ ln( π e ) . (22) Next, w e prop ose the en tropic UP in the OLCT domain. Theorem 3.11. L et f ∈ L 2 ( R ) , and k f k 2 = 1 , then E ( | f | 2 ) + E ( | O A f | 2 ) ≥ ln( π eb ) . (23) Pr o of. Let g ( t ) = f ( t ) e j a 2 b t 2 + j 1 b tτ , the OLCT O A f ( u ) b e rewritten in the form of ( 5 ), and G ( u ) b e define d by (6 ). Since k f k 2 = 1, w e ha v e k g k 2 = k f k 2 = 1 . Hence, b y Prop osition 3.10 , w e get − Z R | g ( t ) | 2 ln | g ( t ) | 2 d t − Z R | G ( ub ) | 2 ln | G ( ub ) | 2 d u ≥ ln( π e ) . (24) Let u ′ = ub in (2 4), w e hav e − Z R | g ( t ) | 2 ln | g ( t ) | 2 d t − 1 b Z R | G ( u ′ ) | 2 ln | G ( u ′ ) | 2 d u ′ ≥ ln( π e ) . (25) Substituting | g ( t ) | = | f ( t ) | , and | G ( u ) | = √ b | O A f ( u ) | in to (25), we obtain − Z R | f ( t ) | 2 ln | f ( t ) | 2 d t − Z R | O A f ( u ) | 2 ln  b | O A f ( u ) | 2  d u ≥ ln( π e ) . Using the fact that k O A f k 2 = k f k 2 = 1 , w e get E ( | f | 2 ) + E ( | O A f | 2 ) = − Z R | f ( t ) | 2 ln | f ( t ) | 2 d t − Z R | O A f ( u ) | 2 ln | O A f ( u ) | 2 d u ≥ ln b Z R | O A f ( u ) | 2 d u + ln( π e ) = ln b + ln( π e ) = ln( π eb ) , whic h completes the pro of. 9 Theorem 3.11 measures the incompatibilit y of measuremen ts in terms of Shannon en trop y . W e nex t demonstrate that Theorem 3 .11 can also imply the Heisen b erg’s UP men tioned in [16]. A t the b eginning, let us in tro duce some notations. Let µ be a probability measure on R . The v ariance of µ is defined as V ( µ ) = inf ξ ∈ R Z R ( t − ξ ) 2 d µ ( t ) . (26) If the integral on the right side of (26) is finite for one v alue of ξ , then it is finite for ev ery ξ . In this case, it can ac hiev e the minimization when ξ is t he mean of µ : M ( µ ) = Z R t d µ ( t ) . F or ρ ∈ L 1 ( R ), if d µ ( t ) = ρ ( t )d t , we sa y ρ is a pro babilit y densit y function, and use the notations M ( ρ ) a nd V ( ρ ) instead of M ( µ ) and V ( µ ), resp ectiv ely . Using the fact k f k 2 = k O A f k 2 , w e know that if f ∈ L 2 ( R ) and k f k 2 = 1, then | f | 2 and | O A f | 2 b oth are probability densit y functions on R . Lemma 3.12. [4, T h e or em 5.1] L et ρ b e a pr ob abi l i ty density function on R with finite varianc e, then E ( ρ ) is wel l defi n e d and E ( ρ ) ≤ 1 2 ln[2 π eV ( ρ )] . (27) Com bining Theorem 3.1 1 and Lemma 3.12 , w e then immediately get the Heisen- b erg’s UP [16] as fo llo ws. Corollary 3.13. L et f ∈ L 2 ( R ) , and k f k 2 = 1 , then V ( | f | 2 ) V ( | O A f | 2 ) ≥ b 2 4 , (28) which implies Z R ( t − ξ ) 2 | f ( t ) | 2 d t Z R ( u − ζ ) 2 | O A f ( u ) | 2 d u ≥ b 2 4 for any f ∈ L 2 ( R ) and any ξ , ζ ∈ R . 4 Conclus ion In this pap er, five differen t forms of UPs asso ciated with the OLCT are prop osed. First, we deriv e Nazaro v’s UP for the OLCT, whic h is a quantitativ e vers ion of UP . It shows that it is not p ossible for a nonzero function f and its OLCT O A f to b oth b e supp orted on sets of finite Leb esgue measure. Second, based on the decreasing 10 prop ert y , w e prop ose tw o UPs in t he OLCT domain: Hardy’s UP and its v arian t- Beurling’s UP . These t w o UPs state that it is imp ossible for a nonzero function f and its OLCT O A f to both decrease v ery rapidly . Finally , w e generalize Pitt’s inequalit y to the OLCT domain, a nd then obta in logarithmic UP for t he OLCT. Moreo v er, with regard to Shannon entrop y , we extend en tropic UP to the OLCT domain. In the future w ork, w e will conside r these UPs for discrete signals. Ac kno wledge men ts The author thanks the referees v ery m uc h for carefully reading the pap er and for elab orate and v aluable suggestions. References [1] W. Bec kner, Pitt’s inequality and the uncertain ty principle. Pro c. Amer. 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