Convolution and correlation theorems for the windowed offset linear canonical transform
aa r X i v : . [ m a t h . G M ] J u l Convolution and correlation theorems for thewindowed offset linear canonical transform*
Wen-Biao, Gao
School of Mathematics and StatisticsBeijing Institute of Technology
Beijing 102488, China
Bing-Zhao, Li
Beijing Key Laboratory on MCAACIBeijing Institute of Technology
Beijing 102488, ChinaEmail address:li [email protected]
Abstract —In this paper, some important properties of thewindowed offset linear canonical transform (WOLCT) such asshift, modulation and orthogonality relation are introduced.Based on these properties we derive the convolution andcorrelation theorems for the WOLCT.
Index Terms —Windowed offset linear canonical transform,Convolution, Correlation
I. I
NTRODUCTION
The offset linear canonical transform (OLCT) [1], [2], [4],[5], [12], [13] is a generalized version of the linear canonicaltransform (LCT) with four parameters ( a, b, c, d ) [6]–[11]. Itis a six parameter ( a, b, c, d, u , w ) class of linear integraltransform. Because of the two extra parameters, time shifting u and frequency modulation w , the OLCT are more generaland flexible than the LCT. It has been widely applied insignal processing and optics [7]–[9], [11].As a mathematical operation, the convolution providessome applications in pure and applied mathematics suchas numerical linear algebra, numerical analysis, and signalprocessing [18], [19]. Correlation is similar to convolutionand it is another useful operation in signal processing, opticsand detection applications [20]–[22]. In some domains suchas the LCT domain [3], [6], [13], Wignar-Ville transformdomain [1], [8], [12] and the OLCT domain [1], [12], theconvolution and correlation operations have been studied.They are the most fundamental and important theorems inthese domains.In [14], [15], [17], the window function based on linearcanonical transform (WLCT) were presented. It is believed tobe a new and important signal processing tool. They obtainedsome main properties such as covariance property, shift,modulation, orthogonality property and inversion formulas.The results of the WLCT have been well applied. Althoughthe windowed offset linear canonical transform (WOLCT)[23] has been proposed, some properties of WOLCT hasnot been studied. The purpose of this paper is to study theWOLCT. Some properties of WOLCT are obtained and its This work is supported by the National Natural Science Foundationof China (No. 61671063), and also by the Foundation for InnovativeResearch Groups of the National Natural Science Foundation of China (No.61421001). convolution and correlation theorems are derived. The resultsare very important application in some fields such as digitalsignal and image processing.In this paper we first review the OLCT. Next, we introducethe definition of the WOLCT, and obtain some importantproperties such as linearity, inversion formula and parity.Finally, we present the convolution and correlation theoremsfor the WOLCT. II. P
RELIMINARY .Let us briefly review some basic properties.The Lebesgue space L ( R ) is defined as the space of allmeasurable functions on R such that k f k L ( R ) = (cid:18)Z R | f ( t ) | d t (cid:19) < ∞ (1)Now we introduce an inner product of the functions f, g defined on L ( R ) is given by h f, g i L ( R ) = Z R f ( t ) g ( t )d t. (2) Definition 1. (OLCT) [1] Let A = ( a, b, c, d, u , w ) be amatrix parameter satisfying a, b, c, d, u , w ∈ R , and ad − bc = 1 . The OLCT of a signal f ( t ) ∈ L ( R ) is defined by O A f ( u ) = O A [ f ( t )]( u ) = R + ∞−∞ f ( t ) K A ( t, u )d t, b = 0 √ de i cd ( u − u ) + iuw × f ( d ( u − u )) , b = 0 (3) where K A ( t, u ) = 1 √ i πb e i a b t − i b t ( u − u ) − i b u ( du − bw ) × e i d b ( u + u ) (4)From definition 1 it can be seen that for case b = 0 theOLCT is simply a time scaled version off multiplied by alinear chirp. Hence, without loss of generality, we assume b = 0 . If u = 0 and w = 0 , the OLCT reduces to the LCT[1], [3], [5], [7].The inverse of an OLCT with parameters A =( a, b, c, d, u , w ) is given by an OLCT with parameters − = ( d, − b, − c, a, bw − du , cu − aw ) . The exactinverse OLCT expression is [16] f ( t ) = O A − ( O A f ( u ))( t ) = e i cd u − iadu w + i ab w × Z + ∞−∞ O A f ( u ) K A − ( u, t )d u, (5)Next, we introduce one of important properties for the OLCT,its generalized Parseval formula [4], [5], as follows: Z R f ( t ) g ( t )d t = Z R O A f ( u )) O A g ( u ))d u (6)III. T HE WINDOWED OFFSET LINEAR CANONICALTRANSFORM
Here we give definition of the WOLCT, then introducesome properties.
Definition 2. (WOLCT) [23] Let φ ∈ L ( R ) \{ } be awindow function. The WOLCT of a signal f ∈ L ( R ) withrespect to φ is defined by V Aφ f ( u, w ) = Z R f ( t ) φ ( t − w ) K A ( t, u )d t (7) where K A ( t, u ) is given by (4). For a fixed w , we have V Aφ f ( u, w ) = O A [ f ( t ) φ ( t − w )]( u ) (8)Using the inverse OLCT to (8), we have f ( t ) φ ( t − w ) = O A − ( V Aφ f ( u, w ))( t )= e i cd u − iadu w + i ab w × Z + ∞−∞ V Aφ f ( u, w ) K A − ( u, t )d u, (9)Some basic properties of the WOLCT are summarized in thefollowing theorem. Property 1 (Linearity) . Let φ ∈ L ( R ) \{ } be a windowfunction and f, g ∈ L ( R ) the WOLCT is a linear operator,namely, (cid:2) V Aφ ( λf + µg ) (cid:3) ( u, w ) = λV Aφ f ( u, w ) + µV Aφ g ( u, w ) (10) for arbitrary constants λ and µ .Proof. This follows directly from the linearity of the productand the integration involved in Definition 2.
Property 2 (Shift) . Let φ ∈ L ( R ) \{ } be a windowfunction and f ∈ L ( R ) . Then we have V Aφ { T t f } ( u, w ) = V Aφ { f } ( u − at , w − t ) × e iat w − i ac t + ict ( u − u ) (11) where T t f ( t ) = f ( t − t ) . Proof. By Definition 2, we have V Aφ { T t f } ( u, w ) = Z R f ( t − t ) φ ( t − w ) K A ( t, u )d t (12)By making the change of variable x = t − t in the aboveexpression, we obtain V Aφ { T t f } ( u, w ) = Z R f ( x ) φ ( x − ( w − t )) × K A ( x + t , u )d x = Z R f ( x ) φ ( x − ( w − t )) Y d x = Z R f ( x ) φ ( x − ( w − t )) × K A ( x, u − at )d x × e iat w − i a b t ( ad − i b t ( u − u )( ad − = V Aφ { f } ( u − at , w − t ) × e iat w − i ac t + ict ( u − u ) (13)where Y = √ i πb e i a b x + i a b t + i ab xt − i b x ( u − u ) × e − i b t ( u − u ) − i b u ( du − bw )+ i d b ( u + u ) .Which completes the proof. Property 3 (Modulation) . Let φ ∈ L ( R ) \{ } be a windowfunction and f ∈ L ( R ) . Then we have V Aφ { M s f } ( u, w ) = V Aφ { f } ( u − bs, w ) × e ibsw − i db s + ids ( u − u ) (14) where M s f ( t ) = f ( t ) e ist .Proof. From Definition 2, it follows that V Aφ { M s f } ( u, w ) = Z R f ( t ) e ist φ ( t − w ) K A ( t, u )d t = Z R f ( t ) φ ( t − w ) Y d t = Z R f ( t ) φ ( t − w ) K A ( t, u − bs )d t × e ibsw − i db s + ids ( u − u ) = V Aφ { f } ( u − bs, w ) × e ibsw − i db s + ids ( u − u ) (15)where Y = √ i πb e i a b t − i b t (( u − bs ) − u ) − i b u ( du − bw ) e i d b ( u + u ) .Which completes the proof. Property 4 (Shift and modulation property) . Let φ ∈ L ( R ) \{ } be a window function and f ∈ L ( R ) . Thenwe have EV Aφ { T t M s f } ( u, w ) = V Aφ { f } ( u − bs − at , w − t ) (16) here E = e i b ( bs + at )( d ( u − u )+ b ( ds + ct ) − bw ) × e i t b (2 u − bs − u ) .Proof. The proof of Property 4 can be achieved by directlycombining Properties 2 and Properties 3.We will prove the inversion formula for the WOLCT bythe connection between the OLCT and the WOLCT. Fromthis theorem, we know that it is possible to restore theoriginal signal f perfectly using the inverse WOLCT asfollows. Property 5 (Inversion formula) . Let φ, ψ ∈ L ( R ) \{ } bewindow function, h ψ, φ i 6 = 0 and f ∈ L ( R ) . Then we get f ( t ) = 1 h ψ, φ i e i cd u − iadu w + i ab w Z R V Aφ f ( u, w ) × K A − ( u, t ) ψ ( t − w )d u d w (17) Proof.
Multiplying both sides of (9) from the right by ψ ( t − w ) and integrating with respect to dw we get Z R f ( t ) φ ( t − w ) ψ ( t − w )d w = e i cd u − iadu w + i ab w × Z R V Aφ f ( u, w ) K A − ( u, t ) × ψ ( t − w )d u d w (18)Using (2), we have f ( t ) = 1 h ψ, φ i e i cd u − iadu w + i ab w Z R V Aφ f ( u, w ) × K A − ( u, t ) ψ ( t − w )dud w (19)which completes the proof.If φ = ψ , then f ( t ) = 1 k φ k e i cd u − iadu w + i ab w Z R V Aφ f ( u, w ) × K A − ( u, t ) φ ( t − w )d u d w (20) Property 6 (Orthogonality relation for WOLCT) . Let φ, ψ ∈ L ( R ) \{ } be window function and f, g ∈ L ( R ) . Then weget h V Aφ f ( u, w ) , V Aψ g ( u, w ) i = h f, g ih ψ, φ i (21) Proof.
From (6) and (8), we get h V Aφ f ( u, w ) , V Aψ g ( u, w ) i = Z R Z R V Aφ f ( u, w ) × V Aψ g ( u, w )d u d w = Z R Z R O A [ f ( t ) φ ( t ′ )]( u ) × O A [ g ( t ) ψ ( t ′ )]( u )d u d w = Z R Z R f ( t ) φ ( t ′ )( g ( t ) ψ ( t ′ ))d t d w = Z R f ( t ) g ( t )( Z R ψ ( t ′ ) φ ( t ′ )d w )d t = h f, g ih ψ, φ i (22)where t ′ = t − w which completes the proof.Based on the above theorem, we may conclude the fol-lowing important consequences. (i) If φ = ψ , then h V Aφ f ( u, w ) , V Aφ g ( u, w ) i = h f, g ik φ k (23)(ii) If f = g , then h V Aφ f ( u, w ) , V Aψ f ( u, w ) i = k f k h ψ, φ i (24)(iii) If f = g and φ = ψ , then h V Aφ f ( u, w ) , V Aφ f ( u, w ) i = k f k k φ k = Z R Z R | V Aφ f ( u, w ) | d u d w (25) Property 7 (Parity) . Let φ ∈ L ( R ) \{ } be a windowfunction and f ∈ L ( R ) . Then we have V AP φ { P f } ( u, w ) = V Aφ { f } (2 u − u, − w ) e i w ( u − u ) (26) where P f ( t ) = f ( − t ) .Proof. From Definition 2, it follows that V AP φ { P f } ( u, w ) = Z R f ( − t ) φ ( − ( t − w )) K A ( t, u )d t = Z R f ( − t ) φ ( − t − ( − w )) K ′ A d t × e i b (2 u − u )( du − bw ) − i d b ((2 u − u ) + u ) × e − i b u ( du − bw )+ i d b ( u + u ) = V Aφ { f } (2 u − u, − w ) e i w ( u − u ) (27)where K ′ A = √ i πb e i a b ( − t ) − i b ( − t )((2 u − u ) − u ) × e − i b (2 u − u )( du − bw )+ i d b ((2 u − u ) + u ) .Which completes the proof. Property 8.
Let φ ∈ L ( R ) \{ } be a window function and f ∈ L ( R ) . Then we have V Aφ { f } ( u, w ) = V Af { φ } ( u − aw, − w ) × e icw ( u − u )+ iaww − i ac w (28) Proof.
From Definition 2, let t − w = t , then V Aφ { f } ( u, w ) = Z R f ( t ) φ ( t − w ) K A ( t, u )d t = Z R φ ( t ) f ( t − ( − w )) 1 √ i πb × e i a b ( t + w ) − i b ( t + w )((2 u − u ) − u ) × e − i b u ( du − bw )+ i d b ( u + u ) d t = Z R φ ( t ) f ( t − ( − w )) 1 √ i πb × e i a b t − i b t (( u − aw ) − u ) × e − i b ( u − aw )( du − bw )+ i d b (( u − aw ) + u ) d t × e i b ( u − aw )( du − bw ) − i d b (( u − aw ) + u ) × e i a b w − i b w ( u − u ) × e − i b u ( du − bw )+ i d b ( u + u ) = V Af { φ } ( u − aw, − w ) × e icw ( u − u )+ iaww − i ac w (29)which completes the proof.IV. C ONVOLUTION AND C ORRELATION T HEOREMS FOR
WOLCTIn this section, we derive convolution and correlationtheorems for WOLCT. efinition 3 (OLCT Convolution) . For any f, g ∈ L ( R ) ,we define the convolution operation ⋆ for OLCT by ( f ⋆ g )( t ) = Z R f ( x ) g ( t − x ) e − i a b x ( t − x ) d x (30)As a consequence of the above definition, we get thefollowing important theorem: Theorem 1 (WOLCT Convolution) . Let φ ∈ L ( R ) \{ } .Then, for every f, g ∈ L ( R ) , we have V Aφ⋆ψ ( f ⋆ g )( u, w ) = B Z R V Aφ f ( m , m ) V Aψ g ( m , w − m ) × e i a b m ( da − m − w ) d m (31) where m = u − a ( w − m ) , m = u − a m , B = √ i πbe i b ( u − a w )( du − bw ) e − i da b w ( a w − u ) − i a b ( u + u ) .Proof. Based on Definition 2 and Definition 3, we get V Aφ⋆ψ ( f ⋆ g )( u, w ) = Z R ( f ⋆ g )( t )( φ ⋆ ψ )( t − w ) K A ( t, u )d t = Z R Z R f ( x ) g ( t − x ) e − i a b x ( t − x ) d x × Z R φ ( r ) ψ ( t − w − r ) e − i a b r ( t − w − r ) d r × K A ( t, u )d t (32)Setting x = x , t = x + x , r = x − m , y = i b u ( du − bw ) , y = i d b ( u + u ) , we get V Aφ⋆ψ ( f ⋆ g )( u, w ) = Z R f ( x ) g ( x ) φ ( x − m ) × ψ ( x − ( w − m )) e − i ab x x × e + i a b x ( w − m )+ i a b mx − i a b m ( w − m ) × √ i πb e i a b ( x + x ) − i b ( x + x )( u − u ) × e − y + y d x d x d m = Z R f ( x ) g ( x ) φ ( x − m ) × ψ ( x − ( w − m )) 1 √ i πb × e i a b x − i b x ( u − a ( w − m ) − u )+ i a b x × e − i b x ( u − am − u ) − i a b m ( w − m ) × e − y + y d x d x d m = B Z R Z R f ( x ) φ ( x − m ) × K A ( x , m )d x × Z R g ( x ) ψ ( x − ( w − m )) × K A ( x , m )d x e i ab m ( da − )( m − w ) d m = B Z R V Aφ f ( m , m ) V Aψ g ( m , w − m ) × e i a b m ( da − m − w ) d m (33)where K A ( x , m ) = √ i πb e i a b x − i b x ( u − a ( w − m ) − u ) × e − i b ( u − a ( w − m ))( du − bw )+ i d b (( u − a ( w − m )) + u ) , K A ( x , m ) = √ i πb e i a b x − i b x ( u − a m − u ) × e − i b ( u − a m )( du − bw )+ i d b (( u − a m ) + u ) . Corollary 1.
If the parameter of WOLCT changes to ( a, b, c, d, u , w ) = ( a, b, c, d, , , then the Theorem 1reduces to convolution theorem as follows: V Aφ⋆ψ ( f ⋆ g )( u, w ) = √ i πbe − i da b w ( a w − u ) − i a b u × Z R G Aφ f ( m , m ) G Aψ g ( m , w − m ) × e i a b m ( da − m − w ) d m (34) where G Aφ f and G Aψ g are the window functions in the LCTdomain of f and g [14], [15], respectively. Corollary 2.
If the parameter of WOLCT changes to ( a, b, c, d, u , w ) = (0 , , − , , , , then the Theorem 1reduces to convolution theorem as follows: V Aφ⋆ψ ( f ⋆ g )( u, w ) = √ i πbe − iuw Z R V Aφ f ( u, m ) × V Aψ g ( u, w − m )d m (35) Definition 4 (OLCT Correlation) . For any f, g ∈ L ( R ) , wedefine the correlation operation ◦ for OLCT by ( f ◦ g )( t ) = Z R f ( x ) g ( x + t ) e i a b x ( x + t ) d x (36)Next, we establish the correlation theorem for theWOLCT. Theorem 2 (WOLCT Correlation) . Let φ ∈ L ( R ) \{ } .Then, for every f, g ∈ L ( R ) , we have V Aφ ◦ ψ ( f ◦ g )( u, w ) = B Z R V AP φ (cid:8)
P f (cid:9) ( m , − m ) × V Aψ g ( m , m ) e − i a b m ( da − m d m (37) where P f ( x ) = f ( − x ) , m = u − a ( w + m ) , m = u + a m , B = √ i πbe i b ( u − a w )( du − bw ) − i da b w ( a w − u ) − i a b ( u + u ) m = w + m Proof.
Based on Definition 3 and Definition 4, we get V Aφ ◦ ψ ( f ◦ g )( u, w ) = Z R ( f ◦ g )( t )( φ ◦ ψ )( t − w ) K A ( t, u )d t = Z R Z R f ( x ) g ( x + t ) e i a b x ( t + x ) d x × Z R φ ( r ) ψ ( r + t − w ) e i a b r ( r + t − w ) d r × K A ( t, u )d t (38)etting x = x , t = x − x , r = x − m , we get V Aφ ◦ ψ ( f ◦ g )( u, w ) = Z R f ( x ) g ( x ) φ ( x − m ) × ψ ( x − m ) × e i ab x x − i a b x m − i a b mx + i a b mm × K A ( x − x , u )d x d x d m = B Z R Z R f ( x ) φ ( x − m ) × K A ( x , m )d x × Z R g ( x ) ψ ( x − m ) K A ( x , m )d x × e − i a b m ( da − m + w ) d m = B Z R V AP φ (cid:8)
P f (cid:9) ( m , − m ) × V Aψ g ( m , m ) e − i a b m ( da − m d m (39)where K A ( x , m ) = √ i πb e i a b ( − x ) − i b ( − x ) × e u − a ( w + m ) − u × e − i b ( u − a ( w + m ))( du − bw )+ i d b (( u − a ( w + m )) + u ) , K A ( x , m ) = √ i πb e i a b x − i b x ( u + a m − u ) × e − i b ( u + a m )( du − bw )+ i d b (( u + a m ) + u ) . Corollary 3.
If the parameter of WOLCT changes to ( a, b, c, d, u , w ) = (0 , , − , , , , then the Theorem 2reduces to correlation theorem as follows: V Aφ ◦ ψ ( f ◦ g )( u, w ) = √ i πbe − iuw Z R V AP φ (cid:8)
P f (cid:9) ( u, − m ) × V Aψ g ( u, w + m )d m (40)V. C ONCLUSION