Convolution, Product and Correlation Theorems for Simplified Fractional Fourier Transform: A Mathematical Investigation

Convolution, Product and Corr elation Theorems for Simplified Fractional Fourier T ransform: A Mathematical I nvestigation Sanjay Kum ar * Department of ECE, Thapar Institute of Engineering and Technology, Patiala, Punjab, India E-mail: (*-Corresponding Author) sanjay.kumar@thapar.edu ___________________________________________________________________________ ___________ Abstract The notion of fractional Fourier tr ansform ( FrFT) has been used and inves tigated for man y years b y various research comm unities, which finds wide spread applications in ma n y diverse f ields of research study . The potential applications in cludes ranging from q uantum physics, harmonic anal ysis, optical information processing, pattern recognit ion t o varied allied areas of si g nal processing. Many si gnificant theorems and properties of the FrFT have been investi gated and applied to many signal processing applications, most important amon g these are convolution, p roduct and correlation th eorems. S till many magnificent resear ch works related to the conventional FrFT lac ks the elegance and si mplicity of the convolution, pro duct and correlation theorems similar to the Euclidean Fourier transform (FT), whic h for convolution theorem states that the FT of the convolution of two fun ctions i s th e product of their respective FT s. The purpose of this paper is to devise th e equivalent elegancy of convolution, product and correlation theorems, as in the case of Euclidean FT . Building on the seminal wor k o f Pei et a l . and the potential of the simplified fractional Fourier transform ( SmFrFT), a d etailed mathem atical investi gation is established to present an elegant definition of convolution, product and correlation theorem s in the SmFrFT domain, along with their associated important properties. I t has been shown that the established theorems a long with their associated properties very nicely generalizes to the classical Euclidean FT. Keywords : Convolution theorem; C orrelation th eorem; Digital si gnal processing; Fractional Fourier transform; Fourier transform; Nonstationary signal processin g ; Produ ct theorem 1. Introduction As it is well-known that the FT is on e of the best and most valuable t ools used in signa l processin g and analysis for centuries. I t finds its diverse application areas in science and engineering [1, 2]. The fractional Fourier trans form ( FrFT) is a generalization o f the Euclidean Fourier transform ( FT), which has found to have several ap plicati ons in the areas of op tics and signal pro cessing [3]. It leads to the g eneralization of the notion of sp ace (or time) and frequ ency do mains, whi ch are central concepts of signal processin g [4- 12]. It is defined via an integral as                                (1) where the transformation kernel         of the FrFT is given by                     !"# $       %    &'( ) * +   &,& )- ./) 0 12   1  34 5 6  7   *     ./)  512  7   %     ./)    51 % 4  2 (2) where ) indicates the rotation angle of the transformed signal in the Fr FT doma in. W hen )  2 5 8 , th e F rFT reduces to the Euc lidean F T, a nd when )  3 , it is the same as t he id entit y operation. It also satisfies the ad ditivity p roperty   9  :     ;   : 9       ;   <:      . The detailed properties were des cribed in [3,4,6]. Thus, it is a ge neralization of th e Euclidean FT and is reg arded as a counter-clockwise rotation of the sign al coordinates around the orig in in the time-frequency (TF) plane with the rotation angle ) [3, 9, 10, 12]. As it is well-known that t he FrFT is able to process n on-stationary or chirp signals better than the Euclidean FT, due t o the reason that a chirp s ignal forms a line or an impulse in the TF plane and thus there exists a fractional transformation order in which such signals are compact [13 ]. Ch irp si gnals are n ot c ompact in th e time or sp atial domain. T hus, one can extract the signal easily in an appropriate (opt imum) fractional F ou rier domain, when it is not possible to separat e the si gnal and noise in the spatial or Fourier frequenc y domain. It is this introduction of e xtra degree of freedom which gives the F rFT a potential improvement ove r the Euclidean FT [3]. In [14], the authors have introduced various simplifi ed forms of the Fr FT, known as the s implified fractional Fourier transform (SmFrFT). The reason for establishing the SmFrFT is that they are simplest f or digital computa tion, optical implementation, graded-index (GRIN) medium implementation, and radar system i mplementation, wi th th e same c apabilities as the conventional Fr FT for designing fractional filters or for fractional correlation. The Sm FrFT poss ess a gr eat potential fo r rep lacing the conventional F rFT in many applications. Pei et al . [14] establishes five types of SmFrFTs that have the sa me capabilit ies as the conventional FrF T for the fractional fil ter desi gn or for fractional co rrelation and simultaneously are sim plest for di gital co mputation, optical implementation, and radar s ystem i mplementation. T hus, Sm FrFTs h ave a great potential to substit ute for the conventional Fr FTs in many real-time appl ications [12, 15-18]. Another dominant advantage that SmFr FT poss ess over t he conventional FrFT is it s less computational co mplexity, as is evident from [14]. In this paper, the main focus is on T ype 1 Sm FrFT, because it is simpler fo r di gital implementation, which was d iscussed in [14]. Man y p roperties of the FrFT are cu rrently w ell-known [19-21], includin g its convolution, product and correlation theorems. However, the convolution, pr oduct and correlation theorems for th e conventional F rFT [1 9-21] do not g eneralize the classical result o f the Euclidean FT [1, 2]. As it well- known that the convolution theorem o f th e Euclidean FT for the functions =    and >    with associated Euclidean FTs, ?  @  and A  @  , respectively is given b y [1, 2] =    B >    C D ?  @  A  @  (3) where C and B denotes the Euclidean FT operation and the convolution operation, respectively. T his convolution theorem can also be written as [1, 2]  =  E  >   * E  E   C D ?  @  A  @  (4) Thus, the convoluti on theorem states t hat the co nvolution of two time-domain functions results in simple multiplication o f their Euclidean FTs in the Euclidean FT domain ― a reall y powerful result. Similar is the case with corr el ation theorem in the Euc lidean FT domain for two complex-valued fu nctions , which is given by [1, 2] = F    G>    C D ? H  *@  A  @  (5) where C , G and  I  H H H denotes the Euclidean FT operatio n, the correlation operation, and the complex conjugate, respectively. This correlation theorem can also be written as [1, 2]  = F  E  >  E %   E   C D ? H  *@  A  @  (6) Thus, the correlation theorem states tha t multiplying th e Euclidean FT of one fu nction with the complex conjugate of the other function gives the Euclidean FT of their correlation. However, the convolution and correlation theorem f or th e conventional F r FT lacks this si mplicity and elegancy in the analytical result [19-21]. Various researchers in th e fracti onal signal processing s ociety [22- 27] have developed different modified versions of these theor ems in the F rFT and l inear c anonical transform (LCT) domains, by utilizing the conventional definition of their transforms. But, still there exists a room for its improvement t o reflect upon the elegancy and the simplicit y o f the theorems. In th is paper, convoluti on, product and correlation theorems are propos ed bas ed the simplified FrFT, which preserves the ele gance and simplicity comparable to th at of t he Euclidean FT, which finds wid espread applications in various all ied research areas of signal processing [15-1 7]. The conventional c onvolution, product an d correlation th eorems and its associated properties are shown to be special cases of the derived result s. The rest of the p aper is or ganized as follows. In Sec tion 2, the p reliminaries of the simplified fractional F ou rier transform (SmFrFT) is presented, f ollowe d by the c onvolution, product an d the correlation theorems associated with SmFrF T i n Sections 3, 4 and 5, respectivel y , along with the analytical proofs of t heir associated p roperties. F i nally, con clusions and th e future s cope of wor k fo r the fractional si gnal processing society is summarized in Section 5. 2 Preliminaries 2.1 The Simplified Fractional Fourier Transform: Definition and Integral Representation The simplified fractional Fourier transform (SmFrFT) of the signal     is represented by [14]:  J         J              J          (7) where   J         K  !"# $*+    %     &'( )- (8) Here,  and   can interchangeably represent time an d fractional fr equency domains. The transform output lies between time and frequenc y domains, ex cept for th e special c ases of )  3 and )  2 5 8 which belongs to FT domain. B ased u pon (7), the S mFrFT can be realized i n a t hree st ep p rocess (s ee Fig. 1), as opposed to the conventional FrFT which is realized by a four step process, as follows: (i) pre-multi plication of t he i nput si g nal b y a line ar chirp wit h the frequenc y modulation ( FM) rate det ermined by the fractional rotation angle ) or the fractional transformation order L , related by )  L2 5 8 with L M N ; (ii) computation of the Euclidean FT ( C ); (iii) post-multiplication by a complex amplitude factor. The inverse SmFrFT is given by [14]:          O  P Q R Q    O  S T R    J         (9) where  J      represents the SmFrFT of the input signal     . 3 Convolution Theorem associated with SmFrFT Theorem 3.1 For any tw o functions = , > M U   N  , let ? J  , A J  denote the S mFrFT of = , > , respectively . The convolution operator of the SmFrFT is defned as  = B >      =  E  >   * E  V WX  E    E   (10) where , V WX  E    O Y  YR    . Then, the SmFrFT of the convolution of tw o complex functions is given by  J  9 = B > ;      K +52 ? J      A J      (11) Proof: From the definition of SmFrFT (7) and the SmFrFT convolution (10), one obtains  J  9 = B > ;       K    9 =    B >   ;   O  R S T < P Q R Q    (12)  J  9 = B > ;       K    Z  =  E  >   * E  V WX  E    E   [   O  RS T < P Q R Q    (13) F or solv ing (13), letting   * E   \  J  9 = B > ;       K     =  E      >  \  O Y]   O    ] ;       K    =  E    O  S T Y< P Q Y Q   E _  K   >  \    O S T ] < P Q ] Q   \ _ K +52 (15) By the definition of SmFrFT, the above expression (15) reduces to  J  9 = B > ;     K +52 ? J    A J     , (16) which proves the theorem in SmFrFT domain. Special case : F or t he Euclidean FT, the rotation angle )  2 5 8 , then the expression (16) reduces to  J  8 9 = B > ;    8   K +52 ? J  8    8  A J  8    8   K +52 ? J  @  A J  @  (17) This me ans that the proposed convolution theor em beha ves similar to the Euclidean FT , i.e., the convolution in the time-domain is equivalent to the multiplication in the simplified fracti onal frequency domain, except for the amplitude factor K +52 and where   8  @ . Some properties associated with the convolution theorem in SmFrFT domain are illustrated below. Property 1 ( Shift Convolution ). Let = , > M U   N  . The SmFrFT of ` a = B > and = B ` a > i s given by  J  9 ` a = B >  ;      K +52 O S T a< P Q a Q   ? J     *  &'( )  A J      (18)  J  9 = B ` a > ;      K +52 O S T a< P Q a Q   ? J      A J     *  &'( )  (19) where, the symbol ` a represents the shift operator of a function by delay  i.e., ` a         *   ,  M N . Proof: The shift convolution operator ` a = B > is given by  ` a = B >       =  E *   >   * E  V WX  E    E   (20) where, V WX  E    O Y  YR    . It implies  ` a = B >       =  E *   >   * E  O Y  YR    E   (21) Now, from the definition of SmFrFT (7), one obtains  J  9 ` a = B >  ;       K    9 ` a =    B >     ;   O  R S T < P Q R Q    (22) Simplifying (22) further, one obtains  J  9 ` a = B >  ;       K    Z  =  E *   >   * E  O Y  YR    E    [   O R S T < P Q R Q    (23) To solve (23), let’s assume   * E   b  J  9 ` a = B >  ;       K    =  E *   O S T Y< P Q Y Q     E _   >  b  O  S T c< P Q c Q     b  J  9 ` a = B >  ;       =  E *   O  S T Y< P Q Y Q     E _  A J      (24) F urther, b y letting  E *    d , and multipl y ing nu merator an d denominator o f () by K +52 , one so lves (24) as  J  9 ` a = B >  ;       K   O S T a< P Q a Q    =  d    O   S T a    e< P Q e Q   d _  A J      _ K +52  J  9 ` a = B >  ;      K +52 O S T a< P Q a Q   ? J     *  &'( )  A J      , which proves the shift convolution property. Similarly, fo r solvin g  J  9 = B ` a >  ;     and utilizing the shift conv olution operator o f function = B ` a > as  =  E  >   * E *   V WX  E    E   , where, V WX  E     O Y  YR    and b ased on the previous steps, one obtains  J  9 = B ` a > ;      K +52 O S T a< P Q a Q   ? J      A J     *  &'( )  , (25) which proves the shift convolution property in SmFrFT domain. Thus, (24) an d (25) indicates that if we apply a linear time dela y to one signal in the t ime domain and convolve it with the another time domain signal, then the Sm FrFT of the convolved signal is identical to the multiplications of the SmF rFTs of the re spective signals, excep t that one of the signal has been shifted in the SmFrFT domain b y an amount dependent on the change in time shift in the time domain, and there is a multiplication with the complex harmonic dependent on the time shift. Special case : F or t he Euclidean FT, the rotation angle )  2 5 8 , then the expression (24) and (25) reduces to  J  8 9 ` a = B >  ;   8   K +52 O  S f Q 8 a ? J  8   8 A J  8   8  i.e, C 9 ` a = B >  ; @   K +52 O ga ? J  @  A J  @  (26)  J  8 9 = B ` a > ;    8   K +52 O  S f Q 8 a ? J  8    8  A J  8    8  i.e, C 9 = B ` a > ; @   K +52 O ga ? J  @  A J  @  (27) This means that the proposed shift convolution pr operty behaves si milar to the Euclidean FT, as is evident from (26) and (27), respectively. Property 2 ( Mod ulation Convolution ). L et = , > M U   N  . Th e SmFrFT of h i = B > and = B h i > is given by  J  Z h i = B >  [     K +52 ? J     * j  A J      (28)  J  Z = B h i >  [     K +52 ? J      A J     * j  (29) where, the s ymbol h i represents the modul ation operator, i.e., the modulat ion b y j of a function     , h i      O iR     , j M N . Proof: The modulation convolution operator h i = B > is given by  h i = B >        O iY =  E  >   * E  V WX  E    E   (30) where, V WX  E    O Y  YR    . It implies  h i = B >        O iY =  E  >   * E  O Y  YR    E   (31) Now, from the definition of SmFrFT (7), one obtains  J  Z h i = B >  [      K    Z h i =    B >     [   O  R S T < P Q R Q    (32) Simplifying (32) further, one obtains  J  Z h i = B >  [      K     =  E    >   * E     O iY <Y  YR     S T R< P Q R Q   E (33) By letting   * E   k , (33) reduces to  J  Z h i = B >  [      K    =  E     O   S T i  Y< P Q Y Q   E _  K    l  k     O S T X< P Q X Q   k  _ K +52 Simplifying further, one obtains  J  Z h i = B >  [     K +52 ? J     * j  A J      , (34) which proves the modulation convolution property in SmFrFT domain. Similarly, for solving  J  Z = B h i > [    and u tilizin g t he modulation convolution o perator of function = B h i > as  =  E  O i  RY  >   * E  V WX  E    E   , wh ere, V WX  E     O Y  YR    and b ased on the previous steps, one obtains  J  Z = B h i >  [     K +52 ? J      A J     * j  , (35) which proves the modulation convolution property in SmFrFT domain. Thus, (34) and (35) indicates that if we apply a linear change i n phase to o ne si gnal in the time d omain and con volve it with the another time domain signal, then the SmFrFT of the convolved sig nal is identical to the mu ltiplications of the SmFrFTs of the respective signals, except that one of the signal has been shifte d in the SmFrFT domain by an amount dependent on the change in phase in the time domain. Special case : In case of FT, (34) and (35) reduces to (for )  2 5 8 )  J  8 Z h i = B >  [   8   K +52 ? J  8    8 * j  A J  8    8   K +52 ? J  @ * j  A J  @  , i.e., C Z h i = B > [  @   K +52 ? J  @ * j  A J  @  (36)  J  8 Z = B h i >  [   8   K +52 ? J  8    8  A J  8    8 * j  C Z = B h i > [  @   K +52 ? J  @  A J  @ * j  (37 ) This means that the p roposed modulation convolution property behaves s imilar to the Eu clidean FT, as is evident from (36) and (37), respectively. Property 3 ( Time- Frequency shift Convolution ). Le t = , > M U   N  . T he SmFrFT of h i ` a = B > and = B h i ` a > is given by  J  Z h i ` a = B > [     K +52O   S T i  a < P Q a Q   ? J     * j *  &'( )  A J      (38)  J  Z = B h i ` a > [     K +52 O   S T i  a< P Q a Q   ? J      A J     * j *  &'( )  (39) where, the s ymbol ` a and h i represents t he shift op erator of a fun ction by del ay  and the modulation operator of a function b y j , i. e., for the function     , ` a         *   ,  M N and h i      O iR     , j M N . Proof: The time-frequency shift convolution operator is given b y  h i ` a = B >       O iY   =  E *   >   * E  V WX  E    E (40) where, V WX  E    O Y  YR    . It implies  h i ` a = B >       O iY   =  E *   >   * E  O Y  YR    E (41) The SmFrFT of (41) is obtained as  J  Z h i ` a = B > [      K    Z h i ` a =    B >     [   O  R S T < P Q R Q    (42) Simplifying (42) further, one obtains  J  Z h i ` a = B > [      K     =  E *       >   * E  O iY <Y  YR     R S T < P Q R Q   E  (43) By letting   * E   m , (43) is simplified as  J  Z h i ` a = B > [      K    =  E *      O iY  S T Y< P Q Y Q   E _  K    >  m    O  S T m< P Q m Q   m _ K +52 (44) Let  E *    n , (44) reduces to  J  Z h i ` a = B > [      K    =  n     O   S T ia    o < P Q o Q   n  _ K +52  _ O S T a<ia < P Q a Q   _  A J      (45) Thus,  J  Z h i ` a = B > [     K +52 O   S T i  a< P Q a Q   ? J     * j *  &'( )  A J      , (46) which proves the time-frequency shift convolution propert y in SmFrFT domain. Similarly, for solving  J  Z = B h i ` a > [    and utilizing the shift and modulation convolution operator of function = B h i ` a > as  =  E  O i  RY  >   * E *   V WX  E    E   , where, V WX  E     O Y  YR    and based on the previous steps, one obtains  J  Z = B h i ` a > [     K +52 O   S T i  a< P Q a Q   ? J      A J     * j *  &'( )  , (47) which proves the time-frequency shift convolution propert y in SmFrFT domain. Special case : In case of FT, (46) and (47) reduces to (for )  2 5 8 )  J  8 Z h i ` a = B > [   8   K +52 O   S f Q 8 i  a ? J  8    8 * j  A J  8    8  , i.e., C Z h i ` a = B >  [  @   K +52 O   gi  a ? J  @ * j  A J  @  (48)  J  8 Z = B h i ` a > [   8   K +52 O   S f Q 8 i  a ? J  8    8  A J  8    8 * j  i.e, C Z = B h i ` a > [  @   K +52 O   gi  a ? J  @  A J  @ * j  (49) This means that the proposed time-frequency shift convolution p roperty b ehaves similar to the E uclidean FT, as is evident from (48) and (49), respectively. 4. Product Theorem associated with SmFrFT Theorem 4.1 For any two functions = , > M U   N  , l et ? J  , A J  denote the SmFrFT of = , > , respectively . We define the product operation associated with SmFrFT as d     =    >    V c     =    >    O P Q R Q   (50) where , V c     O P Q R Q   . Then, the SmFrFT of the product of two fu nctions is given by  J  9 d   ;   J  Z =    >    V c    [      ? J      p  A J      (51) Proof: The function d is in U   N  and thus it s SmFr FT i s given b y (7). To comp ute  J  9 d   ; , express the function d    in terms of its SmFrFT, as follows:  J  9 d   ;  q J        K    Z =    >    V c    [   O  RS T < P Q R Q       K    9 =   ;   >    V c    O  RS T < P Q R Q    (52)  J  9 d   ;  q J        K   O  RS T < P Q R Q      r     O  P Q R Q    O  s T R ? J   t      t  u >    V c     Simplifying further and using V c     O P Q R Q   , one obtains  J  9 d   ;  q J        K   O   S T s T  R < P Q R Q     >      _      ? J   t      t  (53) Solving (53), one gets the following analytical expression  J  9 d   ;  q J            ? J   t      A J     * t    t       ? J      p  A J      , (54) which proves the product theorem in SmFrFT domain. Thus, the product theorem in SmFrFT domain states that the SmF rFT of the product of tw o functions is obtained by conventional convolution between the SmFrFTs of the two functions. 5. Correlation Theorem associated with SmFrFT Theorem 5.1 For an y two complex -valued functions = , > M U   N  , let ? J  , A J  denote the S mFrFT of = , > , respectively . We define the correlation operator of the SmFrFT as  =G>      = F  E  >   % E  V Wv  E    E   (55) where , V Wv  E     O Y  Y ;  K +52 ? J  H H H H  *   A J      (56) Proof: From the definition of SmFrFT (7), one obtains  J  9 =G> ;   K    9 =    G>   ;   O  RS T < P Q R Q     J  9 =G> ;   K    Z  = F  E  >   % E  V Wv  E    E   [   O  RS T < P Q R Q    (57) F or solv ing (57), letting  % E  w  J  9 =G> ;   K     = F  E      >  w  O Yx   O    xY  S T < P Q  xY  Q   E\ (58) Rearranging and noting that the complex-valued function =     =     % + =     , one obtains  J  9 =G> ;   K    = F  E     O  S T Y< P Q Y Q   E _ K +52A J      (59) where, A J        >  w     O  S T x< P Q x Q   (60) Solving (60) further, one obtains  J  9 =G> ;   K     =   E  * + =   E     O  S T Y< P Q Y Q   E _ K +52A J       J  9 =G> ;   K    =   E     O   S T  Y< P Q Y Q   E _ K +52 A J     * +  K    =   E     O   S T  Y< P Q Y Q   E _ K +52 A J      (61) Solving (61), one gets  J  9 =G> ;  K +52 ?  J   *   A J      * + K +52 ?  J   *   A J      or,  J  9 =G> ;  K +52  y ?  J   *   * +?  J   *  z A J       K +52 ? J  H H H H  *   A J       (62) which proves the theorem. Special case : F or t he Euclidean FT, the rotation angle )  2 5 8 , then the expression (62) reduces to  J  8 9 =G> ;  K +52 ? J  8 H H H H H H  *  8  A J  8    8  C 9 =G> ;  K +52 ? J {  *@  A J  @  (63) This mea ns that the proposed correlation theorem behaves si milar t o the Euclidean FT, except for the amplitude factor. Some properties associated with the correlation theorem in SmFrFT domain are illustrated below. Property 1 ( Shift Convolution ). Let = , > M U   N  . The SmFrFT of ` a =G> and =G` a > is given by  J  9 ` a =G>  ;      K +52 O  S T a < P Q a Q   ? H J   *  *  &'( )  A J      (64)  J  9 =G` a >  ;      K +52 O  S T a < P Q a Q   ? H J   *   A J     *  &'( )  (65) where, the symbol ` a represents the shift operator of a function by delay  i.e., ` a         *   ,  M N . Proof: The shift correlation operator ` a =G> is given by  ` a =G >       = F  E *   >   % E  V Wv  E    E   (66) where, V Wv  E     O Y  Y       = F  E *   >   % E  O Y  Y  ;       K    9 ` a =    G>     ;   O  R S T < P Q R Q    (68) Simplifying (68) further, one obtains  J  9 ` a =G>  ;       K    Z  = F  E *   >   % E  O Y  Y  ;       K    = F  E *   O  S T Y< P Q Y Q     E _   >  b  O  S T c< P Q c Q     b (70)  J  9 ` a =G>  ;       = F  E *   O  S T Y< P Q Y Q     E  _  A J      (71) F urther, by lettin g  E *    d , and multipl ying n umerator an d denominator o f (71) by K +52 , one solves (71) as  J  9 ` a =G>  ;       K   O  S T a< P Q a Q    = F  d    O   S T  ;      K +52 O  S T a < P Q a Q   A J      |  K   =   d    O   S T  ;      K +52 O  S T a < P Q a Q   A J     y ? J   *  *  &'( )  * + ? J   *  *  &'( ) z  J  9 ` a =G>  ;      K +52 O  S T a < P Q a Q   A J      _ ? H J   *  *  &'( )  (74) Thus,  J  9 ` a =G>  ;      K +52 O  S T a < P Q a Q   ? H J   *  *  &'( )  A J      , (75) which proves the shift correlation property of the correlation theorem in SmFrFT domain. Similarly, for solving  J  9 =G` a > ;     and utilizin g the shift corr elation o perator of fu nction =G` a > as  = F  E  >   % E *   V Wv  E    E   , wh ere, V Wv  E     O Y  Y  ;      K +52 O  S T a < P Q a Q   ? H J   *   A J     *  &'( )  , (76) which proves the shift correlation property of the correlation theorem in SmFrFT domain. Special case : F or t he FT, the rotation angle )  2 5 8 , then the expression () and () reduces to  J  8 9 ` a =G >  ;    8   K +52 O  S f Q 8 a ? H J  8  *  8  A J  8    8  i.e, C 9 ` a =G> ; @   K +52 O ga ? H J  *@  A J  @  (77)  J  8 9 =G` a >  ;    8   K +52 O  S f Q 8 a ? H J  8  *  8  A J  8    8  i.e, C 9 =G` a > ; @   K +52 O ga ? H J  *@  A J  @  (78) This means t hat the prop osed shift correlation pr operty b ehaves si milar t o t he E uclidean FT, as is evid ent from (77) and (78), except for the amplitude factor. Property 2 ( Modulation Convolution ). Let = , > M U   N  . The SmFrFT of h i =G> and =Gh i > is given by  J  Z h i =G> [     K +52 ? H J   *  * j  A J      (79)  J  Z =Gh i > [     K +52 ? H J   *   A J     * j  (80) where, the s ymbol h i represents the modul ation operator, i.e., the modulat ion b y j of a function     , h i      O iR     , j M N . Proof: The modulation convolution operator h i =G > is g iven b y  h i =G>        O iY = F  E  >   % E  V Wv  E    E   (81) where, V Wv  E     O Y  Y        O iY = F  E  >   % E  O Y  Y [      K    Z h i =    G>     [   O  R S T < P Q R Q    (83) Simplifying (83) further, one obtains  J  Z h i =G> [      K     = F  E    >   % E     O iY <Y  Y [      K    = F  E     O   S T  [     K +52 ? H J   *  * j  A J      , (85) which proves the modulation correlation property of the correlation theorem in Sm FrFT domain. Similarly, for solvin g  J  Z =Gh i > [    and utilizing the modul ation correlation operator of function =Gh i > as  = F  E  O i  R   % E  V Wv  E    E   , where, V Wv  E    O Y  Y [     K +52 ? H J   *   A J     * j  , (86) which proves the modulation correlation property of the correlation theorem in Sm FrFT domain. Special case : In case of FT, (85) and (86) reduces to (for )  2 5 8 )  J  8 Z h i =G>  [   8   K +52 ? H J  8  *  8 * j  A J  8    8   K +52 ? H J  *@ * j  A J  @  , i.e., C Z h i =G>  [  @   K +52 ? H J  *@ * j  A J  @  (87)  J  8 Z =Gh i > [   8   K +52 ? H J  8  *  8  A J  8    8 * j  C Z =Gh i > [  @   K +52 ? H J  *@  A J  @ * j  (88) This means that the pro posed modulation correl ation property behaves s imilar to the Euclidean FT, as is evident from (87) and (88), except for the amplitude factor. Property 3 ( Time-Frequency shift Convolution ). Let = , > M U   N  . The SmFrFT of h i ` a =G> and =Gh i ` a > is given by  J  Z h i ` a =G> [     K +52 O   S T i  a< P Q a Q   ? H J     * j *  &'( )  A J      (89)  J  Z =Gh i ` a > [     K +52 O   S T i  a< P Q a Q   ? H J   *   A J     * j *  &'( )  (90) where, the s ymbol ` a and h i represents t he shift op erator of a fun ction by del ay  and the modulation operator of a function by j , i.e., for the function     , ` a         *   ,  M N and h i      O iR     , j M N . Proof: The time-frequency shift correlation operator is given b y  h i ` a =G>       O iY   = F  E *   >   % E  V Wv  E    E (91) where, V Wv  E     O Y  Y       O iY   = F  E *   >   % E  O Y  Y [      K    Z h i ` a =    G>    [   O  R S T < P Q R Q    (93) Simplifying (93) further, one obtains  J  Z h i ` a =G> [      K     = F  E *       >   % E  O iY<Y  Y [      K    = F  E *      O iY < S T Y< P Q Y Q   E _  K    >  m    O S T m< P Q m Q   m _ K +52 (95) Let  E *    n , (95) reduces to  J  Z h i ` a =G> [      K    = F  n     O   S T i a    o < P Q o Q   n  _ K +52  _  O  S T a< ia< P Q a Q   _ A J      (96) Solving (96) further, by letting the complex-valued function =  n   =   n  % +=   n  , one obtains Thus,  J  Z h i ` a =G> [     K +52 O   S T i  a< P Q a Q   ? H J     * j *  &'( )  A J      , (97) which proves the time-frequency shift correlation property of the correlation theorem in SmFrFT domain. Similarly, for so lving  J  Z =Gh i ` a > [    and utilizing the shift and modulation correlat ion operator of function =Gh i ` a > as  = F  E  O i  R   % E *   V Wv  E    E   , where, V Wv  E    O Y  Y [     K +52 O   S T i  a< P Q a Q   ? H J   *   A J     * j *  &'( )  , (98) which proves the time-frequency shift correlation property of the correlation theorem in SmFrFT domain. Special case : In case of FT, (97) and (98) reduces to (for )  2 5 8 )  J  8 Zh i ` a =G> [  8   K +52 O   S f Q 8 i  a ? H J  8   8 * j  A J  8   8  , i.e., C Z h i ` a =G>  [  @   K +52 O   gi  a ? H J  @ * j  A J  @  (99)  J  8 Z =Gh i ` a > [   8   K +52 O   S f Q 8 i  a ? H J  8  *  8  A J  8    8 * j  i.e, C Z =Gh i ` a > [  @   K +52 O   gi  a ? H J  *@  A J  @ * j  (100) This means that the proposed time-frequency shift corre lation property beh aves similar to the Euc lidean FT, as is evident from (99) and (100), except for the amplitude factor. 6. Conclusions and Future Scope of Work In this paper, an elegant analytical expressions of convolution, product, and correlation of t wo functions is in troduced in the simplified fraction al F ourier transform domain. Th e newl y established convolution, pr oduct, and correlation the orems along with their associated properties ge neralizes very nicely the classical result of E uclidean Fourier transform. The p roposed app roach o ffers the following advanta ges. F ir st it is the first attempt to establish different si gnal processin g theorems, which ver y nicely generalizes to the classical Four ier transform, which was not e arlier attain able with t he conventional definitions of fractional F ou rier transform. Second , it has the adde d advantage of less computational comple xit y [1 4] as compared to the conventional frac tio nal Fourier tran sfor m definitions, which will be benef icial for the reconfigurable implementation for different signal processing applications. As a future work, th e sampling of the bandlimited signals in the SmFrFT domain w ill be investigated based on the de rived convolution t heorem, wit h the establishme nt of the different formula e o f uni form sampling and low pass reconstruction. F urther, the approach of simplified fractional Fourier t ransform could be elabo rated in the l inear canonical t ransform and other angular transforms, which would prove to b e an important mathematical tool for radar and sonar si gnal processin g applications, a long wit h the reconfi gurable implementation for viable signal processing applications. Acknowledgements The w ork was supported by Scie nce and Engineering Research Bo ard (SER B) (No. SB/S3/EECE/0149/2016), Department of Science and Technology (DST), Government of India, India. References [1] A. Antoniou, Digita l Signal Processing: Signal, Systems, and Filters (McGraw-Hill, 2005) [2] R.N. Bracewell, The Fourier Transforms and i ts Applications (McGraw-Hill, 1986) [3] H. Ozaktas, Z. Z alevsky, M.A. Kuta y, The Fractional Fourier Transform with appli cations in Optics and Signal processing (Wiley, New York, 2001) [4] V. Namias, T he fractional orde r Fourier transform and i ts application to quantum me chanics, IMA Journal of Applied Mathematics. 25 (3) 241-265 (198 0) [5] A.C. McBride, F.H. Kerr, On Namias’ fractional F ou rier transforms. IMA Journal of Applied Mathematics. 39 (2) 159-175 (1987) [6] L.B. Almeida, Th e fr actional Fourier transfor m and ti me-frequency representation. I EEE Transactions on Signal Processing. 42 (11) 3084-3093 (1994) [7] S.C Pei, J.J. Ding, Relations b etween fractional o perations and time-frequen cy distributions, and their applications. 49 (8) 1638-1655 (2001) [8] S. Kumar, K. Singh, R. Saxena, Analysis o f Dirichlet and Generalized “Hamming” window functions in the fractional Fo urier transform do mains. Signal Processing. 91 (3) 600-606 (2011) [9] K. Sin g h, S. Kumar, R. Saxena, Caputo-based fractional derivative in fractional Fourier transform domain. IEEE Jo urnal on Emergin g and Selected T opics in Cir cuits and S y stems. 3 (3), 330-337 (2013) [10] S. Kumar, K. Singh, R. Saxena, Closed-form an alytical expression of fractional order differentiation in fractional Fourier transfor m do main. Ci rcuits Systems and Signal Processin g. 32 (4) 1875-1889 (2013) [11] S. Kumar, R. Saxena, K. Singh. Fractional Fourier transform and Fractional-order calculus-based image edge detection. Circuits S y stems and Si gnal Processing. 36 (4) 1493-1513 (2017) [12] S. Kumar. Analysis and desig n of non-recursive digital diff erentiators i n fractional domain for si g nal processing applications. Ph.D. dissertation, Thapar University, Patiala, India, 2014 [13] C. C lemente, J.J. Soraghan, Range Doppler and chirp scaling proc essing of s ynthetic aperture r adar data using the fractional Fourier transform. IET Signal Processin g. 6 (5) 503-510 (2012) [14] S.C. Pei, J.J. Ding, Si mplified fractional Fourier transforms. Journal of the Optical Societ y of America A. 17 (12) 2355-2367 (2000) [15] X. Zhang, L. Liu, J. Ca i, Y. Y ang, A pre-estimation algorithm for LFM si gnal based on Simplified fractional Fourier transform. Journal o f Information and Computational Science. 8 (4) 645-652 (2011) [16] W.Q. W an g, Moving ta rget indication via three-antenna SAR wi th simplified fractional Fourier transform. EURASIP Journal on Advances in Si gnal Processing. 117 1-10 (2011) [17] H.X. B u, X. Bai, R. Tao, Compressed sensing SAR imaging bas ed on sparse representation in fractional Fourier domain. Science China Information Sciences. 55 (8) 1789-1800 (2012) [18] B. Deng, J. Luan, S. Cui, Analysis o f parameter estimation usin g the sampling-type al gorithm if discrete fractional Fourier transform. Defence Technolo gy. 10 (4) 321-327 ( 2014) [19] L.B. Almeida, Product and convolution theorems for t he fractional Fo urier transform . IEEE Signal Processing Letters. 4 (1) 15-17 (1997) [20] A. I. Za y ed, A convol ution and product theorem fo r t he fractional Fourier transfor m. IE EE Signal Processing Letters. 5 (4) 101-103 (1998) [21] R. Torres, P.P. Fine t, Y. Torres, Fractional convolution, fractional correlation and their translation invariance properties. Signal Processing. 90 (6) 1976-1984 (2010) [22] D.Y. Wei, Q. W. Ran, Y.M. Li, J. Ma, L.Y. Tan, A convolution and product theorem for the linear canonical transform. IEEE Signal Processing Letters. 16 (10) 853-856 (2009) [23] Q. W. Ran, H. Z hao, L.Y. Tan, J. Ma, Sampling of bandlimited si gnals in fractional F ourier transform domain. Circuits, S y stems, and Signal Processing. 29 (3) 459-467 (2010) [24] D. We i, Q. Ran, Y. Li, A convolution and correlation theorem for the Linear Canonical t ransform and its application. Circuits, Systems, and Signal Processing. 31 (1) 301-312 (2012) [25] H. Zha o, R. W ang, D. S ong, Recovery of b andlimited sig nals in linear c anonical transform domain from noisy samples, Circuits, Systems, and Signal Processin g. 33 (6) 1997- 2008 (2014) [26] H. Huo, W . Sun, Sa mpling theorems an d error e stimates for random signals in the linear can onical transform domains. Signal Processing. 111 31-38 (2015) [27] X. Zhi, D. Wei, W. Zhang, A generalized convolution theore m for the special affine F ourier transform and its convolution to filteri ng. Optik . 127 (5) 2613-2616 (2016 ) Fig. 1. Simplif ied fractional Fourier transform (SmFrFT) block diagram .