Reduced Order Fractional Fourier Transform A New Variant to Fractional Signal Processing Definition and Properties

Reduced Order Fracti onal Fourier Transform ― A New Variant to Fractional Signal Processing: Definition and Properties Sanjay Kumar Department of ECE, Thapar Institute of Engineering and Technology, Patiala, Punjab, India E-mail: sanjay.kumar@thapar.edu ___________________________________________________________________________ ___________ Abstract In this paper, a new vari ant to fractional s ignal processin g is proposed known as the Reduced Order Fractional Fourier Transform (ROFrFT) . Various properties satisfied by its transformation kernel is derived. The properties associated with th e proposed ROFrFT like shift, modulation, ti me-frequency shift property are also derived and it is s hown mathematically th at when the rotation angle of ROFrFT approaches    , th e proposed ROFr FT reduc es t o the conventional Fourier transform (FT). Also , the ROFrFTs o f various kinds of signals is also derived and it is sho wn that the obtained anal ytical expressions of different ROFrFTs are a reduced form of the conventional fractional F ou rier transform (FrFT). I t is also shown t hat proposed definition of F rFT is e asier to be h and led analytically. F inall y, the convoluti on theorem associated with the proposed ROFrFT i s derived with its various properties like shift convolution, modulation con volution, and time-frequency shift m o dulation properties. It has been shown that with this proposed new de finition of FrFT, the convolution theorem h as been reduced t o multiplication i n the fractional frequenc y domain ― an exciting result, same as that of Euclidean Fourier transform. Keywords Fractional F o urier transform; Fractional Fourier frequency domain f iltering; Fourier transform; Time-frequency signal analysis and processing. 1. Introduction The characteristics of si gnals o f interest determines t he s election of signal processin g techniques to be applied. Various freq uency-based and time-fr equency (TF) b ased te chniques are applied and mentioned in popular literature [1] [2] [3]. In th e con ventional si gnal processing methods, the two natural existi ng variables n amely, time    and frequenc y    are used e xclusi vely a n d ind ependently of each other, whereas in the TF si gnal proces sing methods, these variables are used concurrently [3]. This means that rathe r t han viewing a on e-dimensional signal and some trans form, the TF signal anal ysis and processin g investigates a two-dimensional si gnal, which is obtained from the signal through some TF transform. The conventional frequen cy-based signal processing methods rely on Fourier transfor mation, whereas the TF signal anal ysis and processing involves man y T F trans forms, na mely short-ti me Fourier transform (STFT), wavelet transf orm ( WT), bilinear T F distribution/ Wigner distribution ( WD), Gabor- W i gner distribution (GWD) , fractional F o urier transform (F r FT), linear can o nical transform (L C T), Stockwell transform (ST) and to name a few [3]. The ubiquit y o f the ori ginal F ourie r transfor m ( FT) has pro liferated m any i mportant si gnal and image analysis applications. It is known as a powerful tool which reveals the overall spectral contents by assuming that th e given signal is stationary i n nature. However, this assumption o f st ationarity is an ideal assumption and not p articularly usef ul i n practical applications [4], sinc e most practical si gnals of interest ar e no n- stationary in nature, which have d ynamic frequency content. Thus, it i s necessar y to resort t o the T F representation (TFR) that represents the energy density of a signal simultaneousl y in the TF plane. Thus, the T F tr ansforms are of great interest among signal processing researchers in varied allied real- time app lications due to their natur al decomposition of a sign al in to a function which is localized in both time and frequency. An excell ent survey on t he state of a rt of th e TF si gnal anal y sis and p rocessing and its applications was given in [1] [3] [5] [6]. In the rece n t ye ars, along with the various TF Rs of signals, various researchers are using the concept of fractional Fourier transform ( FrFT), which extends the capabilities of the conventional FT. The FrFT is a generalization of the conv entional FT, which depends on a n additional pa rameter and can be inter preted as a rotation in the TF plane [7] [8]. The FrFT has been extensivel y ap plied for si gnificant applications in di gital signal [9] [10] [11] [12], i mage pro cessing [13] [1 4] [15] [16] as well as in qu antum mechanics and opti cal information processing [17] [18] [19]. It is well-known that the conventional FT can be u sed to adequately represent signals in terms of the sinusoids in the f requency domain, and app lying the FrFT allows t o analyze a given signal in the TF do main and thus, it ca n be used for sig nals having non-sinusoidal basis such as linear chirp si gnals [20]. Th e advantages o f using FrFT-based methods includ es high resistance to noise as w ell as low computational complexity [19] [ 20], its practica l applic ations are expected to g row s ignificantl y i n y ears to come, given that the FrFT offers many adv antages over the conventional Fourier anal ysis, includi ng signal/image r estoration and noise removal [34]. In this paper, a mathematical investigation is done to deter mine a n ew an d n ovel definition of the F rFT ― t he Reduced Order Fractional Fourier Transform (ROFrFT), which could prove b eneficial to fractional signal processi ng applications and it has not been invol ved in any l iterature s o far. Its definitio n with various p r operties associated with it and its transformat ion kernel have been d e rived anal y ti cally. Next, thorough analytical d erivations is don e to dete rmine various anal ytical expressions of different kinds of signals w hich d epends on the rotation angle. Finally, the convolution theo rem associated it is deriv ed with detailed derivations of shift, modulation and time-frequenc y shift properties . The rest o f the paper is or ganized as follows. In Section 2, the prel iminaries of the reduced ord er fractional Fourier tr ansform (RO FrFT) is presented, with various p roperties of its transformation kernel in Section 3 and various prop erties associated with ROFrFT in Section 4 a re investigated along with their analytical proo fs. In Se ction 5, a mathematical invest igation is pr esented for deriving RO FrFTs of di fferent kinds of signals. The new definition o f convolution theorem of RO FrFT which will prove to b e much beneficial for fractional si g nal processing so ciety is p resented in Section 6 and finally conclusions and the future scope of the proposed work for the fractional signal processing society is summarized in Section 7. 2. Reduced Order Fractional Fourier Transform: Definition and Integral Representation As it well known that the fractional Fourier t ransform ( FrFT) is a generalization of th e conventional F ourier tr ansform ( FT), which was introduced fr om analytical aspect b y Namias and appeared i n man y applications in optics field at that time. It w as around 1 990s that its potential capabilit y has a p peared i n si gna l processing societ y. How ever, with the advancem ent o f s cience and technolo gy domains, th e FT has faced many shortcomings in its nature, where its orthogonal basis are sinusoidal in nature. F or processin g practical si gnals, su ch as biomedical si gnals, seis mic data, radar, sona r, audio, video signals, etc. the Fourier t ransformation is n ot capable to extract the us eful information from them and t hat too in non-stationary envir onment, so t here’s a need of sig n al pro cessing t ransformation which co u ld process non-stationary si gnals in non-stationar y environment faithfully. Man y rich literature are available where various signal p rocessing researchers appl y FrFT to solve dail y real life pro blems, whether they app ly it for biomedical, seismic, wireless, radar, sonar, audio, video processing etc. applications. The added ad v anta ge of using FrFT t ool lies in its nature ― the Fr FT is a lin ear t ransformation whe re its orthogonal basis is a l inear f requency modulation (chirp) signal, so it has the capability to pro cess non- stationary si gnals efficientl y as compared to the conventio nal FT. Also, i t is more fl exible and sui table for processing chirp-li ke si gnals due to the additional degree of freedom as co mpared to FT [8], [9], [11 ], [17], [18], [19]. Additionally, t hrough the usage of FT, the whol e s pectrum of the sig n al o f intere st is obtained and it cannot obtain th e local TF characteristics o f the signal, which is essential for non-stationary signal processing. So many no vel si gnal processin g t ransforms h ave be en disco vered by researchers, w hich has their own advantages and limitations [21]. However, it’s been a decade th at th e res earch on FrFT h as proli ferated in varied research areas of communication, radar, s onar, i mage, etc. The FrFT c an be int erpreted as a rotation of the signal in the TF plane by an angle [22]. The time-domain and frequ ency-domain representat ions are the two special cases of the FrFT. A fundamental advanta ge of us ing FrFT for si gnal filtering is t hat the si gnal o f i nterest can b e represented in an y domain w ithin the range of the rotation an gle, rather th an bein g limited to o nly eit her in the time-domain o r in the frequency-domain [22] . Mathematically, the FrFT implements th e o rder parameter  , which acts on th e conventional FT operator. To say, th e   order FrFT represents the   power of th conventional FT operator [8]. The FrFT is a linear o perator with a fracti onal Fourier ord er parameter  or transform para meter  , which co rresponds to the  th fractional power of FT o perator,  , and can be viewed as a counterclockwis e rotation by a fractional order  in the TF plane. The FrFT of a signal     is defined as [9], [17], [18], [19]                                    (1) where        , the t ransformation kernel,            ! "#  $%  & ! ' ( ) *+ , ) ) - "#  ! . ,  "/"  with the transform angle    0 [17], 1 $  23 and   4 denotes the FrFT operator. T he  th Fr F T domain makes an angle   0 with the time do main in the time-frequency plane [12 ]. When      , it conver ges to the classical FT and when    , it will be an identit y operation. It is seen that the FrFT transforms a signal into an intermediate domain betw een time and f requency when  5 6      , where 6 is an integer [23]. Also, i t can be infer red from (1) that the k ernel          is co mposed o f chirp b asis functions with a s weep rate o f 789  . So the FrFT can be interpreted as signal decomposition i nto chirp functions. Thus, due to th e property of concentration of LFM ener gy and it s multi-domain nature, the FrFT has achiev ed a wider acceptance in the DSP community [24]. Based upon (1), it can be seen that th e conventional FrFT can be realized in a four step p rocess as mentioned in [9] [25]. As it was discussed in [26], for the f ractional correlation o peration with the opti cal implementation in the conventional F rFT domain, the ch irp term : ;< = ! $   $ 789 > in the conventional F r FT definition (1) can be removed and one gets a re duced form of the FrFT d efinition that could p rove to be beneficial for the optical information processing society. So motivated b y the res earch potential of [26], t he proposed work emphasize t he use of n ew definition of FrFT which could rev eal ma gnificent p ropert y and could be applied in various scienc e and en g ineerin g applications. Based on a comparison with the conventional FrFT, it is found out that the proposed definition of FrFT ― RO FrFT, is easier t o be handled an alytically, and also all the analytical ex pressions obtained for different properties, convolution theorem and its properties are much easier to obtain anal ytically. The Reduced Order Fractional Fourier transform (ROFrFT) of the signal     is represented by  ?  @    A   ?            B  ?          (2) where B  ?        C 3 2 1 789  :;< D ! $  $ 789  2 1    7E7  F (3) Here,  and   can in terchangeably r epresent time an d frequency domains. The transform o utput lies between ti me and frequenc y do mains, except for the special cases of    and      . Based upon (2), the ROFrFT can be realized in a three step process, which is illustrated in F i g. 1 as follows: (i) pre-multiplication of th e input si gnal b y a linear chi rp with the frequen cy modulation (FM) rate determined by the transform order; (ii) computation of the scaled FT (  ) with a scaling factor of 7E7  ; (iii) post-multiplication by a co mplex amplitude factor. F ig. 1. RO FrFT block diagram. 3. Properties satisfied by the ROFrFT Transfor mation Kernel The transformation kernel of ROFrFT is given by B  ?        C 3 2 1 789  :;< G 1   $ 789  2 1      7E7  H (4) The transformation kernel of ROFrFT satisfies the following properties: (i) Diagonal symmetry : B  ?        B  ?       (5) Proof : The right-hand s ide of (4) becomes B  ?        C 3 2 1 789  :;< D ! $  $ 789  2 1    7E7  F . This shows the equivalence of (4). Hence proved. (ii) Complex conjugate : B   ?        B  ?       I I I I I I I I I I I I I (6) Proof : The left-hand side of (6) becomes B  ?        C 3 J 1 789  :;< G 2 1   $ 789  J 1    7E7  H and the right-hand side of (6) becomes B  ?       I I I I I I I I I I I I I  C 3 J 1 789  :;< G 2 1   $ 789  J 1    7E7  H Hence proved. (iii) Point symmetry : B  ?  2       B  ?     2   (7) Proof: The left-hand side of (7) becomes B  ?  2      C 3 2 1 789  :;< G 1   $ 789  J 1    7E7  H and the right-hand side of (7) becomes B  ?     2   C 3 2 1 789  :;< G 1   $ 789  J 1    7E7  H Hence proved. 4. Properties associated wi th ROF rFT This section derives the important analytical p rop erties o f the RO FrFT in detail. It is seen that t he results are the generalizations of the basic properties of the FT. Theorem 4.1 ( Shift Property ): Let     K L  M  . The ROFrFT of a shift by N K M is given by  ?  O P Q  R       ?  O    S N  R       & ! $  Q ) TUV  S ! . , Q TWT    ?     S  N 78E   (8) Proof: Replacing    b y    S N  in the inte gral re presentation (2) of ROFr FT, one gets  ?  O    S N R       C 3 2 1 789      S N    & X )  ) TUV !  . , TWT   (9) By assuming   S N    Y , i.e.,     Y Z N  one solves (9) as  ?  O    S N R       C 3 2 1 789       Y     & X )   [ ZQ  ) TUV !    [ ZQ  . , TWT    Y (10) Solving further, one gets  ?  O    S N R       & X ) Q ) TUV S! . , Q TWT  C 3 2 1 789       Y     & X )   [  ) TUV  ! . ,   [ SQ TUW   TWT    Y (11) Thus, the ROFrFT of a shift by N K M of a function    S N  is given by  ?  O P Q  R       ?  O    S N R       & X ) Q ) TUV S! . , Q TWT   ?     S N 78E    (12) If  equals    in (12), the ex pression (12 ) reduces t o & S!\Q   ]  , same as that of FT, where ] is th e frequency variable in the FT domain. Theorem 4.2 ( Modulation Property ): Let     K L  M  . The ROFrFT of a modulation by ^ K M is given by  ?  _ ` a  b       ?  _ & S !a     b       ?     Z ^ Ecd   (13) Proof: Fo r the frequenc y shiftin g, replace     by :;<  S1^      in the i ntegral representati on (2) of ROFrFT, one gets  ?  _ & S!a     b      C 3 2 1 789    :;<  S1^         & X )  ) TUV !  . , TWT   (14) Simplifying further,  ?  _ & S!a     b      C 3 2 1 789        & X )  ) TUV ! . ,  TWT  S!a    (15) Solving further, one gets  ?  _ & S!a     b     C 3 2 1 789       & X )  ) TUV !  . , Za Wef    TWT    (16) Thus, the ROFrFT of a modulation by ^ K M of a fun ction & S!a     is given by  ?  _ ` a  b      ?  _ & S!a     b      ?     Z ^ Ecd   (17) If  equals    in (17), the expression (17) reduces to   ] 2 ^  , same as that of FT . Theorem 4.3 ( Time-frequency shift property ): Let     K L  M  . Then we get  ?  _ P Q ` a  b      ?  _    2 N  & !a b      & X ) Q ) TUV ! . , Q TWT  g!aQ  ?     2 N 78E  2 ^ Ecd   (18) Proof: The time-frequen cy shi ft means replacing     by    2 N  & !a in the integral representation ( 2) of ROFrFT.  ?  _    2 N  & !a b     C 3 2 1 789       2 N  & !a    & X )  ) TUV !  . , TWT   (19) Simplifying leads to  ?  _    2 N  & !a b     C 3 2 1 789       2 N  & X )  ) TUV !    . , a Wef   TWT     (20) Now assuming,   2 N    Y , i.e.,     Y J N  one solves (20) as  ?  _    2 N  & !a b     C 3 2 1 789       Y     & X )   [ gQ  ) TUV  !   [ gQ    . , a Wef   TWT    Y (21)  ?  _    2 N  & !a b     & X ) Q ) TUV ! . , Q TWT  g!aQ  C 3 2 1 789       Y     & X )   [  ) TUV  ! [  . , TWT Q TUV  a    Y (22) Simplifying further,  ?  _    2 N  & !a b     & X ) Q ) TUV  !. , Q TWT  g!aQ  C 3 2 1 789    & X )   [  ) TUV !  [  . , Q TUW  a Wef       Y  Thus, the ROFrFT of the derivative of a function    2 N  & !a is given by  ?  _ P Q ` a  b      ?  _    2 N  & !a b      & X ) Q ) TUV ! . , Q TWT g!aQ  ?     2 N 78E  2 ^ Ecd   (23) If  equals    in (23), the expression (23) reduces to & !  \a  Q   ] 2 ^  , same as that of FT. Theorem 4.4 ( Multiplication by hij  kl  ): Let     K L  M  . Then we get  ?  O     78E  m  R      3  n  ?     2 m Ecd   J  ?     J m Ecd   o (24) Proof: From (13) above, one have     & !m  ?pqpr s t t u   ?     2 m Ecd   (25)     & !m  ?pqpr s t t u   ?     J m Ecd   (26) Therefore,     78E  m   ?pqpr s t t u  $ n  ?     2 m Ecd   J  ?     J m Ecd  o (27) F or  equals    in (27), the expression (27) reduces to  $ @   ] 2 m  J   ] J m A , same as that of FT. Theorem 4.5 ( Inversion of Time axis ): Let     K L  M  . Then we get  ?  O   2   R        ?   2    (28) Proof: For deducing th e RO FrF T of t he time inversion axis, replace     by   2  in the integral representation (2) of ROFrFT, one gets  ?  O   2 R      C 3 2 1 789     2     & X )  ) TUV !  . , TWT   (29) Let 2   YY in (29), one obtains  ?  O   2 R      2 C 3 2 1 789       YY     & X )   [[  ) TUV g!    [[  . , TWT    YY (30) or  ?  @   2 A   C 3 2 1 789       YY     & X )   [[  ) TUV !   . ,   [[  TWT    YY Thus,  ?  O   2 R        ?   2   (31) If  equals    in (31), the expression (31) reduces to   2]  , same as that of FT. Theorem 4.6 ( Multiplication property ): Let     K L  M  . Then we get  ?  O     R       1 Ecd      ?         (32) Proof:  ?       C 3 2 1 789       & X )  ) TUV !  . , TWT     (33) Differentiation of () with respect to   gives vw x ,  . ,  v . ,   2  1 7E7   y C 3 2 1 789    @    A   & X )  ) TUV !  . , TWT  z  (34) The factor in the curl y b races of (34) represents the ROFrFT of t he fun ction    . Rearranging (34), o ne obtains  ?  O    R       1 Ecd    v w x ,  . ,  v . , (35) The form (35) results in much si mpler an alytical expression than the conventio nal expressio n [17] [19 ]. If the rotation an gle  equals    , the expression (35) r educes to 1 vw  \  v\ , where ] is the Fourier trans form frequency variable. This shows that the proposed RO FrF T reduces to the conventional FT property. Theorem 4.7 ( Differentiation property ): Let     K L  M  . Then we get  ?  y      z      ' 1   7E7  J 78E      -  ?      (36) Proof: To obtain the R OFrFT of the derivati ve of a f unction, replace    by      in the integral representation (2) and int egrate b y parts, assu ming that    {  , when  { S| [17] (If    is di fferentiable for all  and vanishes as  { S| , then t he RO FrFT of the derivative of t he function can be relat ed to the transform of the undifferentiated function through the use of integration by parts.)  ?  } v~  v •      C 3 2 1 789    v~  v    & X )  ) TUV !   . , TWT   (37) After simplifying, one obtains  ?  } v~  v •      2  1 789    ?  @     A J  1  7E7    ?       (38)  ?  } v~  v •       2  1 789     1 Ecd    vw x ,  . ,  v. , J   1  7E7    ?       (39) Thus, the ROFrFT of the derivative of a function      is given by  ?  } v~  v •      €1  7E7  J 78E  v v . , •  ?      (40) So if  equals    in (40), the expression (40) reduces to 1]    ]  , same as that of FT. Theorem 4.8 ( Mixed product property ): Let     K L  M  . Then we get  ?  y       z      1  Ecd     ‚  $  ?         $ J  1 7E7         ?         J  1 7E7      ?      ƒ (41) Proof: To obtain the ROF r FT of the function  v~  v , use the formulation s of Theorems 4.6 and 4.7, which is recapitulated as  ?  O    R       1 Ecd    v w x ,  . ,  v . , and  ?  } v~  v •      €1   7E7  J 78E  v v. , •  ?      , one easil y obtains  ?  } v~  v •      1 Ecd   v v. , y 1  7E7  J 78E  v v. , z  2  vw x ,  . ,  v. , 2  ?      J 1 Ecd  78E  v ) w x ,  . ,  v . , ) (42) Thus, the ROFrFT of the function  v~  v is given by  ?  } v~  v •      ! $ Ecd    G v ) w x ,  . ,  v. , ) J 1 7E7      vw x ,  . ,  v . , J 1 7E 7     ?     H (43) These properties are listed below in Table 1: Table 1 Properties of ROFrFT Operation Signal „  l  ROFrFT, … † ‡  ˆ ‡  Time shift    S N  & ! $  Q ) TUV  S ! . , Q TWT    ?     S  N 78E   Modulation & S !a      ?     Z ^ Ecd   Time-frequency shift    2 N   & !a & ! $ Q ) TUV   ! . , Q TWT  g !aQ   ?     2 N 78E  2 ^ Ecd   Modulation: Multiplication by hij  kl      78E  m  Multiplication by j‰Š  kl      Ecd  m  3  n  ?     2 m Ecd   J  ?     J m Ecd   o 3 1 n  ?     2 m Ecd   2  ?     J m Ecd  o Multiplication        1 Ecd      ?         Inversion of time axis   2    ?   2    Conjugation  ‹     I ?       Even function     J   2     ?      J  ?   2     Odd function     2   2     ?      2  ?   2     Differentiation       ' 1   7E7  J 78E      -  ?      Mixed product         1  Ecd     ‚  $  ?         $ J  1 7E7         ?         J  1 7E7      ?      ƒ 5. ROFrFT of simple signals The following section outlines the detailed analytical derivations of RO FrFTs of some simple signals. Delta Signal : F or      Œ  2 N  , the integral representation of ROFr FT (2) is given b y  ?        C 3 2 1 789   Œ   2 N  & X )  ) TUV !  . , TWT     Using the delta sifting property [2], one obtains  ?  O Œ  2 N  R       ?        C 3 2 1 789  :;< D ! $ N $ 789  2 1 N   7E7 F (44) Unit Step Signal : F or      3 , its integral representation of R OFrFT (2) is given by  ?        C 3 2 1 789   & X )  ) TUV  ! . , TWT     (45) Solving (45) and by knowing of the fact that  & !  •~ ) gŽ~     = % • >    & ! • • & ! ‘ ) •’ Thus, after simplify (45), one obtains  ?  O 3 R      ?      C   3 J 1 9“d   :; )   ! . , TWT !a      (47) Solving (47) and by knowing of the fact that  & • ~ ) Ž~    = % • >    &  ‘ ) •’ , one obtains  ?  _ & !a b     C   3 J 1 9“ d   :;< D2 ! $ ^ $ 9“d  2 1   $ 7E7    J 1^   E:7  F (48) Exponential Signal multiplied by l : To determine the RO FrFT of       & !a , m ake use of Theo rem 4 .6 (Multiplication property), which is recapitulated below as  ?  O  ”   R       1 Ecd    v• x ,  . ,  v . , , wh ere the function ”     & !a and its ROFrFT – ?      i s given b y  ?  _ & !a b     C   3 J 1 9“ d   :;< D2 ! $ ^ $ 9“d  2 1   $ 7E7    J 1^   E:7  F  ?  _ & !a b     1 Ecd    v v . , } C   3 J 1 9“d   :;< D2 ! $ ^ $ 9“ d  2 1  $ 7E7    J 1^   E:7  F• (49) Solving (49) and after mathematical manipulations, one obtains  ?  _  & !a b     C   3 J 1 9“ d     7E7  2 ^  9“d   :;< D2 ! $ ^ $ 9“d  2 1   $ 7E7    J 1^   E:7 F Linear Chirp Signal : F or      & !—  ) 0$ , its integral representation of RO FrFT (2) is giv en by  ?        C 3 2 1 789    & !—  ) 0$    & X )  ) TUV !  . , TWT    ?        C 3 2 1 789    & =! ˜ ) g X ) TUV  > )   ! . , TWT       (50) Solving (50) and by knowing of the fact that  & • ~ ) Ž~    = % • >    &  ‘ ) •’ , one obtains  ?  _ & !—  ) 0$ b       $%  !gTUV    —gTUV   :;< D 21 TWT  $   g— V™f     $ F (51) Exponential Signal : F or      &  ) 0$ , its integral representation of ROFr FT (2) is giv en by  ?        C 3 2 1 789    &  ) 0$    & X )  ) TUV !  . , TWT    ?        C 3 2 1 789    & = š )  X ) TUV > )   !. , TWT       (52) Solving (52) and by knowing of the fact that  & •~ ) Ž~    = % • >    & ‘ ) •’ , one obtains  ?  _ &  ) 0$ b      ›  :;< D2 $  3 J 1 789     $ F (53) Exponential Signal : F or      & — ) 0$ , its integral representation of ROFr FT (2) is given b y  ?        C 3 2 1 789    & — ) 0$    & X )  ) TUV !  . , TWT   After simplification, one obtains  ?        C 3 2 1 789    & = ˜ )  X ) TUV > )   ! . , TWT       (54) Solving (54) and by knowing of the fact that  & •~ ) Ž~    = % • >    & ‘ ) •’ , one obtains  ?  _& — ) 0$ b      $%  ! TUV    —! TUV   :;< } 2 $   J 1 789   = TWT )  — ) gTUV )  >   $ • (55) Exponential Signal with Time Delay : F or      &  ˜ )  Q  ) , its integral representation of ROFr FT (2) is giv en by  ?        C 3 2 1 789    &  ˜ )   Q  )    & X )  ) TUV !  . , TWT    ?        C 3 2 1 789    & = ˜ )  X ) TUV > )   !. , TWT  —Q   ˜ ) Q )    (56) Solving (56) and by knowing of the fact that  & •~ ) Ž~ œ    = % • >    & ‘ ) •’ œ , one obtains  ?  }&  ˜ )  Q  ) •       $%  ! TUV    —! TUV   :;< } 2  —g! TUV    — ) gTU V )    D TWT )  $   $ J 1N 7E7   2 ! $  N $ 789  F• (57) Exponential nature Signal : F or       &  ) 0$ , its integral representation of ROFr FT (2) is given b y  ?        C 3 2 1 789     &  ) 0$    & X )  ) TUV !  . , TWT    ?        C 3 2 1 789     & = š )  X ) TUV > )   !. , TWT       (58) Solving (58) and by knowing of the fact that   & •~ ) Ž~   2  = % • >    Ž • & ‘ ) •’ , one obtains  ?  _  &  ) 0$ b     21 ›   3 J 1 789   Ecd  :;< } 2 $  3 J 1 789     $ •    (59) Exponential nature Signal with Tim e Delay : F or        2 N  &   Q  ) 0$ , its integral representation of RO FrFT (2) is giv en by  ?        C 3 2 1 789      2 N  &   Q  ) 0$    & X )  ) TUV !  . , TWT   (60) Simplifying (60) further, one obtains  ?       &  • ) )  ž 2 N ž $  (61) where the integrals ž and ž $ are defined as ž  C 3 2 1 789     :;< D2 = $ 2 ! $ 789  >  $ 2  1  7E7  2 N  F    (62) and ž $  C 3 2 1 789    :;< D 2 = $ 2 ! $ 789 >  $ 2  1  7E7  2 N  F    (63) Solving (62) and (63), an d by knowing of the fact that   & •~ ) Ž~   2  = % • >    Ž • & ‘ ) •’ and  & •~ ) Ž~    = % • >    & ‘ ) •’ , one obtains ž  2 ›   = ! . , TWT Q ! TUV  > :;< G $  ! . , TWT Q  )  ! TU V   H (64) and ž $  ›   :;< G $  !. , TWT Q  )  ! TUV   H (65) F rom (64) and (65), (61) becomes  ?       &  • ) ) y 2 ›   = ! . , TWT Q ! TUV  > :;< G $  !. , TWT Q  )  ! TUV   H 2 N  ›   :;< G $  ! . , TWT Q  )  ! TUV   Hz (66) Simplifying further, (66) becomes  ?  _   2 N  &   Q  ) 0$ b     ›  = Q  g! TUV   !$. , TWT   ! TUV   > :;< =2 Q ) $ > :;< G $  !. , TWT  Q  )  ! TUV   H (67) The ROFrFTs of different signals are listed below in Table 2: Table 2 ROFrFTs of some signals Signal „  l  ROFrFT with angle ‡ , … † ‡  ˆ ‡  Ÿ  l 2  C 3 2 1 789  :;< D ! $ N $ 789  2 1  N    7E7  F ¡ C    3 J 1 9“d   :;< n 1 7E7        $ o ¢ S £¤l C    3 J 1 9“d   :;< D 2 ! $ ^ $ 9“d  2 1   $ 7E7     S 1^   E:7  F l¢ S £¤l C    3 J 1 9“d     7E7  Z ^  9“d   :;< G 2 1  ^ $ 9“d  2 1   $ 7E7     S 1^   E:7  H ¢ S £¥ l ¦ 0 ¦  $ %  ! g TUV    S — g TUV   :;< D 2 1 TWT  $    S — V™f      $ F ¢  l ¦ 0 ¦ ›   :;< D 2 $  3 J 1 789       $ F ¢  ¥l ¦ 0 ¦  $ %   ! TUV    —  ! TUV   :;< } 2 $   J 1 789    = TWT )  — ) g TUV )  >    $ • ¢  ¥ ¦  l   ¦  $ %   ! TUV    —  ! TUV   :;< } 2  — g ! TUV    — ) g TUV )    D TWT )  $    $ J 1N 7E7     2 ! $  N $ 789  F • l  ¢  l ¦ 0 ¦ 2 1  ›     3 J 1  789   Ecd     :;< y 2 3   3 J 1 789       $ z   l 2   ¢   l   ¦ 0 ¦ ›   ' N  3 J 1 789   2 1    7E7   3 2 1 789   -  :;< ' 2 N $  - :;< § 3   1   7E7  2 N  $  3 2 1 789   ¨ 6. Convolution Theorem associated with ROFrFT Theorem 6.1 For any two functions  , © K L  M  , let ª ?  , « ?  denote the ROFrFT of  , © , respectively . The convolution operator of the ROFrFT is defned as   ¬ ? ©        N  ©   2 N  - "®  N   N   (68) where , - "®  N    & !Q  Q  TUV  . Then, the ROFrFT of the convolution of two complex functions is given by  ?  O  ¬ ? © R      C ! TUV   ª ?      « ?      (69) Proof: From the definition of ROFrFT 2() and the ROFrFT convolution (68), one obtains  ¯  O  ¬ ? © R      C 3 2 1 789    O     ¬ ? ©   R   & X )  ) TUV  !  . , TWT   (70)  ¯  O  ¬ ? © R      C 3 2 1 789    _    N  ©   2 N  - "®  N    N   b   & X )  ) TUV  !  . , TWT   (71) F or solvin g (71), letting   2 N   °  ¯  O  ¬ ? © R      C 3 2 1 789       N      ©  °  & !Q± TUV  & X )  ± gQ  ) TUV  !   ± gQ  . , TWT  N °  (72) Rearranging and multiplying numerator and denominator of (72) b y C 3 2 1 789  , one obtains  ¯  O  ¬ ? © R      C 3 2 1 789      N    & X ) Q ) TUV  ! Q . , TWT  N ² C 3 2 1 789   ©  °    & X ) ± ) TUV ! ± . , TWT  ° ² C ! TUV  (73) By the definition of ROFrFT, the above expression (73) reduces to  ¯  O  ¬ ? © R      C ! TUV   ª ?      « ?      , (74) which proves the theorem in ROFrFT domain. It is to be noted from (74) that thi s new proposed definition o f FrF T makes th e convolution theo rem more excitin g ― the con volution theorem in R OFrFT do main gets reduced in the same form as that of conventional Euclidean Fourier trans formation. That is to s ay, the convoluti on of two signals of in terest in fractional domain gets reduced to simple multiplication o f their fractional frequenc y transforms. Th is is an added advantage of the pro posed new definiti on of F r FT, which will find widespread applications i n fractional fi lter desi gn in various si gnal p rocessin g s ystems li ke communication, rad ar, sonar, seismic, biomedical processing and to name a few. Special case : F or the Eucli dean FT, the rotation angle      , then the expression (74) reduces to  ¯ % $  O  ¬ ? © R   % $     ª ? % $    % $   « ? % $    % $     ª ?  ]  « ?  ]  (75) This me ans that the proposed convolution theorem behaves similar to the Euclidean F T, i.e., the c onvolution in the t ime-domain is eq uivalent to the multiplication in the reduced fr actional frequency domain, where  % $   ] . Some properties associated with the convolution theorem in ROFrFT domain are illustrated below: Property 1 ( Shift convolution ). Let  , © K L  M  . The ROFrFT of P v  ¬ ? © and  ¬ ? P v © is given by  ?  O P v  ¬ ? © R      C ! TUV   & ! . , v TWT g X ) v ) TUV  ª ?     2  789   « ?      (76)  ?  O  ¬ ? P v © R      C ! TUV   & ! . , v TWT g X ) v ) TUV  ª ?      « ?     2  789   (77) where, the symbol P v represents the shift operator of a function by delay  i.e., P v         2   ,  K M . Proof: The shift convolution operator P v  ¬ ? © is given by  P v  ¬ ? ©        N 2   ©   2 N  - "®  N    N   (78) where, - "®  N    & !Q  Q  TUV  . It implies  P v  ¬ ? ©        N 2   ©   2 N  & !Q  Q  TUV  N   (79) Now, from the definition of ROFrFT (2), one obtains  ?  O P v  ¬ ? © R     C 3 2 1 789    O P v     ¬ ? ©     R   & X )  ) TUV  !  . , TWT   (80) Simplifying (80) further, one obtains  ?  O P v  ¬ ? © R      C 3 2 1 789    _    N 2   ©   2 N  & !Q  Q  TUV  N    b   & X )  ) TUV !   . , TWT   Solve above expression by lettin g   2 N   ” , one obtains  ?  O P v  ¬ ? © R      C 3 2 1 789      N 2   & ! . , Q TWT g X ) Q ) TUV    N  ²   ©  ”  & ! . , ³ TWT  g X ) ³ ) TUV    ”  ?  O P v  ¬ ? © R         N 2   & ! . , Q TWT  g X ) Q ) TUV    N  ²  « ?      (81) F urther, b y l etting  N 2    ´ , and multiplying n umerator and d enominator of (81) b y C 3 2 1 789  , one solves to get  ?  O P v  ¬ ? © R      C 3 2 1 789   & ! . , v TWT g X ) v ) TUV     ´    & !  . , v TUV   TWT µg X ) µ ) TUV  ´ ² « ?      ² C ! TUV   ?  O P v  ¬ ? © R      C ! TUV   & ! . , v TWT g X ) v ) TUV  ª ?     2  789   « ?      , which proves the shift convolution property (76). Similarly, for solvin g  ?  O  ¬ ? P v © R     and utili zing the shift convo lution op erator of function  ¬ ? P v © as    N  ©   2 N 2   - "®  N    N   , wher e, - "®  N     & !Q  Q  TUV  and b as ed on the previous steps, on e obtains  ?  O  ¬ ? P v © R      C ! TUV   & ! . , v TWT g X ) v ) TUV  ª ?      « ?     2  789   , (82) which proves the shift convolution property (77) in RO FrF T domain. Thus, (76) and (77) indicates that if we ap pl y a linear ti me d ela y to one si gnal in the time dom ain a nd fractional convolve it wit h the another time do main si gnal, then the RO FrFT of th e convolved si gna l i s identical to the multiplications of th e ROFrFTs of t he respective signals, except that one of the signal h as been shifted in the ROF r FT domain by 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. Spec ial case : F or the Eucli dean FT, the rotation angle      , then the expression (76) and (77) reduces to  ? % $  O P v  ¬ ? © R   % $    & ! . • )  v ª ? % $    % $   « ? % $    % $   i.e,  O P v  ¬ ? © R ]    & !\v ª ?  ]  « ?  ]  (83)  ? % $  O  ¬ ? P v © R   % $    & ! . • )  v ª ? % $    % $   « ? % $    % $   i.e,  O  ¬ ? P v © R ]    & !\v ª ?  ]  « ?  ]  (84) This means that the prop osed shift convoluti on pr operty behaves si milar to the Euclide an FT, as is evid ent from (83) and (84), respectively. Property 2 ( M odulation convolution ). Let  , © K L  M  . The ROFrFT of ` a  ¬ ? © an d  ¬ ? ` a © is given by  ?  _ ` a  ¬ ? © b      C ! TUV  ª ?     2 ^ Ecd   « ?      (85)  ?  _  ¬ ? ` a © b      C ! TUV   ª ?      « ?     2 ^ Ecd   (86) where, the s ymbol ` a represents the modulation o perator, i.e., the modulation b y ^ o f a function     , ` a      & !a     , ^ K M . Proof: The modulation convolution operator ` a  ¬ ? © is given by  ` a  ¬ ? ©       & !aQ   N  ©   2 N  - "®  N    N   (87) where, - "®  N    & !Q  Q  TUV  . It implies  ` a  ¬ ? ©       & !aQ   N  ©   2 N  & !Q  Q  TUV  N   (88) Now, from the definition of ROFrFT (2), one obtains  ?  _ ` a  ¬ ? © b     C 3 2 1 789    _ ` a     ¬ ? ©     b   & !  . , TWT g X )  ) TUV   (89) Simplifying (89) further, one obtains  ?  _ ` a  ¬ ? © b     C 3 2 1 789       N    ©   2 N     & !aQ g!Q  Q  TUV ! . ,  TWT g X )  ) TUV  N  (90) By lett ing   2 N   ¶ , and m ultiplyin g numerator and denominator of (90) by C 3 2 1 789  , (90) reduces to  ?  _ ` a  ¬ ? © b     C 3 2 1 789      N     & !  . , a Wef   Q TWT  g X ) Q ) TUV  N ² C 3 2 1 789    ·  ¶     & ! . , ® TWT g X ) ® ) TUV  ¶  ² C ! TUV  Simplifying further, one obtains  ?  _ ` a  ¬ ? © b     C ! TUV  ª ?     2 ^ Ecd   « ?      , (91) which proves the modulation convolution property in ROFrFT domain. Similarly, for solvin g  ?  _  ¬ ? ` a © b    and utilizing the modulation convolution o perator of function  ¬ ? ` a © as    N  & !a  Q  ©   2 N  - "®  N    N   , where, - "®  N     & !Q  Q  TUV  and based on the previous steps, one obtains  ?  _  ¬ ? ` a © b     C ! TUV   ª ?      « ?     2 ^ Ecd   , (92) which proves the modulation convolution property in ROFrFT domain. Thus, (85) and (86) indicates that if we apply a linear ch ange in phase to one signal in the time domain and fractional convolv e i t with the another ti me domain s ignal, t hen the RO FrFT of the convolv ed signal is identical to the multiplications of th e ROFrFTs of t he respective signals, except that one of the signal h as been shifted in the RO FrFT domain b y an amount dependent o n the change in phase in the time do main, wit h an amplitude factor dependent on the rotation angle  . Special case : In case of FT, (85) and (86) reduces to (for      )  ? % $  _ ` a  ¬ ? © b  % $    ª ? % $    % $  2 ^  « ? % $    % $    ª ?  ] 2 ^  « ?  ]  , i.e.,  _ ` a  ¬ ? © b  ]   ª ?  ] 2 ^  « ?  ]  (93)  ? % $  _  ¬ ? ` a © b  % $    ª ? % $    % $   « ? % $    % $  2 ^   _  ¬ ? ` a © b  ]   ª ?  ]  « ?  ] 2 ^  (94) This means that the p roposed modulation convolution p roperty behav es similar to th e Euclidean FT, as is evident from (93) and (94), respectively. Property 3 ( Time-Frequency shift convolution ). Let  , © K L  M  . The ROFrFT of ` a P v  ¬ ? © a nd  ¬ ? ` a P v ©  is given by  ?  _ ` a P v  ¬ ? ©  b     C ! TUV   & !  . , a Wef   v TWT g X ) v ) TUV  ª ?     2 ^ Ecd  2  789   « ?      (95)  ?  _  ¬ ? ` a P v © b     C ! TUV   & !  . , a Wef   v TWT g X ) v ) TUV  ª ?      « ?     2 ^ Ecd  2  789   (96) where, the symbol P v and ` a represe n ts the s hift operator of a functio n b y del ay  and th e modulation operator of a function by ^ , i.e., for th e function     , P v         2   ,  K M and ` a      & !a     , ^ K M . Proof: The time-frequency shift convolution operator is given by  ` a P v  ¬ ? ©       & !aQ     N 2   ©   2 N  - "®  N    N (97) where, - "®  N    & !Q  Q  TUV  . It implies  ` a P v  ¬ ? ©       & !aQ     N 2   ©   2 N  & !Q  Q  TUV  N (98) The ROFrFT of (97) is obtained as  ?  _ ` a P v  ¬ ? ©  b     C 3 2 1 789    _ ` a P v     ¬ ? ©    b   & !  . , TWT g X )  ) TUV   (99) Simplifying (99) further, one obtains  ?  _ ` a P v  ¬ ? ©  b     C 3 2 1 789   ¸ ¸   N 2       ©   2 N  & !aQ g!Q  Q  TUV !  . , TWT g ! $  ) TUV  N (100) By letting   2 N   ¹ , (100) is simplified as  ?  _ ` a P v  ¬ ? ©  b         N 2      & !aQ !. , Q TWT  g X ) Q ) TUV  N ² C 3 2 1 789    ©  ¹    & ! . , ¹ TWT  g X ) ¹ ) TUV  ¹ (101) Let  N 2    º , and multipl y i ng numerator and deno minator of (101) by C 3 2 1 789  , (101) reduces to  ?  _ ` a P v  ¬ ? ©  b     C ! TUV      º     & !»  . , a Wef  v TUW   TWT g X ) » ) TUV  º  ² C 3 2 1 789   ² & !. , v TWT  g!av g X ) v ) TUV  ²  « ?      (102) Thus,  ?  _ ` a P v  ¬ ? © b     C ! TUV   & !  . , a Wef   v TWT g X ) v ) TUV  ª ?     2 ^ Ecd  2  789   « ?      , (103) which proves the time-frequency shift convolution propert y in ROF rFT domain. Similarly, for so lving  ?  _  ¬ ? ` a P v © b    and u tilizing the shift an d modulation convolution operator o f function  ¬ ? ` a P v © as    N  & !a  Q  ©   2 N 2   - "®  N    N   , where, - "®  N     & !Q  Q  TUV  and based on the previous steps, one obtains  ?  _  ¬ ? ` a P v © b     C ! TUV   & !  . , a Wef   v TWT g X ) v ) TUV  ª ?      « ?     2 ^ Ecd  2  789   , (104) which proves the time-frequency shift convolution propert y in ROF rFT domain. Special case : In case of FT, (95) and (96) reduces to (for      )  ? % $  _ ` a P v  ¬ ? ©  b  % $    & !  . • )  a  v ª ? % $    % $  2 ^  « ? % $    % $   , i.e.,  _ ` a P v  ¬ ? © b  ]   & !  \a  v ª ?  ] 2 ^  « ?  ]  (105)  ? % $  _  ¬ ? ` a P v © b  % $    & !  . • )  a  v ª ? % $    % $   « ? % $    % $  2 ^  i.e,  _  ¬ ? ` a P v © b  ]   & !  \a  v ª ?  ]  « ?  ] 2 ^  (106) This means th at the pr oposed ti me-frequency shift con volution p rop erty b ehaves similar to the Euclidean F T, as is evident from (105) and (106), respectively. 7. Conclusions and Future Scope of Work In this paper, a new definition of the fractional Fourier tr ansform is introduced, which is considered to be a reduced form of the conventional FrFT ― Reduced Order Fractio nal Fourier Transform (ROFr FT). The mathematical definition along with its various properties is derived and it is shown analytica ll y that its definition gets reduced to the conventional Fourier transfor m definit ion for th e rot ational an gle of    radians. Also, different analytical derivation of various kinds of signals is derived, which is much simpler in mathematical for m as compared to the conventional FrFT definitio n. Finally, t he new definition of the convolution theorem is presented with i ts v arious i mportant properties such as shift, modulation and ti me- frequency shift convolution properties. It is to be noted that the propos ed new def inition of FrFT very nicel y generalizes the convolution theorem same as that of conventional Fourier transformation, which was not possible with the earlier definitions of conventional fraction al Fourier transformation. 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