A Low-Complexity Approach to Computation of the Discrete Fractional Fourier Transform

Transcription

1 Circuits Sst Signal Process (1) 36: DOI 11/s z A Low-Compleit Approach to Computation of the Discrete Fractional Fourier Transform Dorota Majorkowska-Mech 1 Aleksandr Cariow 1 Received: 3 December 15 / Revised: 1 Januar 1 / Accepted: 19 Januar 1 / Published online: 4 Februar 1 The Author(s) 1 This article is published with open access at Springerlinkcom Abstract This paper proposes an effective approach to the computation of the discrete fractional Fourier transform for an input vector of an length This approach uses specific structural properties of the discrete fractional Fourier transformation matri Thanks to these properties, the fractional Fourier transformation matri can be decomposed into a sum of three or two matrices, one of which is a dense matri, and the rest of the matri components are sparse matrices The aforementioned dense matri has uniue structural properties that allow advantageous factorization This factorization is the main contributor to the reduction in the overall computational compleit of the discrete fractional Fourier transform computation The remaining calculations do not contribute significantl to the total amount of computation Thus, the proposed approach allows to reduce the number of arithmetic operations when calculating the discrete fractional Fourier transform Kewords Discrete linear transforms Discrete fractional Fourier transform Eigenvalue decomposition Mathematics Subject Classification MSC 15A4 MSC 15A1 MSC 65Y B Dorota Majorkowska-Mech dmajorkowska@wizutedupl Aleksandr Cariow acariow@wizutedupl 1 Facult of Computer Science and Information Technolog, West Pomeranian Universit of Technolog, Szczecin, Poland

2 Circuits Sst Signal Process (1) 36: Introduction Man discrete fractional transforms, such as discrete fractional cosine and sine transforms [], discrete fractional random transform [1], discrete fractional Hartle transform [3], discrete fractional Hilbert transform [1], discrete fractional Hadamard transform [], and discrete fractional Fourier transform [1] have been found ver useful for signal processing [], digital watermarking [5], image encrption [,1,15], image and video processing [9] Among other transformations, the discrete fractional Fourier transform is perhaps most freuentl used To date, a number of effective algorithms for various discrete fractional transforms have been developed [,16,5] Fractional Fourier transform (FRFT) is a generalization of the ordinar Fourier transform (FT) with one fractional parameter This concept was initiall introduced b Wiener [6] The FRFT was recognized as a transform method b mathematicians in 19 after the work of amias [14] in which the FRFT was introduced as a fractional power of the ordinar FT operator To compute the FRFT b using digital techniues a discrete version of this transform was needed It initiated the work of defining a discrete version of FRFT (DFRFT) Eisting approaches to the definition of the discrete Fourier transform can be categorized into three major tpes The first approach is represented b direct sampling of the FRFT [16,1] It is the least complicated approach and there are uite a few different algorithms that have been developed for the computation of this tpe of DFRFT But these discrete realizations could lose man important properties of the FRFT such as unitarit, reversibilit, additivit; therefore, its applications are ver limited This approach is still used and developed [11] The second approach relies on a linear combination of ordinar Fourier operators raised to different powers [4,4] However, as emphasized in [3], these realizations often produce a result that does not match the result of the continuous FRFT In other words, it is not a discrete version of the continuous transform The third approach is based on the idea of eigenvalue decomposition [3,1,19] This tpe of DFRFT possesses all essential properties which are posed as reuirements for DFRFT such as unitarit, additivit, reduction to discrete Fourier transform (when the power is eual to unit), approimation of the continuous FRFT However, the eigenvalue decomposition approach cannot be written in a closed form and ma be computationall costl The authors are interested in this tpe of DFRFT because currentl there are no fast algorithms created for its realization The work [] described the method to reduce the computational load of the DFRFT b one half In that work authors defined the discrete fractional cosine transform (DFRCT), which is the fractional version of the first tpe discrete cosine transform (DCT-I), and the discrete fractional sine transform (DFRST) the fractional version of the first tpe discrete sine transform (DST-I) The also discovered the relationships between the eigenvectors of the DFRCT and DFRFT, as well as between the eigenvectors of DFRST and DFRFT matrices The authors proved that the eigenvectors of the DFRCT matri of size can be easil obtained from the even eigenvectors of the DFRFT matri of size Similarl, the showed that the eigenvectors of the DFRST matri of size can be obtained from the odd eigenvectors of the DFRFT matri of size + These results allowed them to demonstrate that the DFRFT transform of an even size can be obtained through splitting the input signal into the even and odd parts, proper truncation of

3 41 Circuits Sst Signal Process (1) 36: each part and then computing DFRCT of size / + 1 from the even part and DFRST of size / 1 from the odd part After this operations are performed, the output signals should be etended properl and in the end added (before the addition one of them should be multiplied b the factor determined) This method is uite elegant and allows to reduce the computational cost b about one half, but it works onl for signals of an even length The work [] presented a comparative analsis of the most famous algorithms for the computation of all these tpes of DFRFTs Mathematical Background The normalized DFT matri of size is defined as follows: F w 1 w 1 w w ( ) 1 w 1 w ( 1)( ) w 1 w ( )( 1) w ( 1) (1) where j is the imaginar unit, w e j π, and 1/ is the normalization scaling factor Since F is a smmetric matri and F F I, F is a unitar matri (smbol means Hermitian transpose and I denotes an identit matri) This implies the following properties [6]: (1) all the eigenvalues of F are nonzero and have magnitude one, and () there eists a complete set of -orthonormal eigenvectors, so we can write F Z Z T () where is a diagonal matri of size, whose diagonal entries are the eigenvalues of F and Z is the matri whose columns are normalized mutuall orthogonal eigenvectors of the matri F The eigenvector z (k) corresponds to the eigenvalue λ k Since the matri F satisfies the propert F 4 I, each of its eigenvalues have to fulfil euation λ 4 k 1, so the DFT matri has onl four distinct eigenvalues: 1, 1, j, j The multiplicities of these eigenvalues are well known [13], so for 4 the eigenvalues are degenerated This means that the set of eigenvectors is not uniue For this reason, it is necessar to specif a particular eigenvector set, which will be used in a definition of DFRFT The fractional power of matri, including DFT matri, can be obtained from its eigenvalue decomposition and the power of eigenvalues F a Z a ZT (3) where the fractional parameter a is real Obviousl, for a the DFRFT matri becomes the identit matri, and for a 1 it is transformed into ordinar DFT

4 Circuits Sst Signal Process (1) 36: matri For completeness it should be added that an inverse discrete fractional Fourier transform (IDFRFT) matri for a fractional parameter a is eual to the DFRFT matri for a fractional parameter a, so (F a ) 1 F a (4) The definition (3) of DFRFT was first introduced b Pei and Yeh [1,19] The defined the DFRFT in terms of a particular set of eigenvectors, which constitute the discrete counterpart of the set of Hermite Gaussians functions (these functions are well-known eigenfunctions of FT and the fractional Fourier transform was defined through a spectral epansion in this base [14]) This idea was developed in work [3] An important propert of the eigenvectors of the DFT matri is that the are either even or odd vectors [13] Because of periodicit, the arguments are interpreted modulo (as in the ordinar DFT), so a vector z is even, when its coordinates (in standard basis) satisf the euations z i z i (5) for i 1,,, / ( is the greatest integer less than or eual to the argument ) We are indeing the coordinates of the vector z from to 1 A vector z is odd when its coordinates fulfil the following properties: z and z i z i (6) for i 1,,, / 3 Special Structure of DFRFT Matri In this paper we assume that the set of eigenvectors of the matri F has alread been calculated, as it was shown in [3], and the eigenvectors are ordered according to the increasing number of zero-crossing After normalization the form the matri Z which appears in Es () and (3) We also assume that in the matri, which occurs in these euations, the eigenvalues are arranged in the order of their associated eigenvectors It should be noted that the DFRFT matri calculated from (3) is smmetric because 1 i, j k z i,k λ a k z j,k 1 k z j,k λ a k z i,k j,i () for i, j, 1,, 1 Moreover, the DFRFT matri has additional special properties, which result from the fact that each column of the matri Z is either an even or odd vector One of them is that the first row (with inde ) is an even vector Proposition 1 The first row of the matri F a is an even vector It means that for j 1,,, /, j, j ()

5 41 Circuits Sst Signal Process (1) 36: Proof 1, j z,k λ a k z j,k (9) k 1, j z,k λ a k z j,k (1) k If the column with inde k of the matri Z is an even vector, then z j,k z j,k for an j, hence, z,k λ a k z j,k z,k λ a k z j,k If the column with inde k of the matri Z is an odd vector, then z,k and also z,k λ a k z j,k z,k λ a k z j,k for an j Hence, the right side of E (9) is eual to the right side of E (1) Obviousl, the first column of the matri F a matri is smmetric is an even vector too, because this Proposition A matri which we obtain after removing the first row and the first column from the matri F a is persmmetric It means that for i, j 1,,, 1 Proof i, j j, i (11) 1 i, j z i,k λ a k z j,k (1) k 1 j, i z j,k λ a k z i,k (13) k If the column with inde k of the matri Z is an even vector, then z j,k z j,k for an j, hence, z i,k λ a k z j,k z i,k λ a k z j,k z j,k λ a k z i,k for an i and j If the column with inde k of the matri Z is an odd vector, then z j,k z j,k and also z i,k λ a k z j,k z i,k λ a k ( z j,k) z j,k λ a k z i,k, hence, the right side of E (1) is eual to the right side of E (13) The two aforementioned properties of the matri F a and its smmetr give it a special structure We will write the matri F a as a sum of three or two matrices and these matrices will have special structures as well This trick ma be useful to reduce the number of arithmetical operations when calculating a product of the matri F a b a vector The number of components of the sum depends on whether is even or odd If is even, the matri F a can be written as a sum of three matrices F a A( + B( + C( (14)

6 Circuits Sst Signal Process (1) 36: where A ( B ( C ( f, a ( (,1 f f ( f, 1,, 1,1,1, 1,, 1,1 1,1 1, 1 1, +1 1, 1 1, 1 1, +1 1, +1 1, +1 1, 1 1, 1 1, +1 1, 1 1, 1 1, +1 1, 1 1,1 1, 1, 1, 1,, 1, 1, 1, 1, (15) (16) (1) If is odd, we can write the matri F a as a sum of onl two matrices F a A( + B( (1)

7 414 Circuits Sst Signal Process (1) 36: where A ( B ( f, a (,1 f, 1, 1,1,1, 1, 1,1 1,1 1, 1 1, +1 1, 1 1, 1 1, 1 1, +1 1, +1 1, +1 1, +1 1, 1 1, 1 ( 1, 1 f 1, +1 1, 1 1,1 (19) () Eample 1 presents the structures of DFRFT matrices and their components for a selected even number and a selected odd number Eample 1 The structure of the matri F a for isasfollows: b c d e g e d c c h i j k l m n d i o p r s m F a e j p t u w r l g k u u k e l r w u t p j d m s r p o i c n m l k j i h (1) where the entries: b, c, d, e, g, h, i, j, k, l, m, n, o, p,, r, s, t, u,w, are comple numbers, which are determined b and the fractional parameter a We can write the above matri as a sum F a A( + B ( + C ( ()

8 Circuits Sst Signal Process (1) 36: where A ( B ( C ( b c d e g e d c c d e g e d c h i j l m n i o p r s m j p t w r l l r w t p j m s r p o i n m l j i h k u k u u k u k (3) (4) (5) The structure of the matri F a for is as follows: b c d e e d c c g h i j k l F a d h m n o p k e i n r o j e j o r n i d k p o n m h c l k j i h g We can write this matri as a sum (6) F a A( + B ( ()

9 416 Circuits Sst Signal Process (1) 36: where A ( B ( b c d e e d c c d e e d c g h i j k l h m n o p k i n r o j j o r n i k p o n m h l k j i h g () (9) 4 The Method of DFRFT Computing Suppose ou want to calculate the discrete fractional Fourier transform for the input vector B ( we denote the output vector which is calculated using the formula ( Fa (3) We assume that the matri F a is given To calculate the output vector ( it is necessar to perform comple multiplications and ( 1) comple additions However, if we use decomposition (14) when is an even number or decomposition (1) when is odd, the number of arithmetical operations reuired for calculating the discrete fractional Fourier transform can be significantl reduced We can multipl each component of the sum b the input vector separatel, and finall add the results Let (A, denote the product A (, (B, the product B ( and, if is even, (C, the product C ( First, we will focus on calculating the product of the matri A ( b the input vector (A, A ( (31) If is even, the matri A ( has the form (15) and

10 Circuits Sst Signal Process (1) 36: (A, ( ), +,1 ( ) + +, ,,1, 1,, 1,1 (3) To make this calculation it is necessar to perform 1 comple additions and +1 comple multiplications If is odd, the matri A ( has the form (19) and (A, (, +,1 ( ) + +, 1 1,1, 1, 1, ) To make this calculation it is necessar to perform 1 comple additions and comple multiplications Eample presents the forms of vectors (A, for and Eample For the matri A ( is defined in (3) The product of this matri b the input vector [, 1,, ] T will have the form (A, b + c( 1 + ) + d( + 6 ) + e( ) + g 4 c d e g e d c (33) (34)

11 41 Circuits Sst Signal Process (1) 36: ( b c d e g c d e g (b) Fig 1 Data flow diagrams for calculation of vectors: a (A, and b (A, b c d e c d e For the matri A ( is defined in () The product of this matri b the input vector [, 1,, 6 ] T will have the form (A, b + c( ) + d( + 5 ) + e( ) c d e e d c (35) Figure 1 shows the data flow diagrams for the calculation of vectors (A, and (A, In this paper, the data flow diagrams are oriented from left to right Straight lines in the figures denote the operations of data transfer We use lines without arrows not to clutter the picture Points where lines converge denote summations ote that the circles in this figure show the operations of multiplication b a comple numbers inscribed inside the circles Calculations of (A, can be compactl described b appropriate matri vector procedures If is even, this procedure will be as follows: (A, T (+1) V ( +1 X (+1) (36) where the matri X (+1) is responsible for summing appropriate entries of the input vector: 1 + 1, +,, / 1 + /+1 and it has the form 1 1 ( 1) 1 ( 1) X (+1) ( 1) 1 I 1 ( 1) 1 J 1 1 ( 1) 1 1 ( 1) 1 1 ( 1) 1 ( 1) (3)

12 Circuits Sst Signal Process (1) 36: where the matrices m n and 1 m n are matrices of size m n in which all the entries are eual to or 1, respectivel J k is the echange matri of size k in which all the entries are zero, ecept those on the counter-diagonal going from the upper right corner to the lower left corner and all the counter-diagonal entries are eual to 1, ie The matri V ( form: J k 1 (3) which occurs in E (36) is a diagonal matri, which has the following ( V ( +1 diag,, (,1,, f, ), (,1,, f, (39) The last matri T (+1) which occurs in E (36) has the following form: 1 1 ( +1) 1 ( 1) T (+1) ( 1) ( +1) I 1 ( 1) 1 1 ( +1) 1 ( 1) 1 ( 1) ( +1) J 1 ( 1) 1 (4) If is odd, the matri vector procedure for calculating (A, will have a bit different form (A, where the matrices which occur in (41) are as follows: X T V ( X (41) I J 1 1 (4) ( V ( diag,, (,1,, f, 1,,1,, f (, 1 ) (43) T I 1 J 1 (44)

13 413 Circuits Sst Signal Process (1) 36: The matri vector procedures for calculation (A, and (A, are presented in Eample 3 This eample shows eplicate forms of matrices which occur in these procedures, assuming that the matrices A ( and A ( are defined in (3) and (), respectivel Eample 3 For the matri vector procedure (36) for calculation (A, will have the form (A, T 9 V ( 9 X 9 (45) where the matrices are as follows: X I J (46) V ( 9 diag(b, c, d, e, g, c, d, e, g) (4) T I J For the matri vector procedure (41) for calculation (A, will have the form (A, T V ( X (49) where the matrices are as follows: X 3 1 I 3 J (5) (4)

14 Circuits Sst Signal Process (1) 36: V ( diag(b, c, d, e, c, d, e) (51) T 3 4 I J (5) ow we will focus on the product of the matri B ( (B, b the input vector B ( (53) If is even, the matri B ( has the form (16) We can see that (B, (B, / and also coordinates and / are not involved in this calculation Because of the special structure of the matri B (, the rest of coordinates of the vector (B, are convenient to calculate pairwise: (B, 1 with (B, 1, (B, with (B,,,(B, / 1 with (B, /+1, because for k 1,,,/ 1 we can write [ (B, k (B, k ] [ + + 1,k 1, k 1, k [ 1,k,k, k, k,k ( f k, 1 ] [ ] 1 1 ] [ k, +1 ( f k, +1 k, 1 ] + [ ] 1 +1 (54) All the suare matrices in the above euation have a structure of tpe [ ] v z z v (55) and such a matri can be written as a product [1]: [ ] v z 1 [ ] v + z z v H H v z (56) where H [ ]

15 413 Circuits Sst Signal Process (1) 36: is the Hadamard matri of size If we take into account the above-mentioned relationship, then E (54) will take the following form: [ (B, k (B, k ] ([ 1 H + + [ 1,k + 1, k,k +, k ( + f k, 1 k, +1 1,k ],k, k 1, k ( f k, 1 k, +1 ] [ ] 1 H + 1 [ ] H + H [ 1 +1 ] (5) We should note that the last multiplication of the matri H b the sum of earlier calculated vectors (in parentheses) is eecuted onl once We should also note that when we calculate each subvector of the output vector first we have to multipl the Hadamard matri H b the subvectors: [ 1, 1 ] T, [, ] T,,[ / 1, /+1 ] T,created from pairs of input coordinates This allows us to perform these calculations onl once and not to repeat them man times If is odd, the matri B ( has the form () Looking at this matri we can see that (B, and also coordinate is not involved in this calculation The rest of coordinates of the vector (B, are also convenient to calculate pairwise: (B, 1 with (B, 1, (B, with (B,,,(B, ( 1)/ with (B, (+1)/, because for k 1,,,( 1)/ can write [ ] [ ] (B, k (B, 1,k [ ] 1, k 1 k 1, k 1,k 1 [ ] +,k [, k +, k k, 1 k, +1,k k, +1 k, 1 ] [ ] 1 +1 All the suare matrices in the euation above have the same structure as in (55), so we can write [ ] ([ ] (B, k (B, k (5) 1 H 1,k + 1, k [ ] 1 1,k H 1, k 1 [ ] +,k +, k [ ],k H +, k + + [ ] k, 1 k, +1 1 H (59) k, 1 k, +1 +1

16 Circuits Sst Signal Process (1) 36: ( H H H H H H (b) Fig Data flow diagrams for calculation of vectors: a (B, and b (B, 6 H H H H H H Using the approach presented above, we will show data flow diagrams for calculation of vectors (B, and (B, For and the matrices B ( and B ( are defined in (4) and (9), respectivel Figure shows the data flow diagrams for calculation of vectors (B, and (B, ote that the rectangles in this figure show the operations of multiplication b the matrices inscribed inside the rectangles In Fig a the numbers i are eual to: (h + n)/, 1 (h n)/, (i + m)/, 3 (i m)/, 4 ( j + l)/, 5 ( j l)/, 6, 3, (o + s)/, 9 (o s)/, 1 (p + r)/, 11 (p r)/, 1 4, 13 5, 14 1, 15 11, 16 (t + w)/, 1 (t w)/, where the numbers: h, n,,ware the entries of the matri B ( Similarl, in Fig b the numbers i are eual to: (g + l)/, 1 (g l)/, (h + k)/, 3 (h k)/, 4 (i + j)/, 5 (i j)/, 6, 3, (m + p)/, 9 (m p)/, 1 (n + o)/, 11 (n o)/, 1 4, 13 5, 14 1, 15 11, 16 ( + r)/, 1 ( r)/, where the numbers: g, l,,r are the entries of the matri B ( Calculation of vector (B, can be compactl described b the suitable matri vector procedure If is even, this procedure will be as follows: (B, R ( ) W ( ) ( ) Q ( ( ) U ( ) ( ) M ( ) (6) where the matri M ( ), which is responsible for reordering the coordinates of the input vector and multipling the matri H b the subvectors [ 1, 1 ] T, [, ] T,,[ / 1, /+1 ] T, has the form

17 4134 Circuits Sst Signal Process (1) 36: M ( ) [ [ ] [ ] ] 1 1 ( ) 1 I 1 ( ) 1 J 1 The smbol in the euation shown above denotes the Kronecker product operation The matri U ( ) / ( ) in E (6), which is responsible for replication of the earlier calculated vector has the form (61) U ( ) ( ) 1 1 I (6) The diagonal matri Q ( in E (6) is responsible for multipling the vector ( ) / of the sums and differences of respective entries of the matri (16) b the vector calculated earlier The factor 1/ is also included in this matri It has the form Q ( ( ) ( f ( 1,1 + diag 1, +, 1, 1, 1 ( + f 1, 1, 1,1, 1, 1, +1 ( + f 1, 1 1, +1,, 1, 1, 1,, ( f 1, 1 1, +1,, ( f 1, 1 1, +1 (63) The matri W ( ) ( ) / in E (6) is responsible for summation and has the form [ ] I 1 W ( ) ( ) 1 [1 ] (64) J 1 1 [ 1] The last matri R ( ) in E (6) is responsible for multipling the matri H b the subvectors of length and reordering the coordinates It has the form R ( ) 1 ( ) I 1 ( ) J [1 1] [1 1] To calculate the vector (B, according to the procedure (6), it is necessar to perform ( )/ comple additions From among these additions, are needed to multipl the matri M ( ) b the input vector, (/ )( ) additions are needed to multipl the matri W ( ) ( ) / b earlier obtained vector, and are needed to multipl the matri R ( ) b the last obtained vector The additions (65)

18 Circuits Sst Signal Process (1) 36: which were performed to obtain the matri Q ( were not counted, since this ( ) / matri ma be prepared in advance The calculation of the vector (B, also reuires ( ) / comple multiplications All these multiplications are needed to multipl the diagonal matri Q ( b the appropriate vector ( ) / If is odd, the procedure for calculation of vector (B, will be as follows: (B, R ( 1) W ( 1) ( 1) Q ( ( 1) U ( 1) ( 1) M ( 1) (66) where the matrices that occur in E (66) have the forms M ( 1) [ [ ] [ ] ] 1 1 ( 1) 1 I 1 J (6) U ( 1) 1 ( 1) 1 1 I 1 (6) ( f ( 1,1 + 1, 1, 1,1 1, 1, Q ( ( 1) diag 1, +, 1, 1 1, 1 1,, 1, + 1, +1, + 1, +1, 1, 1, 1, 1, 1 1, +1,, 1, +1 (69) W ( 1) ( 1) [ I 1 J ] [1 ] [ 1] () R ( 1) 1 ( 1) I 1 J 1 [1 1] (1) [1 1] To calculate the vector (B, according to the procedure (66), it is necessar to perform ( + 1)( 1)/ comple additions From among these additions 1 are needed to multipl the matri M ( 1) b the input vector, ( 3)( 1)/ are needed to multipl the matri W ( 1) ( 1) / b earlier obtained vector, and 1are needed to multipl the matri R ( 1) b the last obtained vector This procedure

19 4136 Circuits Sst Signal Process (1) 36: also reuires ( 1) / comple multiplications All these multiplications are needed to multipl the diagonal matri Q ( b the appropriate vector ( 1) / The matri vector procedures for the calculation of (B, for and are presented in Eample 4 This eample also demonstrates the eplicate forms of matrices that occur in these procedures, assuming that the matrices B ( and B ( are defined in (4) and (9), respectivel Eample 4 For the matri vector procedure (6) for calculating (B, will have the form (B, R 6 W 6 1 Q ( 1 U 1 6M 6 () where the matrices which occur in above presented euation are as follows: [ [ ] [ ] ] 1 1 M I J (3) U I ( Q ( h + n 1 diag, h n, i + m, i m, (4)

20 Circuits Sst Signal Process (1) 36: j + l, j l, i + m o + s, o s, p + r p + r, p r, i m,, p r, t + w, t w, j + l, j l, ) (5) [ ] I3 1 W [1 ] J [ 1] (6) R 6 I 3 [1 1] J 3 [1 1] () For the matri vector procedure (66) for calculating (B, will have the form (B, R 6 W 6 1 Q ( 1 U 1 6M 6 () where the matrices which occur in the euation presented above are as follows: [ [ ] [ ] ] 1 1 M I 3 J (9)

21 413 Circuits Sst Signal Process (1) 36: The matri U 1 6 is the same as in E () ( Q ( g + l 1 diag, g l, h + k h k, i + j, i j m + p i j, m p, n + o,, h + k, n + o, n o The matri W 6 1 isthesameasine(), n o, h k,, i + j, + r, r, ) R 6 I 3 [1 1] 1 1 J 3 [1 1] () (1) The last component C ( will focus on the product of the matri C ( For even the matri C ( (C, appears in decomposition (14) onl if is even ow we (C, has the form (1) and b the input vector C ( () 1,, 1, ( 1, ) + + 1, 1, 1, ) ( , (3) To make this calculation it is necessar to perform comple additions and 1 comple multiplications Eample 5 presents the form of the vector (C, for

22 Circuits Sst Signal Process (1) 36: Fig 3 Data flow diagrams for calculation of vector (C, k u k u ( C, ( C, 1 ( C, ( C, 3 ( C, 4 ( C, 5 ( C, 6 ( C, Eample 5 For the matri C ( is defined in (5) The product of this matri b the input vector [, 1,, ] T will have the form (C, k 4 4 u 4 k( 1 + ) + ( + 6 ) + u( ) + 4 u 4 4 k 4 (4) Figure 3 shows the data flow diagram for calculation of vector (C, Calculation (C, can be concisel described b appropriate matri vector procedure This procedure will be as follows: (C, K ( 1) G ( 1 L ( 1) (5) where the matri L ( 1) is responsible for summing the appropriate entries of the input vector: 1 + 1, +,, / 1 + /+1 and it has the form L ( 1) ( 1) 1 I 1 ( ) 1 1 J 1 1 ( ) ( 1) (6) The matri G ( form: 1 that occurs in E (5) is a diagonal matri, which has the following ( ) G ( 1 diag,,, 1,,,,,,, 1,, 1, ()

23 414 Circuits Sst Signal Process (1) 36: The last matri K ( 1) which occurs in E (5) has the form 1 ( ) 1 1 ( ) K ( 1) 1 I ( ) 1 1 ( ) 1 J 1 () The matri vector procedure (5) for calculating (C, for is presented in Eample 6 This eample also shows eplicate forms of matrices which occur in this procedure, assuming that the matri C ( is defined in (5) Eample 6 For the matri vector procedure (5) for calculating (C, will have the form (C, K G ( L (9) where the matrices are as follows: [ ] 3 1 I L J (9) G ( diag(k,, u,, k,, u) (91) K 3 4 I J (9) To obtain the final output vector (, namel, the discrete fractional Fourier trans-, (B, and also (C, if is form defined in (3), we have to add vectors (A, even For even we have (B, (C,, so to obtain ( we do not need to

24 Circuits Sst Signal Process (1) 36: Table 1 umber of additions and multiplications for even (A, Additions 1 Multiplications + 1 (B, (C, Summing Total ( ) 3 ( ) Table umber of additions and multiplications for odd (A, (B, Summing Total Additions 1 Multiplications ( 1)(+1) 1 ( 1) perform an additions Also, since (B, /, it is necessar to perform onl one addition to obtain ( / For other coordinates ( i, where i and i /, we have to perform two additions The whole number of additions when summing vectors (A,, (B, and (C, is eual to ( )+1 3 If is odd we have to add onl two vectors: (A, and (B, Since (B,, we do not need to perform an additions to obtain ( For other coordinates ( i, where i wehavetoperform one addition The whole number of additions when summing vectors (A, and (B, is eual to 1 5 Computational Compleit Let us assess the computational compleit in terms of the number of multiplications and additions reuired to obtain DFRFT Direct calculation of the discrete fractional Fourier transform for an input vector, assuming that the matri F a defined b (3) is given, reuires multiplications of a comple number and ( 1) comple additions If we use the decomposition of the matri F a into three or two matrices, according to (14) or(1), respectivel, then the vectors (A,, (B, and (C, if is even are calculated When, in the end, the are added, the total number of additions and multiplications will be smaller We will evaluate the number of additions and multiplications b calculating vectors (A,, (B, and (C, if is even, using our method These vectors and the total number of the operations, which are needed to obtain the resulting vector (, will be summed These calculations will be performed for an even and odd separatel and the results will be presented in Tables 1 and, respectivel Table 3 presents the number of additions and multiplications, which are necessar for calculating DFRFT transform for the input vector of even length, using the direct method, the method from [] and the method proposed Since the method from work

25 414 Circuits Sst Signal Process (1) 36: Table 3 Comparison of number of arithmetical operations for even length DFRFT Additions Multiplications Direct method Method from [] Proposed method Direct method Method from [] Proposed method Table 4 Comparison of number of arithmetical operations for odd length DFRFT Additions Multiplications Direct method Proposed method Direct method Proposed method [] cannot be used for the input vector of odd length, Table 4 presents the number of additions and multiplications, which are necessar for calculating DFRFT transform for the input vector of odd length, using onl the direct method and the method proposed After the analsis of Tables 3 and 4, it can be seen that the number of multiplications in the method proposed is almost twice as small as that in the direct method of calculating DFRFT This observation is true for vectors of both even and odd lengths The number of additions in our method is also smaller than in the direct method This difference in favour of our method increases with the length of the input vector

26 Circuits Sst Signal Process (1) 36: For an input signal of an even length we can compare our method with the method from [] As it can be seen in Table 3, the number of additions is lower in the method from [], but the number of multiplications is lower in the method proposed 6 Conclusions In this paper, we proposed a new approach to the design of the reduced compleit algorithms for the discrete fractional Fourier transform computation In the approach proposed, the original DFFT matri can be decomposed as an algebraic sum of a dense matri and of one or two another matrices which have man zero entries Thus, the decomposition of the original matri into submatrices can be represented as a sum of the partial products of each submatri b the same input vector The dense matri possesses a uniue structure that allows us to determine an effective factorization of this matri and leads to accelerate computation b reducing the arithmetical compleit of a matri vector product The calculation of the remaining matri vector products reuires onl a small number of arithmetic operations Based on the matri factorization and the Kronecker product, the effective method for the DFRFT computation has been derived For the sake of simplicit, the two eamples, for and, have been considered However, it is clear that the approach proposed is applicable for an arbitrar case (regardless of whether the length of the input vector is odd or even) Acknowledgements Funding was provided b West Pomeranian Universit of Technolog Szczecin (PL) Open Access This article is distributed under the terms of the Creative Commons Attribution 4 International License ( which permits unrestricted use, distribution, and reproduction in an medium, provided ou give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made References 1 A Cariow, Strategies for the snthesis of fast algorithms for the computation of the matri vector products J Signal Process Theor Appl 3(1), 1 19 (14) A Cariow, D Majorkowska-Mech, Fast algorithm for discrete fractional Hadamard transform umer Algorithms 6(3), 55 6 (15) 3 ÇC Candan, MA Kuta, HM Ozaktas, The discrete fractional Fourier transform IEEE Trans Signal Process 4(5), () 4 BW Dickinson, K Steiglitz, Eigenvectors and functions of the discrete Fourier transform IEEE Trans Acoust Speech Signal Process 3(1), 5 31 (19) 5 I Djurović, S Stanković, I Pitas, Digital watermarking in the fractional Fourier transformation domain J etw Comput Appl 4(), (1) 6 PR Halmos, Finite Dimensional Vector Spaces (Princeton Universit Press, Princeton, 194) B Hennell, JT Sheridan, Fractional Fourier transform-based image encrption: phase retrieval algorithm Opt Commun 6(1 6), 61 (3) M Irfan, L Zheng, H Shahzad, Review of computing algorithms for discrete fractional Fourier transform Res J Appl Sci Eng Technol 6(11), (13) 9 Jindal, K Singh, Image and video processing using discrete fractional transforms SIViP (), (14) 1 S Liu, Q Mi, B Zhu, Optical image encrption with multistage and multichannel fractional Fourierdomain filtering Opt Lett 6(16), (1)

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