Spectral Analysis of Generalized Triangular and Welch Window Functions using Fractional Fourier Transform

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1 Online ISSN , Print ISSN ATKAFF 571, Pooja Mohindru, Rajesh Khanna, S. S. Bhatia Sectral Analysis of Generalized Triangular and Welch Window Functions using Fractional Fourier Transform DOI 1.735/automatika UDK [ ]: Original scientific aer The aer resents a new closed-form exression for the fractional Fourier transform of generalized Triangular and Welch window functions. Fractional Fourier Transform FrFT is a arameterized transform having an adjustable transform arameter which makes it more flexible and suerior over ordinary Fourier transform in several alications. It is an imortant tool used in signal rocessing for sectral analysis. The analysis of generalized Triangular and Welch window functions in fractional Fourier domain establishes a direct relationshi between their FrFTs and fractional angle. Based on the mathematical model obtained, it is observed that adjustable sectral arameters of these functions can be obtained by modifying the fractional angle. The various values of sectral arameters such as half main-lobe width, side lobe fall-off rate and maximum side-lobe level with change in order of fractional Fourier transform are also obtained for these functions. Key words: Fractional Fourier transform, Generalized triangular function, Sectral analysis, Welch window, Window function Sektralna analiza ooćene trokutaste i Welchove rozorske funkcije korištenjem frakcijske Fourierove transformacije. U radu je rikazan novi izraz za zatvoreni oblik frakcijske Fourierove transformacije ooćene trokutaste i Welchove rozorske funkcije. Frakcijska Fourierova transformacija FrFT arametrizirana je transformacija s odesivim arametrom transformacije koja je u odreženim rimjenama fleksibilnija i sueriornija u odnosu na uobičajenu Fourierovu transformaciju. Ističe se kao važan alat u obradi signala i sektralnoj analizi. Analiza ooćene trokutaste i Welchove rozorske funkcije u odručju frkacijske Fourierove transformacije usostavlja izravni odnos izmežu FrFT-a i frakcijskog kuta. Koristeći dobiveni matematički model, uočeno je da se odesivi sektralni arametri ovih funkcija mogu izvesti mijenjanjem frakcijskog kuta. Različite vrijednosti sektralnih arametara, kao što su olovica širine sektralnog vrha, stoa snižavanja amlituda viših harmonika ili najveća amlituda viših harmonika, odnosno njihova ovisnost u odnosu na red frakcijske Fourierove transformacije, takožer se mogu odrediti uotrebom ovih funkcija. Ključne riječi: Frakcijska Fourierova transformacija, ooćena trokutasta funkcija, sektralna analiza, Welchov rozor, rozorska funkcija 1 INTRODUCTION Windows are weighting functions that attenuate signals at their discontinuities. The window functions are alied to the time-domain signals and the rocess of multilying the signal with the smoothly ending window function is called windowing technique. Windowing is done to make an infinitely long function finite in length so that frequency content of a signal of interest can be measured. As a result of this, the truncated signal exhibits different sectral characteristics from the original continuous-time signal. domain characteristics of several window functions are analyzed to determine their suitability for a secific alication and to reduce the sectral leakage that results because of selecting a finite time interval signal. Window functions are real, non-zero and time limited functions. They eak in the middle frequencies and decrease to zero at the edges in order to reduce the effects of the discontinuities that results because of finite duration. They are frequently used in various areas of signal rocessing and communications such as seech rocessing, digital filter design, and sectrum estimation. No existing window function is best in all asects [16]. Thus, it is needed to select an aroriate window function according to the requirement of a articular alication on the basis of the erformance features [4], such as the attenuation at the maximum height of a side lobe, generally the first side-lobe the side-lobe level, the rate at which eak of the side-lobes decrease in magnitude side- AUTOMATIKA ,

2 lobe fall-off rate and the main-lobe width of width of main-lobe at -3dB below main-lobe eak. The narrower the main-lobe width, the better will be the frequency resolution; and the lower the side lobe level, the better will be the amlitude accuracy or noise suression. The narrow main-lobe width and reduced side-lobe level are conflicting requirements. When the main lobe width decreases, the remaining energy sreads out to side-lobes thereby increasing sectral leakage. Thus, the roblem lies in deciding which window function is the best to aly on the signal being studied in order to estimate the sectral characteristics of a finite duration signal. There are numerous standard window functions that can be chosen for the revention of sectral leakage in the signal and to rovide the secified side-lobe level [7, 1]. But the reduction of side-lobe leakage due to the alied window function introduces leakage from the exansion of main-lobe in ordinary frequency domain. This reduces sectral resolution and also some gain is lost because of main-lobe sreading. Thus, we need to increase the length of the selected window function to imrove sectral resolution. Alying fractional Fourier transform to the window function, one can imrove sectral resolution with no need of changing the length of the window function [9, 15]. Therefore, comutational time and design time can be saved. This aer resents a new mathematical model for obtaining the FrFT of generalized Triangular and Welch window functions. Based on which, it is found that these functions can be used as an adjustable windows in fractional Fourier domain for doing the signal sectral analysis. By changing the value of fractional order arameter, the sectral arameters such as Half Main-Lobe Width HMLW, Maximum Side-Lobe Level MSLL and Side- Lobe Fall-Off Rate SLFOR of the resulting windows can be controlled. Therefore, trade-off roblem between the frequency resolution and sectral leakage can be easily solved. The variations in sectral arameters of these window functions are studied for different values of fractional order arameter. A lot of sectrum of Welch window function is shown below in Fig. 1 to define these sectral arameters. 1. Half Main-Lobe Width HMLW: It is the frequency at which the Main lobe dros to the eak rile value of the side lobes. 2. Maximum Side-Lobe Level MSLL: It is the largest side lobe level in decibels relative to the main lobe eak gain. 3. Side-Lobe Fall-Off Rate SLFOR: It is the asymtotic decay rate of side-lobe level in decibels er db Gain MSLL HMLW SLFOR Fig. 1. Resonse Log Magnitude Plot of Welch window to define sectral arameters decade/octave of frequency of the eaks of the side lobes. It is roosed in the work that by adjusting fractional order arameter to different values, main-lobe width can be minimized and side-lobe fall-off rate can be raised to maximum. Thus, a choice can be made between detection and resolution. Detection means detecting a desired signal in the resence of noise. Resolution refers to the ability of distinguishing narrowband sectral comonents. 2 FRACTIONAL FOURIER TRANSFORM FrFT is the generalization of ordinary Fourier transform FT that deends on a arameter α and can be interreted as a rotation by an angle α in the time-frequency lane [2, 11, 12]. It is reresented by R α and α=aπ/2, where α is the angle of rotation and a is the fractional order arameter. The time domain and frequency domain are the secial cases of the FrFT domain with α being 2nπ and 2nπ + π/2, resectively, where n is an integer. It can also be viewed as a fractional ower of the Fourier oerator. It is more affluent in theory, more stretchy in alications, and imlementation cost is same as that of comuting FT [13, 14]. The continuous fractional Fourier transform CFrFT [2, 12] of a signal xt reresented along time axis denoted by t, with rotation angle α is comuted as: F α [xt] = X α u = xtk α t, udt, 1 where the transform kernel K α t, u of the FrFT is given 222 AUTOMATIKA ,

3 by: K αt, u = { 1 j cot α ex j t2 +u 2 cot α jut csc α 2π 2 if α is not a multile of π, δt u, if α is a multile of 2π, δt + u, if α + π is a multile of 2π, }, 2 with 1 j cot α C α =, = 1 cot α and q = csc α. 3 2π 2 Here δt denotes Dirac-delta distribution. The Fourier transform is a secial case of FrFT for α = π 2. The FrFT is a unified time-frequency transform, which reveals the characteristics of a signal gradually changing from time domain to frequency domain with the fractional order α changing from to 1. Fractional Fourier Transform FrFT is a arameterized transform with an adjustable transform arameter. It has an extra degree of freedom and can achieve better erformance over ordinary Fourier transform. FrFT has become a very efficient mathematical tool in the various alications like harmonic analysis, signal synthesis, time-frequency analysis, digital watermarking, image encrytion, modulation and multilexing in communications, etc. FrFT holds all the roerties of Fourier transform and therefore, has tremendous otential for imrovement in areas where FT has been used [3, 5, 8]. The develoment of fast algorithms for comutation of FrFT has made alication of FrFT functional in real-time digital signal rocessing. 3 DERIVATION OF FRFT OF GENERALIZED TRIANGULAR FUNCTION The generalized Triangular function denoted by xt for any scaling arameter L, is defined as: 1 t L t L xt = 1 + t L L t. 4 otherwise For arameter value, L = 1, xt is equivalent to Triangular window. The FrFT X α u of generalized Triangular function xt is comuted as follows: Substituting xt in 1 results: X α u = C α ex ju 2 L 1 t L 1 + t L ex jt 2 jqut dt+ ex jt 2 jqut dt. 5 Changing t by t in the first integral, gives: X α u = C α ex ju 2 1 t L where 1 t L Equation 6 can be rewritten as: ex jt 2 + jqut dt+ ex jt 2 jqut dt. 6 X α u = C α ex ju 2 I 1 + I 2, 7 I 1 = I 2 = 1 t L exjt2 + jqutdt, 8 1 t L exjt2 jqutdt. 9 Solving for I 1 searately, the integral can be written as: I 1 = 1 exjt 2 + jqutdt }{{} I 11 1 t ex jt 2 + jqut dt L }{{} I 12 1 Now, first solving for I 12 = t exjt2 + jqutdt, the integral can be rewritten as: I 12 = 1 or I 12 = 1 = 1 1 qu 2 t + jqu jqu exjt 2 + jqutdt, t + jqu exjt 2 + jqutdt jqu exjt 2 + jqutdt d dt {exjt2 + jqut} exjt 2 + jqutdt. Simlifying, I 12 = 1 exjt 2 + jqut L qu 2 ex jt 2 + jqut dt I 11 11a 11b AUTOMATIKA ,

4 = 1 ex jl 2 + jqul 1 qu 2 I c Now, solving the integral I 11 = exjt2 + jqutdt, the following exression results [1]: π jq 2 I 11 = 2 j ex u 2 [ j qu + 2L Erfi ] [ j qu Erfi 12 ], where Erfiz is imaginary error function of z, which is defined in the whole comlex z-lane. By using 11c and 12, solving for 1, one gets: π I 1 = 2 j Erfi 1 L 1 + qu 2L ex jq 2 u 2 [ ] [ ] j qu + 2L j 2 qu Erfi ex jl 2 + jqul By using Erf iz = jerfjz and simlifying 13, one gets: j jq 2 u 2 { I 1 = ex 2L3/2 π 2L + qu [ ] [ 3/2 qu + 2L Erf j 2 + Erf j 3/2 qu ] + jq 2 u 2 2 j qu + 2L } j ex ex. 14 Similarly I 2 can be comuted by relacing qu by qu in equation 14, i.e., j jq 2 u 2 { I 2 = ex 2L3/2 π 2L qu [ ] [ 3/2 qu + 2L Erf j 2 + Erf j 3/2 qu ] + jq 2 u 2 2 j qu + 2L } j ex ex. 15 Solving for equation 7 by using 3, 14 and 15, one gets: 2 exju 2 csc 2α jcot αl+csc αu2 B ex 2 cot α + jcot αl csc αu2 ex 2 cot α X α u = A Erf[ 3/4 csc αu], C 3/4 cot αl+csc αu + Erf[ 2 cot α ] Erf[ 3/4 csc αu] D 3/4 cot αl csc αu Erf[ 2 cot α ] where A = 1 j cot α π cot 3 α 1 L j 4 16 ex ju2 tan α, 17 2 B = 2 2j cot α, 18 C = cot αl + csc αu 2π, 19 D = cot αl csc αu 2π. 2 From equation 16 to 2, it can be seen that FrFT of generalized Triangular function is directly deendent on the fractional angle α. 4 DERIVATION OF FRFT OF WELCH WINDOW The Welch window function denoted by xt, is defined as: { } 1 t 2 1 t 1 xt = 21 elsewhere The FrFT X α u of Welch window is comuted as follows: Substituting xt in 1 results: X α u = C α ex ju t 2 ex jt 2 jqut dt. Equation 22 can be rewritten as: X α u = C α ex ju 2 ex jt 2 jqut dt }{{} I 3 t 2 ex jt 2 jqut dt I First solving for I 4 searately, the integral can be written as: I 4 = t ex jqut t exjt 2 dt. 24 }{{}}{{} u dv 224 AUTOMATIKA ,

5 Let u = t ex jqut and v = t ex jt 2 dt, which gives: du = jqut ex jqut dt + ex jqut dt and v = 1 ex jt 2. Now, solving equation 24 via integrating by arts, one gets: I 4 = t exjt2 jqut jqut 1 exjt 2 jqutdt I 5 = 1 1 ex j jqu + ex j + jqu + I Similarly solving for I 5, the integral can be written as: I 5 = jqu t exjt 2 jqutdt }{{} I 6 exjt 2 jqutdt. 26 I 3 Solving 26 for I 6 in a similar manner as solving for I 12 using 11a: I 6 = 1 exjt 2 jqut 1 + qu 2 exjt 2 jqutdt I 3 = 1 qu ex j jqu ex j + jqu + 2 I By using 26 and 27, solving for I 5, one gets: I 5 = qu ex j jqu ex j + jqu 2 jq 2 u I By using 25 and 28, solving fori 4, one gets: I 4 = 1 ex j jqu + ex j + jqu + qu ex j jqu ex j + jqu 4j2 + 1 jq 2 u 2 1 I Now, solving the integral exjt2 jqutdt the following exression results [1]: I 3 = π jq 2 2 j ex u 2 [ ] j qu + 2 Erfi [ ] j qu Erfi 2. 3 By using 29 and 3, solving for 22 and rearranging, one gets: { exju 2 jq 2 u 2 / X α u = C α ex [ j jqu 8j 3/2 5/2 + jq 2 u 2 / ] j [ 2qu + 2qu ex 2jqu ex 2jqu ] + π4j 2 jq 2 u 2 [ ] j qu Erfi [ ] } j qu Erfi Putting values of C α, and q using 3 in 31 and simlifying, one gets: X α u = A 1 { ex j 2 cot αu 12 [ 2u csc α + 2u csc α ex2ju csc α 2 cot α 2 cot α ex2ju csc α ] [ ] + B 1 Erf 3/4 cot α u csc α 2 cot α [ ] } + Erfi 3/4 cot α + u csc α, 2 cot α 32 where A 1 = 1 1 j cot α 4j 3/2 π cot 3 α, 33 B 1 = π 2j cot 2 α + ju 2 csc 2 α cot α. 34 From equation 32 to 34, it can be seen that FrFT of Welch window function is directly deendent on the fractional angle α. 5 RESULTS AND DISCUSSIONS The lot of Triangular function for.5 t.5 is shown in Fig. 2. The lot of Welch window as a function of time is shown in Fig. 3. The magnitude of FrFT AUTOMATIKA ,

6 1. Amlitude For L =.7 For L =.5 For L = Time sec 8 Fig. 2. The Triangular window function in time-domain Amlitude Time sec Fig. 3. The Welch window in time-domain of these functions i.e. X α u in db is lotted versus frequency u cycles/sec. Figure 4 shows the lot for calculating MSLL for generalized Triangular function by varying scaling arameter L to different values keeing fractional order a =.5. Figures 5 illustrate the lot for calculating MSLL for Triangular function by varying fractional order arameter to different values keeing scaling arameter L =.5. Figure 7 shows the MSLL lot for Welch window for different values of fractional order. The lots for calculating SLFOR for these functions are also shown in Fig. 6 and 8. The continuum of fractional Fourier transform of Triangular and Welch window functions to sinc as the fractional order is varied from to 1 are also shown in Fig. 9 and Fig. 1 resectively. The values of MSLL, HMLW and SLFOR for Triangular function scaling arameter L =.5 and Welch window function are tabulated in Table 1 and Table 2 resectively for various values of fractional order arameter a. It is observed from the Figs. and Tables that sectral arameters of Triangular function and Welch window deend uon the value of fractional order arameter a in time-frequency lane. Main-lobe width of both the func Fig. 4. MSLL lot for generalized Triangular function fractional order a =.5 for values of scaling arameter L =.5,.7 and For a =.2 For a =.5 For a = Fig. 5. MSLL lot for Triangular function scaling arameter L =.5 for fractional orders a =.2,.5 and 1 tions increases regularly with increase in fractional order a. SLFOR for Triangular function shows variation between -8.9 db/octave to -11 db/octave with change in order arameter a. MSLL is also reduced with increase in fractional order a, e.g. MSLL for a =.7 is 25.8 db comared to is db for a = 1. If the value of arameter L is taken to be equal to 1, Fig. 11 shows the sectral arameters of triangular window at fractional order a = 1. From Table 2, the SLFOR for Welch window shows variation between -7.8 db/octave to db/octave with change in order arameter a. MSLL is reduced with increase in fractional order a, e.g. MSLL for a =.7 is 226 AUTOMATIKA ,

7 5 1 For a =.2 For a =.5 For a = For a =.2 For a =.5 For a = Fig. 6. SLFOR lot for Triangular function scaling arameter L =.5 for fractional orders a =.2,.5 and For a =.2 For a =.5 For a = Fig. 7. MSLL lot for Welch window for fractional orders a =.2,.5 and db comared to is db for a = 1. Thus, an otimal domain can be selected in order to make a comromise between increase in main-lobe width and side-lobe level reduction. 6 APPLICATION OF FRFT IN TUNING WIDTH OF THE TRANSITION-BAND The transition bandwidth of window-based FIR filters is roortional to the window main-lobe width, which in turn is inversely roortional to the length of the window function [15]. The transition width can be directly reduced by increasing the window length but at the cost of increased comutations. This aer resents an alternate Fig. 8. SLFOR lot for Welch window for fractional orders a =.2,.5 and 1 Fractional order Parameter Magnitude Fig. 9. The continuum of FrFT function for different values of fractional order a methodology to tune the transition width using FrFT. A triangular window based low-ass FIR filter with cut-off frequency =.5π and length N = 64 is simulated. The above Fig. 12 shows the variability in frequency resonse of the filter with change in value of fractional order arameter. As the fractional order a is reduced from 1 to, the transition width of window based FIR filter can be made narrow. 7 CONCLUSION The mathematical analysis for obtaining the fractional Fourier transform of Triangular function and Welch win- 5 AUTOMATIKA ,

8 Fractional order Parameter Magnitude Fig. 1. The continuum of fractional Fourier transform of Welch window for different values of fractional order a Fig. 11. SLFOR for Triangular window L = 1 at fractional order a = 1. The value of MSLL is db and SLFOR is -12 db/octave Table 1. Characteristics of Triangular function scaling arameter L =.5 for different values of fractional order arameter a Fractional Order α MSLL HMLW db SLFOR db/octave Table 2. Characteristics of Welch window for different values of fractional order arameter a Fractional Order α MSLL db HMLW SLFOR db/octave dow is resented in the aer. The different sectral arameters of these window functions are obtained by changing fractional order to different values. A good window can achieve low side-lobe levels with minimum increase in Fig. 12. The magnitude resonse of Triangular window based low-ass FIR filter for fractional order a =.5,.45 & 1 main-lobe width. For Triangular and Welch window function, the side-lobe reduces but at the exense of HMLW. The analysis reveals that as the fractional order is reduced, main-lobe width can be minimized and SLFOR can be raised to maximum. Thus, it can be concluded that FrFT of Triangular function and Welch window varies directly with the change in fractional angle a. The new derived model for FrFT of these functions clearly shows that and a tradeoff can be made between increase in main-lobe width and reduced side-lobe level which best suites the desired alication. Triangular function and Welch window can be used as adjustable windows in the fractional Fourier domain for estimating the sectrum of a signal so that a choice can be made between amlitude accuracy and sectral resolution. 228 AUTOMATIKA ,

9 Sectral Analysis of Generalized Triangular and Welch Window Functions using Fractional Fourier Transform REFERENCES [1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Grahs and Mathematical Tables, 9th Edn., Dover, New York, , [2] L. B. Almeida, The fractional Fourier transform and timefrequency reresentation, IEEE Trans. Sig. Proc., 42, , [3] C. Caus and K. Brown, Fractional Fourier transform of the Gaussian and fractional domain signal suort, IEE Proceedings -Vision, Image and Signal Processing, vol. 15, no. 2, 99 16, 23. [4] F. J. Harris, On the Use of Windows for Harmonic Analysis with Discrete Fourier Transform, Proceedings of the IEEE, vol. 66, no. 1, 51 83, [5] R. Jacob, Develoment of Time- Techniques for Sonar Alications, Ph.D. Thesis, Nov. 21, [6] S. Kumar, K. Singh and R. Saxena, Analysis of Dirichlet and Generalized Hamming window functions in the fractional Fourier transform domains, Elsevier Signal Processing, 21, [7] C. S. Lessard, Signal Processing of Random Physiological Signals, Morgan and Clayool, U.S.A., , 26. [8] A. Mostayed, S. Z. K. Sajib and S. Kim, Novel Parameter Estimation Method for chir signals using Bowtie chirlet and discrete fractional fourier transform, Int. Journal of Signal Processing, Image Processing and Pattern, 2, 89 18, 29. [9] P. V. Muralidhar, A. S. Srinivasa Rao, S. K. Nayak. Sectral Interretation of Sinusoidal Wave using Fractional Fourier Transfrom Based FIR window Functions, International Review on Comuters and Software, vol 4., no 6., , 29. [1] A. Nuttall, Some windows with very good sidelobe behavior, IEEE Trans. Acoust., Seech, Signal Process., vol. 29, no. 1, 84 91, [11] H. M. Ozaktas, O. Arikan, M. A.Kutay, G. Bozdagi, Digital comutation of the fractional Fourier transform, IEEE Trans. Signal Process. vol. 47, no. 9, , [12] H. M. Ozaktas, M. A. Kutay and Z. Zalevsky, The Fractional Fourier Transform with Alications in Otics and Signal Processing, John Wiley, Chichester, , 21. Pooja Mohindru was born in Bhiwani, India. She received B.E. Honors Degree in Electronics & Communication in 21 from Deenbandhu Chhotu Ram University of Science and Technology, Murthal and M.E. Degree in 23 from Thaar University, Patiala. She has been awarded Ph.D. in E.C.E. from Thaar University in 214. Presently she is working as Assistant Professor with University College of Engineering, Punjabi University, Patiala. Her research area includes Signal rocessing and Digital communication. She has 7 aers in reuted national/international journals. Rajesh Khanna was born in Ambala, India. He received B.Sc. Eng. degree in Electronics & Comm. in 1988 from Regional Engineering College, Kurkshetra and M.E. degree in 1998 from Indian Institute of Sciences, Bangalore. He was with Hartron R&D centre till Until 1999 he was in All India Radio as Assistant Station Engineer. Presently he is working as Professor at the deartment of Electronics & Communication Engineering at Thaar University, Patiala. He has done his Ph.D. in the area of adative antennas for mobile communication. He has ublished 3 research aers in international/national journals. He has accomlished 2 Research Projects. S. S. Bhatia is currently working as Professor, School of Mathematics and Comuter Alications, Thaar University, Patiala, Punjab, India. He received his M.Sc., M.Phil. and Ph.D. in 1985, 1986 and 1994 resectively, majoring Mathematics. He is associated with Thaar University for the last 24 years. He has vast exerience of teaching at UG and PG level to Science and Engineering students at Thaar University. He has ublished over 55 research aers in Journals, International and National Conferences in the areas of Functional Analysis, Reliability Analysis and Image rocessing. He has guided 2 Ph.D. and 7 M.Phil. scholars. He has accomlished 2 UGC Major Research Projects. He is a Life member of Punjab Academy of Sciences, Indian Society of Industrial and Alied Mathematics and Indian Society of Technical Education. [15] S. N. Sharma, R. Saxena and S. C. Saxena, Tuning of FIR filter transition bandwidth using fractional Fourier transform. Elsevier Signal Processing, 87, , 27. AUTHORS ADDRESSES Asst. Prof. Pooja Mohindru, Ph.D. Deartment of Electronics and Communication Engineering, University College of Engineering, Punjabi University, Urban Estate Phase II, IN-1472, Patiala, Punjab, India ooja_citm13@rediffmail.com Prof. Rajesh Khanna, Ph.D. Deartment of Electronics and Communication Engineering, Prof. S. S. Bhatia, Ph.D. School of Mathematics and Comuter Alications, Thaar University, Nabha Road, IN-1474, Patiala, Punjab, India {rkhanna,ssbhatia}@thaar.edu [16] P. Singla, Desired order continuous olynomial time window functions for harmonic analysis, IEEE Trans. Instrum. Meas., vol. 59, no. 9, , 21. Received: Acceted: [13] T. Ran, D. Bing and W. Yue, Research rogress of the fractional Fourier transform in signal rocessing. Science in China: Series F Information Sciences, vol. 49, no. 1, 1 25, 26. [14] E. Sejdic, I. Djurovic, and L. Stankovic, Fractional Fourier transform as a signal rocessing tool: An overview of recent develoments, Elsevier Signal Processing, 91, , 211. AUTOMATIKA ,