FIXED WINDOW FUNCTIONS WITH NEARLY MINIMUM SIDE LOBE ENERGY BY USING FRACTIONAL FOURIER TRANSFORM

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1 International Journal of Advanced Engeerg Technology E-ISSN Research Article FIXED WINDOW FUNCTIONS WITH NEARLY MINIMUM SIDE LOBE ENERGY BY USING FRACTIONAL FOURIER TRANSFORM 1 Rachna Arya, Dr. Kulbir Sgh Address for Correspondence 1 Lecturer (ECED),Shri Ram Muriti Smarak College of Engg Tecnology,bareilly. Asistant Prof. (ECED),Thapar Institute of Engg and Technology, Patiala ABSTRACT In this paper, the Fractional Fourier Transform (FRFT) analyses of fixed wdow functions have been carried out for different values of parameter a. An attempt has been made to study the variation of the parameters Half Ma Lobe Width (HMLW), Side Lobe Fall of Rate (SLFOR), and Maximum Side Lobe Level (MSLL) of fixed wdows with the variation of parameter a the FRFT doma. Here all the fixed wdow functions are compared (i.e. Rectangular wdow, triangular wdow and coe wdow) by decreag the side lobe label energy and creag ma lobe width. The results obtaed confirm the decreag side lobe label energy and creag ma lobe width. 1. INTRODUCTION The FRFT has been found to have several applications the areas of optics [1,, 3] and signal procesg [4, 5]. It also leads to generalization of notion of space (or time) and frequency domas which are central concepts of signal procesg. It depends on a parameter α = aπ / and can be terpreted as a rotation by an angle α the time-frequency plane or decomposition of the signal terms of chirps transforms. It has many applications solution of differential equations, optical beam propagation and spherical mirror resonators, optical diffraction theory, quantum mechanics, statistical optics, optical system design and optical signal procesg, signal detectors, correlation and pattern recognition, space or time-variant filterg, multiplexg [6], signal and image recovery, restoration and enhancement [7, 8], study of space or time-frequency distributions (TFDs) [9]. Wdow functions have been successfully used various areas of signal procesg and communication. Wdow functions have also played an important role the area of signal procesg, such as spectrum estimation, digital filter design, speech procesg, and other fields. There are many differg functional forms, which comprise a subset of classical wdows known to most engeers. A complete review of many wdow functions and their properties was presented by Harris [1]. Wdows are used harmonic analysis to reduce the undesirable effects related to spectral leakage. However, most of the wdow functions usually have a tradeoff between the HMLW and MSLL. Let w (t) represents the wdow function and W ( f ) is its FT. Then, the common properties of all the wdow functions considered can be summarized as: a) w (t) is real, even, non-negative and time limited. b) W ( f ) has a ma lobe at orig and side lobes at both sides. c) w (t) should atta its maximum at t=. d) If the m th derivative of the wdow w (t) is impulsive, then the peak of the side lobe of W ( f ) decays asymptotically as 6m / octave. e) And such that W (f) is of short duration.. FRACTIONAL FOURIER TRANSFORM The FRFT is beg used almost all applications where Fourier transforms were used. One of the reasons is that, the FRFT provides additional degree of freedom to the problem as parameter a gives multidirectional applications various areas of optics and signal procesg particular and physics and mathematics general. An attempt has been made to evaluate the FRFT of the Bartlett and coe wdows. The FT of a function can be considered as a lear differential operator actg on that function. The FRFT generalizes this differential operator by lettg it depend on a contuous parameter a. Mathematically, a th order FRFT is the a th power of FT operator. The FRFT of a function s (t) can be given as: π α exp(. i ) a 4 i i itf F [ s( t)] = S ( f ) = exp( f Cotα exp( f Cotα ) s( t) d t πsα Sα (1) IJAET/Vol.II/ Issue III/July-September, 11/19-5

2 International Journal of Advanced Engeerg Technology E-ISSN Fig 1: FRFT doma Time-Frequency plane where. α= aπ / Let s (t) is the time signal the, time- frequency plane. Its FT is a function of frequency and hence it lies on the vertical axis. Thus by the FT, the representation axis is changed from time to frequency, which corresponds to a counter clockwise rotation over an angle π/ a time frequency plane. When α =π (a=), the representation axis is the reversed time axis, i.e., the time axis rotated over an angle π. When α = 3π/ (a=3), the representation axis is the reversed frequency axis, i.e., the frequency axis rotated over an angle π. It will now be clear that by another rotation over π/, which brgs us back to the origal time axis. Thus all the representations that one can obta by the classical FT correspond to representations on the (orthogonal) axes of time and frequency, possibly with a reversion of the orientation shown Fig 1. In general it can be said that the FRFT of a function is equivalent to a fourstep process: Multiplyg the function with a chirp, takg its FT, aga multiplyg with a chirp and then multiplication with an amplitude factor. 3. FIXED WINDOWS The wdow belongg to this category is havg all the parameters constant or fixed, hence the name. On the basis of function of such wdows, they are aga subdivided to two groups. They are- 1. The wdows generated either with the help of standard functions or by some combation of them.. The wdows can also be derived by truncatg the Fourier series. Their parameters are no doubt constant, but by choog a proper combation of series coefficients a particular manner the wdow s side lobes structure and correspondg ma lobe width can be varied accordgly. These combations IJAET/Vol.II/ Issue III/July-September, 11/19-5 are fixed numbers to get an optimum wdow by this method. Rectangular, Triangular, Coe, Hanng, Hammg, Blackman wdow are some commonly used fixed wdows. Rectangular, triangular, and coe wdows are shown this paper Rectangular Wdow The rectangle wdow is unity over the observation terval, and can be thought of as a gatg sequence applied to the signal function so that they are of fite extent. The Rectangular wdow is defed as: 1 n N 1 w R ( n) = (3.1) otherwise The spectral wdow for the DFT wdow sequence is given by : N 1 θ N W( θ( = exp j θ (3.) 1 θ 3.1. Triangular Wdow or Bartlet wdow: The triangular wdow is defed by: n N n w( n) = N ( 3. 3 ) N w( N n) n N 1 The spectral wdow correspondg to the DFT sequence given : N 1 θ N 4 W( θ( = exp j θ (3.4) N 1 θ 4 This is seen to be equal to the square of the Rectangular wdow DFT frequency response of length N/ Coe Wdow The Coe wdow discrete time doma is defed as

3 International Journal of Advanced Engeerg Technology E-ISSN π n w( n) = cos N n N (3.5) nπ n =,1,... N 1 N The coe wdow time doma and its Fourier Transform is defed by the followg equation respectively 1 w( t ) = cos( π t ) t (3.6) { elsewhewe cos( πf ) W ( f ) = π (1 4 f ) 4. PARAMETERS OF FIXED WINDOW FUNCTION: The parameters of wdow function which are generally used for its evaluation are: Selectivity amplitude ratio (SAR) or (MSLL): This is the peak ripple value of the side lobes and it is evaluated from the lobe magnitude plot of transformed wdow. Selectivity S or Ma Lobe Width (MLW): This is the frequency at which the ma Lobe drops to the peak ripple value of the side lobes. For convenience (HMLW) or S/ is computed. Side Lobe Fall off Rate [SLFOR]: This is the asymptotic decay rate of the side lobe level. The illustrations of the defition of the parameters HMLW (S/), SLFOR, & MSLL (SAR) from the log magnitude plot for Rectangular wdow is given Fig.. 5 RESULTS AND DISCUSSION The FRFT of rectangular, Bartlett and coe wdows for various value of a is shown Fig 3 to 5. The log lear plot for calculatg MSLL and HMLW along with log plot for calculatg SLFOR The value of MSLL, SLFOR and HMLW are tabulated Table 1, & 3 for various value of a. From these figures and Tables it is clear that at a =1, MSLL of Rectangular wdow is -13 down from the ma lobe peak and the side lobes fall off rate -6. per octave,is given for a selected value of a is shown Fig 3,4 & MSLL HMLW SLFO Fig : Log magnitude plot of rectangular Wdow to illustrate the defition of the parameters HMLW (S/), MSLL (SAR) & SLFOR. The value of MSLL, SLFOR and HMLW are tabulated Table 1, & 3 for various value of a. From these figures and Tables it is clear that at a =1, MSLL of Rectangular wdow is -13 down from the ma lobe peak and the side lobes fall off rate -6. per octave,bartlett wdow is - 6 down from the ma lobe peak and the side lobes fall off rate -1. per octave and for coe wdow is -3 down from the ma lobe peak and the side lobes fall off rate is also -1. per octave as the case of FT. As the value of a decreases from 1 to the Ma Side Lobe Energy of fixed wdow starts creag, HMLW decreases from and also decreases SLFOR. Relevant parameters for fixed wdows obtaed by FRFT are listed Table-1 to 3 and show figures at a=.4. Table 1: Parameters of rectangular wdow with variation a. S. No. a MSLL HMLW SLFOR () (bs) (/octave) (a) Rectangular wdow IJAET/Vol.II/ Issue III/July-September, 11/19-5

4 International Journal of Advanced Engeerg Technology E-ISSN (b) Log lear plot (for MSLL) at a =.4 - G a ( c ) Log log plot (for SLFOR) at a =.4 G a i n d B (d ) Log lear plot (for HMLW) at a =.4 Fig 3: Log magnitude plot of Rectangular wdow at a = (a) Triangular wdow IJAET/Vol.II/ Issue III/July-September, 11/19-5

5 International Journal of Advanced Engeerg Technology E-ISSN i n d B G a i n d B (b) Log lear plot (for MSLL) at a = (c) Log log plot (for SLFOR) at a = (d ) Log lear plot (for HMLW) at a =.4 Fig 4: Log magnitude plot of Bartlet wdow at a = (a) coe wdow IJAET/Vol.II/ Issue III/July-September, 11/19-5

6 International Journal of Advanced Engeerg Technology E-ISSN G a i n d B (b) Log lear plot (for MSLL) at a =.5 G a i n d B ( c ) Log log plot (for SLFOR) at a = d ) Log lear plot (for HMLW) at a =.4 Fig 5: Log magnitude plot of coe wdow at a =.4. Table : Parameters of Bartlett wdow with variation a. S. No. a MSLL () HMLW (bs) SLFOR (/octave) Table 3: Parameters of Coe wdow with variation a. S. No. a MSLL () HMLW (bs) SLFOR (/octave) IJAET/Vol.II/ Issue III/July-September, 11/19-5

7 International Journal of Advanced Engeerg Technology E-ISSN CONCLUSION The FRFT doma analysis of Rectangular, Bartlett and coe wdow functions for different values of parameter a had been carried out. The creag value of a reduces the side lobe level which turn broadens the ma lobe reducg resolution. The rectangular, Bartlett and coe wdow can be further concluded that as a is creased from to 1, the MSLL starts buildg up and it goes upto -13 at a=1, the HMLW creases and it goes upto.81 bs and SLFOR creases upto -6 /octave, the bartlet wdow that as a is creased from to 1, the MSLL starts buildg up and it goes upto -6 at a=1, the HMLW creases and it goes upto 1.63 bs and SLFOR creases upto -1 /octave and the coe wdow that as a is creased from to 1, the MSLL starts buildg up and it goes upto -3 at a=1, the HMLW creases and it goes upto 1.35 bs and SLFOR creases upto -1 /octave.this study revels that there is a variation the wdow parameters with variation a and a best optimal solution can be obtaed for a variety of applications. REFERENCE 1. R. N. Bracewell, The Fourier transforms and its applications, McGraw-Hill (1986).. D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their optical implementation-i, J. Opt.Soc. Am. A, 1, (1993) 3. H. M. Ozaktas and David Mendlovic, Fourier transforms of fractional order and their optical terpretation,opt. Commun., 11, (1993). 4. M. F. Edren, M. A. Kutay and H. M. Ozaktas, Repeated filterg consecutive fractional Fourier domas and its applications to signal restoration, IEEE Trans. Signal Procesg, 47, (1999). 5. I. S. Yetik and A. Nehorai, Beamformg ug the fractional Fourier transform, IEEE Trans. Signal Proc-esg, 51, (3). 6. H. M. Ozaktas, B. Barshan, D. Mendlovic and L. Onural, Convolution, filterg and multiplexg frac-tional domas and their relation to chirp and wavelet transforms, J. Opt. Soc. Am. A, 11, (1994). 7. Adolf W. Lohmann, Image rotation, Wigner rotation and the fractional Fourier transform, J. Opt. Soc. Am. A,1, (1993). 8. L. Yu, K. Q. Wang, C. F. Wang and D. Zhang, Iris verification based on fractional Fourier transform, ProcFirst Int. Conf. on Mache Learng and Cybernetics, Beijg, pp (). 9. S. C. Pei and J. J. Dg, Relations between fractional operations and time-frequency distributions and their applications, IEEE Trans. Signal Procesg, 49, (1). 1. F.J. Harris, On the use of wdow functions for harmonic analysis with discrete Fourier transform, Proc. IEEE, Vol.66, January 1978, pp Rajiv saxena and kulbir gh Fractional Fourier Transform: A novel tool for signal procesg J.Indian Inst,Jan-Feb.5 IJAET/Vol.II/ Issue III/July-September, 11/19-5