Time-frequency plane behavioural studies of harmonic and chirp functions with fractional Fourier transform (FRFT)

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1 Maejo International Journal of Science and Technology Full Paper ISSN Available online at Tie-frequency plane behavioural studies of haronic and chirp functions with fractional Fourier transfor (FRFT) Renu Jain, Rajiv Saxena and Rajshree Mishra 3,* School of Matheatics and Allied Sciences (SOMAAS), Jiwaji University, Gwalior (M.P.) India Jaypee Institute of Engineering and Technology (JIET), Guna (M.P.) India 3 School of Matheatics and Allied Sciences (SOMAAS), Jiwaji University, Gwalior (M.P.) India * Corresponding author, e-ail: rajshreeishraa@gail.co Received: 8 June 9 / Accepted: 9 Noveber 9 / Published: 9 Deceber 9 Abstract: The behaviour of haronic and chirp functions was studied on the tiefrequency plane with the help of fractional Fourier transfor (FRFT). Studies were also carried out through siulation with different nubers of saples of the functions. Variations were observed in the axiu side-lobe level (MSLL), half ain-lobe width (HMLW) and side-lobe fall-off rate (SLFOR) of the functions. The paraeters of these functions were copared with a siilar set of paraeters of soe of the popular window functions. It can thus be concluded that in the tiefrequency plane, the chirp function provides better spectral paraeters than those of Boxcar window function with soe particular values of rotational angle. A siilar type of inference can also be drawn for the haronic function in the tie-frequency plane. Of course the rotational angle ight vary in this case and a coparative analysis was carried out with Fejer window and the cosine-tip window functions. This study ay prove to be helpful in replacing these existing window functions in a variety of applications where a particular paraeter or group of paraeters of the haronic and chirp functions are found superior. Keywords: haronic function, chirp function, fractional Fourier transfor (FRFT), window function, spectral paraeters, tie-frequency plane Introduction The Fourier transfor (FT) is the ost frequently and extensively used tool in signal processing and analysis []. Fractional Fourier transfor, generally known as FRFT, is a generalisation of FT with an order paraeter `a'. Matheatically, the a th order FRFT F a is the a th power of ordinary FT operation F. The first order (a=) FRFT is the ordinary FT and the zeroth order FRFT is the identity transforation.

2 46 FRFT has now becoe the ost frequently and extensively used tool in the area of signal processing, analysis and optics. It also finds any applications in differential equations, optical bea propagation, quantu echanics, statistical optics and optical signal processing [-4]. In fact, in every area in which FT and frequency doain concepts are utilised, there exists a provision for the generalisation and iprovisation by applying FRFT. Fractionalisation of the transfor started in 99 by Wiener [5] by solving of the differential equations. Condon in 937 [6] and Bargann in 96 [7] followed his work. Naias in 98 and Mcbride and Kerr in 987 did rearkable work in developing the transfor, its algebra and calculus [8-9]. The definition of continuous-tie FRFT given by Naias is validated to date. In 993 Aleidia cae out with the tie-frequency representation of FRFT []. Aleidia, Zayed and Mustard contributed in developing the properties of FRFT [-3]. On the other hand, Ozaktas and his tea and any other researchers worked on coputation of FRFT and on discrete version of FRFT [4-5]. Due to diversified applications of FRFT any researchers are continuously contributing in developing the transfor, its discrete version and applications [3, 4, 6]. During the last 5 years an enorous aount of research publications have been noticed in this area and FRFT is now established as a very powerful tool for alost all scientific and engineering applications. In spectral analysis of a function, three paraeters are of prie iportance, i.e. the axiu side-lobe level (MSLL), the half ain-lobe width (HMLW) and the side-lobe fall-off rate (SLFOR). An iproveent in these paraeters can iprove the spectral perforance of the function. Figure provides a graphical representation of these paraeters agnitude in db MSLL = -4.64db SLFOR = -.3db HMLW= noralised frequency Figure. Paraeters of haronic function The ain objective of this paper is to study and analyse the behavioural changes of haronic and chirp functions in the fractional Fourier doain. The haronic function is defined by exp ( ix) and the chirp function is defined by exp (-i /4) exp (i x ) where the doain of the functions is a set of real nubers. We observed changes in the above three paraeters of both functions in the FRFT doain, i.e. in tie-frequency plane.

3 46 Defining the Fractional Fourier Transfor (FRFT) The FRFT F a of order a R is a linear integral operator that aps a given function f(x), x R on to f a ( ). R by a f ( ) F ( ) K (, x ) f(x) dx... () a a where K a (, x) is kernel of transfor and defined as follows: x K a (, x ) C exp{ i ( ( x )cot )}... () sin with C where exp( i( sgn(sin ) / 4 / )) i cot... (3) sin a. Equation () is defined only for a, i.e. is not a ultiple of. K a (, x) = ( x ) for a 4 or... (4) and K a (, x) = ( x ) for a 4 or ( )... (5) where is an integer. Since a appears in equations only in the arguent of trigonoetric functions then the definition is periodic in a (or ). Thus we will often liit our attention to the interval a (, ] or (-, ] and soeties a [, 4 ) or [, ). When a is outside the interval a we need siply replace a by it s odulo 4 equivalent. Different cases can be tabulated, as given in Table. Table. Soe iportant cases of FRFT [6] Operation on signal Value of paraeter a Value of a / Kernel FT operator 4 + ( 4 ) / exp( i x ) Parity operator 4 + ( ) ( x ) Identity operator 4 ( x ) Inverse Fourier Transfor (IFT) operator Fractional operator F = F F = P F 4 = I 4 +3 ( 4 3) / exp( i x ) F 3 = F -

4 46 The discrete version of FRFT is known as discrete fractional Fourier transfor (DFRFT) and evaluation of FRFT is done by using DFRFT coputational techniques [5]. The DFRFT of a signal f(x) can be coputed by following four steps: (i) Multiply the function by a chirp (a function whose frequency linearly increases with tie), (ii) Take its Fourier transfor with its arguent scaled by cosec, (iii) Again ultiply with a chirp, and (iv) Multiply with a coplex constant. FRFT is now a widely studied transfor and it is observed that FRFT of a signal exists under the sae conditions as those for FT. The analytical expressions for FRFT of soe coon signals have been calculated as presented in Table. Table. FRFT of standard functions [4] Function FRFT e i / exp(-i u tan ) cos ( u ) e i( / / 4 ) sin e i (cot (u )-u cosec ) ( u ) i( / / 4 ) e i u cot sin e exp( i u ) e i / cos exp( u ) exp(- u ) e -i (tan ( u )-u sec ) exp(i u ) i tan [( tan ) /( tan )] i u cot e FRFT Analysis of Haronic Function Spectral paraeters of the haronic function were observed in fractional Fourier doain through siulation. In one of the studies, the fractional angle was varied fro to π/ keeping the length of the haronic function constant. These siulation studies were ade by using DFRFT coputation techniques.

5 463 Figure shows haronic function in tie doain and Figure 3 shows how the haronic function evolves into its Fourier doain as angle α increases fro to π/. Figure 4 represents the siulation results of haronic function in tie-frequency plane, obtained by varying either the length of function (N) or by changing the value of fractional angle (α). Soe of the siulation results are given in Table 3..5 frft n Figure. FRFT of haronic function at a= agnitudes frft at a=. frft at a=.5 frft at a= n Figure 3. Magnitudes of FRFT of haronic function by varying 'a' It was observed that for sall sizes of signal (6, 3, 64), as fractional angle α increases in the tie-frequency plane, MSLL of the function decreases in a regular anner, while a slight increent is noted in SLFOR. The HMLW of the function is also regular in nature except at a few particular points, and shows increents in the tie-frequency plane (Figure 5-7). For large nubers of saples, the haronic function shows irregular behaviour in general in the tiefrequency plane. In this case, it shows regularity in its behaviour only in a saller part of α doain. During the studies of all the above cases of haronic function in the tie-frequency plane, soe specific values of fractional angle α have been noted, whereat paraeters of the haronic function in FRFT doain are better than those of Fejer window function [7] in the sense of spectral perforance.

6 FRFT of 6 point haronic function FRFT of 3 point haronic function FRFT of 8 point haronic function - FRFT of 64 point haronic function FRFT at a=. FRFT at a=.4 on x-axis - noralised frequency -4 FRFT of 56 point haronic function FRFT at a=.6 FRFT at a=.8 FRFT at a= on y-axisagnitude of FRFT of haronic function in db (decibel) Figure 4. FRFT of haronic function for different sizes of saples Table 3. Paraeters of FRFT of haronic function Fractional angle α * = decibel Saple under consideration MSLL HMLW SLFOR in db * in db * 3π/ π/ π/ π/ π/ π/ π/ π/ π/

7 465 for 6 saples for 3 saples - for 64 sales MSLL fractional variable Figure 5. MSLL of FRFT of haronic function for different sizes of saples S LFOR for 6 saples for 3 saples for 64 saples fractional variable Figure 6. SLFOR of FRFT of haronic function for different sizes of saples for 6 saples for 3 saples for 64 saples. HMLW fractional variable Figure 7. HMLW of FRFT of haronic function for different sizes of saples

8 466 FRFT Analysis of Chirp Function FRFT of the chirp function was obtained and studied through siulation. Figure 8 shows chirp function in tie doain while Figure 9 shows how the function transfors fro tie doain to frequency doain. Figure shows the siulation results. Table 4 represents soe of the siulation results for FRFT of a chirp function. It was observed that in general the chirp function behaves in an irregular anner in tiefrequency plane. Figure indicates its nature in α doain for different sizes of saples and Figure shows changes in MSLL of the function in fractional doain. Variation in MSLL increases in α doain when the nuber of saples is increased. In this case regular increents were noted only in a little part of α doain close to the angle α = π/. HMLW of the function in general increases regularly with slight irregular variation for sall saples, as shown in Figure, while SLFOR of the sae function also shows variation in an irregular anner but varies between -3 to -6 db. Large nubers of saples produced two specific results as discussed later frft n Figure 8. FRFT of chirp function at a= FRFT at a =. FRFT at a =.5 agnitudes gnit ude.5.5 FRFT at a = n Figure 9. Magnitudes of FRFT of chirp function by varying 'a'

9 FRFT of 6 point chirp function FRFT of 3 point chirp function FRFT of 64 point chirp function FRFT of 56 point chirp function FRFT at a=. FRFT at a=.4 FRFT at a=.6 FRFT at a=.8 FRFT at a= FRFT of 8 point chirp function on x-axisnoralised frequency on y-axisagnitudes of chirp FRFTs in db (decibel) Figure. FRFT of chirp function for different sizes of saples Table 4. Paraeters of FRFT of chirp function * = decibel Fractional angle α Saples under consideration MSLL (in db * ) HMLW SLFOR (in db * ) π/ π/ π/ π/ π/ π/ π/ π/

10 MSLL in db MSLL for 6 saples MSLL for 3 saples MSLL for 64 saples fractional variable Figure. MSLL of FRFTs of chirp function for different sizes of saples HMLW 4 3 HMLW for 6 saples HMLW for 3 saples HMLW for 64 saples fractional variable Figure. HMLW of FRFTs of chirp function for different sizes of saples Discussion It can be concluded that in the case of haronic function, changes in HMLW, SLFOR and MSLL are in general regular in nature for sall saples, but irregularity is observed when the size of saples is increased in the FRFT doain. Two specific results for haronic function for α = 4π/ and 9π/ were obtained during siulation studies and analysis, as shown in Figures 3-4. Paraeters of these results were copared with those of cosine tip window function and Fejer window function. Superiority in results can be observed fro Table 5. Frequency doain representation of haronic function is included with its shifted version for coparison. These specific results can replace the window functions to generate a variety of applications and can also be applied according to the requireent of the spectral paraeters.

11 469 The behaviour of chirp function changes with the variation of fractional angle α with respect to the nuber of saples in tie-frequency plane. This siulation studies have generated two iportant results with α = 3π/ and α = 7π/ for chirp function in fractional Fourier doain. Like the haronic function, chirp function is not a window function but paraeters of these two results are far better than the frequency doain paraeters of Boxcar window function. Figure 5 shows db (decibel) plot of chirp function at α = 3π/ in tie-frequency plane. Figure 6 shows db plot of the sae function at α = 7π/. Coparative paraeters are shown in Table 5. These two plots prove to be a better replaceent for Boxcar window function and can be used as a tool for specific applications. agnitude in db - -5 a d agnitude in db noralised frequency 5 noralised frequency Figure 3. Magnitude of FRFT of haronic function Figure 4. Magnitude of FRFT of haronic at a =.8 function at a =.9 agnitude in db agnitude in db - -5 a - d noralised frequency noralised frequency Figure 5. Magnitude of FRFT of chirp function Figure 6. Magnitude of FRFT of chirp function at a =.3 at a =.7

12 47 Table 5. Coparative paraeters Function Fractional angle α MSLL (in db) HMLW SLFOR (in db) Haronic Cosine tip window Haronic Fejer window Chirp Boxcar window Chirp Conclusions The FRFT doain analysis of haronic and chirp functions has been carried out and a few cases of these functions are observed to show better spectral perforances in coparison to the existing weight functions. This clearly indicates that haronic and chirp functions with soe specific values of FRFT with order `a' provide a better candidate in applications wherever these weight functions are used. References. R. N. Bracewell, "The Fourier Transfor and Its Applications", nd Edn., McGraw Hill, New York, A. W. Lohann, "Iage rotation, Wigner rotation and the fractional order Fourier transfor", J. Opt. Soc. A. A, 993,, D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transfor and their optical ipleentation: I", J. Opt. Soc. A., A, 993,, H. M. Ozaktas, M. A. Kutay and Z. Zalevsky, "The Fractional Fourier Transfor with Applications in Optics and Signal Processing", John Wiley and Sons, New York,. 5. N. Wiener, "Heritian polynoial and Fourier analysis", J. Math. Phys. MIT, 99, 8, E. U. Condon "Iersion of the Fourier transfor in a continuous group of functional transforations", Proc. Nat. Acad. Sci. USA, 937, 3, V. Bargann, "On a Hilbert space of analytic function and an associated integral transfor, Part I", Coun. Pure Appl. Math., 96, 4, 87-4.

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