Generalization of Fixed and Flexible Window Functions

Muralidhar P. V *   S.K. Nayak **
* Aditya Institution of Technology and Management, Tekkali, India.
** Berhampur University, Berhampur, India.

Abstract

The Fractional Fourier Transform (FRFT) is the generalization of the classical Fourier Transform (FT). The FRFT was introduced about seven decades ago as literature reveals. It appears that is was remained largely unknown to the signal image processing community to which it may be potentially useful. Here, the authors have introduces a novel derivation for FRFT to extract the spectral parameters like Maximum Side Lobe level Attenuation (MSLA), Half Band Width(HBW), Side Lobe Fall Of Ratio (SLFOR) of Dirichlet and Blackman window functions. And also an attempt is made to get the spectral characteristics of all existing windows like Dirichlet, Bartlett, Hanning, Bohman etc., with generalized equation which consists of FRFT of Dirichlet, Blackman window functions using Alaloui operator.

Keywords :

Introduction

In order to reduce the effects of spectral leakages in Harmonic analysis, windows are used [1-6]. Window functions are successfully used in the areas like interpolation factors to design Anti-Imaging filters, speech processing systems, digital filter design and beam forming [7,8]. Windows are also useful to solve reconstructive errors which are objective functions to design the prototype filters [9]. Windows are essentially applicable in spectral analysis of signals [10,11].According to [8], the parameter of FRFTi.e could not hold good for generalization of FRFT to FT [12]. In this proposed Derivatrion of FRFT, an attempt is made to study the variations of window parameters like HBW, MSLA and MSLFOR by different values of fluid parameter of FRFT to FT at . This paper is organized as follows. Section 1 gives an overview of FRFT, mathematical model of different windows like dirichlet and Blackman window functions. Section 2 presents generalization of different window functions using the proposed formula in equation (34). Conclusion is discussed in the last Section.

1. Fractional Fourier Transform

Fractional Fourier Transform is widely used in Quantum Mechanics and Quantum Optics [13]. Fractional Fourier Analysis can obtain the mixed time and frequency components of signals [14]. It finds various applications like pattern recognisition with some spatial distortion, Image representation, compression and noise removal in signal processing [15-17]. FRFT is used for the Interpretation of sinusoidal signals and design of Digital FIR Filters [18-19].

The continuous–time Fractional Fourier Transform of a signal is defined through an interval [3]

[1]

Where the transform kernel of the FRFT is given by

 

 

[2]

Where indicates rotation of angle of the Transformed signal for FRFT.

1.1 Dirichlet Window Function

The mathematical analysis of Dirichlet window using FT is carried out in the following section. The mathematical characteristic equation of Dirichlet window is

[3]

= 0 otherwise.

Now solving FRFT for equation-(3) yields

Substituting equation-(3) in equation-(1) and applying limits results to

[4]

Let

[5]

Then equation-(4) becomes

[6]

Substituting equation-(3) in equation-(6) results to

[7]

Now multiplying both sides with , you will get

[8]
[9]

Now Integrating and Applying limits on both sides

[10]

Now applying limits on both sides, results to

[11]
[12]
[13]

Let

[14]

According to [17]

[15]

Substituting equation(15) in equation(14) and integrating we get

[16]
[17]
[18]

Finally

[19]

Thus equation(19) is the FRFT based Dirichlet window.

1.2 Four Term Blackman-harris Window Function

The Expression for four Term Blackman-Harris window is [20]

[20]

= 0 otherwise.

[21]
[22]

Then equation(21) becomes

[23]

Substitute equation-(20) in equation-(23) then

[24]

Equation(24) is divided into four parts like I3 ,I4 ,I5 ,I6 where

[25]
[26]
[27]
[28]

Now solving for I3

[29]

The Mathematical Derivation of I3 is same as Dirichlet window derived in section 1.1

According to that

[30]

Similar mathematical analysis of above derivation we could get ,

[31]
[32]
[33]

Finally

[34]

Equation(33) is the FRFT based 4 Term Blackman-Harris window.

2. Generalization of Windows Based on Fractional Fourier Transform

Section 1 gives the Mathematical derivations of Dirichilet and Blackman-Harris windows based on FRFT. By using these two windows, we proposed other windows like Hamming, Hanning, Bohman etc. by varying a of FRFT and Al-aloui parameter [20] shown in equation(34) which is described as FRFT of lowest MSLA(Dirichilet window) and FRFT of highest possible MSLA(here we considered Blackman-Harris window) with Al-aloui operator.It could also give spectral parameters of all exising windows like Bartlett,Hanning,Hamming ,Bohman etc.including lowest MSLA Dirichilet window and Highest possible MSLA Blackman-Harris window which are Tabulated in Table1 along with their spectral responses plotted as shown in Figures 1 to 3.

(35)

Figure 1. Spectral Responses of different windows based on 'a' value between 0.9218 to 1

Figure 2. Spectral Responses of different windows based on 'a' value between 0.8650 to 0.9130

Figure 3. Spectral Responses of different windows based on 'a' value between 0.8620 to 1

Table 1. Spectral Parameters of various windows using Fractional Fourier Transform

Conclusion

An attempt is made such that all existing windows like Bartlett, Hanning, Hamming, Bohman etc. spectral parameters are derived by blending of FRFT of Dirichlet window and FRFT of Blackman-harris window with Al-aloui operator as shown in Table 1 and from the graphs Figure 1 to Figure 3, and to understand the methodology of this paper is modelled as shown in Figure 4. This formula also generalized spectral analysis of forthcoming windows by simply replacing highest possible MSLA with newely possible highest MSLA of forthcoming windows.

Figure 4. Generalization of all window functions using proposed formula

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