Sharpening the response of an FIR filter using Fractional Fourier Transform

Sharpening the response of an FIR filter using

Fractional Fourier Transform

Somesh Chaturvedi 1, Girish Parmar 2, Pankaj Shukla 3,

Department of Electronics Engineering, Rajasthan Technical University, Kota-324022, India

Abstract- In this paper we have implemented FIR filter with the help of Kaiser Window and Fractional Fourier Transform (FRFT). The window shape parameter is tuned for the transition band by considering linear phase FRFT Finite Impulse Response (FIR) filter. Here FRFT of Kaiser Window is taken and convolved with the response function for tuning purposes of the transition band which makes effective transition band. This proposed method includes the change of parameters of Kaiser window by which other windows like Rectangle, Bartlett, Hamming Blackman and Hanning windows are generated by using FRFT. The efficiencies of this method in terms of main lobe and side ripples are better than the above mentioned windows under Fourier transform.

Index Terms—Fourier Transform, Fractional Fourier transform, Kaiser window, FIR filter.

I. Introduction

Now-a-days Fractional Fourier transform has been widely used as tool in signal processing [1], quantum mechanics and quantum optics [2-5], pattern recognisation [6] and study of time frequency distribution [7]. As the mathematics reveals that the generation of the Fourier transform is the fractional Fourier transform [FRFT], which are explained in [2,4]. The FRFT can be interpreted as the rotation of angle α in the time frequency plane. The basic properties of FRFT as, when rotation angle α = π/2 corresponds to the classical Fourier transform, α = 0 corresponds an Identity operator and when we apply FRFT on a signal, the signal decomposes into chirps i.e., complex exponentials with linearly varying instantaneous frequencies.

Since FIR filter of linear phase, inherent stability, negligible quantization noise and can be efficiently implanted in multi-rate digital signal processing applications, so it is used frequently in many areas of signal processing then the infinite impulse response (IIR) filters. In general FIR filters can be derived with the help of windows, sampling and optimal polynomials [8-9]. The design of filters with Fourier transform provides the greatest mismatch occurring as the edges of the transition band. The ringing occurs at the edges because of the finite series Fourier transform cannot produce the sharp edges at the transition band.

So suitable windows are to be taken to produce an effective transition band. Kaiser window function provides maximum side lobe decreases and tuning in dBs. The parameter β (Bessel’s coefficient) of Kaiser Window we can achieve different type of windows like Rectangular, Bartlett, Hanning, Hamming, Blackman window. By combining Kaiser window with FRFT we can get the better results than the normal Fourier transform cases. The organization of the paper is as follows:

The section II deals with Fractional Fourier transform, section III deals with design of windows using Kaiser, section IV deals with implementation of tuning procedure using convolution. Section V deals with simulation and results and section VI deals with the conclusion.

II. Fractional Fourier Transform

FRFT is a linear operator that corresponds to the rotation of the signal through an angle which is not a multiple of π/2, i.e. it is there presentation of the signal along the axis u making an angle α with the time axis.

i). Definition

The FRFT is defined with the help of the transformation kernel Kα as [5-6] :

( ) 2

( , ) ( ) 2

1 cot

2

t u if is a multiple of K t u t u if is a multiple of

j

if is not a multiple of α

α

π

α π

π

α

_α

_π

π

  _{∂ −}      = ∂ + +    −       (1)

Another useful form of writing the square root factor preceding the transformation kernel; Kα can be obtained by using [9]:

1 cot

2 2 sin

j

j α jeα

π π α

− ₌ −

(2)

The FRFT is defined using this kernel as FRFT of order α of x (t) denoted by

X

_α

( )

u

x t K t u dt

( )

_α

( , )

∞

−∞

=

_∫

(3)

where;

2 2

( cot / 2) ( cot / 2) cos ( )

1 cot

( ) ;

2 ( )

( ) ; 2

( ) ; 2

j u j t jut ec dt

j

e x t e

X u if is not a multipleof

x t if is a multiple of

x t if is multiple of

α α α

α

α

π

α

π

α

π

α π

π

∞ − −∞  ₋        =      − +    

∫

(4)

ii). Computation of the FRFT

The FRFT of a signal x(t) can be computed by the following steps :

1. product by a chirp-chirps are functions whose frequency is linearly increasing with time.

2. a Fourier transform with its argument scaled by cosec(α). 3. another product by a chirp.

4. a product by a complex constant.

Let, Fα denotes the operator corresponding to the FRFT of angle; α. Under this notation, some of the important properties of the FRFT [7-9] operator are listed below:

(a) Identity operator. Fo is the identity operator. The FRFT of order α = 0 is the input signal itself. The FRFT of order α= 2π corresponds to the successive application of the ordinary Fourier transform 4 times and therefore also acts as the identity operator, i.e. Fo = Fπ/2 = I:

(b) Fourier transform operator. Fπ/2 is the Fourier transform operator. The FRFT of order α= π/2 gives the Fourier transform of the input signal.

(c) Successive applications of FRFT. Successive applications of FRFT are equivalent to a single transform whose order is equal to the sum of the individual orders. Fα(Fβ)= Fα+β

(d) Inverse. The FRFT of order - α is the inverse of the FRFT of order a since F-α(Fα)= Fα-α = Fo= I:

iv). Applications of FRFT

The FRFT, being just an extension of the classical Fourier transform, can be used effectively in all situations where the Fourier transform is presently being used. Some gains can be expected in most of these applications because of the additional degree of freedom (angle of rotation) that the FRFT provides us with.

III. DESIGN OF WINDOWS USING KAISER

In a Kaiser window, the side lobe level can be controlled with respect to the main lobe peak by varying a parameter α .The width of the main lobe can be varied by adjusting the length of the filter. The Kaiser window function is given by:

0

0

( )

1

,

( )

2

( )

0 ,

k

I

N

for n

I

n

otherwise

β

α

ω

−



≤



=







(5)

Where α is an independent variable determined by Kaiser. The parameter β is expressed by:

0.5 2

2

1

1

n

N

β α

=





−



_



_





−













(6)

The modified Bessel function of the first kind,

I x

₀

( )

, can be computed from its power series expansion given by:

2

0

1

1

( ) 1

! 2

k k

x

I x

k

∞ =



_{ }



= +



_{ }



 









∑

(7)

=

( )

(

) (

( )

)

2 3

2 2

2

2 2 2

0.25

0.25

0.25

1

...

(2!)

1!

3!

x

x

x

+

+

+

+

(8)

10

1

20 log

1

p p

p

A

=

+ ∂

− ∂

dB

(9)

10

20 log

s s

A

= −

∂

dB

(10)

The transition bandwidth is:

s p

F

f

f

∆ = −

(11)

The different values of β emulates the different window functions which is expressed in table-1 TABLE-I β VERSUS WINDOW FUNCTION

β Window Function

0 Rectangular

1.33 Bartlett

3.86 Hanning

4.86 Hamming

7.04 Blackman

3 Kaiser

In this paper an attempt has been done for evalued all windows using FRFT on Kaiser window

The modified impulse response is computed using:

1

( )

( )

( ).

2

k d

N

h n

=

w n h n

for n

≤

−

(12)

The main lobe width, the peak side lobe level can be varied by varying the parameters α and N. The side lobe peak can be varied by varying the parameter α. The window shape parameter provides a convenient continuous control over the fundamental window trade-off between side-lobe level and main-lobe width. Larger the value β provides lower side-lobe levels, but at the price of a wider main lobe. Widening the main lobe reduces frequency resolution when the window is used for spectrum analysis.

IV. IMPLEMENTATION OF TUNING PROCEDURE

An FIR digital filter operation is a linear convolution of the finite duration impulse response with the input signal sequence

x n

( )

. The filtering operation in the frequency domain is done using the FFT/IFFT algorithms [11]. Impulse

response,

h n

( )

for Kaiser window [12]. FIR filter is given by

h n

( )

=

w n h n

_k

( )

_d

( )

where

h n

_d

( )

are the desired or

ideal impulse response, and

w n

_k

( )

, the Kaiser window sequence.

In this scheme, in place of impulse response coefficients, window function coefficients

w n

k

( )

are computed

Results for method of tuning using Kaiser for different window value shown in fig: 1-6.

1) Rectangular window (β=0)

Figure1 Frequency response of Kaiser window for β=0 at different values of α

2) Bartlett window (β=1.33)

3) Hanning window (β=3.86)

Figure 3 Frequency response of Kaiser window for β=3.86 at different values of α

4) Hamming window (β=4.86)

Figure 5 Frequency response of Kaiser window for β=7.04 at different values of α

6) Kaiser window (β=3)

Figure 6 Frequency response of Kaiser window for β=3 at different values of α

VI. CONCLUSION

In this paper we implemented FRFT Rectangular ,Bartlett, Hanning, Hamming and Blackman window from FRFT Kaiser window as the parameter ‘β’ changed for different values β=0,1.33,3.86,4.86,7.04 and 3.

FIR filters should have less transition region and less ripples in the pass and stop band. This can be achieved by

reducing the main lobe width and side lobe level. From this work we can get the optimal FIR filters by using FRFT, As ‘α’ decreases from 1 to 0.1 main lobe width decreases .So we can reduces the transition region.

series. In this work, variability in the transition band characteristics has been achieved by designing the filter by convolving the FRFT of window function. This approach does not require a redesign involving the computation of impulse response coefficients as decreasing the FRFT angle is observed to be analogous to increasing the filter order. Computational burden can thus be reduced by this method as sharp transitions are obtained on line by simply reducing the FRFT order.

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