Fractional Shift Invariant System in the Linear Canonical Transform Domain

Fractional Shift Invariant System in the Linear Canonical Transform Domain

A. S. Gudadhe, P. P. Thakare

Department of Mathematics, Govt. Vidarbha Institute of Science and Humanities, Amravati

Abstract

Keywords: Linear Canonical transform, shift invariant system, impulse response, canonical convolution

Abstract:

Shift invariant systems are especially important in signal processing, image processing. Fourier transform plays an important role in dealing with shift invariant system. Gudadhe had extended this concept to fractional domain and apply it to fractional Fourier transform.

Here we extend the concept of shift invariant system in the Linear Canonical transform domain.

1. Introduction

   Linear Canonical transform (LCT) which is the generalized of Fractional Fourier transform (FrFT) of time and frequency plane is defined by,

   = = , ,

   where the kernel is,

   (u, t)= 1

   2

   (1)

   2

   1

   2

   , 0

   = 2 (), = 0 (1)

   and , , , are real numbers with = 1. The kernel can also be viewed as a 2×2 matrix

   = with determinant = 1.

   This transform is linear but not shift invariant. Gudadhe and Thakare [3] had defined

   2 1 1 1

   unitary canonical shift operator = 2

   which shifts the support of signal in the time-frequency plane by the amount .

   Linearity and time invariance are the basic properties of system. Many physical processes can be modeled by linear time invariant systems which can be analyzed in details. Moreover the complete characterization of linear shift invariant system can be developed in terms of its response to a unit impulse.

   As mentioned in [4], one of the most important properties of Fourier transform with regard to its use in dealing with linear time invariant system, is its effect on convolution operator.

   It is derived that if , are input and output of linear time invariant system with impulse response then = , where denotes convolution operator given by

   =

   (2)

   Then the convolution theorem of Fourier transform gives

   = (3)

   where , are Fourier transform of , resp. Moreover is Fourier transform of impulse response i.e. = .

   Thus (3) gives =

   This result is consistent with shifting property of Fourier transform. That is if

   = then

   =

   In this paper a new linear canonical invariant system is formulated which will give similar type of consistency with the corresponding shifting property of Linear Canonical transform as shown above for conventional Fourier transform.

   It is known that if LCT of is denoted by

   then

   2

   2

   = 2 1

   1 1

   Simple calculation shows that right hand side of above equation can also be expressed as

   1 2 + 2

   2 1

   1 1

   =

   2

   2

   2

   2

   =

   2 1

   2 + 2

   2 2 2 2 2 +

   2

   2

   2

   2

   2

   2

   2

   2

   = 2

   (4)

2. Convolution for Linear Canonical Transform

   [2] gives new definition for canonical type convolution in terms of usual convolution .

   2

   For any function , let us define the functions by =

   functions f and g, we define the convolution operation by

   . For any two

   = =

   2

   (5)

   Let = and , , denote the Linear Canonical transform of , ,

   respectively. Then

   =

   2

   (6)

   This can be considered as generalization of (3) since the coefficient term is of unit modulus.

3. Linear Canonical shift invariant system

   We define linear canonical shift invariant system as canonical convolution of two functions, one

   is and other is , where is impulse response at

   That is =

   By using convolution theorem as given in (6) for canonical domain we have

   =

   2 2

   2

   2 + 2

   =

   2

   2

   2

   = 2

   (7)

   Note that this result coincides with the shifting property for Linear Canonical transform in its new form as given in (4).

   Thus output of the linear canonical shift invariant system is given by canonical convolution of

   the functions and , where is the impulse response.

4. Conclusion

A new definition of linear canonical shift invariant system is introduced. This definition generalized the idea of linear shift invariant system in the Linear Canonical transform domain.

References

1. Gudadhe A S: Generalization of linear shift invariant system in the fractional domain, Int. Journal of Math. Analysis, Vol.3, 2009, no. 21, 1027-1030.

2. Gudadhe A S: Product and Convolution Theorem for the Generalized Linear Canonical Transform, Vidarbha Journal of Science, Vol.7, No. 3-4, 2012, 26-30.

3. Gudadhe A S: Unitary and Hermitian Linear Canonical Operators (communicated), JOSA

4. Oppenhein A V et al: Signals and systems, Reprint at Prentice Hall of India, New Delhi,(1987).