COMPARATIVE STUDY OF DST INTERPOLATION APPROACH OF FRACTIONAL ORDERS

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like GrunwaldLetnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.


INTRODUCTION
Fractional order of differentiation are more mysterious because they have no obvious geometric interpretation [1]- [4]. This subject started becoming more popular when it was realized that, compared to Frick's laws of diffusion, leads to the derivatives and integrals with half order for calculating the certain electrochemical problem is more convenient and economical.
Digital fractional order based differentiator applications is applicable in biomedical signal processing, digital signature verification, sharpness of images in digital image processing, neural networks, collection of real-time data using cloud computing etc [4]- [10]. D β f (x) = d β f (x)/dx β is a β th derivative order for a function f (x). If β takes positive integral value then we get ordinary derivatives, otherwise it is known as fractional order derivative such that Re(β ) > 0.
In this section, design approach of differentiator will be discussed of non-integer order derivatives definitions as Grunwald-Letnikov, Weyl's and Conformable derivative. In section 2, the different derivatives definitions are discussed. In section 3, transfer function of DST-IV is determined using various non-integer derivatives further, we also determine the transfer function of DST-I, DST-II and DST-III [11] [12]. In section 4, computative problems and comparative analysis is discussed and at last, conclusion discussed based on DST interpolation approach.

DEFINITIONS OF VARIOUS FRACTIONAL ORDER DERIVATIVES
2.1. Grunwald-Letnikov fractional derivative. Fractional derivative of a function f (t) of order Re (β ) > 0 using Grunwald-Letinkov definition.
The symbol of gamma function is denoted Γ(·) Theory of gamma function is generalizing the factorial function of natural numbers.
D β e at = a β e at (2) exists for all f ∈S and all β with Re(β ) > 0. Where S is the class of all functions f which are infinitely differentiable everywhere. β = n − ν ν > 0 and the integer with smallest value is denoted by n and n always greater than ν. If f is a function, not necessarily of class S, for which W −β f (t) exists and has n continuous derivatives; W β is represented by Weyl's derivative.
Where E n = (−1) n d n dt n W β e at = a β e at (6) t>0 i.e. for all values of t, β ∈ (0, 1). If an condition exists for a function g which is β -differentiable with in the range (0, b), with condition b > 0 and lim t→0 + D β (g(t)). Its expression can be define as If function g is β -differentiable then the conformable fractional derivative of some elementary function is 3. DESIGN METHOD FOR VARIOUS FRACTIONAL ORDER DERIVATIVES USING DST-IV 3.1. Design method for Grunwald-Letnikov fractional order derivative using DST-IV.
Suppose we have a signal f (t) in continuous-time domain and signal f (t) are sampled and converted into f (0), f (1), · · · , f (P − 1) i.e. finite-time sequence/discrete-time sequence. Then DST-IV function is defined as After putting the value eq.(13) into eq. (14), we get Put t in place m in the previous equation, here t represent continuous-time and m represent discrete-time.
f (t) is a interpolated signal for the continuous-time domain and b(n,t) is a basis interpolated Apply Grunwald-Letnikov fractional derivative definition of β th order on equation (16) From eq. 18 fractional derivative of basis interpolated function is D β b(n,t) = 2 P ∑ P−1 k=0 π(k+ 1 2 ) P β sin π(n+ 1 2 )(k+ 1 2 ) P sin π(t+ 1 2 )(k+ 1 2 ) P + πβ 2 (19) Putting the value of D β b(n,t) into the eq.(18) The ideal frequency response of digital differentiator and its transfer function approximates equal Delay value denotes by I and the equation of the FIR filters system function is When an input signal u(m) is passed through a system FIR filter then its output generate u(m − 1), u(m − 2), ..., u(m − P + 1) samples with equal amount of delay in each of the input signal.
The output of the FIR filter is The filter coefficients h(υ) is determined from the eq.(20), when y(m) approximately equal to For solving this problem an index mapping technique is used The eq.(26) can be simplified after linking eq.(20) and eq.(24) FIR filter coefficients is determined after equating eq.(21) into (29), With the help of window techniques, we can modify the coefficients of FIR filter. So, in this paper we are using Hanning window and it's transfer function is defined below as Modified coefficients of FIR filter using window techniques is The system performance of the digital fractional order differentiator can be evaluated for DST-IV method with the help of integral error squares formula in frequency domain.
Above expression in term of E is used for checking performance of designing approach of digital fractional order differentiator.

Design method for
Weyl's fractional derivative using DST-IV. Similarly for DST-IV method the system transfer function using Weyl's fractional order derivative is With the help of window techniques, we can modify the coefficients of FIR filter. In this paper we are using Hanning window Modified coefficients of FIR filter using window techniques is

Design method For Conformable fractional derivative using DST-IV. Similarly for
DST-IV method the system transfer function using Conformable fractional order derivative is With the help of window techniques, we can modify the coefficients of FIR filter.In this paper we are using Hanning window and it's transfer function is defined below as Modified coefficients of FIR filter using window techniques is Orders With the help of above given error table, we can determine which digital fractional order differentiator (DFOD) will be suited for our proposed mehtod.In DST-IV case, the order from onward β = 0.3 to β = 0.9 the size of error for Grunwald-Letinkov based digital fractional order differentiator are smaller than Conformable and Weyl's based digital fractional order differentiator.Size of error for fractional order onward β = 0.7 to β = 0.9 Conformable based DFOD is smaller than the Weyl's based DFOD.     DST-III method system transfer function is given below: The transfer function of DST-III using Grunwald-Letnikov fractional order derivative is The transfer function of DST-III using Weyl's fractional order derivative is The transfer function of DST-III using Conformable fractional order derivative is     DST-II method system transfer function is given below The transfer function of DST-II using Grunwald-Letnikov fractional order derivative is The transfer function of DST-II using Weyl's fractional order derivative is The transfer function of DST-II using Conformable fractional order derivative is Orders On the basis of above given error table, we can determine which DFOD will be suited for our proposed design mehtod.In DST-II case, the order from onward β = 0.3 to β = 0.9 the size of error for Grunwald-Letinkov based DFOD are smaller than other DFOD.For order β = 0.7 size of error for weyl's DFOD is smaaler than Conformanble DFOD, while for order β = 0.9 size of error for Conformable based DFOD is smaller than the weyl's based DFOD. Fig. 9,10,11 and 12 shows the frequency response.    DST-I method system transfer function is given below The transfer function of DST-I using Grunwald-Letnikov fractional order derivative is The transfer function of DST-I using Weyl's fractional order derivative is The transfer function of DST-I using Conformable fractional order derivative is Orders On the basis of above given error table, we can determine which DFOD will be suited for our proposed design mehtod.In DST-I case, the order from onward β = 0.3 to β = 0.9 the size of error for Grunwald-Letinkov based DFOD are smaller than other DFOD.Onwards β = 0.7 to β = 0.9 size of error for Conformable based DFOD is smaller than the weyl's based DFOD.

CONCLUSION
In this paper, comparative analysis of DSTs interpolation approach of Grunwald-Letinkov, Weyl's and Conformable DFOD with respect to ideal response are presented. On the basis of computative problems we conclude that, Grunwald-Letinkov FOD with DST-IV is well suited for optimal design values P = 100, I = 50, λ 1 = 0.9. Weyl's DFOD perform better than Conformable DFOD in case of DST-I,DST-II, DST-III and DST-IV.For order β = 0.9, the size of error for Conformables DFOD is smaller than Weyls DFOD in case of all DST aprroach.
In future our interest is to design a digital fractional order differentiators for other fractional derivatives with DCTs/DSTs. We are also interested in extending interpolation approach to multidimensional DCTs/DSTs.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.