International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 8– No.1, December 2014 – www.ijais.org 29 FIR Linear Phase Fractional Order Digital Differentiator Design using Convex Optimization Simranjot Singh Research Scholar E.C.E.D. Thapar University Kulbir Singh Associate Professor E.C.E.D. Thapar University ABSTRACT In this paper, design of linear phase FIR digital differentiators is investigated using convex optimization. The problem of differentiator design is first described in terms of convex optimization with different optimization variables’ options, taken one at a time. The method is then used to design first order low pass differentiators and results are compared with Salesnick’s technique and Parks McClellan algorithm. The designed FIR low pass differentiator has improvement in transition width and flexibility to optimize different parameters. The concept of low pass differentiation is further generalized to fractional order differentiators. Fractional order differentiators are designed by using minmax technique on mean square error. Design examples demonstrate easy design procedure and flexibility in the process as well as improvement over existing fractional order differentiators in terms of mean square error in passband. Finally, fractional order differentiators are designed and used for texture enhancement of color images. Better texture enhancement than existing filtering approaches is established based on average gradient and entropy values. General Terms FIR Filter, Fractional order differentiator, Low pass differentiator, Full band differentiator, Image Enhancement. Keywords FIR Filter, Fractional order differentiator, Low pass differentiator, Full band differentiator, Image Texture Enhancement. 1. INTRODUCTION FIR filters are characterized by following equation          where  is input signal and  is impulse response. So, they are linear systems described by convolution relation in input and output [1], [2]. Its frequency response is given by           The response can also be written as vector product, as in [3]        The vector      and  contains real valued filter coefficients. FIR filters have the advantage of linear phase if filter coefficients are symmetric or anti-symmetric about its center point. The frequency response of linear phase filter can be expressed as            where      is amplitude response of the filter, such that            here   is half of impulse response and       The parameters   and  depend on length of impulse response and type of FIR linear phase filter [2]. These parameters for the four types of filters are given by TABLE 1: Properties of the four types of FIR filters      Type I N/2 0   Type II N/2 -         Type III N/2 0  Type IV N/2 -         Digital differentiator is a very important tool in signal processing applications [4]. Higher order differentiation is used in biomedical signal processing, radar and sonar, image processing, velocity and acceleration measurement etc. [5]. Low pass digital differentiators are used to avoid unwanted amplification of noise, as in case of full band ones [6]. The higher order case of low pass differentiators’ design becomes significant as they suffer from noise amplification at higher frequencies even more, because of their exponentially increasing gain. In recent years, fractional operators are investigated extensively in all engineering fields. They have gathered attention due to their application to fractal dimension of science [7]. The frequency response of a fractional order low pass differentiator with order  is given by