For this given specification 10 KHz of sampling frequency is chosen. Equiripple filter design method is opted. Equiripple method provides same tolerance as that of Kaiser Window with less number of filter orders. Filter is designed using ‘fdatool’ of the MATALAB software. | Chapter 1 Mathematical Review Electronic signals are complicated phenomena and their exact behavior is impossible to describe completely. However simple mathematical models can describe the signals well enough to yield some very useful results that can be applied in a variety of practical situations. Furthermore linear systems and digital filters are inherently mathematical beasts. This chapter is devoted to a concise review of the mathematical techniques that are used throughout the rest of the book. Exponentials and Logarithms Exponentials There is an irrational number usually denoted as e that is of great importance in virtually all fields of science and engineering. This number is defined by lim 1 - oo X Unfortunately this constant remains unnamed and writers are forced to settle for calling it the number e or perhaps the base of natural logarithms. The letter e was first used to denote the irrational in by Leonhard Euler 1707-1783 so it would seem reasonable to refer to the number under discussion as Euler s constant. Such is not the case however as the term Euler s constant is attached to the constant y defined by y lim N x N 1 V - - loge N i n I The number e is most often encountered in situations where it raised to some real or complex power. The notation exp x is often used in place of ex since 1 2 Chapter One the former can be written more clearly and typeset more easily than the latter especially in cases where the exponent is a complicated expression rather just a single variable. The value for e raised to a complex power z can be expanded in an infinite series as co exp z X f i1-3 n on- The series in converges for all complex z having finite magnitude. Logarithms The common logarithm or base-10 logarithm of a number x is equal to the power to which 10 must be raised in order to equal x y log10 x o x 10y The natural logarithm or base-e logarithm of a number x is equal to the power to which e must be raised in
