 two zeroes (2.107) The complex functions a (w) and B (w) are given by Equations 2.113 and 2.114. (2.113) This is used to define the transfer function, H(z), of the system, which may be written as in Equation 2.108 where H(z) is given by Equation 2.109. (2.114) (2.108) A vector interpretation of the frequency response for a linear zeros (2.109) system with three poles and two zeros is shown in Figure 2.10 and Equations 2.115 and 2.116. (2.115) From the transfer function of a system, the frequency response is found by evaluating the transfer function at z=eji, where the angular frequency W = 2rf. angle Factorising the above transfer function as in Equation 2.110, (2.116) where the zeroes and poles, Z. and P may be complex numbers, and evaluating this function at z=ej", = we obtain the associated fre- quency response as in Equations 2.111 and 2.112. 2.9 The inverse z-transform b(2-z))(z-2)...(2-2) (2.110) In the section describing the Fourier transform, it was straightfor- ward to relate x(n) to X(k) through the definition of the integral transform, and vice versa. In this section we will consider methods (2.111) of obtaining the inverse z-transform, that is, to obtain x(nT) given X(z). The formal method is derived using Cauchy's integral theorem. From the definition of the z-transform, multiplying both sides by (2.112) zm-1 and integrating around a closed contour in the z-plane, gives Equation 2.117. 2/16 Fourier analysis (2.117) Replacing x(n) by its inverse Fourier transform, gives Equation 2.121. (2.121) If the path of integration encloses the origin, then from Cauchy's integral theorem, the right hand side is zero except when m = n, in which case it is 2j. Therefore Equation 2.118 can be obtained, which may be evalu- Interchanging the order of the integral and the summation, and ated using the residue theorem. using the definition of Equation 2.122 gives Equation 2.123, where means convolution of the non-truncated Fourier transform (2.118) X(ej") with the rectangular window transform (ej()). The remaining meth