f the z-transforms of h(k) and x(n) as in Equation 2.104. ((z) = H (z)X(z) (2.104) The first term represents a contribution from any initial condition- S, and if x(n) = 0 for n < 0, the z-transform of a delayed sequence is Other important properties of the z-transform are shown in Table just z-PX (z), where p is the delay. This is of course, analogous to the shift theorem for the Laplace transform. In both signal processing applications and linear systems theory, linear recursive operators are used extensively, examples in both areas being the design of recursive digital filters amd feedback 2.8.3 Convolution summation property systems in control, respectively, and the general input-output rela- tionship for such a linear recursive system can be written in the form From the linear system theory point of view, this represents one of shown in Equation 2.105. the most valuable properties. Consider the definition of the convol- ution of an input sequence x(n), with the impulse response h(k) of a (2.105) linear system, to yield an output y(n), as in Equation 2.101. (2.101) y(m)= Taking the z-transform of both sides of the expression, and inter- changing the order of summation we obtain Equation 2.106. Taking the double sided z-transform of this equation gives Equa- tion 2.102. (2.102) (2.106) Again, changing the order of summation, and defining p==n-k, we Using the shift theorem, this expression can be written as in obtain Equation 2.103. Equation 2.107. Table 2.2 Properties of the z-transform Property Time series z-transform Region of convergence rex</2/<rax rcy</2/<ray Linearity tax(n)+hy(n) aX(z)+hy(z) At least max ( rcx, rcy)</2/<min(rax, ray) Time shift x(n-m) rex</Z</ax Convolution X(z)Y(z) At least max (k)y(n-k) k=-00 Exponential X(a-1) alra</<</a/r multiplication nx(n) -2dd(2) </z/<ra multiplication - zz dz Product x(n)y(n) rcy</z/<ra Correlation X(z-1)((2) (k)y(n+k) The inverse z-transform 2/15 1 m (z) Re(z) Figure 2.10 Geometrical constructionof the frequency response for a linear system with three poles and