quation 2.90. (2.90) (2.92) If, as before, we sample this continuous signal x(t), we obtain the sampled signal X (t) given by Equation 2.91, with x(t) = 0 for t ( 0. Making the substitution gives Equation 2.93. (2.91) z = e T (2.93) Linear system y(n) = =h(n)*x(n) H (z) = Z (h(n)) Y(z) = H (z)X(z) Figure 2.9 Transfer function relationship for linear systems Linear time-invariant digital systems 2/13 Table 2.1 Commonly used z-transform pairs x (n) Region of convergence 8 (n) All Z /z/>1 z - 1 a"u(n) -a"u(-n-1) na"u(n) /z/>/a/ (z-a)2 -na" (u( - n) (z-a)2 (z-a)k 2 - 1 2 a<z<a- ( 2 - a ) ( z - 1/a ) sin(nw)u(n) zsinw /z/>1 Z - 2z cos w + 1 cos(nw) (u(n) z(z-cosw) /2/>1 22 - 2z cos w + 1 a" sin (nw ) u (n) az sin w a" cos (nw) u (n) z(z-acosw) Z - 2az cos w + a Using the definition of Equation 2.94 gives Equation 2.95 which 2.8.2 Delay property is the z-transform of the sampled sequence, x(nT) and is also the Laplace transform of the sampled signal X st Consider a sampled sequence that has been delayed by p samples, and define the delayed sequence to be. x' (nT), as in Equation 2.97. X(z) = X (s) (2.94) x'(nT) = x (nT-pT) (2.97) (2.95) X(2) - Ex(nt)z* The z-transform of this delayed sequence is then as in Equation 2.98. Before deriving an expression for the transfer function and fre- quency response of a general linear system, a few of the properties x(2)= of the z-transform need to be explained. (2.98) 2.8.1 Linearity The z-transform of the sum of two sequences multiplied by arbitrary constants is the sum of the z-transforms of the individual sequences, This may be written in terms of the undelayed sequence, and with a redefinition of the summation index in the second term, we obtain as in Equation 2.96, where Z represents the z-transform operator and X(z) and Y(z) are the z-transforms of the sequences x(n) and y(n) Equation 2.99. respectively. (2.99) (2.96) 2/14 Fourier analysis Therefore, the z-transform of a delayed sequence is given by (2.103) Equation 2.100. (2.100) This can therefore be written as the product o