invariant systems. the impulse response of the system as in Equation 2.85. The z-transform of any data sequence x(n), is defined as in Equation 2.88. (2.85) (2.88) X(2)= k=-00 A system is linear if and only if the system's response to the sum In this definition, Z is a continuous complex variable and X(z) is of two signals, each multiplied by arbitrary scalar constants, is equal referred to as the 'two-sided z-transform', since both positive and to the sum of the system's responses to the two signals taken negative values of the index k are allowed. separately, as in Equation 2.86. As an example of a z-transform, let us consider x(n) to be samples of an exponential function such and e-ak for (n) = k() and a> 0. The z-transform of this sampled signal is given by =aR[x(m)]+a,Rlx,(n)] (2.86) Equation 2.89, which is a simple rational function of a and z. Since the signal is zero for k < 0, we have a one sided transform in this A system is time-invariant if the response to a shifted version of case. Table 2.1 shows some common sampled sequences and their the input is identical to a shifted version of the response based on z-transforms. the unshifted input, and a system operator R is time-invariant if Equation 2.87 holds for all values of m. (2.89) 8[x(n-m)]=z^R[x(n)] (2.87) In linear systems analysis, the concept of a transfer function is The operator z-m represents a signal delay of m samples. fundamental and is defined simply as the transform of the output of 2/12 Fourier analysis M(0) - x(0) M(1) - x(8) M(2) x(4) M(3) _x(12)( M(4) x(2) M(5) _x(10)C M(6) x(6) M(7) _x(14)C M(8) x(1) M(9) -x(9) M(10)_x(5) X(10) M(11)_x(13) X(11) M(12)_x(3) X(12) M(13)_x(11) X(13) M(14)_x(7) X(14) M(15)-x(15) X(15) Figure 2.8 Flow graph of a 16 point DIT FFT the system divided by the transform of the input. The Laplace The Laplace transform of the sampled signal X (t) is then as in transform is an important tool that is used in continuous systems Equation 2.92. theory, and for a continuous signal x(t), this transform is defined as in E