Example 7: arbitrary signals z -transform Inverse z -transform Any signals can be represented in the z-domain: 12345678910-2-3-4 k y(k)y(k) 5 z -transform Inverse z -transform 12345678910-2-3-4 k y(k)y(k) 3 2 1
Stable and Causal Systems Re Im 1 The system G(z) is stable if all the roots (i.e., d i ) of the denominator are in the unit circle of the complex plane.
Stable and Causal Systems Re Im 1 The system G(z) is causal if the number of poles is greater than that of zeros (i.e., M N ).
Example 8: a non-causal filter Suppose a transfer function is given By applying the inverse z-Transform Therefore, the system is causal if
Example 9: open-loop controller Suppose the dynamic equation of a system is Approximating the dynamic equation by The transfer function from U(z) to Y(z) is 1/2
Example 9: open-loop controller A promising control algorithm is 2/2 However, the control algorithm is non-causal.
Frequency Response of H(z) The z-transform of a function x(k) : The Fourier-transform of a function x(k) : (Recall: Similarity of the z-Transform and Fourier Transform) The frequency response is obtained by setting where T is the sampling period.
Example 10: frequency response of a low pass filter Suppose a lowpass filter 1/2 By substituting for z, The magnitude is
2/2 Since, Example 10: frequency response of a low pass filter
IIR Filters and FIR Filters An IIR (Infinite Impulse Response) filter is A FIR (Finite Impulse Response) filter is