Frequency Response of Discrete-time LTI Systems Prof. Siripong Potisuk

Transfer Functions Let x[n] be a nonzero input to an LTI discrete-time system, and y[n] be the resulting output assuming a zero initial condition. The transfer function, denoted by H(z), is defined: Can be determined by taking the Z-transform of the governing LCCDE and applying the delay property The system’s impulse response:

BIBO Stability BIBO = Bounded-input-bounded-output A linear time-invariant (LTI) discrete-time system with transfer function H(z) is BIBO stable if and only if the poles of H(z) satisfy That is, the poles of a stable system, whether simple or multiple, must all lie strictly within the unit circle in the complex z-plane Marginally unstable  one or more simple poles on the unit circle

Ex. Consider a 2 nd order discrete-time LTI system with (a) Determine the transfer function of the system and comment on the stability of the system. (b) Determine the zero-state response due to a unit-step input and the DC gain of the system.

For a discrete-time LTI system, the frequency response is defined as Frequency Response

In terms of transfer function, The frequency response is just the transfer function evaluated along the unit circle in the complex z-plane. Re(z) Im(z)  H(e j  ) periodic in  with period 2  1

For H(z) generated by a difference eq. with real coefficients,

Ex. Consider a 2 nd order discrete-time system with Plot the magnitude and phase responses of the system. Determine also the DC and the high-frequency gain.

Effects of Pole & Zero Locations A zero at indicates that the filter will fully reject spectral component of input at Effects of a zero located off the unit circle depends on its distance from the unit circle. A zero at origin has no effect. A pole on the unit circle means infinite gain at that frequency. The closer the poles to the unit circle, the higher the magnitude response.

Ex. Roughly sketch the magnitude response of the system with

Ex. Roughly sketch the magnitude response of the system with

For a given choice of H(e j  ) as a function of , the frequency composition of the output can be shaped: - preferential amplification - selective filtering of some frequencies

Ex. Consider a 1 st order IIR digital filter with (a) Determine c such that the system is BIBO stable. (b) Without plotting the magnitude response of the system, determine the type of this filter. (c) Verify the answer in (b) using MATLAB.