Discrete-time Signal Processing Lecture 5 (Transform analysis of LTI)

1. Husheng Li, UTK-EECS, Fall 2012 Discrete-time Signal ProcessingLecture 5 (Transform analysis of LTI)

2. Time domain: . Frequency domain: Characterization of LTI

3. The frequency response of LTI is given by Magnitude response: Phase response: Frequency response of LTI

4. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use ARG as the wrapped phase and arg as the continuous phase. Discontinuity

5. We define the group delay as An ideal delay system causes a linear phase response. The group delay represents a convenient measure of the linearity of the phase. Group delay

6. Illustration of group delay

7. A LTI can be written as The system function is given by from which we can derive the zeros and poles. LTI with difference equations

8. The condition of stability is equivalent to the condition that the ROC of H(z) includes the unit circle. If the system is causal, the impulse response h(n) must be right-sided sequence. Causality and stability are not necessarily compatible requirements. Stability and causality

9. For a system with system function H(z), its inverse system has function 1/H(z). In the inverse system, the poles and zeros will be swapped. There are two classes of LTIs: At least one nonzero pole (IIR) No poles (FIR) Midterm: Oct.16, 2012 Homework: 5.1, 5.2, 5.3 Inverse systems and impulse response

10. Magnitude gain: . Phase response: arg[H(w)]. The left figure shows the frequency response of 1st-order systems Frequency response

11. If the magnitude of the frequency response an the number of poles and zeros are known, there are only finite number of choices for the associated phase. We need to use Relationship between magnitude and phase

12. A system for which the frequency-response magnitude is a constant, referred to as an all-pass system, passes all of the frequency components of its input with constant gain or attenuation. All pass system

13. If the system is stable and causal, the poles must be inside the unite circle, but no restrictions are put for the zeros. For certain classes of problems, it is useful to impose an additional restriction that the inverse system also be stable and causal. Such systems are referred to as minimum-phase systems. Properties: Minimum phase-lag property Minimum group-delay property Minimum energy-delay property Minimum-phase system