Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit...
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![Page 1: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/1.jpg)
DISCRETE-TIME SIGNAL PROCESSINGLECTURE 5 (TRANSFORM ANALYSIS OF LTI)
Husheng Li, UTK-EECS, Fall 2012
![Page 2: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/2.jpg)
CHARACTERIZATION OF LTI
Time domain: .
Frequency domain:
![Page 3: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/3.jpg)
FREQUENCY RESPONSE OF LTI
The frequency response of LTI is given by
Magnitude response:
Phase response:
![Page 4: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/4.jpg)
DISCONTINUITY
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.
![Page 5: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/5.jpg)
GROUP DELAY
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.
![Page 6: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/6.jpg)
ILLUSTRATION OF GROUP DELAY
![Page 7: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/7.jpg)
LTI WITH DIFFERENCE EQUATIONS
A LTI can be written as
The system function is given by
from which we can derive the zeros and poles.
![Page 8: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/8.jpg)
STABILITY AND CAUSALITY
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.
![Page 9: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/9.jpg)
INVERSE SYSTEMS AND IMPULSE RESPONSE
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
![Page 10: Husheng Li, UTK-EECS, Fall 2012. The principle value of the phase response will exhibit discontinuities when viewed as a function of w. We use.](https://reader035.fdocuments.in/reader035/viewer/2022070414/5697c0201a28abf838cd24d1/html5/thumbnails/10.jpg)
FREQUENCY RESPONSE
Magnitude gain: . Phase response:arg[H(w)]. The left figure
shows the frequency response of 1st-order systems
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RELATIONSHIP BETWEEN MAGNITUDE AND PHASE
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
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ALL PASS SYSTEM
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.
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MINIMUM-PHASE SYSTEM
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