DISCRETE-TIME SIGNAL PROCESSINGLECTURE 5 (TRANSFORM ANALYSIS OF LTI)

Husheng Li, UTK-EECS, Fall 2012

CHARACTERIZATION OF LTI

Time domain: .

Frequency domain:

FREQUENCY RESPONSE OF LTI

The frequency response of LTI is given by

Magnitude response:

Phase response:

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.

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.

ILLUSTRATION OF GROUP DELAY

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.

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.

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

FREQUENCY RESPONSE

Magnitude gain: . Phase response:arg[H(w)]. The left figure

shows the frequency response of 1st-order systems

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

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.

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