Chapter 5 transform analysis of linear time-invariant system

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Chapter 5 transform analysis of linear
time-invariant system
5.1 the frequency response of LTI system 5.2
system function 5.3 frequency response for
rational system function 5.4 relationship between
magnitude and phase 5.5 all-pass system 5.6
minimum-phase system 5.7 linear system with
generalized linear phase
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5.1 the frequency response of LTI system
magnitude-frequency characteristic
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transform curve from linear to log magnitude
log magnitude
linear magnitude
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phase-frequency characteristic
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Figure 5.7
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understand group delay
Figure 5.1
EXAMPLE
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Figure 5.2
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5.2 system function
Characteristics of zeros and poles (1)take
origin and zeros and poles at infinite into
consideration, the numbers of zeros and poles are
the same. (2)for real coefficient, complex zeros
and poles are conjugated, respectively. (3)if
causal and stable, poles are all in the unit
circle. (4)FIRhave no nonzero poles, called
all-zeros type, steady IIRhave
nonzero pole if no nonzero zeros , called
all-poles type
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5.3 frequency response for rational system
function
1.formular method
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2. Geometrical method
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EXAMPLE
magnitude response in w near zeros is minimum,
there are zeros in unit circle, then the
magnitude is 0 magnitude response in w near
poles is maximumzeros and poles counteracted
each other and in origin does not influence the
magnitude.
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EXAMPLE
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EXAMPLE
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3.matlab method
B1 A1,-0.5 figure(1) zplane(B,A) figure(2
) freqz(B,A) figure(3) grpdelay(B,A,10)
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5.4 relationship between magnitude and phase
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EXAMPLE
Pole-zero plot for
,H(z) causal and stable, Confirm the poles and
zeros
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5.5 all-pass system
Zeros and poles are conjugate reciprocal For real
coefficient, zeros are conjugated , poles are
conjugated.
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EXAMPLE
Y
N
Y
Y
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Characteristics of causal and stable all-pass
system
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5.6 minimum-phase system
inverse system
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explanation (1)not all the systems have inverse
system? (2)inverse system may be
nonuniform? (3)the inverse system of causal and
stable system may not be causal and stable?
the condition of both original and its
inverse system causal and stable
zeros and poles are all in the unit circle,such
system is called minimum-phase system,
corresponding hn is minimum-phase sequence?
poles are all in the unit circle,
zeros are all outside the unit circle, such
system is called maximum-phase system?
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minimum-phase and all-pass decomposition If H(z)
is rational, then
poles outside the unit circle
zeros outside the unit circle
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Application of minimum-phase and all-pass
decomposition Compensate for amplitude distortion
Figure 5.25
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Minimum-phase system and some all-pass system in
cascade can make up of another system having the
same magnitude response, so there are infinite
systems having the same magnitude response.
Properties of minimum-phase systems
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(3)minimum energy-delay(i.e. the partial energy
is most concentrated around n0)
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EXAMPLE
????
maximum phase
minimum phase
Systems having the same magnitude response
Figure 5.30
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minimum phase
Figure 5.31
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Figure 5.32
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5.7 linear system with generalized linear phase
5.7.1 definition 5.7.2 conditions of
generalized linear phase system 5.7.3 causal
generalized linear phase (FIR)system
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5.7.1 definition
Systems having constant group delay
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EXAMPLE
ideal delay system
EXAMPLE
differentiatormagnitude and phase are all linear
physical meaning all components of input signal
are delayed by the same amount in strict linear
phase system ,then there is only magnitude
distortion, no phase distortion. it is very
important for image signal and high-fidelity
audio signal to have no phase distortion. when
B0, for generalized linear phase, the phase in
the whole band is not linear, but is linear in
the pass band, because the phase PI only
occurs when magnitude is 0, and the magnitude in
the pass band is not 0.
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EXAMPLE
square wave with fundamental frequency 100 Hz
linear phase filter lowpass filter with cut-off
frequency 400Hz
nonlinear phase filter lowpass filter with
cut-off frequency 400Hz
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Generalized linear phase in the pass band is
strict linear phase
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Generalized linear phase in the pass band is
strict linear phase
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5.7.2 conditions of generalized linear phase
system
Or
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Meven
Modd
Figure 5.35
Mnot integer
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EXAMPLE
Mnot integer
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EXAMPLE
determine whether these system is linear
phase,generalized or strict?a and ß?
(1)
(2)
(3)
(4)
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5.7.3 causal generalized linear phase (FIR)system
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Magnitude and phase characteristics of the 4
types
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(No Transcript)
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I
II
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III
IV
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Characteristic of zeros commonness
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Characteristic of every type
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characteristic of magnitude get from
characteristic of zeros
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Application of 4 types of linear phase system
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summary
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requirement concept of magnitude and phase
response, group delay transformation among
system function, phase response and difference
equation concept of all-pass, minimum-phase and
linear phase system and characteristic of zeros
and poles minimum-phase and all-pass
decomposition conditions of linear phase
system , restriction of using as filters
key and difficulty linear phase system
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exercises
5.17 complementarityminimum-phase and all-pass
decomposition 5.21 5.45 5.53
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the first experiment
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problem 1(D) problem 11 problem 13(C) problem
22(A) problem 24(A)(C)