Fundamentals of
Measurement Technology
(13)
Prof,Wang Boxiong
5.4.3 Types of filters
1,Low-pass filters
Fig,5.40 Low-pass filters
5.4.3 Types of filters
RC lower-pass filter
io
o ee
dt
deRC
11)( sseesH
i
o
(5.63)
(5.64)
where the frequency
RCf?2
1?
corresponds to the point of amplitude
attenuation of –3dB,and is the upper cut-off
frequency.
5.4.3 Types of filters
For simple first-order
systems,the attenuation
is quite gradual with
frequency,6dB/octave,
By adding more,stages”
(see Fig.5.42(a))the
sharpness of cutoff may
be increased,But the
disadvantage,loading
effects should be take
into account.Fig,5.41 Amplitude and phase characteristics of the first-order
low-pass filters
5.4.3 Types of filters
Many electrical filters are active devices based
on op-amp technology,Passive filters have very
low noise,require no power supplies,and have
a wide dynamic range; Active filters are much
more adjustable and versatile,can cover very
wide frequency ranges,have very high input
and very low output impedances,and can be
configured for simple switching from low-pass
to high-pass and combination for band-pass or
band-reject behavior.
5.4.3 Types of filters
Fig.5.44(b) illustrates a,state-variable filter”
with electrically adjustable parameters,The
state-variable filter provides three simultaneous
outputs,a low-pass,a high-pass,and a band-pass:
1/2/ 122
nni
ep
ssse
e
1/2/ /1 0 0 22 2222
nn
c
i
hp
ss
sVCRs
e
e
1/2/
/20
22
nn
C
i
bp
ss
sVRCs
e
e
(5.66)
(5.67)
(5.68)
5.4.3 Types of filters
Fig,5.42 Sharper-cutoff low-pass filters(1)
5.4.3 Types of filters
Fig,5.42 Sharper-cutoff low-pass filters(2)
5.4.3 Types of filters
Many methods can be adopted in filter design to
raise the filter order,four basic designing methods,
Butterworth,Chebyshev,Bessel,Cauer or elliptical
filters,With the increase in filter order,the transition
band become steeper,the attenuation increases,thus
the filtering effect strengthened.
Theoretically,it is possible to cascade several RC
networks to enhance filter order and to accelerate
the attenuation,In practice,loading effect between
different cascaded stages must be considered,To
solve the loading effect,a better way is to use
operational amplifiers to construct active filters.
5.4.3 Types of filters
RC-network is connected to the input terminal of
an op-amp,The filter’s cut-off frequency is then,
RCf c?2
1
2?
and its gain
1
1 RRK F
Connecting a high-pass network to the feedback-
loop at the amplifier yields a low-pass filter,As
shown in Fig,5.43 (b),the filter has a cut-off
frequency,
CRf Fc?2
1
2?
and its gain
1R
RK F?
5.4.3 Types of filters
Fig,5.43 First-order active low-pass filter
5.4.3 Types of filters
Fig,5.44 Active second-order low-pas filter and voltage-controlled state-variable filter
5.4.3 Types of filters
2,High-pass filters
1 s ssH
jjjH 1
21
jH
1a r c t g?
(5.72)
(5.73)
(5.74)
(5.74)
The cut-off frequency:
RCf c?2
1
1?
5.4.3 Types of filters
Fig,5.45 High-pass filters
5.4.3 Types of filters
① Multiple-loop negative-feedback network
(Double-ladder structure)
Fig.5.48 shows the circuit,where all
components are expressed in admittance,
According to Kirchhoff’s law:
213214111 YeYeeYeeYee yx
52321 YeeYee y
,for Node 1
,for Node 2
(5.77)
(5.78)
As e2 is a virtual ground,so e2=0,then the transfer
function ex and ey:
5.4.3 Types of filters
Different filters can be obtained by substituting
Y1-Y5 with the resistors and capacitors.
Assuming,Y1,Y3 and Y4 are resistors,and Y2
and Y5 are capacitors,then,Y1=1/R1,Y2=C2s,
Y3=1/R3,Y4=1/R4,and Y5=C5s,Substituting
them into Eq.(5.80),the transfer function for it:
4343215
31
YYYYYYY
YY
sE
sE
sH
x
y
(5.80)
52434312
2
52431
4
1
)
111
(
)(
CCRRRRRC
s
s
CCRRR
R
sH
(5.81)
5.4.3 Types of filters
A second-order
low-pass filter,
whose DC gain:
1
4
R
RK
and the cut-off
frequency:
5243
1
CCRRc
(5.82)
(5.83)
Fig,5.48 Multiple-loop negative-feedback network
5.4.4 Applications
To obtain special filtering effects,different filters
or filter groups are often connected in series or in
parallel.
① Series-connection
To enhance filtering effects,two band-pass
filters with the same center frequency are often
connected in series.
② Parallel-connection
often used in spectrum analysis and the extraction
of special frequency components from signals,
5.4.4 Applications
A general way is to make the upper cut-off
frequency of the former filter equal the lower
cut-off frequency of the latter one,All the
filters in a group must have the same gain.
Fig,5.50 Band-widths arrangement of the group of band-pass filters over
the analyzed frequency range of signal
5.4.4 Applications
The center frequency of a band-pass filter,f0:
1221 cco fff (5.86) arithmetic average:
geometrical average:
21 cco fff
(5.87)
bandwidth of a band-pass filter:
12 cc ffB
(5.88)
also known as –3dB-bandwidth or half-power
band-width.
Relative bandwidth or the percentage bandwidth,b:
%100
of
Bb (5.89)
5.4.4 Applications
Relationship between the relative bandwidth
and the quality factor:
Qb
1? (5.90)
In spectrum analysis,a group of band-pass filters
with their center frequencies changed in steps is
required,when the center frequency changes,the
bandwidth of each filter takes on values
according to a defined rule,
5.4.4 Applications
a) Constant-percentage-band-width filters
c o n s t a n t%1 0 0
0
12
0
f fffBb cc
b) Constant-bandwidth filters
c o n s t a n t12 ＝cc ffB
A constant-percentage-bandwidth filter is often
realized by use of an octave band-pass filter,
whose upper and lower frequencies have the
following relationship:
12 2 cnc ff?
(5.91)
5.4.4 Applications
Fig,5.51 Ideal constant-percentage-bandwidth filter and constant-bandwidth filter
(a) Constant-percentage-bandwidth filter
(b) Constant-bandwidth filter
5.4.4 Applications
since
022 2 ff
n
c?
From Eq.(5.91),and
021 2 ff
n
c
Q
fffB
cc
0
12 ＝
,then
22 22
nn
of
Bb (5.92)
The following relationships can be thus obtained:
%78.5%56.11%16.23%7.70
12
1
6
1
3
11
b
n
12 2 ono ff?
(5.93)
Eq,(5.92) and (5.93) are the basis for filter designs.
5.4.4 Applications
③ Application of band-pass filters in signal
frequency analysis
a) Parallel-connected multi-channel filters
Fig,5.52 Parallel-connected multi-channel filters
Disadvantage,when the analyzed frequency
range is large,more filters are required.
5.4.4 Applications
b) Frequency-scanning type
Frequency-canning analyzer utilizes only one
band-pass filter with an adjustable center
frequency.
Fig,5.53 Frequency-scanning spectrum analyzer
The signal for adjusting the center frequency
can be generated with a saw-tooth wave
generator,
5.4.4 Applications
Disadvantage,a filter requires a certain rise time
Te for its operation,and Te is inversely
proportional to the bandwidth B,In addition,
change in center frequency also needs time,
therefore the bandwidth of the filter in the
analyzer can’t be made too narrow.
c) Heterodyne type
Fig,5.54 Principle block diagram of heterodyne frequency analyzer
5.4.4 Applications
Letting the input signal
tntfUtfUtx
i
iiisss
1
2s i n2s i n
(5.94)
where the first term is the frequency
component being analyzed,expressed as us.
sss tfU2s in
Assuming that the carrier-signal generator provides
a sine wave um with a frequency fm:
tfUu mmm?2s in?
(5.95)
When um is mixed with us,then:
5.4.4 Applications
The mixed signal consists of two parts,a
frequency-summed signal with a frequency of
fm+fs,and a frequency-subtracted signal with a
frequency of fm-fs,When the frequency of signal
generator,fm,makes the following equality hold:
ssmsm
ssmsm
msssmsm
tffUU
tffUU
tftfUUuu
2c o s
2
1
2c o s
2
1
2s i n2s i n
(5.96)
0fff sm that is sm fff 0
(5.97)
5.4.4 Applications
Then only the frequency-summed signal
can pass the filter,so the filter output u0 contains
the information on the amplitude Us and the
phase φs of the input signal us.
ssmsm tffUU2c o s21
Since the bandwidth of filter is B,so only the
signal components whose frequency is within
the frequency range (f0-B/2,f0+B/2) can pass,
after having been mixed and lying in the pass-
band ( ).
20
Bf?
5.4.4 Applications
As f0 is usually higher than the maximum
frequency of the signal x(t),all the frequency
components outside the bandwidth can be
totally rejected.
If the bandwidth analyzed is Ba=fH-fL,then the
adjusted frequency range fm is:
)()( 00 HL ffff ——
5.4.4 Applications
d) Tracking filters
Fig,5.55 Principle of variable-frequency tracking filter
5.4.4 Applications
e) Other types of filtering
I,Filtering by signal correlation
The cross-correlation of two sinusoidal
signals is defined as:
2
2
00 s i ns i n0
T
Txy t d tmBtAR
（ 5.105）
For m=n,TABR
xy 2)0(?
（ 5.106）
for m≠n, 00?xyR （ 5.107）
Cross-correlation is often used to extract useful
information from the noise-contaminated signals.
5.4.4 Applications
Fig,5.59 Block diagram of the principle of spectrum analyzer based on
filtering by correlation
5.4.4 Applications
II,Filtering by statistical averaging
All the filters mentioned above are of the
frequency-selective type and,of course,
require that the desired and spurious signals
occupy different portions of the frequency
spectrum,When signal and noise contain
the same frequencies,such filters are
useless,A basically different scheme may
be employed usefully under such
circumstances if the following is true:
5.4.4 Applications
1,The noise is random.
2,The desired signal can be caused to repeat
itself.
If one adds up the ordinates of several samples
of the total signal at like values of abscissa
(time),the desired signal will reinforce itself,
This will occur even if the frequency content
of signal and noise occur in the same part of
the frequency spectrum,Thus theoretically the
noise can be eliminated to any desired degree
by adding a sufficiently large number of
signals.
Measurement Technology
(13)
Prof,Wang Boxiong
5.4.3 Types of filters
1,Low-pass filters
Fig,5.40 Low-pass filters
5.4.3 Types of filters
RC lower-pass filter
io
o ee
dt
deRC
11)( sseesH
i
o
(5.63)
(5.64)
where the frequency
RCf?2
1?
corresponds to the point of amplitude
attenuation of –3dB,and is the upper cut-off
frequency.
5.4.3 Types of filters
For simple first-order
systems,the attenuation
is quite gradual with
frequency,6dB/octave,
By adding more,stages”
(see Fig.5.42(a))the
sharpness of cutoff may
be increased,But the
disadvantage,loading
effects should be take
into account.Fig,5.41 Amplitude and phase characteristics of the first-order
low-pass filters
5.4.3 Types of filters
Many electrical filters are active devices based
on op-amp technology,Passive filters have very
low noise,require no power supplies,and have
a wide dynamic range; Active filters are much
more adjustable and versatile,can cover very
wide frequency ranges,have very high input
and very low output impedances,and can be
configured for simple switching from low-pass
to high-pass and combination for band-pass or
band-reject behavior.
5.4.3 Types of filters
Fig.5.44(b) illustrates a,state-variable filter”
with electrically adjustable parameters,The
state-variable filter provides three simultaneous
outputs,a low-pass,a high-pass,and a band-pass:
1/2/ 122
nni
ep
ssse
e
1/2/ /1 0 0 22 2222
nn
c
i
hp
ss
sVCRs
e
e
1/2/
/20
22
nn
C
i
bp
ss
sVRCs
e
e
(5.66)
(5.67)
(5.68)
5.4.3 Types of filters
Fig,5.42 Sharper-cutoff low-pass filters(1)
5.4.3 Types of filters
Fig,5.42 Sharper-cutoff low-pass filters(2)
5.4.3 Types of filters
Many methods can be adopted in filter design to
raise the filter order,four basic designing methods,
Butterworth,Chebyshev,Bessel,Cauer or elliptical
filters,With the increase in filter order,the transition
band become steeper,the attenuation increases,thus
the filtering effect strengthened.
Theoretically,it is possible to cascade several RC
networks to enhance filter order and to accelerate
the attenuation,In practice,loading effect between
different cascaded stages must be considered,To
solve the loading effect,a better way is to use
operational amplifiers to construct active filters.
5.4.3 Types of filters
RC-network is connected to the input terminal of
an op-amp,The filter’s cut-off frequency is then,
RCf c?2
1
2?
and its gain
1
1 RRK F
Connecting a high-pass network to the feedback-
loop at the amplifier yields a low-pass filter,As
shown in Fig,5.43 (b),the filter has a cut-off
frequency,
CRf Fc?2
1
2?
and its gain
1R
RK F?
5.4.3 Types of filters
Fig,5.43 First-order active low-pass filter
5.4.3 Types of filters
Fig,5.44 Active second-order low-pas filter and voltage-controlled state-variable filter
5.4.3 Types of filters
2,High-pass filters
1 s ssH
jjjH 1
21
jH
1a r c t g?
(5.72)
(5.73)
(5.74)
(5.74)
The cut-off frequency:
RCf c?2
1
1?
5.4.3 Types of filters
Fig,5.45 High-pass filters
5.4.3 Types of filters
① Multiple-loop negative-feedback network
(Double-ladder structure)
Fig.5.48 shows the circuit,where all
components are expressed in admittance,
According to Kirchhoff’s law:
213214111 YeYeeYeeYee yx
52321 YeeYee y
,for Node 1
,for Node 2
(5.77)
(5.78)
As e2 is a virtual ground,so e2=0,then the transfer
function ex and ey:
5.4.3 Types of filters
Different filters can be obtained by substituting
Y1-Y5 with the resistors and capacitors.
Assuming,Y1,Y3 and Y4 are resistors,and Y2
and Y5 are capacitors,then,Y1=1/R1,Y2=C2s,
Y3=1/R3,Y4=1/R4,and Y5=C5s,Substituting
them into Eq.(5.80),the transfer function for it:
4343215
31
YYYYYYY
YY
sE
sE
sH
x
y
(5.80)
52434312
2
52431
4
1
)
111
(
)(
CCRRRRRC
s
s
CCRRR
R
sH
(5.81)
5.4.3 Types of filters
A second-order
low-pass filter,
whose DC gain:
1
4
R
RK
and the cut-off
frequency:
5243
1
CCRRc
(5.82)
(5.83)
Fig,5.48 Multiple-loop negative-feedback network
5.4.4 Applications
To obtain special filtering effects,different filters
or filter groups are often connected in series or in
parallel.
① Series-connection
To enhance filtering effects,two band-pass
filters with the same center frequency are often
connected in series.
② Parallel-connection
often used in spectrum analysis and the extraction
of special frequency components from signals,
5.4.4 Applications
A general way is to make the upper cut-off
frequency of the former filter equal the lower
cut-off frequency of the latter one,All the
filters in a group must have the same gain.
Fig,5.50 Band-widths arrangement of the group of band-pass filters over
the analyzed frequency range of signal
5.4.4 Applications
The center frequency of a band-pass filter,f0:
1221 cco fff (5.86) arithmetic average:
geometrical average:
21 cco fff
(5.87)
bandwidth of a band-pass filter:
12 cc ffB
(5.88)
also known as –3dB-bandwidth or half-power
band-width.
Relative bandwidth or the percentage bandwidth,b:
%100
of
Bb (5.89)
5.4.4 Applications
Relationship between the relative bandwidth
and the quality factor:
Qb
1? (5.90)
In spectrum analysis,a group of band-pass filters
with their center frequencies changed in steps is
required,when the center frequency changes,the
bandwidth of each filter takes on values
according to a defined rule,
5.4.4 Applications
a) Constant-percentage-band-width filters
c o n s t a n t%1 0 0
0
12
0
f fffBb cc
b) Constant-bandwidth filters
c o n s t a n t12 ＝cc ffB
A constant-percentage-bandwidth filter is often
realized by use of an octave band-pass filter,
whose upper and lower frequencies have the
following relationship:
12 2 cnc ff?
(5.91)
5.4.4 Applications
Fig,5.51 Ideal constant-percentage-bandwidth filter and constant-bandwidth filter
(a) Constant-percentage-bandwidth filter
(b) Constant-bandwidth filter
5.4.4 Applications
since
022 2 ff
n
c?
From Eq.(5.91),and
021 2 ff
n
c
Q
fffB
cc
0
12 ＝
,then
22 22
nn
of
Bb (5.92)
The following relationships can be thus obtained:
%78.5%56.11%16.23%7.70
12
1
6
1
3
11
b
n
12 2 ono ff?
(5.93)
Eq,(5.92) and (5.93) are the basis for filter designs.
5.4.4 Applications
③ Application of band-pass filters in signal
frequency analysis
a) Parallel-connected multi-channel filters
Fig,5.52 Parallel-connected multi-channel filters
Disadvantage,when the analyzed frequency
range is large,more filters are required.
5.4.4 Applications
b) Frequency-scanning type
Frequency-canning analyzer utilizes only one
band-pass filter with an adjustable center
frequency.
Fig,5.53 Frequency-scanning spectrum analyzer
The signal for adjusting the center frequency
can be generated with a saw-tooth wave
generator,
5.4.4 Applications
Disadvantage,a filter requires a certain rise time
Te for its operation,and Te is inversely
proportional to the bandwidth B,In addition,
change in center frequency also needs time,
therefore the bandwidth of the filter in the
analyzer can’t be made too narrow.
c) Heterodyne type
Fig,5.54 Principle block diagram of heterodyne frequency analyzer
5.4.4 Applications
Letting the input signal
tntfUtfUtx
i
iiisss
1
2s i n2s i n
(5.94)
where the first term is the frequency
component being analyzed,expressed as us.
sss tfU2s in
Assuming that the carrier-signal generator provides
a sine wave um with a frequency fm:
tfUu mmm?2s in?
(5.95)
When um is mixed with us,then:
5.4.4 Applications
The mixed signal consists of two parts,a
frequency-summed signal with a frequency of
fm+fs,and a frequency-subtracted signal with a
frequency of fm-fs,When the frequency of signal
generator,fm,makes the following equality hold:
ssmsm
ssmsm
msssmsm
tffUU
tffUU
tftfUUuu
2c o s
2
1
2c o s
2
1
2s i n2s i n
(5.96)
0fff sm that is sm fff 0
(5.97)
5.4.4 Applications
Then only the frequency-summed signal
can pass the filter,so the filter output u0 contains
the information on the amplitude Us and the
phase φs of the input signal us.
ssmsm tffUU2c o s21
Since the bandwidth of filter is B,so only the
signal components whose frequency is within
the frequency range (f0-B/2,f0+B/2) can pass,
after having been mixed and lying in the pass-
band ( ).
20
Bf?
5.4.4 Applications
As f0 is usually higher than the maximum
frequency of the signal x(t),all the frequency
components outside the bandwidth can be
totally rejected.
If the bandwidth analyzed is Ba=fH-fL,then the
adjusted frequency range fm is:
)()( 00 HL ffff ——
5.4.4 Applications
d) Tracking filters
Fig,5.55 Principle of variable-frequency tracking filter
5.4.4 Applications
e) Other types of filtering
I,Filtering by signal correlation
The cross-correlation of two sinusoidal
signals is defined as:
2
2
00 s i ns i n0
T
Txy t d tmBtAR
（ 5.105）
For m=n,TABR
xy 2)0(?
（ 5.106）
for m≠n, 00?xyR （ 5.107）
Cross-correlation is often used to extract useful
information from the noise-contaminated signals.
5.4.4 Applications
Fig,5.59 Block diagram of the principle of spectrum analyzer based on
filtering by correlation
5.4.4 Applications
II,Filtering by statistical averaging
All the filters mentioned above are of the
frequency-selective type and,of course,
require that the desired and spurious signals
occupy different portions of the frequency
spectrum,When signal and noise contain
the same frequencies,such filters are
useless,A basically different scheme may
be employed usefully under such
circumstances if the following is true:
5.4.4 Applications
1,The noise is random.
2,The desired signal can be caused to repeat
itself.
If one adds up the ordinates of several samples
of the total signal at like values of abscissa
(time),the desired signal will reinforce itself,
This will occur even if the frequency content
of signal and noise occur in the same part of
the frequency spectrum,Thus theoretically the
noise can be eliminated to any desired degree
by adding a sufficiently large number of
signals.
