Fundamentals of
Measurement Technology
(12)
Prof,Wang Boxiong
5.3 Modulation and demodulation
Modulation is a process of controlling or
changing a parameter (amplitude,frequency,or
phase) of a high-frequency oscillation by use of
a measurand,usually a low-frequency signal,
Controlled parameter is the amplitude of the
high frequency oscillations,amplitude
modulation,or AM,
Frequency varied with the measurand,
frequency modulation,or FM.
Controlled parameter is the phase of the
oscillations,phase modulation,or PM,
5.3 Modulation and demodulation
The measurand,usually a low-frequency signal
controlling the high frequency oscillation,is
called modulating signal,the high-frequency
oscillation used for carrying the low-frequency
signal is known as carrier,and the resulting
signal from the modulation process,still a
high-frequency oscillation,is known as
modulated signal,
In time domain,modulation is a process in which
a certain parameter of the carrier is varied with
the modulating signal,while in frequency
domain,it is a process of phase-shifting,
5.3 Modulation and demodulation
Demodulation is the reverse process of
modulation,in which the original low-
frequency signal,the measurand,is recovered
from the modulated signal,
MODEM has wide applications in engineering.
5.3.1 Amplitude modulation and demodulation
1,Amplitude modulation(AM)
Amplitude modulation,To multiply a high-
frequency signal (carrier) with the measurand
(modulating signal) so that the carrier
amplitude is varied with the measurand.
tftxtx m 02c o s)()( the Fourier transform pair:
)(*)()()( fYfXtytx?,accordingly,we have:
000 *21*)(212c o s fffXfffXtftx (5.25)
5.3.1 Amplitude modulation and demodulation
Fig,5.14 Principle of AM
5.3.1 Amplitude modulation and demodulation
The product of signal x(t) and the carrier is
equivalent in frequency domain to a process of
shifting the spectrum of x(t) to the position of
carrier frequency f0 (Fig,5.14(b)),that is,the
AM process is a process of frequency-shifting.
Sinusoidal signal as the modulating signal
Let the modulating signal x(t)=As sinωst,and
the carrier signal y(t)= Ac sin ωct,The modulated
signal is then
tAtAtytxx ccssm s i ns i n (5.26)
5.3.1 Amplitude modulation and demodulation
where
As=amplitude of signal
ωs=frequency of signal
Ac=amplitude of carrier
ωc=frequency of carrier
The frequency ωc is greater (usually considerably
greater) than ωs,Attention should be paid to that
the modulated signal is in phase with the carrier
for the positive half cycle of the modulating
signal,whereas the modulated signal is in
opposite phase with the carrier for negative half
cycle of the modulating signal.
5.3.1 Amplitude modulation and demodulation
Fig,5.15 AM of sine-wave signal
5.3.1 Amplitude modulation and demodulation
The frequency spectrum can be obtained using
trigonometric identity:
c o s21c o s21s i ns i n
(5.27)
ttAAx scsccsm c o sc o s2
(5.28)
2s i n22s i n2 tAAtAAx sccssccsm
(5.28)
An amplitude-modulating device is actually a
multiplier,In practice,a bridge is often used as
the device,which employs a high-frequency
oscillating source as the carrier signal,and the
output of the bridge is the modulated wave ey.
ye
5.3.1 Amplitude modulation and demodulation
Fig,5.16 shows another example of AM using a strain-gage bridge.
Fig,5.16 Application of amplitude modulation (1)
5.3.1 Amplitude modulation and demodulation
Fig,5.16 Application of amplitude modulation (2)
5.3.1 Amplitude modulation and demodulation
2,Demodulation of amplitude modulation
Demodulation methods for amplitude modulation:
synchronizing demodulation,phase-sensible
demodulation and rectifying-detection.
① Synchronizing demodulation
In time-domain,there exists the following
relation:
tftxtxtftftx 000 4c o s2122c o s2c o s
(5.34)
The synchronizing demodulation is simple in
method,but it requires a linear multiplier of good
quality to avoid signal distortion.
5.3.1 Amplitude modulation and demodulation
Fig,5.19 Principle of synchronizing demodulation
5.3.1 Amplitude modulation and demodulation
② Rectifying detection
Applying a DC bias A to the modulating
signal,the biased signal has a positive
voltage magnitude(Fig.5.20(a)).
The modulated signal xm(t),after the amplitude
modulation,will have an envelope with the
same shape as the original signal,Through
simple (half-wave or full-wave) rectification
and filtering,it is possible to recover the
original signal,The added DC bias must be
removed accurately after the rectification.
5.3.1 Amplitude modulation and demodulation
Fig,5.20 Modulated waveform with a bias
(a) the bias is large enough (b) the bias is not large enough
5.3.1 Amplitude modulation and demodulation
Phase-sensitive demodulation or detection can be
used to identify the polarities of modulating
signals,That an ac signal will change its
polarities when it passes through zero,and the
phase of the modulating wave also has a 180o-
jump with respect to the carrier,so that both
the amplitude and the phase of the original
signal can be preserved.
5.3.1 Amplitude modulation and demodulation
Fig,5.21 Diode-phase-sensitive detector and its principle
5.3.1 Amplitude modulation and demodulation
It should be noted though
that the above result can be
obtained only when the
output voltage of the
secondary stage of the
transformer B must be larger
than that of the secondary
stage of the transformer A,
One example of the
application of phase-
sensitive detection is the
linear variable-differential
transformer (LVDT).
Fig,5.22
5.3.1 Amplitude modulation and demodulation
Fig,5.23 Phase-sensitive demodulation using linear variable-differential transformer (1)
5.3.1 Amplitude modulation and demodulation
Fig,5.23 Phase-sensitive demodulation using linear variable-differential transformer (2)
5.4.1 Filters Introduction
Filtering is the process of rejecting or attenuating
unwanted components of a measurand while
permitting the desired components to pass,
Devices capable of performing filtering are
called filters.
Ways for filtering can be classified into input
and output filtering,A system’s input-output
relationship can be illustrated by use of
Fig.5.32
5.4.1 Filters Introduction
Fig,5.32 Generalized input-output configuration
5.4.1 Filters Introduction
Fig,5.33 General principle of filtering (a) input filtering (b) output filtering
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (1)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (2)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (3)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (4)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (5)
5.4.1 Filters Introduction
Fig,5.35 Basic filter types
5.4.1 Filters Introduction
① Low-pass filter,the flat part of the
characteristic curve from the frequency 0 to f2
is called pass-band,Components of frequencies
higher than f2 are attenuated.
② High-pass filter,the pass-band is form f1 to,
Components of frequencies higher than f1 can
pass and those lower than f1 are blocked.
③ Band-pass filter,contrary to the band-pass
filters,the stop-band is between f1 to f2,and the
components of frequencies between the stop-
band are rejected and other components can
pass,
5.4.2 General characteristics of filter
For an ideal linear system:
tfjeAfH 020 (5.55)
where A0 and t0 are constants,If the frequency
response of a filter has the following form:
o t h e r
ffeAfH ctfj
0
02
0
(5.56)
then the filter is an ideal low-pass filter.
5.4.2 General characteristics of filter
Fig,5.36 Amplitude and phase
characteristics of ideal low-pass filter
Fig,5.37 Impulse response of ideal low-
pass filter
5.4.2 General characteristics of filter
① Impulse response of ideal low-pass filter
)2(s in2)( 0 tfcfAth cc
00?tif,then the equation changes to
00 2s i n2 ttfcfAth cc(5.57)
The impulse response of an ideal low-pass filter
and its waveform extend over the entire time axis,
but its output exists before the input δ(t) is
applied,it appears even when t<0,So the ideal
low-pass filter (and the high-pass,band-pass as
well as all ideal filters) cannot be realized,
5.4.2 General characteristics of filter
② Unit-step response of ideal low-pass filter
For a unit-step input
00
0
2
1
01
)(
t
t
t
tx
an ideal low-pass filter has the following response
tfsi
dthu
txthty
c
22
2
1
*
(5.58)
5.4.2 General characteristics of filter
In Fig.5.58:
dtt ttfsi tfc c )(20 s in)](2[
the sinusoidal integral,expressed in si(x):
xd e f dxsi 0 s in
5.4.2 General characteristics of filter
Fig,5.38 Unit-step response of ideal low-pass filter
(a) without phase-lag,
00?t
(b) with phase-lag,
00?t
5.4.2 General characteristics of filter
5.4.2 General characteristics of filter
Conclusion:
the rise time T0 for the unit-step response of a
low-pass filter is inversely proportional to its
bandwidth B,their product is a constant,
c o n s t a n t0?BT
(5.59)
This condition also holds for other (high-pass,
band-pass,band-rejection) filters,
Calculating with Eq,(5.58) gives,
c
ab ftt
61.0 (5.60)
The bandwidth of a filter represents its frequency
resolution,The narrower the bandwidth,the higher
its resolution,
5.4.2 General characteristics of filter
③ Characteristic parameters of actual filters
Fig,5.39 Amplitude characteristics of the ideal and the actual band-pass filters
5.4.2 General characteristics of filter
a) Cut-off frequencies
The frequencies at which amplitude is equal to
)3(20 dBA?
(fc1 and fc2 in Fig.5.39).
the point a cut-off frequency corresponds to is
called the half-power point.
b)Band-width
Defined as the frequency interval between the
lower and the upper frequency,
12 cc ffB
also known as the (-3dB)-bandwidth,in Hz,
represents the resolving capability of a filter.
5.4.2 General characteristics of filter
c) Ripple magnitude
The perturbation of the amplitude in pass-
band,d,and d should be as small as possible.
d) Quality factor (Q-value)
For a band-pass filter,its quality factor Q is
defined as the ratio of its center frequency f0
to its bandwidth,that is,
B
fQ 0?
5.4.2 General characteristics of filter
e)Octave selectivity
The octave selectivity is the attenuated value
of the amplitude between the upper cut-off
frequency and the frequency or between the
lower cut-off frequency and the frequency,
that is,the attenuated amount for an octave,of
frequency change,expressed in decibel (dB),
f) Filter factor (rectangular coefficient)
The filter factor λ:
dB
dB
B
B
3
60
(5.62)
For an ideal filter,λ=1,
Measurement Technology
(12)
Prof,Wang Boxiong
5.3 Modulation and demodulation
Modulation is a process of controlling or
changing a parameter (amplitude,frequency,or
phase) of a high-frequency oscillation by use of
a measurand,usually a low-frequency signal,
Controlled parameter is the amplitude of the
high frequency oscillations,amplitude
modulation,or AM,
Frequency varied with the measurand,
frequency modulation,or FM.
Controlled parameter is the phase of the
oscillations,phase modulation,or PM,
5.3 Modulation and demodulation
The measurand,usually a low-frequency signal
controlling the high frequency oscillation,is
called modulating signal,the high-frequency
oscillation used for carrying the low-frequency
signal is known as carrier,and the resulting
signal from the modulation process,still a
high-frequency oscillation,is known as
modulated signal,
In time domain,modulation is a process in which
a certain parameter of the carrier is varied with
the modulating signal,while in frequency
domain,it is a process of phase-shifting,
5.3 Modulation and demodulation
Demodulation is the reverse process of
modulation,in which the original low-
frequency signal,the measurand,is recovered
from the modulated signal,
MODEM has wide applications in engineering.
5.3.1 Amplitude modulation and demodulation
1,Amplitude modulation(AM)
Amplitude modulation,To multiply a high-
frequency signal (carrier) with the measurand
(modulating signal) so that the carrier
amplitude is varied with the measurand.
tftxtx m 02c o s)()( the Fourier transform pair:
)(*)()()( fYfXtytx?,accordingly,we have:
000 *21*)(212c o s fffXfffXtftx (5.25)
5.3.1 Amplitude modulation and demodulation
Fig,5.14 Principle of AM
5.3.1 Amplitude modulation and demodulation
The product of signal x(t) and the carrier is
equivalent in frequency domain to a process of
shifting the spectrum of x(t) to the position of
carrier frequency f0 (Fig,5.14(b)),that is,the
AM process is a process of frequency-shifting.
Sinusoidal signal as the modulating signal
Let the modulating signal x(t)=As sinωst,and
the carrier signal y(t)= Ac sin ωct,The modulated
signal is then
tAtAtytxx ccssm s i ns i n (5.26)
5.3.1 Amplitude modulation and demodulation
where
As=amplitude of signal
ωs=frequency of signal
Ac=amplitude of carrier
ωc=frequency of carrier
The frequency ωc is greater (usually considerably
greater) than ωs,Attention should be paid to that
the modulated signal is in phase with the carrier
for the positive half cycle of the modulating
signal,whereas the modulated signal is in
opposite phase with the carrier for negative half
cycle of the modulating signal.
5.3.1 Amplitude modulation and demodulation
Fig,5.15 AM of sine-wave signal
5.3.1 Amplitude modulation and demodulation
The frequency spectrum can be obtained using
trigonometric identity:
c o s21c o s21s i ns i n
(5.27)
ttAAx scsccsm c o sc o s2
(5.28)
2s i n22s i n2 tAAtAAx sccssccsm
(5.28)
An amplitude-modulating device is actually a
multiplier,In practice,a bridge is often used as
the device,which employs a high-frequency
oscillating source as the carrier signal,and the
output of the bridge is the modulated wave ey.
ye
5.3.1 Amplitude modulation and demodulation
Fig,5.16 shows another example of AM using a strain-gage bridge.
Fig,5.16 Application of amplitude modulation (1)
5.3.1 Amplitude modulation and demodulation
Fig,5.16 Application of amplitude modulation (2)
5.3.1 Amplitude modulation and demodulation
2,Demodulation of amplitude modulation
Demodulation methods for amplitude modulation:
synchronizing demodulation,phase-sensible
demodulation and rectifying-detection.
① Synchronizing demodulation
In time-domain,there exists the following
relation:
tftxtxtftftx 000 4c o s2122c o s2c o s
(5.34)
The synchronizing demodulation is simple in
method,but it requires a linear multiplier of good
quality to avoid signal distortion.
5.3.1 Amplitude modulation and demodulation
Fig,5.19 Principle of synchronizing demodulation
5.3.1 Amplitude modulation and demodulation
② Rectifying detection
Applying a DC bias A to the modulating
signal,the biased signal has a positive
voltage magnitude(Fig.5.20(a)).
The modulated signal xm(t),after the amplitude
modulation,will have an envelope with the
same shape as the original signal,Through
simple (half-wave or full-wave) rectification
and filtering,it is possible to recover the
original signal,The added DC bias must be
removed accurately after the rectification.
5.3.1 Amplitude modulation and demodulation
Fig,5.20 Modulated waveform with a bias
(a) the bias is large enough (b) the bias is not large enough
5.3.1 Amplitude modulation and demodulation
Phase-sensitive demodulation or detection can be
used to identify the polarities of modulating
signals,That an ac signal will change its
polarities when it passes through zero,and the
phase of the modulating wave also has a 180o-
jump with respect to the carrier,so that both
the amplitude and the phase of the original
signal can be preserved.
5.3.1 Amplitude modulation and demodulation
Fig,5.21 Diode-phase-sensitive detector and its principle
5.3.1 Amplitude modulation and demodulation
It should be noted though
that the above result can be
obtained only when the
output voltage of the
secondary stage of the
transformer B must be larger
than that of the secondary
stage of the transformer A,
One example of the
application of phase-
sensitive detection is the
linear variable-differential
transformer (LVDT).
Fig,5.22
5.3.1 Amplitude modulation and demodulation
Fig,5.23 Phase-sensitive demodulation using linear variable-differential transformer (1)
5.3.1 Amplitude modulation and demodulation
Fig,5.23 Phase-sensitive demodulation using linear variable-differential transformer (2)
5.4.1 Filters Introduction
Filtering is the process of rejecting or attenuating
unwanted components of a measurand while
permitting the desired components to pass,
Devices capable of performing filtering are
called filters.
Ways for filtering can be classified into input
and output filtering,A system’s input-output
relationship can be illustrated by use of
Fig.5.32
5.4.1 Filters Introduction
Fig,5.32 Generalized input-output configuration
5.4.1 Filters Introduction
Fig,5.33 General principle of filtering (a) input filtering (b) output filtering
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (1)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (2)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (3)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (4)
5.4.1 Filters Introduction
Fig,5.34 Examples of filtering (5)
5.4.1 Filters Introduction
Fig,5.35 Basic filter types
5.4.1 Filters Introduction
① Low-pass filter,the flat part of the
characteristic curve from the frequency 0 to f2
is called pass-band,Components of frequencies
higher than f2 are attenuated.
② High-pass filter,the pass-band is form f1 to,
Components of frequencies higher than f1 can
pass and those lower than f1 are blocked.
③ Band-pass filter,contrary to the band-pass
filters,the stop-band is between f1 to f2,and the
components of frequencies between the stop-
band are rejected and other components can
pass,
5.4.2 General characteristics of filter
For an ideal linear system:
tfjeAfH 020 (5.55)
where A0 and t0 are constants,If the frequency
response of a filter has the following form:
o t h e r
ffeAfH ctfj
0
02
0
(5.56)
then the filter is an ideal low-pass filter.
5.4.2 General characteristics of filter
Fig,5.36 Amplitude and phase
characteristics of ideal low-pass filter
Fig,5.37 Impulse response of ideal low-
pass filter
5.4.2 General characteristics of filter
① Impulse response of ideal low-pass filter
)2(s in2)( 0 tfcfAth cc
00?tif,then the equation changes to
00 2s i n2 ttfcfAth cc(5.57)
The impulse response of an ideal low-pass filter
and its waveform extend over the entire time axis,
but its output exists before the input δ(t) is
applied,it appears even when t<0,So the ideal
low-pass filter (and the high-pass,band-pass as
well as all ideal filters) cannot be realized,
5.4.2 General characteristics of filter
② Unit-step response of ideal low-pass filter
For a unit-step input
00
0
2
1
01
)(
t
t
t
tx
an ideal low-pass filter has the following response
tfsi
dthu
txthty
c
22
2
1
*
(5.58)
5.4.2 General characteristics of filter
In Fig.5.58:
dtt ttfsi tfc c )(20 s in)](2[
the sinusoidal integral,expressed in si(x):
xd e f dxsi 0 s in
5.4.2 General characteristics of filter
Fig,5.38 Unit-step response of ideal low-pass filter
(a) without phase-lag,
00?t
(b) with phase-lag,
00?t
5.4.2 General characteristics of filter
5.4.2 General characteristics of filter
Conclusion:
the rise time T0 for the unit-step response of a
low-pass filter is inversely proportional to its
bandwidth B,their product is a constant,
c o n s t a n t0?BT
(5.59)
This condition also holds for other (high-pass,
band-pass,band-rejection) filters,
Calculating with Eq,(5.58) gives,
c
ab ftt
61.0 (5.60)
The bandwidth of a filter represents its frequency
resolution,The narrower the bandwidth,the higher
its resolution,
5.4.2 General characteristics of filter
③ Characteristic parameters of actual filters
Fig,5.39 Amplitude characteristics of the ideal and the actual band-pass filters
5.4.2 General characteristics of filter
a) Cut-off frequencies
The frequencies at which amplitude is equal to
)3(20 dBA?
(fc1 and fc2 in Fig.5.39).
the point a cut-off frequency corresponds to is
called the half-power point.
b)Band-width
Defined as the frequency interval between the
lower and the upper frequency,
12 cc ffB
also known as the (-3dB)-bandwidth,in Hz,
represents the resolving capability of a filter.
5.4.2 General characteristics of filter
c) Ripple magnitude
The perturbation of the amplitude in pass-
band,d,and d should be as small as possible.
d) Quality factor (Q-value)
For a band-pass filter,its quality factor Q is
defined as the ratio of its center frequency f0
to its bandwidth,that is,
B
fQ 0?
5.4.2 General characteristics of filter
e)Octave selectivity
The octave selectivity is the attenuated value
of the amplitude between the upper cut-off
frequency and the frequency or between the
lower cut-off frequency and the frequency,
that is,the attenuated amount for an octave,of
frequency change,expressed in decibel (dB),
f) Filter factor (rectangular coefficient)
The filter factor λ:
dB
dB
B
B
3
60
(5.62)
For an ideal filter,λ=1,
