5.2 Applications of DSB-SC - Department of Electrical Engineering

EEE3086F
Signals and Systems II
2014
A.J. Wilkinson
[email protected]
http://www.ee.uct.ac.za
Department of Electrical Engineering
University of Cape Town
A.J.Wilkinson, UCT
DSB-SC Applications
EEE3086F Signals and Systems II
504 Page 1 April 14, 2014
5.2 Applications of DSB-SC
5.2.1 USE of DSB-SC in Stereo FM Radio
5.2.2 DSB-SC Quadrature Multiplexing and
Demultiplexing
5.2.3 Synchronous Detection
5.2.4 The Chopper Amplifier
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DSB-SC Applications
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5.2.1 USE of DSB-SC in Stereo FM Radio
(for transmitting stereo audio signals on the
same FM carrier frequency)
A.J.Wilkinson, UCT
EEE3086F Signals and Systems II
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DSB-SC Applications
FM Broadcast Radio (88-108 MHz)
 FM radio channel spacing is 200 kHz; bandwidth <200kHz.
 Originally, FM radio was “mono” i.e. single sound channel;
 Stereo hifi music was combined into a single mono signal by adding left
and right channels (L+R), and fed into an FM modulator, which is a
voltage controlled oscillator (VCO).
 At the FM demodulator, the L+R signal is recovered, amplified and fed
into a speaker.
FM
Modulator
Audio L(t )

LR
FM
VCO
LR
demod.
Audio R (t )
Mono FM Radio
A.J.Wilkinson, UCT
DSB-SC Applications
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USE of DSB-SC in Stereo FM Radio
 One application of DSB-SC arises in stereo transmission of FM radio.
 To convert FM radio to stereo, with minimum change to the existing mono
standard, the standard was later extended in such a way that the original mono
signal (L(t)+R(t)) is transmitted, together with a difference signal (L(t)-R(t)).
 At the receiver, the individual L and R channels can be reconstructed from the
L+R and L-R signals by adding or subtracting them
i.e.
L = [(L-R) + (L+R)]/2
R = [(L+R)-(L-R)]/2
 The approach used, involves assembling a baseband spectrum consisting of:
(1) the (L+R) positioned between 0-15 kHz
(2) The (L-R) positioned next to it using DSB-SC modulation on a sub-carrier
frequency of 38 kHz . The DSB-SC signal occupies a bandwidth of 30 kHz,
extending from 23 to 53 kHz (see illustration).
(3) An additional digital data stream is provided for in the range 59-75 kHz
(used for “RDS” messages that can be displayed on modern FM car radio LCD
displays).
A.J.Wilkinson, UCT
EEE3086F Signals and Systems II
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DSB-SC Applications
“Baseband spectrum”
LR
0
Subsidiary
Pilot Carrier
Audio
(mono.)
15 19 23
DSB  SC
L −R
Lower
Comms (RDS)
L −R
Upper
38
53 59
Complete spectrum
assembled for transmission
via frequency modulation
onto a carrier.
A.J.Wilkinson, UCT
75
kHz
FM
Modulator
DSB-SC Applications
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 In order to allow error-free synchronous demodulation at the receiver, the receiver
requires the 38 kHz carrier signal, with the correct phase.
 Unfortunately there is not much space in the small gap at 38 kHz to allow a BPF to
separate a 38 kHz signal. Instead the trick used is to transmit a 19 kHz ‘pilot carrier’.
This is positioned in the gap between 15 and 23 kHz (see illustration), to allow easy
separation by filtering at the receiver.
 At the receiver, the 38 kHz signal is recreated from the 19 kHz signal using a
‘frequency doubler’ circuit. This can then be used to synchronously demodulate the
(L-R) signal.
Frequency Doubler
19 kHz
 
2
BPF
38 kHz
@ 38 kHz
square
A.J.Wilkinson, UCT
EEE3086F Signals and Systems II
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DSB-SC Applications
Assembly of baseband spectrum for Stereo FM Radio
L
R


LR

 LR
 
DSB-SC


A.J.Wilkinson, UCT


38kHz
cos  s t

To FM
modulator
2
frequency
38kHz divider
19kHz
DSB-SC Applications
Attenuator
Pilot
carrier
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Implementation of stereo FM Radio
LR
LR
 DSB  SC
cos 0t
38 kHz
2
FM Modulator
∑
VCO
LPF

FM
Dem.
DSB  SC
Dem.
19 kHz
Assembly
of baseband
spectrum for
modulation onto
carrier via FM
A.J.Wilkinson, UCT
LR

LR
∑


∑
2L
2R
Recovery of L and R
channels from baseband
spectrum.
DSB-SC Applications
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5.2.2 DSB-SC Quadrature Multiplexing and
Demultiplexing
(for transmitting two signals on the same
carrier frequency; this requires
synchronous demodulation)
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DSB-SC Applications
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Quadrature Multiplexing
 Previously we considered demodulation of DSB-SC with a phase error 
f (t )

ϕ (t)

LPF
1
f t cosθ 
2
cos(c t   )
cos c t
 The output is multiplied by the factor cos  where  1  cos   1
 Note that ifor deg, we get zero output. This property can
be exploited to transmit two signals on the same carrier band.
This technique is known as quadrature multiplexing.
 The word “quadrature” refers to the fact that in phasor form, sin t and
cost are 90 degrees out of phase i.e. the vectors are perpendicular. The
word multiplexing refers to the sharing of a single information channel
by two (or more) signals.
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DSB-SC Applications
Quadrature Multiplexing Block Diagram
f 1 (t )
cos ωc t
×
+
cos ωc t
e 1 (t )=
×
ϕ (t )
LPF
∑
f 2 (t )
e 2 (t )=
+
×
sin ωc t
1
f (t )
2 1
×
LPF
1
f (t )
2 2
sin ωc t
 The orthogonality of sines and cosines is exploited to transmit and
receive two different DSB-SC signals simultaneously, on the same
carrier frequency.
 Show that each signal can be recovered by synchronous detection of the
received signal using carriers of the same frequency but in phase
quadrature.
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DSB-SC Applications
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Maths Identities
 To analyse the quadrature demultiplexer, we use the trig
identities
1 1
2
cos A= + cos 2A
2 2
1 1
2
sin A= − cos 2A
2 2
sin 2A=2sin Acos A
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DSB-SC Applications
Synchronous Demodulation of Quadrature Multiplexed Signals
Quadrature multiplexed signal:
 (t )  f1 (t ) cos c t  f 2 (t ) sin c t
Multiplication by cos gives (upper arm):
2
φ(t )cos ωc t= f 1 (t ) cos ωc t + f ( t )sin ωc t cos ωc t
2
=
1
1
1
f 1 ( t )+ f 1 ( t )cos 2ωc t + f 2 (t )sin 2ω c t
2
2
2
Multiplication by sin gives (lower arm):
2
φ(t )sin ωc t= f 1 (t )cos ωc t sin ω c t + f 2 (t )sin ωc t
1
1
1
= f 1 ( t )sin 2 ωc t+ f 2 ( t )− f 2 ( t )cos 2 ωc t
2
2
2
The low-pass filter, removes the high frequency terms at 2c, so
e 1 ( t )=
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1
f (t )
2 1
e 2 ( t )=
DSB-SC Applications
1
f (t )
2 2
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5.2.3 Synchronous Detection
(used in instrumentation)
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DSB-SC Applications
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Synchronous Detection
 Another use for quadrature demodulation arises in instrumentation
applications in which we wish to measure the magnitude and phase of a
sinusoidal signal, relative to a reference sinusoid.
 Applications include:
1. measurement of the transfer function of a two port device
2. measurement of complex impedance
 Synchronous detection can also be used very effectively for characterizing a
“device under test”, when the output signal is very weak and buried in noise.
 A sinusoidal signal is injected into the device under test, and the output signal
is determined via the synchronous detection technique.
 In situations where the attenuation of a system is to be measured (e.g.
attenuation of a laser light beam through a gas), it is often better to inject a
sinusoidally varying signal (i.e. by modulating the light source), as opposed to
a DC measurement (light source on for the duration of the measurement). DC
measurements are subject to drift and low frequency “1/f noise” in the light
sensor and opamp circuitry. Synchronous detection is therefore often used.
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DSB-SC Applications
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Synchronous Detection
 Let us say we have device for which we wish to characterize
the frequency response, i.e. determine its transfer function
cos(ω 0 t )
A cos (ω0 t +θ )
Device under test
H ω= Aωe j θ ω
A= A(ω0 )
θ =θ (ω0 )
 Inject a sinusoidal signal of frequency , and then use
synchronous demodulation on the output signal to recover the
phasor components “I” and “Q” at the output.
 The amplitude A and phase  can be obtained from the I and
Q signals at the outputs.
Homework: prove output is as claimed.
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EEE3086F Signals and Systems II
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DSB-SC Applications
Some information on sinusoids
 To derive the synchronous detection block diagram, think of
the output signal as a complex phasor (which stores the mag
and phase relative to the input sinusoid): A e jθ
 The input signal and output signals are:
1 jω t 1 − jω t
v in (t )=cos(ω0 t )= e + e
2
2
1
j (ω t+ θ) 1
− j(ω t + θ)
v 0 (t )= A cos (ω 0 t +θ )= A e
+ Ae
2
2
0
0
0
0
 The trick is to multiply the output by e− jω t and pass the
product through a low pass filter (to remove the −2 ω0 term):
1
1
− jω t
− j (2 ω t +θ )
jθ 1
jθ
[v 0 (t )e
]LPF =[ A e + A e
] = Ae
2
2
2
LPF
0
0
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0
DSB-SC Applications
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 Alternatively,
jθ
A e =2[v 0 (t)e
− jω0 t
] LPF =[v 0 (t )(2 cos(ω0 t )− j 2 sin(ω0 t ))]LPF
=[v 0 (t )(2 cos(ω0 t))] LPF + j [v 0 (t )(−2 sin (ω0 t))] LPF
= I + jQ
Q= Asin θ
A
θ
I = A cos θ
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DSB-SC Applications
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Decomposing a Sinusoid in its 'In-Phase' and 'Quadrature'
components
 A sinusoid with amplitude A and phase θ can be decomposed into
the sum of cos and sin components:
v 0 (t )= A cos (ω 0 t +θ )= A[ cos(ω0 t ) cos θ−sin (ω0 t )sin θ ]
= A cos θ cos(ω0 t )− A sin θ sin (ω0 t )
=a cos (ω 0 t )+b sin (ω0 t )
where a= A cos θ
and b=−A sin θ
 Multiplying v0(t) by cos(t) and low-pass filtering extracts a.
Multiplying v0(t) by sin(t) and low-pass filtering extracts b.
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DSB-SC Applications
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Synchronous Detection for measuring Magnitude and Phase
Oscillator
~
cos 0t
Device A cos(0t   ) where A≥0
Under
Test
Buffer
(amplifier )

2 cos 0t
LPF
2 cos 0t
I
In  Phase
Component
 900
 2 sin 0t

A.J.Wilkinson, UCT
DSB-SC Applications
LPF
Q
Quadrature
Component
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Synchronous Detection
 Note: The low pass filter (LPF) is chosen to have long time
constant to remove noise and also to eliminate the high
frequency component at 2ω0
 Output:
 Similarly:
A.J.Wilkinson, UCT
I (t )=[ A cos( ω0 t +θ ) × 2 cos(ω0 t)] LPF
=[ A cos θ + A cos( 2 ω0 t +θ )] LPF
= A cos θ
Q (t)=[ A cos(ω0 t +θ ) × (−2 sin (ω0 t))] LPF
=[− A sin(−θ )− A sin ( 2 ω0 t +θ )]LPF
= Asin θ
DSB-SC Applications
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Synchronous Detection
 From I and Q, we can easily extract A and 
 To get amplitude (note A >= 0):
82
2
2
2
2
I +Q = A cos θ + Asin θ= A
2
2
⇒ A= √ I +Q
The complex signal
I+jQ is essentially
the phasor representation
of A cos(0t   )
2
 To get phase:
Q= Asin θ
Q A sin θ
=
=tan θ
I A cos θ
Q
⇒ θ =arctan
for I>0;
I
(add  for I<0, Q>=0)
A
θ
I = A cos θ
(subtract  for I<0, Q<0)
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DSB-SC Applications
Frequency Domain Illustration of Synchronous Detection
 In synchronous detection, both the signal and noise are translated to baseband. The LPF
serves to eliminate the 20 component, and to bandlimit the noise (that has been
translated to baseband). The bandwidth of the LFP determines both the noise on the
output, and also the dynamics of the measurement (recall that the rise time of the step
response of a LPF is proportional to 1/B)
 The LPF can be thought of as implementing the equivalent of a very narrow bandwidth
BPF. Practically, it is much easier to build a LPF at DC to limit the noise, than a narrowband BPF at 0. Problems will also occur if the narrow BPF drifts off the centre
frequency. A LPF does not suffer from the drift effect – the passband is always on 0 Hz.
π Ae
Analysis
of the I
channel.
A.J.Wilkinson, UCT
−ω 0
−2 ω 0
− jθ
π Ae
jθ
noise
ω0
LPF
0
DSB-SC Applications
2 ω0
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Opamp Synchronous Detector Circuit
 Implements ‘multiplication by a square wave’ +1, -1
10K
Input + noise
A cos(0t   )
10K
LPF
R
10K
Ref square
wave - same
phase as
cos(0t )
A.J.Wilkinson, UCT
V0  A cos 
C
Noise at
output is
bandlimited
by the LPF
JFET
switch
switch open => follower
switch closed => inverter
DSB-SC Applications
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Opamp Synchronous Detector Circuit
 Demodulation via multiplication by a square wave works
because the fundamental component of the square wave mixes
with the input signal (difference frequency creates the DC
output).
 The other harmonics of the square wave create non-DC
components that are removed by the LPF.
 The demodulator is usually preceded by a BPF that ensures
that no input frequencies exist at the frequencies of the
harmonics (otherwise these too would be mixed to DC).
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DSB-SC Applications
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Demodulation with a square wave
Fourier transform Ae  j
of input sinusoid.
Ae j
 0
1
⊗
2π
0
0

Harmonics
of square wave
Convolve
a1 / 5
 50
 a1 / 3
 30
a1
 0
a1
0
 a1 / 3
0
a1 / 5
50
30

Components at fundamental mix with input to create
DC component. Other mixing products removed by LPF.
 60
A.J.Wilkinson, UCT
 40
 20
0
20
60
40
DSB-SC Applications

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5.2.4 The chopper amplifier
(avoids 1/f noise in amplifier circuits)
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DSB-SC Applications
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Chopper Amplifier Principles
 Practice amplifiers are subject to the effects of low frequency
1/f noise with a PSD:
1
S n (ω)∝
|f|
 This limits the SNR of DC and low frequency amplifiers.
PSD of
amplifier
noise.
1 noise region
f
S(f)
lower noise density
0
A.J.Wilkinson, UCT
Freq. f
1 kHz
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DSB-SC Applications
Chopper Amplifier Principles
 Basic Principle
 Avoid low frequency
1
noise by translating signal to a
f
higher frequency (using DSB-SC modulation).
 Then do the amplification with gain K.
 Translate back to baseband using DSB-SC demodulation
f (t)

BPF
K

LPF
Kf (t )
cos c t or PT (t )
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DSB-SC Applications
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A Practical Realization of the Chopper Amplifier
1
Bandpass
amplifier
@ c
2 Gain   K
f (t )
5
3
4
 Kf (t )
6
Low  pass
filter
cos c t
Chopper
F ( )
f (t )
t
A.J.Wilkinson, UCT

0
EEE3086F Signals and Systems II
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DSB-SC Applications
Chopper Amplifier equivalent diagram
f (t )

PT (t )
BPF

−K
LPF
2
 
1 2
2 π
Kf  t 
1  PT (t )
 Note:
 In this figure, modulation by multiplication with PT (t )
PT (t )
Square wave
1
0
A.J.Wilkinson, UCT
t
DSB-SC Applications
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Chopper Amplifier Principles
 Amplifier is an inverting amplifier of gain  K
 Therefore demodulate with inverted carrier
i.e. use 1  PT (t )
 Demodulating chopper waveform
1  PT (t ) 1
0
t
NB: Fourier series analysis reveals that the fundamental
component is 2
sin ω0 t 
π
A.J.Wilkinson, UCT
EEE3086F Signals and Systems II
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DSB-SC Applications
Chopper Amplifier Waveforms
Bandpass
filter
F12 ( )
f12 (t )
t
3c
 3c
−ωc 0
ωc

Chopper Signal
 F34 ( )
f 34 (t )
t

−ωc 0 ωc
Amplified Signal
(inverted)
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DSB-SC Applications
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Chopper Amplifier Waveforms
Low  pass
filter
F56 ( )
f 56 (t )
t
 3c
 c 0
3c 
c
Demodulation  LPF
1 2
2 π
2
( ) Kf (t )=0 . 20 Kf ( t )
1 2
2 π
2
( ) K F (ω )
t

0
Final Output
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DSB-SC Applications
EEE3086F Signals and Systems II
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 Chopper amplifiers are used for very low noise DC amplifier
applications.
 In the previous decades (say <1980), chopper circuits used a mechanical
relay as a switch, switching at several tens of Hz.
 Today, chopper amplifiers for DC amplification are available as
complete integrated circuits – using FET technology as a switch.
 Chopper amplifiers are not so common nowadays, as the performance of
modern low noise operational amplifiers has improved over the years.
 One might still want a chopper amplifier if the application required
avoidance of long term opamp drift in the ‘voltage offset’ parameter.
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DSB-SC Applications
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EEE3086F
Signals and Systems II
End of handout
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DSB-SC Applications
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