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ERG2310A-II p. II-1
Modulation Techniques
Modulation: translates an information-bearing signal (message signal) to a new spectral location (frequency domain)
Frequency (f)
a(t) ↔ A(f)
-fw 0 +fw
Baseband signal
A( f-fc )
(fc –fw) fc (fc+fw) Frequency (f)0
Bandpass signal
• Communication channels bandpass transfer (frequency) response
⇒ translates the message signal to be within the channel transfer response
1013Hz5x1014 HzOptical
2GHz100GHzMillimeterwave
100MHz5GHzMicrowave
2MHz100MHzVHF
100kHz5MHzShortwave Radio
2kHz100kHzLongwave Radio
BandwidthCarrier frequencyFrequency Band
Selected frequency bands
BW
f0f
H(f)
fc: carrier frequency
• Facilitates antenna reception
ERG2310A-II p. II-2
Modulation Techniques
If more than one message signal utilizes a channel
modulation allows translation of different signals to different spectral locations
multiplexing allows two or more message signals to be transmitted by a single transmitter (frequency division multiplexing)
desired modulated signal can be selected by a receiver
f
A1(f)
f
A2(f)
f
A3(f)
modulator
modulator
modulator
f
A1c(f)
f
A3c(f)
f
A2c(f)
f
demodulatorf
channel
f
A2c(f)
f
Low-pass filter
demodulator
Low-pass filter
f
A3c(f)
fc1
fc3
fc2
fc2
fc3
fc3
fc2
fc1
fc3fc2fc1
ERG2310A-II p. II-3
Frequency Translation
Recall: If )()( 2c
tfj ffAeta c −↔π)()( fAta ↔ , then (Fourier transform pairs)
Consider a message signal x(t) , which is bandlimited to the frequency range 0 to Wand has its Fourier transform is X(f) , is multiplied by cos (2π fc t) .
{ } [ ])()(21)2cos()( ccc ffXffXtftx −++=ℑ π
Baseband signal Frequency translated signal
ERG2310A-II p. II-4
Recovery of Baseband Signal
To recovery the baseband signal, we can simply multiply the translated signal with cos (2π fc t).
[ ] )4cos(2)(
2)()2(cos)()2cos()2cos()( 2 tftxtxtftxtftftx cccc ππππ +==
We obtain the baseband signal x(t) and a signal whose spectral range extends from (2fc-W) to (2fc+W). As fc >> W, the extra signal is removed by a low-pass filter.
Frequency translated signal
Baseband signal
[ ])2()2(21)(
21
cc ffXffXfX ++−+
ERG2310A-II p. II-5
Analog (Continuous-Wave) Modulation
A parameter of a high-frequency sinusoidal carrier is varied proportionallyto the message signal x(t) .
[ ])(cos)()( tttAts c φω +=General modulated signal:
ωc : carrier frequency
A(t) : instantaneous amplitude
φ(t) : instantaneous phase deviation
When A(t) is linearly related to the modulating (message) signal
Amplitude modulation (AM)
When φ(t) is linearly related to the modulating signal
Phase modulation (PM)
When time derivative of φ(t) is linearly related to the modulating signal
Frequency modulation (FM)
[ ] )(cos)()( txttAts oc ∝+= A(t) where φω
[ ] )()(cos)( txttAts cc ∝+= (t) where φφω
[ ] )()(cos)( txttAts cc ∝+=dt
(t)d where φφω
*FM & PM are commonly called Angle Modulation
ERG2310A-II p. II-6
Analog (Continuous-Wave) Modulation
Unmodulated carrier frequency
Message signal
Amplitude-modulated signal
Angle-modulated signal (frequency-modulated)
ERG2310A-II p. II-7
Analog (Continuous-Wave) Modulation
Message signal
Unmodulated carrier
Phase-modulated signal
Frequency-modulated signal
ERG2310A-II p. II-8
Amplitude Modulation
The envelope of the modulated carrier has the same shape as the message signal.
The amplitude of the carrier wave [ Ac cos(2π fct) ] varies linearly with the baseband message signal x(t).
The standard form of an amplitude-modulated (AM) signal is given by:
[ ] )2cos()(~1)( tftxmAts cac π+=
Where is the normalized message signal and ma is called the modulation index.
x(t) s(t) envelope
Ac
)(~ tx
ERG2310A-II p. II-9
Amplitude Modulation: DSB-LC
[ ]
)cos()(~)cos()2cos()(~)2cos(
)2cos()(~1)(
ttxmAtAtftxmAtfA
tftxmAts
caccc
caccc
cac
ωωππ
π
+=+=
+=AM signal:
cc fπω 2= where
Double-sideband –large carrier (DSB-LC)
[ ] [ ])()(2
)()(2
)( ccac
ccc ffXffX
mAffff
AfS −+++−++= δδ
ERG2310A-II p. II-10
Amplitude Modulation: DSB-LC[ ] )2cos()(~1)( tftxmAts cac π+=
Ac
Distorted signal !
So, Ac has to be large enough or we have to control the modulation index ma.
ERG2310A-II p. II-11
ma < 1 ma = 1 ma > 1Ac(1+ma)
Ac(1-ma)Ac
Amplitude Modulation: DSB-LC[ ] )2cos()(~1)( tftxmAts cac π+=
x(t)
+DSB-LC
Effect of modulation index ma :
Define modulation depth =c
c
AA−maxA
For a sinusoidal message signal, Amax=Ac(1+ma), thus the modulation depth is ma .
ERG2310A-II p. II-12
Amplitude Modulation: DSB-LC
Product modulator:
[ ] )2cos()(~1)( tftxmAts cac π+=Generation of DSB-LC signal:
Square-law modulator: + filter
vin vout
x(t)
cosωct
Nonlinear device
s(t)221 ininout vavav +=ttxv cin ωcos)( +=
4444 34444 21)(
1
21
22
221 cos)(21cos)()(
ts
ccout ttxaaatatxatxav ωω
++++= where Ac=a1 and ma=2a2/a1
Chopper/rectifier modulator:
+
-
+
-
Bandpass filter at ωc
vo(t)x(t)
Accosωct
ωc+
-
+
-
Bandpass filter at ωc
vo(t)x(t)
Accosωct
ERG2310A-II p. II-13
Amplitude Modulation: DSB-LC
The chopper or rectifier can generate a periodic waveform whose fundamental frequency is ωc rad/sec.
The periodic signal p (t) can be represented as
∑∞
−∞=
=n
tjnn
cePtp ω)(
.)()()( ∑∞
−∞=
=n
tjnn
cetfPtptf ω
Applying the frequency translation property of the Fourier transform, we get
.)()}()({ ∑∞
−∞=
−=ℑn
cn nFPtptf ωω
Chopper/rectifier modulator:
Consider p(t)f(t):
Let f(t) = Accosωct + x(t)
ERG2310A-II p. II-14
Amplitude Modulation: DSB-LC
Demodulation of DSB-LC signal:
By envelope detector: the diode cuts off the negative part of the DSB-LC signal while RC acts as a lowpass filter to retrieve the envelope.
cfRCW /1/1 >>>> where W is the message signal bandwidth
ERG2310A-II p. II-15
Amplitude Modulation: DSB-LC[ ] )2cos()(~1)( tftxmAts cac π+=DSB-LC signal:
Consider the average power of s(t) :
[ ]
[ ]
[ ] [ ]
[ ][ ]
sbc
xxac
cac
caaaac
caac
cac
s
PP
txPPmA
ttxmA
ttxmtxmtxmtxmA
ttxmtxmA
ttxmA
tsP
2
)(~12
0)2cos(0~)(~12
)2cos()(~2)(~1)(~2)(~12
)2cos(1(21)(~2)(~1
)(cos)(~1
)(
222
222
22222
222
222
2
+=
=+=
==+=
+++++=
+++=
+=
=
where
as and (t)x if
ω
ω
ω
ω
Thuscxaxacsbcc PPmPmAPAP 2222
21
41
21 === ;
where Psb : average power per sideband
ssbsccsbxaa PPPPPPPmtxm212
2111)( 2 ≥−=⇒≤⇒≤⇒≤ For
At least 50% of total transmitted power resides in the carrier term which conveys no information wasteful of power
ERG2310A-II p. II-16
Amplitude Modulation: DSB-LC
Fraction of total transmitted power contained in the sidebands is:
xa
xa
cxac
cxa
sbc
sb
PmPm
PPmPPPm
PPP
2
2
2
2
122
+=
+=
+=µ
If x(t) is a single sinusoid, i.e. cosωmt, then
Thus, and is known as the transmission efficiency of DSB-LC AM system.
Example:
21)(2 == txPx
2
2
2 a
a
mm+
=µ
A given AM (DSB-LC) broadcast station transmits an average carrier power output of 40kW and uses a modulation index of 0.707 for sine-wave modulation. Calculate (a) the total average power output; (b) the transmission efficiency; and(c) the peak amplitude of the output if the antenna is represented by a 50-ohm resistive load.
Solution: (a) The total average power output is ).2/1(2 2acsbcs mPPPP +=+=
For ma = 0.707, .50)4/11(40 kW=+=sP
(b) The transmission efficiency is %.205.25.0
)707.0(2)707.0(
2
2
==+
=µ
(c) Consider .1042,2
622
×==⇒= cc RPARAP
The peak amplitude of the output is V.3414)1( =+ Ama
ERG2310A-II p. II-17
Amplitude Modulation: DSB-SCThe “wasted” carrier power in DSB-LC can be eliminated by setting ma=1 and suppressing the carrier.
Thus the modulated signals becomes
In this case, the carrier frequency component is suppressed, thus it is called double-sideband suppressed-carrier modulation (DSB-SC) .
Its spectral density is:
)2cos()(~)( tftxAts cc π=
[ ])()(2
)( ccc ffXffXA
fS −++=
Average Power of the modulated signal:
xc
xxc
cc
s
PA
txPPA
ttxA
tsP
2
22
222
2
41
)(~21
)(cos)(~)(
=∴
=
==
=
=
sb
sb
P
2P
where
ω
ERG2310A-II p. II-18
Amplitude Modulation: DSB-SC
A signal spectrum can be translated an amount ± ωc rad/sec in frequency by multiplying the signal with any periodic waveform whose fundamental frequency is ωc rad/sec.
The periodic signal p (t) can be represented as
∑∞
−∞=
=n
tjnn
cePtp ω)(
.)()()( ∑∞
−∞=
=n
tjnn
cetxPtptx ω
Applying the frequency translation property of the Fourier transform, we get
.)()}()({ ∑∞
−∞=
−=ℑn
cn nXPtptx ωω
Consider p(t)x(t):
ERG2310A-II p. II-19
Amplitude Modulation: DSB-SC
Example:A periodic signal consists of the exponentially decreasing waveform e-at, 0 ≤ t < T, repeated every T seconds. A given signal f(t) is multiplied by this periodic signal. Determine an expression describing the spectrum and the time waveform of the resulting amplitude-modulated signal if all components except those centered at ±ωc, ωc = 2π/T, are discarded.
Solution: The Fourier series for the given periodic signal can be written as
∑∞
−∞=
=n
tjnn
cePtp ,)( ω where ∫ +−==
−−−T
c
aTtjnat
n jnae
Tdtee
TP c
0.111
ωω
The spectrum of the product p(t)f(t) is
.)(1)(1)1(1
ω+ω
ω−+ω−ω
ω+− −
cc
cc
aT Fja
Fja
eT
The corresponding terms in the Fourier series are
.)(1)(1)1(1
ω−
+ω+
− ω−ω− tj
c
tj
c
aT cc etfja
etfja
eT
Combining yields the time waveform
),cos()(12022
θ+ωω+
− −
ttfa
eT c
c
aT
where ),/(tan 10 acω−=θ − ./2 Tc π=ω
ERG2310A-II p. II-20
Amplitude Modulation: DSB-SC
Generation of DSB-SC signal by balanced modulator:
AM Modulator
AM Modulator
+
)(21 tx
)(21 tx−
ttxA cc ωcos)(211
+
ttxA cc ωcos)(211
−
+
-tAtx cc ωcos)(
tA cc ωcos
ERG2310A-II p. II-21
Amplitude Modulation: DSB-SC
Demodulation of DSB-SC signal:Assuming that the transmitted signal is
ttxts cωcos)()( =
To demodulate the signal, we have
ttxtxttxtts
c
cc
ωωω
2cos)()(cos)(cos)(
21
21
2
+==
Taking the Fourier transform of both sides, we get
)2()2(
)(}cos)({
414121
c
c
c
XXXtts
ωωωω
ωω
−+++
=ℑ
ERG2310A-II p. II-22
Amplitude Modulation: DSB-SCConsider a small frequency error, ∆ω, and a phase error, θ0, are introducedin the locally generated carrier signal at the receiver. The signal at the receiver becomes
].)2cos[()(])cos[()(
])cos[(cos)(])cos[()(
021
021
00
θωωθω
θωωωθωω
+∆+++∆=
+∆+=+∆+
ttxttx
tttxtts
c
ccc
After passing via the low-pass filter, the output is
].)cos[()()( 021 θω +∆= ttxteo
Phase error and frequency error results in undesirable distortion. In somecases, they vary randomly, resulting in unacceptable performance.
Remedy:Using a synchronized oscillator to recover the original signal f(t) from themodulated signal φ(t). (Synchronous detection, or coherent detection)
ERG2310A-II p. II-23
Amplitude Modulation: DSB-SC
The original signal x(t) can be recovered from the modulated signal s(t) by multiplyings(t) by cosωct (i.e. synchronous detection).
The same circuits as those used for modulation can be used for demodulation withthe following minor differences.
1. Since the desired output spectrum is centered about ω=0 and therefore a low-passfilter is needed at the output.
2. The oscillator in the demodulator must be synchronized to the oscillator in the demodulator to achieve proper demodulation.
This is usually accomplished by either a direct connection if the modulator and demodulator are in close proximity or by supplying a sinusoid displaced in frequency but related to the modulator-oscillator frequency. The sinusoid is called a “pilot carrier”.
ERG2310A-II p. II-24
Amplitude Modulation: DSB-SC
Pilot Carrier SystemsIt is a common method used in DSB-SC modulation to maintain synchronizationbetween modulator and demodulator.In this case, a sinusoidal tone whose frequency and phase are related to thecarrier frequency is generated and is sent outside the pass-band of the modulatedsignal so it will not alter the frequency response capability of the system.
A tuned circuit in the receiver detects the tone, translate it to the proper frequency,and uses it to correctly demodulate the DSB-SC signal.e.g. Stereo-multiplex system
cosωct÷2 frequency
dividerAtten.
x
∑
∑
∑
L
R
L-R
L+R
38kHz 19kHz
+
+
+
-
+
+
+ To transmitter
L-R (lower sideband)
L-R (upper sideband)
L+R
0 15 19 23 38 53
Audio (mono) DSB-SC
f in kHz
Spectrum used for stereo multiplexing before transmission
Pilo
t ca
rrie
r
ERG2310A-II p. II-25
Amplitude Modulation: DSB-SCPhase-Locked Loop (PLL)In pilot tone system, phase-locked loop is used to synchronize one sinusoidal to another.
A simplified phase-locked loop stereo demodulator.
ERG2310A-II p. II-26
Amplitude Modulation: QAM
Quadrature MultiplexingUsing the orthogonality of sines and cosines , it is possible to transmit andreceive two different signals simultaneously on the same frequency.
Thus, each signal can be recovered by synchronous detection of the receivedsignal using carriers of the same frequency but in phase quadrature.
ttfttfts cc ωω sin)(cos)()( 21 +=
ttfttftfttfttftts
cc
cccc
ωωωωωω
2sin)(2cos)()(cossin)(cos)(cos)(
221
121
121
22
1
++=+=
ttftfttfttftttftts
cc
cccc
ωωωωωω
2cos)()(2sin)(sin)(sincos)(sin)(
221
221
121
221
−+=+=
In the low-pass filter, all terms at 2ωcare attenuated, yielding
).()(),()(
221
2
121
1
tftetfte
==
x
xf1(t)
f2(t)
cosωct
sinωct
+
+∑ s(t)
x
x ½ f1(t)
½ f2(t)
cosωct
sinωct
LPF
LPF
ERG2310A-II p. II-27
ERG2310A-II p. II-28
Frequency Division Multiplexing (FDM)
Frequency-division multiplexing is the positioning of signal spectra in frequency such that each signal spectrum can be separated out from all the others by filtering.
ERG2310A-II p. II-29
Frequency Division Multiplexing (FDM)
Example: commercial radio and television receiver
ERG2310A-II p. II-30
Intermediate Frequency (IF)
Heterodyning means the translating or shifting in frequency.
In the heterodyne receiver the incoming modulated signal is translated in frequency, thus occupying an equal bandwidth centered about a new frequency, known as an intermediate frequency (IF), which is fixed and is not dependent on the received signal center frequency.
The signal is amplified at the IF before demodulation.
If this intermediate frequency is lower than the received carrier frequency but above the final output signal frequency, it is called a superheterodyne receiver.
ERG2310A-II p. II-31
Intermediate Frequency (IF)
Advantage:The amplification and filtering is performed at a fixed frequency regardless ofstation selection.
Disadvantage:Image-frequency problem
Two ways to solve this problemi. Choose the intermediate frequency as
high as possible and practical.ii. Attenuate the image frequency before
heterodyning.
The intermediate frequency chosen must be free from other strong transmissionsor otherwise the receivers will amplify these spurious signals as they leak into the high-gain IF stages.
ERG2310A-II p. II-32
Intermediate Frequency (IF)
Example:A given radar receiver operating at a frequency of 2.80 GHz and using the super-heterodyne principle has a local oscillator frequency of 2.86 GHz . A second radarreceiver operates at the image frequency of the first and interference results.
(a) Determine the intermediate frequency of the first radar receiver.(b) What is the carrier frequency of the second receiver?(c) If you were to redesign the radar receiver, what is the minimum intermediate
frequency you would choose to prevent image-frequency problems in the 2.80-3.00 GHz radar band?
Solution:
(a) MHz.60GHz80.2GHz86.2 =−=−= cLOIF fff
(b) GHz.92.2GHz12.0GHz80.22 =+=+= IFcIMAGE fff
(c) 100MHz.GHz;20.0GHz80.2GHz00.32 ≥=−=−≥ IFMINMAXIF ffff