RF Transceiver Module Design
Chapter 1
From Basics to RF Transceivers
李健榮 助理教授
Department of Electronic Engineering
National Taipei University of Technology

Outline
• Definition of dB
• Phasor
• Modulation
• Transmitter Architecture
• Demodulation
• Receiver Architecture
• From Fourier Transform to Modulation Spectrum
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Definition of dB
• , where
• Power gain
• Voltage gain
• Power (dBW)
• Power (dBm)
• Voltage (dBV)
• Voltage (dBuV)
( )dB 10 log G= ⋅ ( )aG
b
=
2
1
10 log
P
P
 = ⋅  
 
2
1
20 log
V
V
 = ⋅  
 
( )10 log
1-W
P= ⋅
( )10 log
1-mW
P= ⋅
( )20 log
1-Volt
V= ⋅
( )20 log
1- V
V
µ= ⋅
Relative
(Ratio, unitless, dB)
Absolute
(Have unit, dBW, dBm, dBV…)
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In some textbooks, phasor may be
represented as
Euler’s Formula
• Euler’s Formula states that: cos sinjx
e x j x= +
( ) ( ) ( )
{ } { }cos Re Rej t j j t
p p pv t V t V e V e eω φ φ ω
ω φ +
= ⋅ + = ⋅ = ⋅
( )cos sin
def
j
p p pV V e V V jφ
φ φ φ= ⋅ = ∠ = +• Phasor :
Don’t be confused with Vector which is commonly denoted as .A
phasor
A real signal can be represented as:
V
V
( ) ( )cospv t V tω φ= ⋅ +
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Euler’s Trick on the Definition of e
2 3
lim 1 1
1! 2! 3!
n
x
n
x x x x
e
n→∞
 
= + = + + + + 
 
…
x jx=
( ) ( )
2 3 2 4 3 5
1 1
1! 2! 3! 2! 4! 3! 5!
jx jx jxjx x x x x
e j x
   
= + + + + = − + − + + − + − +   
   
… … …
• Euler played a trick : Let , where 1j = −
1
lim 1
n
n
e
n→∞
 
= + 
 
6/33
2 4
cos 1
2! 4!
x x
x = − + − +…
3 5
sin
3! 5!
x x
x x= − + − +…
cos sinjx
e x j x= +
cos sinjx
e x j x−
= −
cos
2
jx jx
e e
x
−
+
=
sin
2
jx jx
e e
x
j
−
−
=
• Use and
we have
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Coordinate Systems
x-axis
y-axis
x-axis
y-axis
P(r,θ)
θ
r
P(x,y)
2 2
r x y= +
1
tan
y
x
θ −
=
cosx r θ=
siny r θ=
Cartesian Coordinate System Polar Coordinate System
(x,0)
(0,y)
( )cos ,0r θ
( )0, sinr θ
Projection
on x-axis
Projection
on y-axis
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Sine Waveform
x-axis
y-axis
P(x,y)
x
y
r
θ θθ
y
θ
0 π/2 π 3π/2 2π
Go along the circle, the projection on y-axis results in a sine
wave.
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x
θ
0
π/2
π
3π/2
Cosine Waveform
x-axis
y-axis
θ
Go along the circle, the projection
on x-axis results in a cosine wave.
Sinusoidal waves relate to a Circle
very closely.
Complete going along the circle to
finish a cycle, and the angle θ
rotates with 2π rads and you are
back to the original starting-point
and. Complete another cycle again,
sinusoidal waveform in one period
repeats again. Keep going along the
circle, the waveform will
periodically appear.
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Complex Plan (I)
It seems to be the same thing with x-y plan, right?
• Carl Friedrich Gauss (1777-1855) defined the complex plan.
He defined the unit length on Im-axis is equal to “j”.
A complex Z = x + jy can be denoted as (x, yj) on the complex plan.
(sometimes, ‘j’may be written as ‘i’which represent imaginary)
Re-axis
Im-axis
Re-axis
Im-axis
P(r,θ)
θ
r
P(x,yj)
2 2
r x y= +
1
tan
y
x
θ −
=
cosx r θ=
siny r θ=
(x,0j)
(0,yj)
( )cos ,0r θ
( )0, sinr θ
( )1j = −
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Complex Plan (II)
Re-axis
Im-axis
1
Every time you multiply something by j, that thing will rotate 90 degrees.
1j = − 2
1j = − 3
1j = − − 4
1j =
1*j=j
j
j*j=-1
-1
-j
-1*j=-j -j*j=1
(0.5,0.2j)
(-0.2, 0.5j)
(-0.5, -0.2j)
(0.2, -0.5j)
• Multiplying j by j and so on:
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Sine Waveform
Re-axis
Im-axis
P(x,y)
x
y
r
θ θθ
y = rsinθ
θ
0 π/2 π 3π/2 2π
To see the cosine waveform, the same operation can be applied to trace out
the projection on Re-axis.
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Phasor Representation (I) – Sine Basis
( ) ( ) { } { }sin Im Imj j t j j
sv t A t Ae e Ae eφ ω φ θ
ω φ= + = =
Re-axis
Im-axis
P(A,ϕ)
y = Asinϕ
θ
0 π/2 π 3π/2 2π
ϕ
tθ ω=
Given the phasor denoted as a point on the complex-plan, you should know it
represents a sinusoidal signal. Keep this in mind, it is very important!
time-domain waveform
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Phasor Representation (II) – Cosine Basis
( ) ( ) { } { }cos Re Rej j t j j
sv t A t Ae e Ae eφ ω φ θ
ω φ= + = =
Re-axis
Im-axis
P(A, ϕ)
y = Acos ϕ
θ
0 π/2 π 3π/2 2π
ϕ
tθ ω=
time-domain waveform
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Phasor Representation (III)
( ) ( ) { }1
1 1 1 1sin Im j j t
v t A t Ae eφ ω
ω φ= + =
Re-axis
Im-axis
P(A1, ϕ1)
ϕ1
P(A2, ϕ2)
P(A3, ϕ3)
θ
0 π/2 π 3π/2 2π
tθ ω=
A1sin ϕ1
( ) ( ) { }2
2 2 2 2sin Im j j t
v t A t A e eφ ω
ω φ= + =
( ) ( ) { }3
3 3 3 3sin Im j j t
v t A t A e eφ ω
ω φ= + =
A2sin ϕ2
A3sin ϕ3
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Phasor Everywhere
• Circuit Analysis, Microelectronics:
Phasor is often constant.
• Field and Wave Electromagnetics, Microwave Engineering:
Phasor varies with the propagation distance.
• Communication System:
Phasor varies with time (complex envelope, envelope, or
equivalent lowpass signal of the bandpass signal).
( ) ( )5cos 1000 30sv t t= + 5 30sV = ∠
( ) ( ) ( ) ( ) ( )
{ }, cos cos Re j x t j x t
v x t A x t B x t Ae Beβ ω β ω
β ω β ω − − +
= − + + = +
( ) j x j x
V x Ae Beβ β−
= +
( ){ }Re j t
V x e ω
=
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Modulation
• Why modulation?
Communication
Bandwidth
Antenna Size
Security, avoid Interferes, etc.
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
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Amplitude Modulation
( ) ( ) cos2m BB cs t s t A f tπ= ⋅
Baseband real signal
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Amplitude-modulated signal
(AM signal)
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Frequency Modulation
( ) ( ){ }cos 2m c f BBs t A f K s t tπ  = + ⋅ 
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
Baseband real signal
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Frequency-modulated signal
(FM signal)
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Phase Modulation
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
( ) ( )cos 2m c p BBs t A f t K s tπ = + 
( )cos 2 c BBA f t tπ φ= +  
Baseband real signal
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Phase-modulated signal
(PM signal)
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Linear Modulation
( ) ( ) ( )cos 2m BB c BBs t A t f t tπ φ= ⋅ +  
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
Baseband real signal
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Linear-modulated signal
( )BBs t ( ) ( ), ?BB BBA t tφ
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Linear Modulation
• Consider a modulated signal
( ) ( ) ( ) ( ) ( )
{ }2
cos 2 Re c BBj f t t
m BB c BB BBs t A t f t t A t e
π φ
π φ
+  
= ⋅ + = ⋅  
( ) ( )
( ) ( ) ( ){ }2 2
Re Re cos sinBB c cj t j f t j f t
BB BB BB BBA t e e A t t j t e
φ π π
φ φ = ⋅ = ⋅ +   
( ) ( ) ( )
( ) ( ) ( )cos sinBBj t
l BB BB BB BBs t A t e A t t j tφ
φ φ= ⋅ = ⋅ +  
( ) ( ) ( ) ( ) ( ) ( )cos sinBB BB BB BBA t t jA t t I t jQ tφ φ= ⋅ + ⋅ = +
( ) ( ) ( ) ( ){ }Re cos2 sin2m c cs t I t jQ t f t j f tπ π= + ⋅ +  
( ) ( )cos2 sin 2c cI t f t Q t f tπ π= −
Time-varying phasor (information in both amplitude and phase)
( )BBs t : real
( )ls t : complex
Modulated signal is the linear combination of I(t), Q(t), and the carrier. Thus the linear modulator
is also called “I/Q Modulator,” and it is an universal modulator.
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Linear Modulator
• The modulator accomplishes the mathematical operation.
( ) ( ) ( ) ( ) ( ){ }Re cos sin cos2 sin 2m BB BB BB c cs t A t t j t f t j f tφ φ π π= ⋅ + +  
( ) ( ) ( ) ( )cos cos2 sin sin 2BB BB c BB BB cA t t f t A t t f tφ π φ π= −
( ) ( )cos2 sin 2c cI t f t Q t f tπ π= −
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
( )I t
cos ctω
sin ctω
( )Q t
( )ms t
+
− 90
( )I t
cos ctω
( )Q t
( )ms t
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I component Q component
I channel Q channel
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Transmitter Architecture (I)
• Linear Transmitter
90
( )I t
cos ctω
( )Q t
( )ms t
Power Amplifier
(PA)
Antenna
Baseband
Processor
90
cos ctω
( )ms t
Power Amplifier
(PA)
Antenna
Matching /
BPF
Matching
( )I t
( )Q t
Baseband
Processor
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Transmitter Architecture (II)
• Polar Transmitter
( ) ( ) ( ) ( ){ } ( ) ( )
{ }22
cos 2 Re Re c BBc
j f t tj f t
m BB c BB l BBs t A t f t t s t e A t e
π φπ
π φ
+  
= ⋅ + = ⋅ = ⋅  
( )BBA t
cos ctω
( )ms t
Switching-mode
PA
Antenna
Phase
Modulator
Matching
( )BBA t
( )BB tφ
Baseband
Processor
Amplitude
Modulator
• Linear regulator
• PWM modulator
• Class-S modulator
• Linear modulator to generate PM signal
• Frequency synthesizer or PLL-based PM modulator
• Analog scheme: EER
( )
{ }2
Re c BBj f t t
e
π φ+  
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Linear Demodulation
( ) ( ) ( ) ( ) ( )cos 2 cos2 sin2m BB c BB c cs t A t f t t I t f t Q t f tπ φ π π= ⋅ + = −  
( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1
cos2 cos 2 sin2 cos2 cos4 1 sin4 sin0
2 2
m c c c c c cs t f t I t f t Q t f t f t I t f t Q t f tπ π π π π π= − ⋅ = ⋅ + − ⋅ +
( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1
sin2 cos2 sin2 sin 2 sin4 sin0 1 cos4
2 2
m c c c c c cs t f t I t f t f t Q t f t I t f t Q t f tπ π π π π π− = − + = − ⋅ + + ⋅ −
( ) ( ) ( )cos4 sin 4
2 2 2
c c
I t I t Q t
f t f tπ π
 
= + − 
 
( ) ( ) ( )sin4 cos4
2 2 2
c c
Q t I t Q t
f t f tπ π
 
= − + 
 
?
Receiver
( )ms t ( )BBs t
Received modulated signal:
Multiplied by “cosine”:
Multiplied by “−−−− sine”:
High-frequency components
(should be filtered out)
High-frequency components
(should be filtered out)
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Linear Demodulator
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
LPF
LPF
( )I t
( )Q t
( )ms t
LPF
LPF
90
cos ctω
( ) ( ) ( )
( ) ( )BBj t
l BBs t A t e I t jQ t
φ
= ⋅ = +
( ) ( ) ( )2 2
BBA t I t Q t= +
( )
( )
( )
1
tanBB
Q t
t
I t
φ −
=
Baseband
Processing
Original Information (or data)
( )I t
( )Q t
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Receiver Architecture
• Linear Receiver (direct conversion)
90
( )I t
cos ctω
( )Q t
( )ms t
Low Noise Amplifier
(LNA)
Baseband
Processor
LPF
LPF
Matching /
BPF
90
( )I t
cos ctω
( )Q t
( )ms t
Low Noise Amplifier
(LNA)
Baseband
Processor
LPF
LPF
Matching
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Digital Modulation
• I(t) and Q(t) for digital transmission?
• Assume that I(t) and Q(t) are pulses at TX, I(t) and Q(t)
waveforms will be recovered ideally after the demodulation
process, of course, pulses.
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( ) ( ) ( )cos2 sin 2m c cs t I t f t Q t f tπ π= −
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
LPF
LPF
( )I t
cos ctω
sin ctω−
( )Q t
TX RX
Assume the LPF has a
sufficiently wide bandwidth to
recover the pulse waveform.
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• A Quadrature Phase Shift Keying (QPSK) signal is a good
example.
Quadrature Phase Shift Keying
( )I t
( )Q t
( )1,1
( )1, 1−( )1, 1− −
( )1,1−
cA+
cA+cA−
cA−
Constellation
( )I t
cos ctω
sin ctω−
( )Q t
S/P
Converter
Binary
Baseband
Data( )I t
( )Q t
Binary
Baseband
Data
bT
2 bT
t
S/P Converter
The linear combination shows that the QPSK signal
has 4 different phase states (1 symbol = 2 bits = 4
states).
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Symbol

• Phase transition in QPSK signal due to simultaneous transition
of I(t) and Q(t).
Phase Transition
( )I t
( )Q t
S/P
Converter
Binary
Baseband
Data
( )I t
( )Q t
t
( )I t
( )Q t
( )1,1
( )1, 1−( )1, 1− −
( )1,1−
cA+
cA+cA−
cA−
Constellation
Constant
envelope
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Bandwidth Consideration (I)
• A rectangular waveform has many frequency components
covering within a wide bandwidth. For many reasons, the
modulation spectrum occupying such a wide bandwidth is not
preferable.
( ) ( ) ( )cos2 sin 2m c cs t I t f t Q t f tπ π= −
( )I t
cos ctω
sin ctω−
( )Q t
TX
f
f
f
( )mS f
How to limit the
bandwidth?
cf
cf
f
0
0
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Bandwidth Consideration (II)
• Is that good to limit the bandwidth at passband?
( )I t
cos ctω
sin ctω−
( )Q t
TX
BPF
cf
f
channel
f
( )I t
cos ctω
sin ctω−
( )Q t
BPF2
BPF1
BPFn
channel
f
channelchannel
Requiring many BPFs for each channel is impractical
selector
cf
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Bandwidth Consideration (III)
• Limit the bandwidth at baseband – Pulse Shaping
( )I t
cos ctω
sin ctω−
( )Q t
LPF
LPF
f
t
0
f
t
0
f0
f0
cf
f
The low-pass filter is use to shape the
waveform, thus called “pulse shaping
filter,” or “shaping filter.”
Constant envelope
(before shaping)
180 18090
Time-varying envelope
(after shaping)
Smooth the sharp transition
( )1,1
( )1,1−
( )Q t
( )I t
( )1, 1−( )1, 1− −
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Band-limiting and Inter Symbol Interference (I)
• Nyquist Filter :
Produce pulse shapes with no ISI at each sampling instant
Brick-wall LPF
IFT
Sinc shape
Raised Cosine Filter
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Band-limiting and Inter Symbol Interference (II)
• Gaussian Filter :
Time domain response is Gaussian as well, it exhibits no overshot or ringing in the
time domain. This smooth well-behaved impulse response results in very little ISI
Reduce bandwidth : Smaller BT causes even faster spectral roll-off , but this has a
price, smaller BT cause more ISI.
Relative bandwidth BT = (filter BW) / (Bite rate)
Power spectra of MSK and GMSK Signals for varying BT
BT=0.2
BT=0.25
BT=0.3
Impulse response
MSK : BT is infinite
(no filter)
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Optimum Receiver
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
LPF
LPF
( )I t
cos ctω
sin ctω−
( )Q t
LPF
LPF
TX RX
Matched Filter
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Picture of a Practical Transceiver
( )I t
cos ctω
sin ctω−
( )Q t
LPFFilter
Filter
DAC
DAC LPF
Digital Processor
Digital pulse shaping filter
Waveform recovery filter
for DAC
Data (bits)
Waveform
(digital, M-bits)
Quantized Waveform (analog) Recovered Waveform (analog)
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
LPF
LPF
Filter
Filter
Digital Processor
Matched filter or
correlator
ADC
ADC
Demodulated Waveform (analog)
Sampled Waveform
(digital)
Decision
Decision
Data (bits)
Mixing spurs remover
Encoder and decoder are not
included here for simple.
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Modulation in Frequency Domain
• Fourier Series Representations
• Non-periodic Waveform and Fourier Transform
• Spectrum of a Real Signal
• AM, PM, and Linear Modulated Signal
• Concept of Complex Envelope
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Fourier Series Representations
• There are three forms to represent the Fourier Series of a
periodic signal :
Sine-cosine form
Amplitude-phase form
Complex exponential form
( ) ( )0 1 1
1
cos sinn n
n
x t A A n t B n tω ω
∞
=
= + +∑
( ) ( )0 1
1
cosn n
n
x t C C n tω φ
∞
=
= + +∑
( ) 1jn t
n
n
x t X e ω
∞
=−∞
= ∑
( )x t
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t
x(t)
t
t
t
( )X jω
ω
1f 13 f 15 f
.etc
T1
1 1C φ∠
2 2C φ∠
3 3C φ∠
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Sine-Cosine Form
( )0 0
area under curve in one cycle
period T
1 T
A x t dt
T
= =∫
( ) 10
2
cos , for 1 but not for 0
T
nA x t n tdt n n
T
ω= ≥ =∫
( ) 10
2
sin , for 1
T
nB x t n tdt n
T
ω= ≥∫
is the DC term
(average value over one cycle)
• Other than DC, there are two components appearing at a given
harmonic frequency in the most general case: a cosine term
with an amplitude An and a sine term with an amplitude Bn.
(A complete cycle can also be noted
from )~
2 2
T T−
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Amplitude-Phase Form
( ) ( )0 1
1
cosn n
n
x t C C n tω φ
∞
=
= + +∑
( ) ( )0 1
1
sinn n
n
x t C C n tω θ
∞
=
= + +∑
2 2
n n nC A B= +
• The sum of two or more sinusoids of a given frequency is
equivalent to a single sinusoid at the same frequency.
• The amplitude-phase form of the Fourier series can be
expressed as either
or
0 0C A= is the DC term
is the net amplitude of a given component at frequency nf1,
since sine and cosine phasor forms are always
perpendicular to each other.
where
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Complex Exponential Form (I)
1
1 1cos sinjn t
e n t j n tω
ω ω= +
1
1 1cos sinjn t
e n t j n tω
ω ω−
= −
1 1
1cos
2
jn t jn t
e e
n t
ω ω
ω
−
+
=
1 1
1sin
2
jn t jn t
e e
n t
j
ω ω
ω
−
−
=
cos sinjx
e x j x= +
cos sinjx
e x j x−
= −
cos
2
jx jx
e e
x
−
+
=
sin
2
jx jx
e e
x
j
−
−
=
Recall that
• Euler’s formula
1
nω is called the positive frequency, and 1
nω− the negative frequency
From Euler’s formula, we know that both positive-frequency and negative-
frequency terms are required to completely describe the sine or cosine
function with complex exponential form.
Here
1jn t
e ω
1jn t
e ω−
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Complex Exponential Form (II)
1 1jk t jk t
k kX e X eω ω−
−+ ( )where kkX X− =
( ) 1jn t
n
n
x t X e ω
∞
=−∞
= ∑
( ) 1
0
1 T
jn t
nX x t e dt
T
ω−
= ∫
• The general form of the complex exponential form of the
Fourier series can be expressed as
where Xn is a complex value
• At a given real frequency kf1, (k>0), that spectral representation
consists of
The first term is thought of as the “positive frequency” contribution, whereas the second is the
corresponding “negative frequency” contribution. Although either one of the two terms is a
complex quantity, they add together in such a manner as to create a real function, and this
is why both terms are required to make the mathematical form complete.
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Period Becomes Infinite
T 2T 3T 4T 5T
( )x t
f
nX
T 2T
T
T
f
nX
f
nX
f
nX
Single pulse T → ∞
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Fourier Transform
( ) ( )X f F x t=   F ( ) ( )1
x t F X f−
=   F
( ) ( ) j t
X f x t e dtω
∞
−
−∞
= ∫
( ) ( ) j t
x t X f e dfω
∞
−∞
= ∫
• Fourier transformation and its inverse operation :
• The actual mathematical processes involved in these operations
are as follows:
2 fω π=
• The Fourier transform is, in general, a complex function
and has both a magnitude and an angle:
( )X f
( ) ( ) ( )
( ) ( )j f
X f X f e X f f
φ
φ= = ∠
( )X f
f
For the nonperiodic signal, its spectrum is continuous, and, in
general, it consists of components at all frequencies in the
range over which the spectrum is present.
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Modulation Spectrum (I)
• From Euler’s Formula :
• AM signal (DSB-SC)
cos
2
jx jx
e e
x
−
+
=
A “real signal” is composed of positive and negative frequency components.
( ) ( )cos2m cs t A t f tπ=
Two-sided amplitude frequency spectrum
( ) ( )2 1000 2 10001
50cos 2 1000
2
j t j t
t e eπ π
π × − ×
× = +
2525
0 Hz 1 kHz1 kHz−
f
One-sided amplitude frequency spectrum
50
0 Hz 1 kHz
( )50cos 2 1000tπ ×
f
t( ) ( )BBs t A t=
f
f
cf0 Hzcf−
0 Hz
USBLSB
USBLSBLSBUSB
cos2 cf tπ
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“real signal”
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Phase
Modulator
Modulation Spectrum (II)
t( )BBs t
f
0 Hz
USBLSB
cos2 cf tπ
( ) ( )2 2
2 2
c cj t j tj f t j f tA A
e e e e
φ φπ π− −
= +
( ) ( )( )cos 2m cs t A f t tπ φ= +
( )
{ } ( )
{ }2 2
Re Rec c
j f t t j t j f t
A e A e e
π φ φ π+  
= ⋅ = ⋅
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“real signal”
f
cf0 Hzcf−
USBLSBLSBUSB
“complex”“complex” “real”
• PM signal
Complex conjugate
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Modulation Spectrum (III)
I/Q
Modulator
t( )BBs t
f
0 Hz
USBLSB
cos2 cf tπ
( ) ( ) ( ) ( )2 2
2 2
c cj t j tj f t j f tA t A t
e e e eφ φπ π− −
= +
( ) ( ) ( )( )cos 2m cs t A t f t tπ φ= +
( ) ( )
{ }2
Re cj t j f t
A t e e
φ π
= ⋅
“real signal”
• I/Q modulated signal
( )I t
( )Q t
f
cf0 Hzcf−
USBLSBLSBUSB
“complex”“complex” “real”
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Complex conjugate
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Concept of the Complex Envelope (I)
• Bandpass real signal :
( ) ( ) ( )( )
( ) ( ) ( ) ( )2 2
cos 2
2 2
c cj t j tj f t j f t
m c
A t A t
s t A t f t t e e e e
φ φπ π
π φ − −
= + = +
( ) ( )
( ) ( )2 21 1
2 2
c cj t j tj f t j f t
A t e e A t e eφ φπ π− −
= +
( )ls t ( )ls t∗
( )lS f∗
( )lS f
Complex timed value
Spectrum
( ) ( )
( ) ( )2 21 1
2 2
c cj t j tj f t j f t
A t e e A t e eφ φπ π− −
= +
( ) 2 cj f t
ls t e π
⋅ ( ) 2 cj f t
ls t e π−∗
⋅
( )l cS f f∗
− −( )l cS f f−
Complex timed value
Spectrum
( ) ( ) ( )
1
2
m l c l cS f S f f S f f∗
 = − + − − 
f
cf0 Hzcf−
USBLSBLSBUSB
( )
1
2
l cS f f−( )
1
2
l cS f f∗
− −
Spectrum of the bandpass signal
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Concept of the Complex Envelope (II)
• Equivalent low-pass signal (complex envelope):
f
0 Hz
( )lS f
cfcf−
( ) 21
2
cj f t
ls t e π
⋅( ) 21
2
cj f t
ls t e π−∗
⋅
( ) ( ) ( )
( ) ( )j t
ls t A t e I t jQ t
φ
= = +
( ) ( ) ( )
1
2
m l c l cS f S f f S f f∗
 = − + − − 
f
cf0 Hzcf−
USBLSBLSBUSB
( ) ( )
1
2
I t jQ t+  
Spectrum of the bandpass signal
( ) ( )
1
2
I t jQ t−  
( )ms t
( ) ( ) ( )
( ) ( )BBj t
ls t A t e I t jQ t
φ
= = +
complex envelope
( ) ( ) ( ) ( ) ( ) 2
cos 2 Re cj t j f t
m cs t A t f t t A t e e
φ π
π φ  = ⋅ + = ⋅    
( ) ( ){ }2
Re cj f t
I t jQ t e π
= +  
complex envelope
carriercarrier2 cj f t
e π
carrier
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Summary
• In this chapter, the phasor was introduced to manifest itself in
the mathematical operation for communication engineering.
• A modulated signal is a linear combination of I(t), Q(t), and
the carrier. This mathematical combination can be realized
with a practical circuitry, say, “modulator.”
• The demodulation is the decomposition of the modulated
signal, which is the reverse process to recover the baseband
signal I(t) and Q(t).
• The modulated signal can be viewed as a complex envelope
carried by a sinusoidal carrier. With this equivalent lowpass
form to represent a bandpass system, the mathematical
analysis can be easily simplified.
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