Chapter 6:
Modulation Techniques
for Mobile Radio
School of Information Science
and Engineering, SDU
Outline
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Frequency versus Amplitude Modulation
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Amplitude Modulation (AM)
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Angle Modulation
Digital Modulation
Line Coding
Pulse Shaping Techniques
Geometric Representation of Modulation Signal
Linear Modulation Techniques
Constant Envelope Modulation
M-ary signaling
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What is modulation
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Modulation is the process of encoding
information from a message source in a
manner suitable for transmission
It involves translating a baseband message
signal to a bandpass signal at frequencies
that are very high compared to the baseband
frequency.
Baseband signal is called modulating signal
Bandpass signal is called modulated signal
Modulation Techniques
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Modulation can be done by varying the
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Amplitude
Phase, or
Frequency
of a high frequency carrier in accordance with the
amplitude of the message signal.
Demodulation is the inverse operation:
extracting the baseband message from the
carrier so that it may be processed at the
receiver.
Analog/Digital Modulation
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Analog Modulation
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The input is continues signal
Used in first generation mobile radio systems
such as AMPS in USA.
Digital Modulation
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The input is time sequence of symbols or pulses.
Are used in current and future mobile radio
systems
Goal of Modulation
Techniques
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Modulation is difficult task given the hostile
mobile radio channels
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Small-scale fading and multipath conditions.
The goal of a modulation scheme is:
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Transport the message signal through the radio
channel with best possible quality
Occupy least amount of radio (RF) spectrum.
Frequency versus Amplitude
Modulation
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Frequency Modulation (FM)
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Most popular analog modulation technique
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Amplitude of the carrier signal is kept constant (constant envelope
signal), the frequency of carrier is changed according to the
amplitude of the modulating message signal; Hence info is carried in
the phase or frequency of the carrier.
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Has better noise immunity:
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Performs better in multipath environment
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Small-scale fading cause amplitude fluctuations as we have seen earlier.
Can trade bandwidth occupancy for improved noise performance.
§
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atmospheric or impulse noise cause rapid fluctuations in the amplitude of the received
signal
Increasing the bandwith occupied increases the SNR ratio.
The relationship between received power and quality is non-linear.
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Rapid increase in quality for an increase in received power.
Resistant to co-channel interference (capture effect).
Frequency versus Amplitude
Modulation
Amplitude Modulation (AM)
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Changes the amplitude of the carrier signal
according to the amplitude of the message signal.
All info is carried in the amplitude of the carrier
There is a linear relationship between the received
signal quality and received signal power.
AM systems usually occupy less bandwidth then FM
systems.
AM carrier signal has time-varying envelope.
Amplitude Modulation
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The amplitude of high-carrier signal is varied
according to the instantaneous amplitude of the
modulating message signal m(t).
m(t)
AM Modulator
Carrier Signal : Ac
sAM(t)
cos( 2πf c t )
Modulating Message Signal : m (t )
The AM Signal :
s AM (t ) = Ac [1 + m(t )] cos( 2πf c t )
Modulation Index of AM Signal
For a sinusoidal message signal m(t ) = Am cos(2πf m t )
Index is defined as:
k=
Am
Ac
SAM(t) can also be expressed as:
s AM (t ) = Re{ g (t )e j 2πf c t }
where
g (t ) = Ac (1 + m(t )]
g(t) is called the complex envelope of AM signal.
AM Modulation/Demodulation
Source
Sink
Wireless
Channel
Demodulator
Modulator
Baseband Signal
with frequency
fm
(Modulating Signal)
Bandpass Signal
with frequency
fc
(Modulated Signal)
fc >> fm
Original Signal
with frequency
fm
AM Modulation - Example
20
15
1/fmesg
10
5
0
-5
1/fc
-10
-15
-20
0
2
4
6
8
10
Message signal : m (t ) = 2 + 2 cos( t )
Carrier signal : Ac cos( 2πf c t ) = 4 cos(10t )
s AM (t ) = Ac [1 + m (t )] cos( 2πf c t )
AM Signal : s AM (t ) = 4[1 + 2 + 2 cos( t )] cos( 2πf c t )
12
14
10
fc =
≅ 1.6 Hz
2π
1
f mesg=
= 0.16 Hz
2π
Angle Modulation
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Angle of the carrier is varied according to the
amplitude of the modulating baseband signal.
Two classes of angle modulation techniques:
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Frequency Modulation
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Instantaneous frequency of the carrier signal is varied
linearly with message signal m(t)
Phase Modulation
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The phase θ(t) of the carrier signal is varied linearly with the
message signal m(t).
Angle Modulation
FREQUENCY MODULATION
t


s FM (t ) = Ac cos(2πf c t + θ (t )] = Ac cos 2πf c t + 2πk f ∫ m( x)dx 
−∞


kf is the frequency deviation constant (kHz/V)
If modulation signal is a sinusoid of amplitude Am, frequency fm:
s FM (t ) = Ac cos( 2πf c t +
k f Am
fm
sin( 2πf mt )]
PHASE MODULATION
sPM (t ) = Ac cos[2πf ct + kθ m(t )]
kθ is the phase deviation constant
FM Example:
4
0
-4
+
0.5
Message signal
FM Signal
Carrier Signal
1
-+
1.5
m (t ) = 4 cos( 2πt )
s (t ) = cos[2π 8t + 4 sin( 2πt )]
cos( 2π 8t )
2
FM Index
βf =
k f Am
W
∆f
=
W
W: the maximum bandwidth of the modulating signal
∆f: peak frequency deviation of the transmitter.
Am: peak value of the modulating signal
Example: Given m(t) = 4cos(2π4x103t) as the message signal and
a frequency deviation constant gain (kf) of 10kHz/V;
Compute the peak frequency deviation and modulation index!
Answer:
fm=4kHz
∆f = 10kHz/V * 4V = 40kHz.
βf = 40kHz / 4kHz = 10
Spectra and Bandwidth of FM
Signals
An FM Signal has 98% of the total transmitted power in a RF bandwidth BT
Carson’s Rule
BT = 2( β f + 1) f m
BT = 2∆f
Upper bound
Lower bound
Example:
Analog AMPS FM system uses modulation index of Bf = 3 and fm = 4kHz.
Using Carson’s Rule: AMPS has 32kHz upper bound and 24kHz lower
bound on required channnel bandwidth.
FM Demodulator
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Convert from the frequency of the carrier
signal to the amplitude of the message
signal
FM Detection Techniques
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Slope Detection
Zero-crossing detection
Phase-locked discrimination
Quadrature detection
Slope Detector
Vin(t)
Limiter
V1(t)
Differentiator
V2(t)
Envelope
Detector
Vout(t)
t


v1 (t ) = V1 cos[2πf c t + θ (t ) ] = V1 cos  2πf c t + 2πk f ∫ m ( x ) dx 
−∞


dθ 
dv1 (t )

sin (2πf c t + θ (t ) )
v 2 (t ) =
= −V1  2πf c +

dt
dt 

d


vout (t ) = V1  2πf c + θ (t ) 
dt


= V1 2πf c + V1 2πk f m (t )
Proportional to
the priginal Message Signal
Digital Modulation
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The input is discrete signals
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Time sequence of pulses or symbols
Offers many advantages
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Robustness to channel impairments
Easier multiplexing of variuous sources of information:
voice, data, video.
Can accommodate digital error-control codes
Enables encryption of the transferred signals
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More secure link
Digital Modulation
The modulating signal is respresented as a time-sequence of symbols
or pulses.
Each symbol has m finite states: That means each symbol carries n bits
of information where n = log2m bits/symbol.
...
0
1
2
3
One symbol
(has m states – voltage levels)
(represents n = log2m bits of information)
T
Modulator
Factors that Influence Choice of
Digital Modulation Techniques
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A desired modulation scheme
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Provides low bit-error rates at low SNRs
§
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Performs well in multipath and fading conditions
Occupies minimum RF channel bandwidth
§
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Power efficiecny
Bandwidth efficieny
Is easy and cost-effective to implement
Depending on the demands of a particular
system or application, tradeoffs are made
when selecting a digital modulation scheme.
Power Efficiency of
Modulation
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Power efficiency is the ability of the modulation technique to
preserve fidelity of the message at low power levels.
Usually in order to obtain good fidelity, the signal power needs
to be increased.
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Tradeoff between fidelity and signal power
Power efficiency describes how efficient this tradeoff is made
 Eb
Power Efficiency : η p = 
 N0
required at the receiver input for certain
Eb: signal energy per bit
N0: noise power spectral density
PER: probability of error

PER 

Bandwidth Efficiency of
Modulation
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Ability of a modulation scheme to accommodate
data within a limited bandwidth.
Bandwidth efficiency reflect how efficiently the
allocated bandwidth is utilized
R
Bandwidth Efficiency : η B = bps/Hz
B
R: the data rate (bps)
B: bandwidth occupied by the modulated RF signal
Shannon’s Bound
There is a fundamental upper bound on achievable bandwidth efficiency.
Shannon’s theorem gives the relationship between the channel
bandwidth and the maximum data rate that can be transmitted over this
channel considering also the noise present in the channel.
Shannon’s Theorem
η B max
C
S
= = log 2 (1 + )
B
N
C: channel capacity (maximum data-rate) (bps)
B: RF bandwidth
S/N: signal-to-noise ratio (no unit)
Tradeoff between BW Efficiency
and Power Efficiency
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There is a tradeoff between bandwidth
efficiency and power efficiency
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Adding error control codes
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Improves the power efficiency
§ Reduces the requires received power for a particular bit
error rate
Decreases the bandwidth efficiency
§ Increases the bandwidth occupancy
M-ary keying modulation
§
§
Increases the bandwidth efficiency
Decreases the power efficiency
§ More power is requires at the receiver
Example:
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SNR for a wireless channel is 30dB and RF
bandwidth is 200kHz. Compute the theoretical
maximum data rate that can be transmitted over this
channel?
Answer:
S
= 10
N
 30 dB 
 10 


C = B log 2 (1 +
S
) = 2 x10 5 log 2 (1 + 1000 ) = 1.99 Mbps
N
Noiseless Channels and Nyquist
Theorem
For a noiseless channel, Nyquist theorem gives the relationship
between the channel bandwidth and maximum data rate that can be
transmitted over this channel.
Nyquist Theorem
C = 2 B log 2 m
C: channel capacity (bps)
B: RF bandwidth
m: number of finite states in a symbol of transmitted signal
Example: A noiseless channel with 3kHz bandwidth can only transmit
maximum of 6Kbps if the symbols are binary symbols.
Power Spectral Density (PSD) of
Digital Signals and Bandwidth
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What does signal bandwidth mean?
Answer is based on Power Spectral Density (PSD)
of Signals
For a random signal w(t), PSD is defined as:
W (f)2 

Pw( f ) = lim  T

T →∞
T


WT ( f ) is th fourier transform of wT (t )
T
T

w
(
t
)
t
−
<
<

wT (t ) = 
2
2
0
elsewhere
Fourier Analysis
Joseph Fourier has shown that any periodic function F(f) with period T,
can be constructed by summing a (possibly infinite) number of sines and
cosines.
∞
a0
+ ∑ a n cos( nωT t ) + bn sin( nωT t )
2 n =1
2π
ωT =
= 2πf 0
T
F (t )=
Such a decomposition is called Fourier series and the coefficients are
called the Fourier coefficients.
A line graph of the amplitudes of the Fourier series components can be
drawn as a function of frequency. Such a graph is called a spectrum or
frequency spectrum. f0 is called the fundemantal frequency.
The nth term is called nth harmonic. The coefficients of the nth harmonic
are an and bn.
Fourier Analysis
The coefficients can be obtained from the periodic function F(t) as follows:
T
2
a0 = ∫ F (t ) dt
T 0
T
2
an = ∫ F (t ) cos nϖ T tdt ,
T 0
n = 1,2,...
T
2
bn = ∫ F (t ) sin nϖ T tdt ,
T 0
n = 1,2,...
Example: A Periodic Function
Find the fourier series of the periodic function f(x), where
One period of f(x) is defined as: f(x) = x, -π < x < π
T=2π
π
−π
0
−π
T = 2π
π
2π
Example: Its Fourier
Approximation
Domain: [-π, π]
4
1 harmonic
3
4
x
2*sin(x)
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
x
2*(sin(x)-sin(2*x)/2)
-4
-3
-2
-1
0
4
3
2 harmonics
3
1
2
3
-2
-1
4
x
2*(sin(x)-(sin(2*x)/2)+(sin(3*x)/3))
3 harmonics
-3
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
1
2
3
x
2*(sin(x)-(sin(2*x)/2)+(sin(3*x)/3)-(sin(4*x)/4))
4 harmonics
3
2
0
-4
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Example: Frequency Spectrum
Magnitude : an2 + bn2
2.5
For First 10 harmonics
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
Harmonics
Each harmonic corresponds to a frequency that is multiple of the fundamental
frequency
Complex Form of Fourier
Series
By substituting Euler’s Expresssion into Fouries expansion:
e
jθ
= cos θ + j sin θ
It can be shown that the following is true:
F (t ) =
∞
jnωT t
c
e
∑ n dt
n = −∞
1
cn =
T
T
2
− jn ωT t
F
(
t
)
e
dt ,
∫
−
n = ... − 2,−1,0,1,2,....
T
2
cn are the complex Fourier coefficients
Digital Modulation - Continues
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Line Coding
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1
Base-band signals are represented as line codes
0
Tb
1
0
1
0
1
V
0
Tb
V
Unipolar
NRZ
Bipolar
RZ
-V
V
Manchester
NRZ
-V
Tb
Pulse Shaping Techniques
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Effect of bandlimited channel:
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How to mitigate ISI?
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Rectangular pulses?
Pulses will spread in time
Each symbol will interfer succeeding symbols (ISI)
Increase the symbol error probability (SER).
increase the channel bandwidth (usually infeasible)
design pulse shapes carefully (desirable & popular)
Spectral shaping is usually done through baseband or IF
processing.
Objectives of pulse shaping techniques:
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Minimize the ISI
Minimize the spectral width of a modulated digital signal
Pulse Shaping Techniques
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Nyquist Criterion for ISI Cancellation
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rectangular "brick-wall" filter
Raised Cosine Rolloff Filter
Gaussian Pulse-shaping Filter
Nyquist Criterion for ISI
Cancellation
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Nyquist was the first to solve the problem of overcoming ISI while
keeping the transmission bandwidth low.
To nullify the effect of ISI, the overall response of the communication
system (including transmitter, channel, and receiver) should be
designed as follows:
K , n = 0
heff (nTs ) = 
 0, n ≠ 0
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Rectangular "brick-wall" filter (ideal low-pass filter)
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Minimize the spectral width (preferable)
Strong side lobes in time domain (sensitive to timing error)
sin (π t Ts )
heff (t ) =
π t Ts
1  f 
H eff ( f ) = Π  
fs  fs 
Nyquist Criterion for ISI
Cancellation
H eff ( f )
f
Raised Cosine Rolloff Filter
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The most popular pulse shaping filter used in
mobile communications.
Transfer function
α is the rolloff factor which ranges between 0 and 1.
Raised Cosine Spectrum
Raised Cosine Rolloff Filter
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Impulse response of the cosine rolloff filter
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A raised cosine filter belongs to the class of filters which satisfy
the Nyquist criterion.
As the rolloff factor α increases, the bandwidth of the filter also
increases, and the time sidelobe levels decrease in adjacent
symbol slots. This implies that increasing α decreases the
sensitivity to timing jitter, but increases the occupied bandwidth.
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Raised Cosine pulses
Spectrum of Raised Cosine
pulse
Gaussian Pulse-shaping Filter
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Non-Nyquist techniques for pulse shaping.
Particularly effective when used in
conjunction with Minimum Shift Keying (MSK)
modulation.
Transfer function
The parameter α is related to B, the 3-dB bandwidth of the baseband
gaussian shaping filter. As α increases, the spectral occupancy of
the Gaussian filter decreases and time dispersion of the applied
signal increases
Gaussian Pulse-shaping Filter
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The impulse response of the Gaussian filter
Geometric Representation of
Modulation Signal
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Digital Modulation involves
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Choosing a particular signal waveform for transmission for
a particular symbol or signal
For M possible signals, the set of all signal waveforms are:
S = {s1 (t ), s2 (t ),..., s M (t )}
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For binary modulation, each bit is mapped to a
signal from a set of signal set S that has two signals
We can view the elements of S as points in vector
space
Geometric Representation of
Modulation Signal
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Vector space
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We can represented the elements of S as linear combination of
basis signals.
The number of basis signals are the dimension of the vector
space.
Basis signals are orthogonal to each-other.
Each basis is normalized to have unit energy:
∞
E = ∫ φi2 (t ) dt = 1
−∞
φi (t ) is the i th basis signal.
Example
2 Eb
s1 (t ) =
cos(2πf ct )
Tb
0 ≤ t ≤ Tb
2 Eb
s2 (t ) = −
cos(2πf c t ) 0 ≤ t ≤ Tb
Tb
2
φ1 (t ) =
cos(2πf ct )
Tb
S=
{ E φ (t ),−
b 1
Two signal
waveforms to
be used for
transmission
The basis signal
Q
}
Eb φ1 (t )
− Eb
Eb
Constellation Diagram
Dimension = 1
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Constellation Diagram
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Properties of Modulation Scheme can be
inferred from Constellation Diagram
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Bandwidth occupied by the modulation increases as
the dimension of the modulated signal increases
Bandwidth occupied by the modulation decreases as
the signal_points per dimension increases (getting
more dense)
Probability of bit error is proportional to the distance
between the closest points in the constellation.
§
Bit error decreases as the distance increases (sparse).
Linear Modulation Techniques
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Classify digital modulation techniques as:
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Linear
§
§
§
§
§
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The amplitude of the transmitted signal varies linearly with
the modulating digital signal, m(t).
They usually do not have constant envelope.
More spectral efficient.
Poor power efficiency
Examples: QPSK,OQPSK.
Non-linear
Binary Phase Shift Keying
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Use alternative sine wave phase to encode bits
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Phases are separated by 180 degrees.
Simple to implement, inefficient use of bandwidth.
Very robust, used extensively in satellite communication.
s1 (t ) = Ac cos(2π f c t + θ c )
binary 1
s2 (t ) = Ac cos(2π f c t + θ c + π ) binary 0
Q
0
State
1
State
BPSK Example
Data
Carrier
Carrier+ π
BPSK waveform
1
1
0
1
0
1
Quadrature Phase Shift Keying
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Multilevel Modulation Technique: 2 bits per symbol
More spectrally efficient, more complex receiver.
Two times more bandwidth efficient than BPSK
Q
01 State
00 State
11 State
10 State
Phase of Carrier: π/4, 2π/4, 5π/4, 7π/4
4 different waveforms
1.5
1
0.5
0
-0.5
-1
-1.50
1.5
1
0.5
0
-0.5
-1
-1.50
cos+sin
11
0.2 0.4 0.6 0.8
1
10
cos-sin
0.2 0.4 0.6 0.8
1
1.5
1
0.5
0
-0.5
-1
-1.50
1.5
1
0.5
0
-0.5
-1
-1.50
-cos+sin
01
0.2 0.4 0.6 0.8
1
00
-cos-sin
0.2 0.4 0.6 0.8
1
Constant Envelope Modulation
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Amplitude of the carrier is constant,
regardless of the variation in the modulating
signal
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Better immunity to fluctuations due to fading.
Better random noise immunity
Power efficient
They occupy larger bandwidth
Examples: FSK, MSK, GMSK
Frequency Shift Keying (FSK)
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The frequency of the carrier is changed according
to the message state (high (1) or low (0)).
s1 (t ) = A cos( 2πf c + 2π∆f )t 0 ≤ t ≤ Tb (bit = 1)
s2 (t ) = A cos( 2πf c − 2π∆f )t 0 ≤ t ≤ Tb (bit = 0)
Continues FSK
s (t ) = A cos( 2πf c + θ (t ))
s (t ) = A cos( 2πf c t + 2πk f
t
∫ m( x)dx)
−∞
Integral of m(x) is continues.
FSK Example
Data
1
FSK
Signal
1
0
1
Combined Linear and Constant
Envelope Modulation Techniques
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Modem modulation techniques exploit the fact that digital
baseband data may be sent by varying both the envelope and
phase (or frequency) of an RF carrier.
M-ary signaling: two or more bits are grouped together to form
symbols, where M=2n (n is an integer).
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MASK
MPSK
MFSK
MQAM
M-ary modulation schemes achieve better bandwidth efficiency
at the expense of power efficiency.
M-ary Phase Shift Keying (MPSK)
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Modulated signal
2 Es
M

si (t ) =
cos  2π fc t +
( i − 1)  , 0 ≤ t ≤ Ts , i = 1, 2,..., M
Ts
2π


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Probability of symbol error in an AWGN channel
 Es
 Eb log 2 M
π 
π 
Pe ≤ 2Q 
Q
sin
=
2
sin


 N


M
N0
M
0


8-PSK Signal Constellation
M-ary Quadrature Amplitude
Modulation (QAM)
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Modulated signal
2 Emin
2 Emin
si (t ) =
⋅ ai cos ( 2π f c t ) +
⋅ bi sin ( 2π f c t )
Ts
Ts
, 0 ≤ t ≤ Ts , i = 1, 2,..., M
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Probability of symbol error in an AWGN channel
1   2 Emin

Pe ≅ 4 1 −
 Q  N
M  

0

3Eav
1  

=
4
1
−
Q


  ( M − 1) N
M  

0




16-QAM Signal Constellation