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Abstract of the courseContent of the course
Course requirements and references
Mobile Communications
Instructor: Nguyen Le Hung
Email: [email protected]; [email protected]
Department of Electronics & Telecommunications Engineering
Danang University of Technology, University of Danang
Mobile Communications Course Information 1
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Abstract of the courseContent of the course
Course requirements and references
Abstract of the course
This undergraduate course helps students to understandmathematical fundamentals and practical transmissiontechniques in 4G mobile communications (i.e., WiMAX,LTE).
The course lecture notes also provide some possibleresearch directions (in 4G mobile broadbandcommunications) that can be considered for final-yearprojects of undergraduate students.
Mobile Communications Course Information 2
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Abstract of the courseContent of the course
Course requirements and references
Content of the course
Chapter 1: IntroductionHistory & development of 1G/2G/3G/4G networks.Promises and future trendsCellular mobile communications
Chapter 2: Mobile wireless channel models.Path lossShadowingMultipath fading channels
Chapter 3: Physical-layer transmission techniques.Digital modulationsPerformance of digital modulations over fading channelsOrthogonal frequency division multiplexing (OFDM)fundamentals
Mobile Communications Course Information 3
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Abstract of the courseContent of the course
Course requirements and references
Course requirements and references
Pre-requisite: Basis knowledge of statistics, stochasticprocesses and digital communications systems.
Class lecture notes: based on the following references:
Gordan L. Stuber, Principles of Mobile Communication,
Second Edition, 2002A. Goldsmith, Wireless Communications, Cambridge2005.Recent IEEE journal and conference papers.
Course assessment:
Exercises and/or projects: 20%
Midterm exam: 30%
Final exam: 50%
Mobile Communications Course Information 4
O li
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OutlineIntroduction
Cellular mobile communications
Chapter 1: Introduction to Mobile
Communications
Nguyen Le Hung
Mobile Communications Chapter 1: Introduction to Mobile Communications 1
O tli
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OutlineIntroduction
Cellular mobile communications
Outline of Chapter 1
1 IntroductionDevelopment of mobile communication systemsMobile broadband technology evolution
Promises and future trends
2 Cellular mobile communicationsSystem model
Frequency reuseCellular concept
Mobile Communications Chapter 1: Introduction to Mobile Communications 2
Outline Development of mobile communication systems
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OutlineIntroduction
Cellular mobile communications
Development of mobile communication systemsMobile broadband technology evolutionPromises and future trends
Development of mobile communications systems
time
code
frequency
code
space
FDMA (1G)e.g., AMPS ~ 1980s
TDMA (2G)e.g., GSM ~ 1990s
OFDM, SDMA (4G)e.g., WiMAX, LTE
2010s
CDMA (3G)e.g., W-CDMA ~ 2000s
frequency
time
time
~ 1 Gbps (stationary),
~ 100 Mbps (mobile)
frequency
frequency
~ 14 Mbps (downlink),
~ 5.8 Mbps (uplink)~ 50 Kbps
A new signal dimension will be exploited in 5G ?
Mobile Communications Chapter 1: Introduction to Mobile Communications 3
Outline Development of mobile communication systems
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OutlineIntroduction
Cellular mobile communications
Development of mobile communication systemsMobile broadband technology evolutionPromises and future trends
OFDM versus FDMA
Frequency
Mobile Communications Chapter 1: Introduction to Mobile Communications 4
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Outline Development of mobile communication systems
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OutlineIntroduction
Cellular mobile communications
Development of mobile communication systemsMobile broadband technology evolutionPromises and future trends
Promises and future trends
multimedia services: Voice, Video distribution, Real-time videoconferencing, Data, for both businessand residential customers:
Explosive traffic growth
Internet growth, VoIP, VideoIP, IPTV
Cell phone popularity worldwide
Ubiquitous communication for people and devices Emerging systems opening new applications
Unified network: Single distributed network,multiple services, packet architecture
Extracted from Digital Communication lecture notes, McGill Uni.
Mobile Communications Chapter 1: Introduction to Mobile Communications 6
Outline System model
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IntroductionCellular mobile communications
yFrequency reuseCellular concept
System model of cellular mobile communications
BTS
LTE/LTEAdvanced
Single Cell
Multicell
approach using
game theory
Uplink (SCFDMA),
limited feedback design
Downlink (OFDMA)
SingleUserMultiuser
Precoding(SDMA)
Multihop
Relay
BTS
BTS
BTS
BTS
Inter
cellinterferenceIntercellinterference
Singleuser/Multihop:
Channel Estimation,
Synchronization (CFO),
Channel Coding, ...
Network Controller STBC with highspeed users
(large Doppler spread)
Cognitive radio
Space Time Block Code: STBC; Peakto
Average Power Ratio: PAPR; V
OFDM
Usercooperation
(cooperative/multihop
communications)
Mobile Communications Chapter 1: Introduction to Mobile Communications 7
Outline System model
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IntroductionCellular mobile communications
yFrequency reuseCellular concept
Frequency reuse
The available spectrum is partitioned among the base stations(BSs).
A given frequency band is reused at the closest possibledistance under a certain requirement of co-channelinterference.
Smaller cells have a shorter distance between reusedfrequencies = an increased spectral efficiency.
Microcells are of great importance in improving spectralefficiency.
Under frequency-reuse, users in geographically separated cellssimultaneously employ the same carrier frequency.
Mobile Communications Chapter 1: Introduction to Mobile Communications 8
Outline System model
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IntroductionCellular mobile communications
Frequency reuseCellular concept
Cellular concept
The cellular layout of a conventional cellular system is quiteoften described by a uniform grid of hexagonal cells or radiocoverage zones.
In practice the cells are not regular hexagons, but instead aredistorted and overlapping areas.
The hexagon is an ideal choice for representing macrocellularcoverage areas, because it closely approximates a circle andoffers a wide range of tessellating reused cluster sizes.
A tessellating reuse cluster of size N can be constructed if
= 2 + + 2, (1)
where and are non-negative integers and . It followsthat the allowable cluster sizes are = 1, 3, 4, 7, 9, 12, . . ..
Mobile Communications Chapter 1: Introduction to Mobile Communications 9
Outline System model
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IntroductionCellular mobile communications
Frequency reuseCellular concept
Cellular concept: Multicell layout with frequency-reuse
3-cell 4-cell
7-cell
Macrocellular deployment
with 7-cell clusters
Macrocellular deployment
with 3-cell clusters
Mobile Communications Chapter 1: Introduction to Mobile Communications 10
Introduction
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Wireless channel modeling
Chapter 2: Wireless Channel models
Mobile Communications Chapter 2: Wireless Channel models 1
IntroductionWi l h l d li
Multipath wireless propagationP h l h d i d f di
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Wireless channel modeling Path loss, shadowing and fading
Multipath wireless propagation
reflection and diffraction
Extracted from Digital Communication lecture notes, McGill Uni.
Mobile Communications Chapter 2: Wireless Channel models 2
IntroductionWi l h l d li
Multipath wireless propagationP th l h d i d f di
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Wireless channel modeling Path loss, shadowing and fading
Path loss, shadowing and fading
The characteristic of (mobile) wireless channel is the variations of
the channel strength over time and frequency.
The variations can be divided into two types:Large-scale fading is yielded by:
path loss of signal as a function of distance andshadowing by large objects such as buildings and hills.
Small-scale fading is yielded by the constructive and destructiveinterference of the multiple signal paths between transmitter andreceiver.
Mobile Communications Chapter 2: Wireless Channel models 3
IntroductionWireless channel modeling
Multipath wireless propagationPath loss shadowing and fading
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Wireless channel modeling Path loss, shadowing and fading
An example of path loss, shadowing and fading
0 50 100 150 200 250 300 350-150
-140
-130
-110
-100
-90
-80
-70
-60
-50
ReceivedPower[dBm]
Traveled distance [m]
Pathloss
Fading +
Shadowing +
Pathloss
Shadowing +
Pathloss
Mobile Communications Chapter 2: Wireless Channel models 4
IntroductionWireless channel modeling
Multipath wireless propagationPath loss shadowing and fading
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Wireless channel modeling Path loss, shadowing and fading
An example of path loss, shadowing and fading (cont.)
0
K (dB)
Pr
P(dB)
t
log (d)
Path Loss Alone
Shadowing and Path Loss
Multipath, Shadowing, and Path Loss
Mobile Communications Chapter 2: Wireless Channel models 5
IntroductionWireless channel modeling
Path loss modelsShadowingF di h l d l
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Wireless channel modelingFading channel model
Path loss models
It is well known that the received signal power decays with the
square of the path length in free space.
More specifically, the received envelope power is
= 4
2, (1)
where:
is the transmitted power, and are the transmitter and receiver antenna gains,respectively is the radio path length.
Mobile Communications Chapter 2: Wireless Channel models 6
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IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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gFading channel model
Path loss models (cont.)
The path loss is defined by
() = 10 log10
= 10 log104
42
sin22 (4)
Several useful empirical models for macrocellular systems have beenobtained by curve fitting experimental data.
Two of the useful models for 900 MHz cellular systems are:
Hatas model based on Okumuras prediction method and
Lees model.
Hatas empirical model is probably the simplest to use. Theempirical data for this model was collected by Okumura in the cityof Tokyo.
Mobile Communications Chapter 2: Wireless Channel models 8
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Fading channel model
Okumura-Hata models
With Okumura-Hatas model, the path loss between two isotropic
BS and MS antennas is
() =
+ log10() for urban area
+ log10() for suburban area + log10()
for open area
(5)
where
= 69.55 + 26.16log10() 13.82log10() () = 49.9
6.55 log10()
= 5.4 + 2 (log10(/28))2
= 40.94 + 4.78 (log10())2 18.33 log10()
Mobile Communications Chapter 2: Wireless Channel models 9
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Fading channel model
Okumura-Hata models (cont.)
and
=
[1.1log10
() 0.7] 1.56 log10() + 0.8 for medium or small city8.28 [log
10(1.54)]
2 1.1 for 200MHz
3.2 [log10
(11.75)]2 4.97 for 400MHz
for large city
(6)
Okumura-Hatas model is expressed in terms of:
the carrier frequency: 150 1000(MHz),
BS antenna height: 30 200(m),
the mobile station (MS) height: 1 10(m),
the distance: 1 20(km).
Mobile Communications Chapter 2: Wireless Channel models 10
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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g
Numerical results of Okumura-Hata models
1 5 10 15 20100
150
200
250
300
350
400
450
Pathloss(dB)
Distance d (km) under scale of log10
urban area
suburban area
open area
Figure 1: Path loss for = 1.5m, = 50m, = 900MHz.
Mobile Communications Chapter 2: Wireless Channel models 11
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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g
Shadowing
A signal transmitted through a wireless channel will typically
experience random variation due to blockage from objects in thesignal path, giving rise to random variations of the received power ata given distance.
Such variations are also caused by changes in reflecting surfaces andscattering objects.
Thus, a model for the random attenuation due to these effects isalso needed. Since the location, size, and dielectric properties of theblocking objects as well as the changes in reflecting surfaces andscattering objects that cause the random attenuation are generallyunknown, statistical models must be used to characterize this
attenuation.The most common model for this additional attenuation islog-normal shadowing.
Mobile Communications Chapter 2: Wireless Channel models 12
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Shadowing (cont.)
Empirical studies have shown that has the following log-normal
distribution:
() =2
2exp
10log10 2 (dBm)
22
() =2
2 exp10log10 (dBm)22
where:
and denote the mean envelop and mean squared levels ofreceived signal (where the expectation is taken over the pdf of the
received envelope). stands for standard deviation. (dBm) = 30 + 10[log10
2
] (dBm) = 30 + 10[log10 ]
Mobile Communications Chapter 2: Wireless Channel models 13
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Shadowing (cont.)
Sometimes is called the local mean because it represents the
mean envelope level where the averaging is performed over adistance of a few wavelengths that represents a locality.
This model has been confirmed empirically to accurately model thevariation in received power in both outdoor and indoor radiopropagation environments
Mobile Communications Chapter 2: Wireless Channel models 14
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Fading channel model
Two MainMultipaths
LocalScattering
The complex transmitted signal can be expressed by
() = Re
()2
. (7)
Over a multipath ( physical paths) propagation channel, thereceived signal can be obtained by
() =
1
=0()( ()) + (). (8)
Mobile Communications Chapter 2: Wireless Channel models 15
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Fading channel model (cont.)
Substituting (7) into (8) yields the following
() = Re
1=0
() ( ()) 2(())
+ ()
= Re1
=0
() (()) 2+ ()
= Re()2
+ ()
As a result, the received baseband signal can be determined by
() =()( ()) + (). (9)
where () is the receiver (thermal) noise signal.
Mobile Communications Chapter 2: Wireless Channel models 16
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Wireless channel modeling (cont.)
The next step in creating a useful channel model is to convert the
continuous-time channel to a discrete-time channel.We take the usual approach of sampling theorem.
Assuming that the input waveform is band-limited to , thebaseband equivalent can be represented by
() =
sinc( ), (10)
where = (/) and sinc() sin()
.
This representation follows from the sampling theorem, which says
that any waveform band-limited to /2 can be expanded in termsof the orthogonal basis functions sinc( ) with coefficients bysamples (taken uniformly at integer multiples of 1/)
Mobile Communications Chapter 2: Wireless Channel models 17
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Wireless channel modeling (cont.)
As a result, the baseband received signal can be determined by
() =
()
sinc (( ()) ) + ()
=
()sinc (( ()) ) + ().
The sampled outputs at multiples of 1/ is (/) then
=
(/)sinc ( (/)) + (/).
(11)
Mobile Communications Chapter 2: Wireless Channel models 18
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Wireless channel modeling (cont.)
Let
then one can have
=
(/)sinc ( (/)) + (/)
Then, the discrete-time channel model can be given by
=
, + (/) (12)
where , =
(/)sinc ( (/))This simple discrete-time signal model is widely used inphysical-layer transmission techniques in OFDM systems (e.g., WiFi,
WiMAX, LTE)
Mobile Communications Chapter 2: Wireless Channel models 19
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Examples of transmitted baseband signal
01
00 10
11
I+11
1
+1
Q b0b1
0 1
I+11
1
+1
Q
b011 10
11 11 10 11
10 10
I+11
1
+1
Qb
0
b1
b2
b3
+3
11 01
11 00 10 00
10 01+3
00 10
00 11 01 11
01 10
00 01
00 00 01 00
01 013
3
BPSK
QPSK
16-QAM
Over multipath channels, the received signal at MS is:
=
, + (/) (13)
It is noted that multipath fading gains , (channel impulseresponse) is time-variant (depend on time index ).
Mobile Communications Chapter 2: Wireless Channel models 20
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
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Channel estimation in mobile communications
Source
encoder
Channel
encoder
Digital
modulation
Channel
Source
decoder
Channel
decoder
Digital
demodulation
S
h
r= Sh + n
Pilot
S
Data
S
Data
S
Pilot
S
Data
S
Data
S
Pilot
S
h h h h h h
Mobile Communications Chapter 2: Wireless Channel models 21
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
C
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Literature Review of Channel Estimation in Wireless
Communications
Detection/decoding
in communicationsRx signal
vectorTx signal
matrixCIR
vectorRx noise
vector
Noncoherent Coherentwithout using CSI- per ormance
loss
use CSI
re uire Channel Estimation CE
(CSI)
r= Sh + n
with channel parameters as:
Deterministic unknowns Random variables
Fisher approaches: Bayesian approaches:
, , , ,
Multipath fading channel (freq. selective) in multi-carriertransmissions (e.g.,OFDM)
Time-invariant (quasi-static) Time-variant (Time-selective)
Perfect
Synch.
Imperfect
Synch.
Channel Estimation (CE)
Blind Pilot Semi-blind
Joint CE and Synch.
Semi-blind
Perfect
Synch.Imperfect
Synch.
Channel Estimation (CE)
PilotJoint CE and Synch.
Semi-blindPilot
Pilot design to minimize:
MSE CRLB
Pilot design to minimize: Pilot design to minimize:
MSE BCRLB
Turbo-based
Decision-direct.
Mobile Communications Chapter 2: Wireless Channel models 22
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
Ti i h i d bil d f k /h
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Time-variant path gain , under mobile speed of 5 km/h
0 1 2 3 4 5 6 7 8 9 10 11 12 13 141
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Time (in OFDM symbol duration)
Absolutevalueofamplitudeo
fonepathgainhl Mobile user speed = 5 km/h,
fc
= 2 GHz,
128FFT, CP length = 10,fs
= 1.92 MHz,
2 time slots in LTE are considered,Jakes model is considered.
pilot OFDM symbol
for channel estimation
Mobile Communications Chapter 2: Wireless Channel models 23
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
d bil d f 50 k /h
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, under mobile speed of 50 km/h
0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.95
1
1.05
1.1
1.15
Time (in OFDM symbol duration)
Abso
lutevalueofamplitudeo
fonepathgainhl
Mobile user speed = 50 km/h,fc
= 2 GHz,
128FFT, CP length = 10,fs
= 1.92 MHz,
2 time slots in LTE are considered,Jakes model is considered
Data OFDM symbol
Mobile Communications Chapter 2: Wireless Channel models 24
IntroductionWireless channel modeling
Path loss modelsShadowingFading channel model
d bil d f 300 k /h
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, under mobile speed of 300 km/h
0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.8
0.9
1
1.1
1.2
1.3
Time (in OFDM symbol duration)
Absolutevalueofamplitudeofonefadinggainhl Mobile user speed = 300 km/h,
fc
= 2 Ghz, 128FFT, CP length = 10, fs
= 1.92 Mhz,
2 time slots in LTE are considered,Jakes model is considered.
Data OFDM symbol
Mobile Communications Chapter 2: Wireless Channel models 25
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union Bound
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y yPassband modulation
Chapter 3: Physical-layer transmission techniques
Section 3.1: Digital modulations
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 1
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union Bound
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Passband modulation
Outline of the lecture notes
1 Digital modulation techniquesAdvantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
2 Signal Space AnalysisRationalSignal and system modelGeometric representation of signals
Practical examplesSignal space representation
3 Receiver Structure and Sufficient StatisticsGeneral resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
4 Error Probability Analysis and the Union Bound
Error probabilityThe union bound on error probability
5 Passband modulationGeneral principlesAmplitude and phase modulationPulse amplitude modulation (MPAM)Phase shift keying (MPSK)Quadrature amplitude modulation (MQAM)
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 2
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union Bound
Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
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Passband modulation
Advantages over analog modulation
The advances over the last several decades in hardware anddigital signal processing have made digital transceivers muchcheaper, faster, and more power-efficient than analogtransceivers.
More importantly, digital modulation offers a number of otheradvantages over analog modulation, including:
higher data rates,powerful error correction techniques,resistance to channel impairments,
more efficient multiple access strategies, andbetter security and privacy.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 3
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundP b d d l i
Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
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Passband modulation
Advantages over analog modulation (cont.)
Digital transmissions consist of transferring information in theform of bits over a communications channel.
The bits are binary digits taking on the values of either 1 or 0.These information bits are derived from the information
source, which may be a digital source or an analog source thathas been passed through an A/D converter.
Both digital and A/D converted analog sources may becompressed to obtain the information bit sequence.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 4
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundP b d d l ti
Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
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Passband modulation
Main considerations in digital modulation techniques
Digital modulation consists of mapping the information bitsinto an analog signal for transmission over the channel.
Detection consists of determining the original bit sequencebased on the signal received over the channel.
The main considerations in choosing a particular digitalmodulation technique are:
high data ratehigh spectral efficiency (minimum bandwidth occupancy)high power efficiency (minimum required transmit power)
robustness to channel impairments (minimum probability of biterror)low power/cost implementation
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 5
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
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Passband modulation
Typical types of digital modulation techniques
Often the previous ones are conflicting requirements, and thechoice of modulation is based on finding the technique thatachieves the best tradeoff between these requirements.
There are two main categories of digital modulation:
amplitude/phase modulationfrequency modulation
Frequency modulation typically has a constant signal envelopeand is generated using nonlinear techniques, this modulationis also called constant envelope modulation or nonlinear
modulation
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 6
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
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Passband modulation
Typical types of digital modulation techniques (cont.)
Amplitude/phase modulation is also called linear modulation.
Linear modulation generally has better spectral propertiesthan nonlinear modulation, since nonlinear processing leads tospectral broadening.
However, amplitude and phase modulation embeds theinformation bits into the amplitude or phase of thetransmitted signal, which is more susceptible to variationsfrom fading and interference.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 7
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques
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Passband modulation
Typical types of digital modulation techniques (cont.)
In addition, amplitude and phase modulation techniquestypically require linear amplifiers, which are more expensiveand less power efficient than the nonlinear amplifiers that canbe used with nonlinear modulation.
Thus, the general tradeoff of linear versus nonlinearmodulation is one of better spectral efficiency for the formertechnique and better power efficiency and resistance tochannel impairments for the latter technique.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 8
Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Passband modulation Signal space representation
Rational
Digital modulation encodes a bit stream of finite length intoone of several possible transmitted signals.
Intuitively, the receiver minimizes the probability of detectionerror by decoding the received signal as the signal in the set of
possible transmitted signals that is closest to the one received.Determining the distance between the transmitted andreceived signals requires a metric for the distance betweensignals.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 9
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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g p p
Rational (cont.)
By representing signals as projections onto a set of basisfunctions, we obtain a one-to-one correspondence between theset of transmitted signals and their vector representations.
Thus, we can analyze signals in finite-dimensional vector
space instead of infinite-dimensional function space, usingclassical notions of distance for vector spaces.
In this section we show:
how digitally modulated signals can be represented as vectorsin an appropriately-defined vector space, and
how optimal demodulation methods can be obtained from thisvector space representation.
This general analysis will then be applied to specificmodulation techniques in later sections.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 10
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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g p p
Transmitted signal
Transmitter Receiver +
n(t)
AWGN Channel
s(t)i 1 K
m ={b ,...,b } ^1 K
m ={b ,...,b }^ ^r(t)
Figure 1: Communication system model over AWGN channel (i.e., aspecial case of wireless channel).
Consider a communication system model as shown in theabove figure.
Every seconds, the sytem sends = log2 bits ofinformation through the channel for a data rate of = /bits per second (bps).
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 11
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Transmitted signal (cont.)
There are = 2 possible sequences of bits and each bitsequence of length comprises a message = {1,...,} , where = {1,...,} is the set ofall such messages.
The message has probability of being selected fortransmission, where
=1 = 1.
Suppose that message is to be transmitted over theAWGN channel during the time interval [0, ). Since thechannel is analog, the message must be embedded into ananalog signal for channel transmission.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 12
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Transmitted signal (cont.)
Therefore, each message is mapped to a uniqueanalog signal () = {1(),...,()} where () isdefined on the time interval [0, ) and has energy
= 0
2 (), = 1,...,. (1)
When messages are sent sequentially, the transmittedsignal becomes a sequence of the corresponding analog signalsas follows
() =
( ). (2)
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 13
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Transmitted and received signals
In the aforementioned model, the transmitted signal is sentthrough an AWGN channel where a white Gaussian noiseprocess () of power spectral density /2 is added to formthe received signal
() = () + (). (3)
T0 2T 3T 4T
s (t)1 1 1
2s (tT)
s (t2T) s (t3T)
s(t)
...
m1
m1
m1
m2
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 14
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Received signal
Given (), the receiver must determine the best estimate ofwhich () was transmitted during each transmissioninterval [0, ).
This best estimate of () is mapped to a best estimate of
the message () and the receiver produces this bestestimate = 1,..., of the transmitted bit sequence.The goal of the receiver design in estimating the transmittedmessage is to minimize the probability of message error
==1
( = sent) ( sent) (4)over each time interval [0, ).
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 15
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Introduction
By representing the signals {(), = 1,...,} geometrically,one can solve for the optimal receiver design in AWGNchannels based on a minimum distance criterion.
Note that, wireless channels typically have a time-varying
impulse response in addition to AWGN. We will consider theeffect of an arbitrary channel impulse response on digitalmodulation performance in the next sections.
The basic premise behind a geometrical representation ofsignals is the notion of a basis set.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 16
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Basis function representation of signals
Specifically, using a Gram-Schmidt orthogonalizationprocedure, it can be shown that any set of real energysignals = {1(),...,()} defined on [0, ) can berepresented as a linear combination of real
orthogonal basis functions {1(),...,()}.We say that these basis functions span the set .Each signal {() } can be represented by
() =
=1 ,(), 0 < , (5)where
, =
0
()() (6)
is a real coefficient representing the projection.Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 17
Digital modulation techniquesSignal Space Analysis
Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Basis function representation of signals (cont.)
These basis functions have the following property0
()() =
1 = ,
0 = . (7)
The basis set consists of the sine and cosine functions
1() =
2
cos (2) (8)
and2() =
2
sin(2) . (9)
where
2 is used to obtain
0
2 () = 1, = 1, 2.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 18
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Basis functions in linear passband modulation techniques
With these basis functions, one only obtain an approximationto (7), since
0
21() =2
0
0.5 [1 + cos (4)] = 1+sin (4)
4(10)
The numerator in the second term of (10) is bounded by 1,and for 1 the denominator of this term is very large.As a result, this second term can be neglected.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 19
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
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Basis functions in linear passband modulation (cont.)
With these basis functions, one can have0
1()2() =2
0
0.5sin(4) = cos(4)
4 0
(11)
where the approximation is taken as an equality as 1.With the basis set 1() =
2/ cos (2) and
2() =
2/ sin (2), the basis function representation(5) corresponds to the complex representation of () interms of its in-phase and quadrature components with anextra factor of2/ as follows
() = ,1
2
cos (2) + ,2
2
sin(2) . (12)
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 20
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
( )
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Basis functions in linear passband modulation (cont.)
In practice, the basis set may include a baseband pulse-shapingfilter () to improve the spectral characteristics of the transmittedsignal:
() = ,1() cos (2) + ,2() sin (2) (13)
where the simplest pulse shape that satisfy (7) is the rectangularpulse shape () =
2/ , 0 < .
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 21
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
D fi i i d i i l i
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Definitions used in signal space representation
We denote the coefficients {,} as a vectors = [,1,...,,] which is called the signalconstellation point corresponding to the signal ().
The signal constellation consists of all constellation points
{s1, ..., s}.Given the basis functions {1(),...,()} there is aone-to-one correspondence between the transmitted signal() and its constellation point s.
The representation of () in terms of its constellation points is called:
its signal space representation andthe vector space containing the constellation is called thesignal space.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 22
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Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
D fi iti d i i l t ti ( t )
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Definitions used in signal space representation (cont.)
With this signal space representation we can analyze theinfinite-dimensional functions () as vectors s infinite-dimensional vector space 2.This greatly simplifies the analysis of the system performance
as well as the derivation of the optimal receiver design.Signal space representations for common modulationtechniques like MPSK and MQAM are two-dimensional(corresponding to the in-phase and quadrature basisfunctions).
In order to analyze signals via a signal space representation,we need to use some definitions for the vector characterizationin the vector space .
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 24
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
D fi iti d i i l t ti ( t )
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Definitions used in signal space representation (cont.)
In particular, the length of a vector in is defined as
s =
=12,. (14)
The distance between two signal constellation points s and sis thus
s s =
=1
(, ,)2
= 0 (() ())2 .(15)
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 25
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
Rational
Signal and system modelGeometric representation of signalsPractical examplesSignal space representation
D fi iti d i i l t ti ( t )
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Definitions used in signal space representation (cont.)
Finally, the inner product (), () between two realsignals () and () on the interval [0, ) is defined as
(), () =
0()(). (16)
Similarly, the inner product s, s between two real vectors is
s, s = ss =
0()() = (), (). (17)
It is noted that two signals are orthogonal if their innerproduct is zero.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 26
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Receiver structure and sufficient statistics
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Receiver structure and sufficient statistics
Given the channel output () = () + (), 0 < , wenow investigate the receiver structure to determine whichconstellation point s or, equivalently, which message , wassent over the time interval [0, ).
A similar procedure is done for each time interval[ , ( + 1)).
We would like to convert the received signal () over eachtime interval into a vector, since it allows us to work infinite-dimensional vector space to estimate the transmitted
signal.
However, this conversion should not compromise theestimation accuracy. For this conversion, consider the receiverstructure shown in the next figure.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 27
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Receiver structure and sufficient statistics (cont )
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Receiver structure and sufficient statistics (cont.)
)()()( tntstri
T
dt0()
T
dt0
()
111, rnsi
)(1 tI
)(tN
I
NNNirns ,
Find ii
mm
As shown in the above figure, the components of signal andnoise vectors are determined by
, = 0
()(), (18)
and
=
0
()(). (19)
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 28
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Receiver structure and sufficient statistics (cont )
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Receiver structure and sufficient statistics (cont.)
We can rewrite () as
() ==1
(, + ) () + () ==1
() + (),
(20)where = , + and () = () =1 ()denotes the remainder noise.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 29
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Proofs of sufficient statistics for optimal detection
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Proofs of sufficient statistics for optimal detection
If we can show that the optimal detection of the transmittedsignal constellation point s given received signal () does notmake use of the remainder noise (), then the receiver canmake its estimate of the transmitted message as afunction ofr = (1,...,) alone.In other words, r = (1,...,) is a sufficient statistic for ()in the optimal detection of the transmitted messages.
Let exam the distribution of r. Since () is a Gaussian
random process, if we condition on the transmitted signal() then the channel output () = () + () is also aGaussian random process and r = [1,...,] is a Gaussianrandom vector.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 30
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Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Proofs of sufficient statistics for optimal detection (cont )
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Proofs of sufficient statistics for optimal detection (cont.)
Thus, conditioned on the transmitted constellation s, the sare uncorrelated and, since they are Gaussian and also
independent. Moreover,
2
= 0/2.
We have shown that, conditioned on the transmitted
constellation s, is a Gauss-distributed random variable thatis independent of , = and has mean , and variance0/2.
Thus, the conditional distribution of r is given by
(rs sent) ==1
( ) = 1(0)
/2exp
10
=1
( ,)2 .(24)
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 32
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient Statistics
Error Probability Analysis and the Union BoundPassband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Proofs of sufficient statistics for optimal detection (cont )
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Proofs of sufficient statistics for optimal detection (cont.)
It is also straightforward to show that [()s] = 0 forany , 0 < . Thus, since conditioned on s and ()are Gaussian and uncorrelated, they are independent.
Also, since the transmitted signal is independent of the noise,
, is independent of the process ().We now discuss the receiver design criterion and show it is notaffected by discarding ().
The goal of the receiver design is to minimize the probability
of error in detecting the transmitted message givenreceived signal ().
To minimize = ( = ()) = 1 ( = ()), wemaximize (
= ()).
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 33
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Proofs of sufficient statistics for optimal detection (cont )
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Proofs of sufficient statistics for optimal detection (cont.)
Therefore, the receiver output
given received signal ()
should correspond to the message that maximizes
( sent()).Since there is a one-to-one mapping between messages andsignal constellation points, this is equivalent to maximizing ( sent()).
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 34
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Proofs of sufficient statistics for optimal detection (cont.)
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Proofs of sufficient statistics for optimal detection (cont.)
Recalling that () is completely described by = (1,...,)and (), we have
(s sent()) = ((,1,...,,) sent(1,...,, ()))= ((,1,...,,) sent, (1,...,), ())
((1,...,), ())
= ((,1,...,,) sent, (1,...,)) (())
((1,...,)) (())
= ((,1,...,,) sent(1,...,)) . (25)where the third equality follows from the fact that the () isindependent of both (1,...,) and of (,1,...,,).
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 35
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Proofs of sufficient statistics for optimal detection (cont.)
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oo s o su c e t stat st cs o opt a detect o (co t )
This analysis shows that (1,...,) is a sufficient statistic for() in detecting , in the sense that the probability of error
is minimized by using only this sufficient statistic to estimatethe transmitted signal and discarding the remainder noise.
Since r is a sufficient statistic for the received signal (), wecall r the received vector associated with ().
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 36
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Decision regions
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g
As aforementioned, the optimal receiver minimizes errorprobability by selecting the detector output
that maximizes
the probability of correct detection1
= ( sentr received).In other words, given a received vector r, the optimal receiverselects = corresponding to the constellation s thatsatisfies
(s
r) > (s
r) ,
= (26)
where (sr) (s sentr received) for the sake ofnotational simplicity.
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 37
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
Decision regions(cont.)
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g ( )
Thus, the decision regions (1,...,) corresponding to(s1, ...,s) are the subsets of the signal space anddefined by
= (r : (sr) > (sr) , = ) . (27)Once the signal space has been partitioned by decisionregions, for a received vector r , the optimal receiveroutputs the message estimate = The receiver processing consists of ) computing the receivedvector r from (), ) finding which decision region contains r, and ) outputting the corresponding message .
Mobile communications-Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 38
Digital modulation techniques
Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound
Passband modulation
General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion
An example on decision regions
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p g
This p