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Mathematics for mobile generationsYrjö Neuvo
Executive Vice President, CTO, NMPMember of the Nokia Group
Executive Board
Diderot mathematical Forum November 22, 2001
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Outline • An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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Outline• An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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050
100150200250
300350
400450
1995 1996 1997 1998 1999 2000
Europe / AfricaAPACAmericas
The #1 consumer electronics industry
Mobile phone market volume worldwide
1 Billion users:1st half 2002
For comparison:6,3M PDA's shipped
first half 2001
Moore's law describes technology evolution• Microelectronics evolve exponentially:
• Performance doubles every 18 months• Size and price do not increase
• The price of computers has diminished 20% annually for 40 years already
• Moore's law is expected to stay valid at least the next decade or two
• What does this really mean?• Already in 2003, an average western home will contain 380
microprocessors (Dataquest)• The processing capacity of an average processor: today at
insect brain level, mouse brain 2010, human brain 2020 … (R. Kurtzweil: The age of spiritual machines)
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Moore's law illustrated -transistors per chip estimate
0
200
400600
800
1000
12001400
1600
1999 2001 2003 2006 2009 2012
Year of first shipment
No.
of
tran
sisto
rs (m
illio
ns)
Source: Semiconductor Industry Association roadmap
Evolution of transistors per chip
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Metcalfe's law and the power of networking
• The utility and benefit of networks increases with the number of nodes and users
• Real growth of a network or application starts only when the usefulness is proven during the initial phases
• The first solution to reach critical mass wins - de-facto standards are a result of this
• Winning networking technologies shape life around the globe
Utility and value = users^2
Users
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0
10
20
30
40
50
60
0% 10% 20% 30% 40% 50% 60% 70% 80%
Mobile Penetration
SMSs
/sub
s/m
onth
FinlandNorway
Germany
Italy
Portugal
GreeceUK
France
Sweden
Spain
Metcalfe's law in action -SMS growth in Europe
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The challenge of predicting the unpredictable • The future is unpredictable
• Trends and directions can be sensed and guessed - but also influenced
• With suitable mental aids the future can be unfogged (slightly…)
• Moore's law can give directions on what is possible
• Metcalfe's law can give hints on how technologies and habits will be adopted
• Roughly 5 years is a critical limit for predictions
• The impact of new phenomena is overestimated in the short term, but underestimated in the long term
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What is the current transition about?
• Packet radio and 3G technology• From scarce to adequate capacity• Removing the technical constraints
for human-centered services
• Internet• Internet becomes invisible, a
platform for using personalized applications and services
• Ubiquitous networking
• Multimedia• From 'Listen to what I say' to 'See
what I mean!'
• From technology to behaviour
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GSM dataCircuit switched
First data-based services appear
Main application: voice
New applications:SMS, later WAP
Technologies: GSM, HSCSD
Data rates: 9.6 - 14.4 - 43.2 kbps
Apps & SW: mainly closed
Standardization: official bodies (ETSI)
Digital mobile industry phases:from past to present
GSMCircuit Switched
Traditional telecom business
Application: Voice
Technologies: GSM (and other cellular protocols)Data rate: 9.6 kbps
Apps & SW: closed
Standardization: official bodies (ETSI)
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… and from the transition onwards2.5G & 3G Circuit Switched & Packet Radio
Adapting packet-switched non-realtime servicesApplications: Voice and servicesNew applications:
WAP browsing over GPRS, Multimedia messaging, email,always-on Internet connection, Rich calls, location-based services, etc
Technologies: GSM, HSCSD, GPRS, 3G radio, Bluetooth, WLAN, Symbian, ...Data rates: Up to 384 kbpsApps & SW: both open and closed(SIM toolkit, WAP, Java, 3rd party SW for Symbian)UI: Menu-based & Micro browser WAP/XHTMLEnhanced interoperabilityStandardisation: 3GPP & Industry Fora
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Outline• An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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Mathematical heritage for mobile networks
• Electromagnetic theory Maxwell equations:
• Traffic theoryErlang B formula:
Erlang C formula:
• Information theoryChannel Capacity:
( )( )∑ =
= m
n
n
m
mn
mp
0!
!
µλ
µλ
+=
NS
BC 1log
ερ
0
)1( =Ε⋅∇ 0)2( =Β⋅∇t∂Β∂
−=Ε×∇)3(t
j∂Ε∂
+=Β×∇ µεµ000
)4(
∑−
= −+−
=1
0 )1(!)(
!)(
1)1(!
)(m
n
mn
m
Q
mm
nmm
mpρ
ρρρρ
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Historical milestones of technology and mathematics leading to cellular systems
1844
TelegraphMorse
1870
ElectromagnetismMaxwell
1888
ElectromagneticwavesHertz
1896
WirelesstelegraphyMarconi
1904
Electron tube Fleming
1925
RadarAppletonBarnett
1948
TransistorBardeenBrattainShockley
1981
Analogcellular
SystemsNMT andAMPS arelaunched
The age ofDigital
CellularSystems isstarting, firstGSM call in
Helsinki
1876
TelephoneBell
1917
1947
CellularsystemconceptAT&T
1958
IntegratedcircuitsTexas
Instruments
1971
The firstmicro
processorIntel 4004
FourierAnalysisFourier
InformationTheory
ShannonSamplingTheoryNyquist
1928
SpectralAnalysisWiener
1930
Algorithms andcomputation
Turing
1936 1948
EstimationTheoryWiener
1942
1940
First concepts forspread spectrum
systems
1822
TeletrafficTheoryErlang
Digital signalprocessor
TexasInstruments
1983 1991
CodingTheory
Hamming
1950
FastFourier
TransformCooleyTukey
1965
MarkovChain
StochasticProcessMarkov
1900
PoissonProcessPoisson
1837
1933
FMmodulationArmstrong
GaloisField
Galois
1846
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1G characteristics
• Analog techniques
• Frequency modulation (FM)
• Many regional systems: NMT, AMPS, …
• Idea of cellular network
• Mobility: handovers, limited roaming
• Multiple access technique: FDMA
• Car telephone
• Speech calls
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Mathematics in 1G
• Mathematics was not used as extensively as in later generations
• The legacy from radio technology and telephone networks was a necessary but not sufficient prerequisite for development of 1G
• Radio interface• Typical problem: How much bandwidth is needed to support transfer
of speech ?• Some mathematical methods needed for synchronization, receiver
techniques, demodulation, interference, modelling, filters
• Network planning• Typical problem: Where to place base stations to provide full and
uniform coverage for moving users ?• Capacity planning: How many cells are needed to serve all users in a
densely populated area?• Optimization techniques
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Outline• An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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2G characteristics
• Digital information transmission also in radio interface
• One widely deployed system: GSM
• Global roaming
• Multiple access technique: TDMA
• Data services possible
• Flexible service addition
• Increased capacity
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Mathematics for 2G
• Legacy from information theory becomes applicable
• Revolution in radio interface --> many new mathematical methods
• digital source coding of speech• linear predictive coding
sampled sequence
estimate
The difference together with coefficients is encoded and transmitted instead of
E.g. Levinson-Durbin algorithm used to compute recursively
• unequal error protection
,...2,1,0,ˆ1
== −=
∑ nxax kn
p
kkn
ka
,...2,1,0, =nxn
nxnn xx −ˆ
ka
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Mathematics for 2G (cont'd)
• Radio channel can be modelled as linear system:
• In practice the signals and impulse response are discretized and the convolution is modelled as a linear filter
.,)(
,),(,)(
,),()()(
valuedcomplexarefunctionsallandsignaloutputpasslowtr
responseimpulsetcsignalinputpasslowtu
where
dtctutr
==
=
−= ∫∞
∞−
τ
τττ
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Mathematics for 2G (cont'd)
• The impulse response is modelled as a tapped delay line model, and each tap is modelled as a stochastic process with Rayleigh or Riceandistribution
Rayleigh distribution
Rician distribution
The function I0 is the zero order modified Bessel function ofthe first kind.
.0,)( 2
2
22 ≥=
−xwheree
xxf
xσ
σ
.0,)()( 202
2
2
22
≥=+−
xwheresx
Iex
xfsx
σσσ
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Mathematics for 2G (cont'd)• channel coding
• error-correcting codes: convolutional codes• finite field F• An [n,k] linear block code over F is a k-dimensional subspace of F n.
• An (n,k) convolutional code is a k-dimensional subspace of F(D)n
where F(D) is the field of rational functions over F• decoding by Viterbi algorithm
• Example (Rate ½ code). Denote the generator matrix of half rate CC code by
• Then encoding by means simply multiplication of an information stream :
)()()( 2/1 DGDiDi ⋅→
( ) ( )4343 + ++ 1 + + 1== DDDDDDGDGDG :)()(:)( 10)2/1(
)(2/1 DGk
k k DiDi ∑∞
==
0)(
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Mathematics for 2G (cont'd)
• Convolutional codes in GSM specification (quote):
• The class 1 bits are encoded with the 1/2 rate convolutional code defined by the polynomials:
• G0 = 1 + D3+ D4
• G1 = 1 + D + D3+ D4
• The coded bits {c(0), c(1),..., c(377)} are then defined by:• c(2k) = u(k) + u(k-3) + u(k-4)• c(2k+1) = u(k) + u(k-1) + u(k-3) + u(k-4) for k =0,1,...,188
u(k) = 0 for k < 0
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Mathematics for 2G
• Error detection: CRC (Cyclic redundancy codes)
here r(D) is the remainder polynomial when dividing i(D)Dn-k by g(D)
In GSM: g(D) = D8 + D4 + D3 + D2 + 1
• Error check by dividing the received polynomial by the generator g(D): If no errors then the remainder is zero.
)()()( DrDDiDi kn −⋅→ −
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Mathematics for 2G (cont'd)
• modulation techniques
• security• encryption in GSM:
ciphertext = plaintext ⊕ A5( key, frame )
• authentication: one-way functions are used
easy to computex f(x)
infeasible to compute
• channel estimation• equalization
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Mathematics for 2G (cont'd)
• Modelling of channels and receiver structures• stochastics
• Algorithm development• suboptimal algorithms
• Evolution on core network structure; digital switches imply new mathematical methods
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Outline• An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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2½G characteristics
• Packet data
• Systems: GPRS
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Mathematical trends for 2½G
• Traffic model changes towards Internet model --> affects network planning
• Complexity increased• More advanced coding schemes
• Reed-Solomon codes• The cyclic code associated to the generator polynomial (divides in
) is the ideal
• In RS (EDGE) code the generator is
• Berlekamp-Massey decoder
• Non-real time services: ARQ protocols
[ ]( )[ ] }1))(deg(,)(|)1mod()()({
)1/()(
−≤∈−⋅=
−⋅
kDiDFDiDDgDi
DDFDg
qn
nq
)(Dg 1−nD [ ]DFq
[ ]DFaDDg qi
i
∈−= +
=∏ )(:)( 122
11
0
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Quote from GSM EDGE specification:– 3.11.2.2 Reed Solomon encoder
The block of 584 information bits is encoded by shortened systematic Reed Solomon (RS) code over Galois field GF(28). The Galois field GF(28) is built as an extension of GF(2). The characteristic of GF(28) is equal to 2.
The code used is systematic RS8 (85,73), which is shortened systematic RS8(255,243) code over GF(28) with the primitive polynomial p(x)=x8+x4+x3+x2+1. The primitive element a is the root of the primitive polynomial, i.e.
a8 = a4 + a3 + a2 + 1.
Generator polynomial for RS8(255,243) code is:
g(x)=; that results in symmetrical form for the generator polynomial with coefficients given in decimal notation
g(x)= x12 +18x11 + 157x10 + 162x9 + 134x8 + 157x7 + 253x6 + 157x5 + 134x4 + 162x3 + 157x2 + 18x + 1
where binary presentation of polynomial coefficients in GF(256) is {a7, a6, a5, a4, a3, a2, a, 1}.
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Outline• An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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3G characteristics
• Global system: UMTS
• Multiple access technique: WCDMA
• More computing power in terminals and network elements
• Core network evolution towards IP
• Multimedia
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Mathematics for 3G
• The idea of CDMA is based on linear algebra. By simplifying it can be described as follows:
orthogonal vectors (one per user) : c1 , c2 ,…, cn
information bit (for user k) ik is spread, i.e. multiplied by vector ck
Base station transmits total signal:
User k computes the correlation of received T and vector ck :
∑=
=n
kkkiT
1
c
kk iT =c,
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Mathematics for 3G (cont'd) Walsh-Hadamard sequences (of length )
• rows of each matrix are orthogonal• Hadamard sequences are used in WCDMA for downlink. They cannot
be used for uplink (lack of synchronization).• Long scrambling codes are used
• in uplink to separate users• in downlink to separate sectors / cells• Long codes are based on Gold sequences • Remark: A well-known upper bound for Gold sequence correlations is a corollary of
Riemann’s hypothesis for function fields of curves proved by Andre Weil 1948.
• CDMA detection structures
−
=nn
nnn HH
HHH2,
1111
2
−
=H( ) ,11 =H
k2
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Mathematics for 3G (cont'd)
• CDMA detection structures
• WCDMA affects the network design; complex resource management problems
• continuous power control and adjustments per terminal (in WCDMA once in 0.67 millisecond)
• RF design: integral equations
• speech recognition
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Mathematics for 3G (cont'd)
• Turbo codes : The idea is to have two parallel concatenated (very simple) convolutionalcodes with internal interleaver between constituent encoders
xk
xk
zk
Turbo codeinternal interleaver
z’k
D
DDD
DD
Input
Output
x’k
1st CC encoder
2nd CC encoder
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Mathematics for 3G (cont'd)
• security: more complex cryptographic algorithms
• IP routers introduced to mobile networks --> math needed in e.g.• header compression• ARQ• traffic models• TCP error correcting
• New services• location services --> fast geometric algorithms• mobile commerce --> digital signatures, public key cryptographyRSA digital signatures (m=message; s= signature):
s ≡ md ( mod n ) ; n = pq; p,q prime numbersm ≡ me ( mod n ) ed ≡ 1 ( mod ϕ(n) )
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Outline• An industry in transition
• Mathematics from past to 1G
• Mathematics for 2G - current bread and butter
• Mathematics for 2½G - the transition starts
• Mathematics for 3G
• … and still more mathematics
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General trends in mathematics for mobile
• Mathematical modelling more and more important• virtual prototypes• complex modelling before physical implementations• system specifications• requirement for increased bitrates
• Increased processing power makes more complex algorithms feasible• sometimes also mathematically simpler but heavier algorithms can
be taken into use
• Management of complex software systems
• Algebraic methods
• Algorithm development
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2G
3G
FutureSystems
0.1 1 10 100 1000Data Rate (Mbps)
Mob
ility
The evolution goes on
• It is already time forresearch communities to look beyond 3G
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Concluding remarks
• Mathematics in a central position
• Are mathematicians too modest?
• Welcome to come closer and take a more active role in the industry
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