yrjö neuvo executive vice president, cto, nmp member of ... for mobile... · member of the nokia...

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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






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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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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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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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• 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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• 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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• 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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• 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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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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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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• 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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• 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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• 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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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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yrjö neuvo executive vice president, cto, nmp member of ... for mobile... · member of the nokia...

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