stanford - winlabrohit negi ' & $ % dela y constrained c hannels-mo del final goal: design...

Stanford University Rohit Negi

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Power Control Strategies for Delay Constrained Channels

Rohit Negi

Ph.D. Candidate

STAR Laboratory

Stanford University


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Outline of Talk

� Background on fading communication channels

� Delay Constrained channels

� Power Control Strategies for such channels

� Extensions, issues

� A sketch of other areas of research


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Wireless communication channel

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Base station

localscatterers

mobile handset

reflector

Figure 1: A typical outdoor wireless system


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Modeling a fading channel

� Simplest case - channel is characterized by ampli�cation scalar factor g(t)

(`channel gain')

� User handset movement ) g(t) uctuates

think `stationary waves'

� ) `fading channel'

� Rate of variation depends on user speed

� Assume moderate rate of variation ) `Block-fading channel' [McEliece

'84]


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Idealization - `Block-fading channel'

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channelfading

time t

g(t)

Figure 2: Illustration of channel

fading

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A block consists of several

symbolsg(t)

block 1 block 2 block n time t

Key : Assume g(t) constant during each blockFigure 3: Block fading channel

model


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Channel capacity - revisited

� The most well-known Shannon capacity idea (which suits wireline channels

admirably)

\What is the largest error-free data rate C that can be supported by a �xed

channel?"

� Valid for a speci�ed transmission power

� But for wireless channels, capacity idea is unclear because channel is

time-varying


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Capacity for wireless channels

� Need to rede�ne capacity for wireless channels

� New de�nitions valid under di�erent practical scenarios

� Concentrate on capacity de�nitions

{ ergodic capacity (classical)

{ expected capacity [Cover '72]

{ outage capacity [e.g. Shamai '94]


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Ergodic, Expected and Outage Capacities

� Think of random variable I(X;Y=g(t)) as \error free data rate, given a

certain transmission strategy"

� Ergodic capacity - useful for fast fades

Cerg = maximum E[I(X;Y=g(t))]

� Several results exist on this. e.g. [Goldsmith '97]

� Expected capacity - same expression as above, but no ergodicity

� Few results. e.g. [Shamai '97]

� Outage capacity - use for slow fades.

Outage capacity is Cout = R0 at error probability Perr if

minimum Prob[I(X;Y=g(t)) < R0] = Perr

� Newer idea. Few results. e.g. [Biglieri '97]


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Key ideas until now

� Idea of fading channel

� Idea of channel capacity, which depends on

{ channel

{ Allowed transmission power (more power ) more capacity)

{ Practicalities, such as channel variation, delay requirements, etc.

� Idea of ergodic, expected and outage capacities for fading channels


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Outline of Talk







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Need for delay constraint

� Practical applications require data transmission within speci�ed

time-window ) delay constraint

� Typical requirement: Need to transmit R0KT bits of data within a time

interval of KT , with a maximum allowed power of KP0

� e.g. Voice traÆc allows a 20 millisec window, within which average data rate

should be at least 8000 bits/sec

� Other applications: real-time video, ...

� Rest of talk will introduce problem formally, and show solutions

� Introducing and solving this problem has been a key contribution of my

research


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Delay constrained channels - model

� Final goal: design optimum transmission schemes, so as to meet delay

requirement of application

� Abstract out key problem by choosing appropriate channel model

� For this, impose delay constraint on block-fading channel

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channelfading

time t

g(t)

Figure 4: Illustration of channel

fading

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A block consists of several

symbolsg(t)

block 1 block 2 block n time t

Delay constraint of K = 2 blocks

Figure 5: Delay constrained chan-

nel model


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Mathematical model

� K blocks of data, each with T symbol transmissions

� Average data rate achievable in K blocks RK = 1K

PK

i=1 log(1 + Pigi)

� Goal is: maximize E[ �(KRK) ]

by choosing fP1; P2; : : : ; PKg appropriately

� Note 1: �(x) is chosen based on practical requirements

� Note 2: Block-fading model allows clean problem speci�cation

� Note 3: [Biglieri '98] has solved problem when all gis are known

simultaneously

� Note 4: Pi is a function of fg1; g2; : : : ; gig only (causal)


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Model choices

Choice of �(x):

� �(x) = x ) expected capacity notion!

� �(x) = 1F (x � KR0) ) outage capacity notion!

� ergodic capacity notion not useful

Choice of power constraint [Biglieri '97]:

�

1K

PK

i = 1 Pi � P0 ) `short-term constraint'

�

1K

PK

i = 1E[Pi] � P0 ) `long-term constraint' (more general)


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Optimum transmission strategies

� Optimum transmission ) choose

{ signaling distribution - optimally chosen gaussian here, and then

{ transmit power fP1; P2; : : : ; PKg

� Thus, problem reduces to chooosing fP1; P2; : : : ; PKg causally

� Qualitatively: Blast away power now, or conserve it for later?

� Assumptions: noise is AWGN, T is large (low coding error), gi's are i.i.d., gi's

are known at transmitter causally, and probability distribution of gi is known


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Outline of Talk







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Desired algorithm

= KPpowerallowed

rate

g1 g2 gK

Remainingallowedpower

Choosepower

P

Transmitrate

log(1+P

Choosepower

P

Transmitrate

log(1+P

2

Block 2 Block KBlock 1

0

Choosepower

P1

Transmitrate

log(1+P g1)1 )KgK

K

)2g2

achieved

= 0

AchievedR 1rate =

Figure 6: Algorithm must specify optimum power P �n

for each g;R; P in the for-

ward direction


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Proposed Dynamic program solution (optimal)

OptimizePK

OptimizePK-1

OptimizeP1

Calculateθ

CalculateθΚ

Calculateθ

Optimizeψ

OptimizeψΚ

Optimizeψ

Κ−11

1 Κ−1

)))( ((

g, R, Pg, R, Pg, R, P

Initialize

ψ

µK+1

(R)=

Block KBlock K-1Block 1


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Algorithm description

� Algorithm: For n = 1; : : : ;K

At time n, choose

P�

n

(gn; R; P ) = argmax

0�Pn�P

n+1(R+ log(1 + Pngn); P � Pn) and

�n+1(gn; R; P ) = n+1(R+ log(1 + P�

n

gn); P � P�

n

) , compute

n(R;P ) =

gnE[ �n+1(gn; R; P ) ] for short-term constraint

n(R;P ) = min

gnE[ �n+1(gn; R; ~P (gn))] for long-term (1)

K+1(R;P ) = �(R) initialization

where the minimization in (1) is over all functions ~P (gn) � 0 such that

E[ ~P (gn)] = P .


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Specializing solution to Expected capacity

� Choose �(x) = x

� Then, dynamic program solves problem:

maximize E[KX

i=1log(1 + Pigi) ] with appropriate power constraint

� Long term constraint: solution reduces to [Goldsmith '97] `time-water�lling'

algorithm

� Short term constraint:

{ Linearity of �(x) ) functions P �n

(gn; R; P ) and n(R;P ) reduce to

P�

n

(gn; P ) and n(P ) respectively

{ At low power levels (SNR), get `pick one block and transmit' scheme.

Performance similar to selection diversity strategy


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Simulation

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

number of blocks K

capa

city

rat

io

−40 dB−30 dB

−20 dB−10 dB

−5 dB0 dB

0 5 10 15 20 25 30 35 40 45 500.95

1

1.05

1.1

number of blocks K

capa

city

rat

io

5 dB

10 dB

20 dB

Figure 7: Capacity ratio of proposed algorithm to constant power transmission,

in a Rayleigh fading channel


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Specializing solution to outage capacity

� Choose �(x) = 1F (x � KR0)

� Then, dynamic program solves problem:

minimize Prob[KX

i=1log(1 + Pigi) < KR0 ] with power constraint

� Long term constraint solution: As K !1, reduces to [Goldsmith '97]

`time-water�lling' algorithm

� Short term constraint: At low power levels (SNR), get same performance as

selection diversity!

� Key point: Do not need variable rate transmission. Simply transmit at rate

R0, and only vary power level!


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Outage capacity - simulation

0 5 10 15 20 2510−5

10−4

10−3

10−2

10−1

100

101

SNR in dB

outa

ge p

roba

bilit

y

K = 1K = 2K = 3K = 5

Figure 8: Outage probability with short term constraint algorithm for R0 = 3 and

various K. Solid lines: optimum algorithm; dotted lines: no power adaptation


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Outage capacity - simulation

0 5 10 15 20 2510−5

10−4

10−3

10−2

10−1

100

101

SNR in dB

outa

ge p

roba

bilit

y

K = 1K = 2K = 3K = 5

Figure 9: Outage probability for R0 = 3 and various K. Solid lines: long term

constraint algorithm; dotted lines short term constraint algorithm


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Key ideas to take away

� New idea of delay constrained fading channel (application speci�c)

� Idealization of problem using block-fading model

� Problem speci�cation based on practicalities - choice of �(x), power

constraint

� Solution of problem involves dynamic programs.

� Application of solution to obtain expected capacity and outage capacity

� Earlier researchers have only looked at two unlikely extremes : K !1

(ergodic capacity), and K = 1


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Outline of Talk







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Other channel models

Matrix/Frequency-selective channels:

� a.k.a. Multi-antenna, frequency-selective

� Instead of log(1 + Pigi) functions, we now have arbitrary rate functions

r(Hi; Pi). Use these in the algorithms

Markovian channel :

� When the channels is not i.i.d. block-fading, but Markovian (a more

realistic case)

� Will need to incorporate states of the channel as extra states in the

program


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Non-zero decoding error probability, and practical codes

� Translating idealization back to the real world

� When T is �nite, and practical codes are used, one can rede�ne an outage

event to mean the following

1. total rate supported by channel for those K blocks is < KR0 or

2. decoding error probability in any of the K blocks > allowed BER

� Now, rede�ne r = f(Hi; Pi; Perr), i.e. the code rate that results in error

probability < Perr. This can be calculated from code tables

� Then, the speci�ed dynamic program solves the delay constraint problem

optimally


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Computational issues

Convergence analysis: as K !1, can show convergence to ergodic capacity

algorithm [Goldsmith '97]

Discretization analysis: in practice, any dynamic program will need to work

with discrete values of R;P; g. How much does one lose due to this?

� A novel analysis was done for the dynamic programs described

� Rigorous and useful bounds were derived for the case of uniform

discretization of R;P , and geometric discretization of g


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Stationarity issues

� Problem speci�ed and solved requires forming a `super-frame' of K blocks,

numbered f1; 2; : : : ;Kg

� But, some applications may not be able to coordinate with such a

super-frame

� So, specify a `stationary performance' problem. Given the sequence of

channel power gains fgi; i = : : : ;�1; 0; 1; : : :g, and given the target rate R0,

and power P0

minimize averagef P[

tXi=�1

e��(t�i) log(1 + Pigi) < R0] g such that

averagefPtg � P0 power constraint

� Novel idea of using a `discounted rate' variable to simulate K blocks in a

stationary fashion

� Problem solved using linear programming

� Randomized power control turns out to be optimal!


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Future work

� Unknown channel statistics (adaptive learning)

� Handling channel variation within block

� Simpler algorithms for Markovian channels

� Simpler algorithms for Multi-access channels


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Outline of Talk







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Blind OFDM Symbol Synchronization

� OFDM is a widely used as a digital broadcast standard

� In OFDM, each `symbol' consists of a sequence of transmissions

� Identifying the beginning of each symbol, is called symbol synchronization

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transmittedsignal

one OFDM symbol

time t

channeldistortion

cyclic prefix is destroyed

receivedsignal

time t

cyclic prefix

Figure 10: Problem of OFDM symbol synchronization


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Blind OFDM Symbol Synchronization - continued

� Derived blind algorithm for OFDM symbol synchronization

� As opposed to then previous algorithms in literature e.g. [van de Beek '97],

this algorithm guarantees correct synchronization asymptotically

� Uses insight into the ranks of certain autocorrelation matrices for OFDM

transmission


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Space-time Coding for Outdoor Wireless Channels

� Space-time codes [Calderbank '97], are novel `trellis' codes that o�er high

diversity and coding gain, in a multiple-transmit antenna environment

� Pros - Good for indoor channels, which have dense scatterers

� Cons - Not for outdoor channels, which have narrow multipaths

� Showed how signal processing yields a certain `channel partition' for outdoor

channels, so that space-time codes can be transmitted over the subchannels

of the partition

� Used the insight into the separability of signal processing and coding, which

occurs for space-time codes (with A. Maleki)


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Other miscellaneous research

� Space-time codes for CDMA (with A. Dabak,S. Hosur)

� Using partial Maximum-likelihood decoding to improve diversity

performance of BLAST [Foschini '96] (with W. Choi)

� Looking at the suitability of turbo-codes for space-time applications (with S.

Vishwanath)

� Narrowband interference cancellation in VDSL (with D. Pal)

� `Soft' interference cancellation for DMT. Multiuser detection in a highly

undersampled channel. (with J. Fan,K. Cheong,W. Choi,N. Wu)


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Future research directions

� Signal Processing and Coding problems in communications

� Fading channels, antenna arrays, channel estimation

� Transmission optimization, `turbo-decoding' methods

� Integrating queueing theory into physical layer design

� Using information theory ideas in other �elds


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Acknowledgments

� J. CioÆ

� A. Maleki, W. Choi, S. Vishwanath

� K. Cheong, J. Fan, N. Wu

� S. Hosur, A. Dabak

� M. Charikar, B. van Roy

stanford - winlabrohit negi ' & $ % dela y constrained c hannels-mo del final goal: design...

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