Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Stanford University Rohit Negi
'&
$%
Power Control Strategies for Delay Constrained Channels
Rohit Negi
Ph.D. Candidate
STAR Laboratory
Stanford University
Page 1
Stanford University Rohit Negi
'&
$%
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
Page 2
Stanford University Rohit Negi
'&
$%
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
Page 3
Stanford University Rohit Negi
'&
$%
Wireless communication channel
����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������
����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������
������������������������
������������������������
��������
��������
����������������������������
����������������������������
�������������������������������������������������������
�������������������������������������������������������
������������������������������������������������
����������������������������������������
����������������������������������������
������������������������������������������������
����������������������������������������
����������������������������������������
��������������������
Base station
localscatterers
mobile handset
reflector
Figure 1: A typical outdoor wireless system
Page 4
Stanford University Rohit Negi
'&
$%
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]
Page 5
Stanford University Rohit Negi
'&
$%
Idealization - `Block-fading channel'
��������
�������� ����������
channelfading
time t
g(t)
Figure 2: Illustration of channel
fading
��������������
����������������������������
������������������������������
���������������������������������������������
���������������������������������������������
������������������������������������������������
������������������������������������������������
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
Page 6
Stanford University Rohit Negi
'&
$%
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
Page 7
Stanford University Rohit Negi
'&
$%
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]
Page 8
Stanford University Rohit Negi
'&
$%
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]
Page 9
Stanford University Rohit Negi
'&
$%
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
Page 10
Stanford University Rohit Negi
'&
$%
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
Page 11
Stanford University Rohit Negi
'&
$%
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
Page 12
Stanford University Rohit Negi
'&
$%
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
��������
channelfading
time t
g(t)
Figure 4: Illustration of channel
fading
����������
����������������������������
������������������������������
���������������������������������������������
���������������������������������������������
������������������������������������������������������������������������
������������������������������������������������������������������������
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
Page 13
Stanford University Rohit Negi
'&
$%
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)
Page 14
Stanford University Rohit Negi
'&
$%
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)
Page 15
Stanford University Rohit Negi
'&
$%
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 knownPage 16
Stanford University Rohit Negi
'&
$%
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
Page 17
Stanford University Rohit Negi
'&
$%
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
Page 18
Stanford University Rohit Negi
'&
$%
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
Page 19
Stanford University Rohit Negi
'&
$%
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 .
Page 20
Stanford University Rohit Negi
'&
$%
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
Page 21
Stanford University Rohit Negi
'&
$%
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
Page 22
Stanford University Rohit Negi
'&
$%
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!
Page 23
Stanford University Rohit Negi
'&
$%
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
Page 24
Stanford University Rohit Negi
'&
$%
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
Page 25
Stanford University Rohit Negi
'&
$%
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
Page 26
Stanford University Rohit Negi
'&
$%
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
Page 27
Stanford University Rohit Negi
'&
$%
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
Page 28
Stanford University Rohit Negi
'&
$%
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
Page 29
Stanford University Rohit Negi
'&
$%
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
Page 30
Stanford University Rohit Negi
'&
$%
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!
Page 31
Stanford University Rohit Negi
'&
$%
Future work
� Unknown channel statistics (adaptive learning)
� Handling channel variation within block
� Simpler algorithms for Markovian channels
� Simpler algorithms for Multi-access channels
Page 32
Stanford University Rohit Negi
'&
$%
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
Page 33
Stanford University Rohit Negi
'&
$%
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
���
���
����
���������
�����
��������
��������
transmittedsignal
one OFDM symbol
time t
channeldistortion
cyclic prefix is destroyed
receivedsignal
time t
cyclic prefix
Figure 10: Problem of OFDM symbol synchronization
Page 34
Stanford University Rohit Negi
'&
$%
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
Page 35
Stanford University Rohit Negi
'&
$%
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)
Page 36
Stanford University Rohit Negi
'&
$%
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)
Page 37
Stanford University Rohit Negi
'&
$%
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
Page 38
Stanford University Rohit Negi
'&
$%
Acknowledgments
� J. CioÆ
� A. Maleki, W. Choi, S. Vishwanath
� K. Cheong, J. Fan, N. Wu
� S. Hosur, A. Dabak
� M. Charikar, B. van Roy
Page 39