The necessity and sufficiency of anytime capacity for stabilization

27 Pages·2012·0.4 MB·English

The necessity and suf(cid:2)ciency of anytime capacity for stabilization of a linear system over a noisy
communication link
Part I: scalar systems
Anant Sahai and Sanjoy Mitter
[email protected], [email protected]
Abstract
We review how Shannon’s classical notion of capacity is not enough to characterize a noisy communication channel if the
channel is intended to be used as part of a feedback loop to stabilize an unstable scalar linear system. While classical capacity
is not enough, another sense of capacity (parametrized by reliability) called (cid:147)anytime capacity(cid:148) is shown to be necessary for
the stabilization of an unstable process. The required rate is given by the log of the unstable system gain and the required
reliability comes from the sense of stability desired. A consequence of this necessity result is a sequential generalization of the
Schalkwijk/Kailath scheme for communication over the AWGN channel with feedback.
Incasesofsuf(cid:2)cientlyrichinformationpatternsbetweentheencoderanddecoder,adequateanytimecapacityisalsoshownto
besuf(cid:2)cientfortheretoexistastabilizingcontroller.Thesesuf(cid:2)ciencyresultsarethengeneralizedtocaseswithnoisyobservations,
delayedcontrolactions,andwithoutanyexplicitfeedbackbetweentheobserverandthecontroller.Bothnecessaryandsuf(cid:2)cient
conditions are extended to continuous time systems as well. We close with comments discussing a hierarchy of dif(cid:2)culty for
communication problems and how these results establish where stabilization problems sit in that hierarchy.
Index Terms
Real-time information theory, reliability functions, error exponents, feedback, anytime decoding, sequential coding, control
over noisy channels
A. Sahai is with the Department of Electrical Engineering and Computer Science, U.C. Berkeley. Early portions of this work appeared in his doctoral
dissertationandafewotherresultswerepresentedatthe2004ConferenceonDecisionandControl.
Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. Support for S.K. Mitter was provided by the
Army Research Of(cid:2)ce under the MURI Grant: Data Fusion in Large Arrays of Microsensors DAAD19-00-1-0466 and the Department of Defense MURI
Grant:ComplexAdaptiveNetworksforCooperativeControlSubaward#03-132andtheNationalScienceFoundationGrantCCR-0325774.
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 1
The necessity and suf(cid:2)ciency of anytime capacity for stabilization of a linear system over a noisy
communication link
Part I: scalar systems
I. INTRODUCTION We are far from the (cid:2)rst to attempt to bring together
informationandcontroltheory.In[7],Ho,Kastner,andWong
For communication theorists, Shannon’s classical channel
drew out a detailed diagram in which they summarized the
capacity theorems are not just beautiful mathematical results,
then known relationships among team theory, signaling, and
they are useful in practice as well. They let us summarize
informationtheoryfromtheperspectiveofdistributedcontrol.
a diverse range of channels by a single (cid:2)gure of merit:
Rather than taking such a broad perspective, we instead
the capacity. For most non-interactive point-to-point com-
ask whether Shannon’s classical capacity is the appropriate
munication applications, the Shannon capacity of a channel
characterization for communication channels arising in dis-
provides an upper bound on performance in terms of end-to-
tributed control systems. Our interest is in understanding the
end distortion through the distortion-rate function. As far as
fundamental relationship between problems of stabilization
distortionisconcerned,allthatmattersisthechannelcapacity
and problems of communication.
and the nature of the source. Given enough tolerance for
Tatikonda’s recent work on sequential rate distortion theory
end-to-end delay, the source can be encoded into bits and
provides an information-theoretic lower-bound on the achiev-
those bits can be reliably transported across the noisy channel
able performance ofa control systemover a channel. Because
if the rate is less than the Shannon capacity. As long as
this bound is sometimes in(cid:2)nite, it also implies that there
the source, distortion, and channel are well-behaved[1], [2],
is a fundamental rate of information production, namely the
there is asymptotically no loss in separating the problems of
sum of the logs of the unstable eigenvalues of the plant,
sourceandchannelcoding.Thisprovidesajusti(cid:2)cationforthe
that is invariantly attached to an unstable linear discrete-time
layered architecture that lets engineers isolate the problem of
process [8], [9]. This particular notion of rate was justi(cid:2)ed
reliable communication from that of using the communicated
by showing how to stabilize the system over a noiseless
information.Recentadvancesincodingtheoryhavealsomade
feedback link with capacity greater than the intrinsic rate for
it possible to approach the capacity bounds very closely in the unstable process.3 Nair et al. extended this to cover the
practical systems.
case of unbounded disturbances and observation noise under
In order to extend our understanding of communication to suitableconditions[10],[11].Inadditiontonoiselesschannels,
interactive settings, it is essential to have some model for the results were extended for almost-sure stabilization in the
interaction. Schulman and others have studied interaction in context of undisturbed4 control systems with bounded initial
thecontextofdistributedcomputation[3],[4].Theinteraction conditions being stabilized over certain noisy channels [12].
thereisbetweencomputationalagentsthathaveaccesstosome We had previously showed that it is possible to stabilize
private data and wish to perform a global computation in a persistentlydisturbedcontrolledGauss-Markovprocessesover
distributed way. The computational agents can only commu- suitable power-constrained AWGN (Additive White Gaussian
nicate with each other through noisy channels. In Schulman’s Noise) channels[13], [14] where it turns out that Shannon
formulation, capacity is not a question of major interest capacity is tight and linear observers and controllers are
since constant factor slowdowns are considered acceptable.1 suf(cid:2)cient to achieve stabilization [15]. In contrast, we showed
Fundamentally, this is a consequence of being able to design that the Shannon capacity of the binary erasure channel
all the system dynamics. The rich (cid:2)eld of automatic control (BEC) is not suf(cid:2)cient to check stabilizability and introduced
provides an interactive context to study capacity requirements the anytime capacity as a candidate (cid:2)gure of merit [16].
since the plant dynamics are given, rather than something that Following up on our treatment of the BEC case, Martins et
can be designed. In control, we consider interaction between al. have studied more general erasure-type models and have
anobserverthatgetstoseetheplantandacontrollerthatgets alsoincorporatedboundedmodeluncertaintyintheplant[17].
to control it. These two can be connected by a noisy channel. There is also related work by Elia that uses ideas from robust
Shannonhimselfhadsuggestedlookingtocontrolproblems control to deal with communication uncertainty in a mixed
for more insight into reliable communication [5]. continuous/discretecontext,butrestrictingtolinearoperations
[18], [19]. Basar and his students have also considered such
(cid:147)::: can be pursued further and is related to a
duality between past and future2 and the notions of problems and have studied the impact of a noisy channels on
boththeobservationsandthecontrols[20].Theareaofcontrol
control and knowledge. Thus we may have knowl-
with communications constraints continues to attract attention
edge of the past and cannot control it; we may
and the reader is directed to the recent September 2004 issue
control the future but have no knowledge of it.(cid:148)
3Thesequentialrate-distortionboundisgenerallynotattainedevenathigher
1Furthermore, such constant factor slowdowns appear to be unavoidable ratesexceptinthecaseofperfectlymatchedchannels.
whenfacingtheverygeneralclassofinteractivecomputationalproblems. 4In seminal work [12], there is no persistent disturbance acting on the
2Thedifferingrolesofthepastandfuturearemadeclearin[6]. unstableplant.
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 2
process.Despitehavingtoobserveandencodetheexactsame
-
StabilizationProblems CommunicationProblems closed-loopprocess,theobserverinternaltothecontrolsystem
with overNoisyChannels requires a channel as good as that required to communicate
NoisyFeedbackChannels (cid:27) withNoiselessFeedback
the unstable open-loop process. This seemingly paradoxical
situation illustrates what can happen when the encoding of
information and its use are coupled together by interactivity.
Fig.1. The(cid:147)equivalence(cid:148)betweenstabilizationovernoisyfeedbackchannels
In this paper (Part I), the basic equivalence between feed-
and reliable communication over noisy channels with feedback is the main
resultestablishedinthispaper. back stabilization and reliable communication is established.
The scalar problem (Figure 2) is formally introduced in
Section II where classical capacity concepts are also shown
of IEEE Transactions on Automatic Control and the articles to be inadequate. In Section III, it is shown that adequate
therein for a more comprehensive survey. feedback anytime capacity is necessary for there to exist an
Manyoftheissuesthatariseinthecontrolcontextalsoarise observer/controller pair able to stabilize the unstable system
for the conceptually simpler problem of merely estimating an across the noisy channel. This connection is also used to give
unstable open-loop process5, across a noisy channel. For this asequentialanytimeversionoftheSchalkwijk/Kailathscheme
estimation problem in the limit of large, but (cid:2)nite, end-to-end for the AWGN channel with noiseless feedback.
delays, we have proved a source coding theorem that shows
that the distortion-rate bound is achievable. Furthermore, it Section IV shows the suf(cid:2)ciency of feedback anytime
is possible to characterize the information being produced capacity for situations where the observer has noiseless ac-
by an unstable process [23]. It turns out that such processes cess to the channel outputs. In Section V, these suf(cid:2)ciency
produce two qualitatively distinct types of information when results are generalized to the case where the observer only
it comes to transport over a noisy channel. In addition to the has noisy access to the plant state. Since the necessary and
classicalShannon-typeofinformationfoundintraditionalrate- suf(cid:2)cientconditionsaretightinmanycases,theseresultsshow
distortion settings6, there is an essential core of information the asymptotic equivalence between the problem of control
thatcapturestheunstablenatureofthesource.Whileclassical with (cid:147)noisy feedback(cid:148) and the problem of reliable sequential
Shannon reliability suf(cid:2)ces for the classical information, this communication with noiseless feedback. In Section VI, these
unstable core requires anytime reliability for transport across results are further extended to the continuous time setting.
a noisy channel.7 As also discussed in this paper, anytime Finally, Section VII justi(cid:2)es why the problem of stabilization
reliability is a sense of reliable transmission that lies between ofanunstablelinearcontrolsystemis(cid:147)universal(cid:148)inthesame
Shannon’s classical (cid:15) sense of reliable transmission and his sense that the Shannon formulation of reliable transmission
zero-error reliability (cid:0)[25]. In [23], we also review how the of messages over a noisy channel with (or without) feedback
sense of anytime reliability is linked to classical work on is universal. This is done by introducing a hierarchy of
sequential tree codes with bounded delay decoding.8 communication problems in which problems at a given level
are equivalent to each other in terms of which channels are
Thenewfeatureincontrolsystemsistheiressentialinterac-
good enough to solve them. Problems high in the hierarchy
tivity. The information to be communicated is not a message
arefundamentallymorechallengingthantheonesbelowthem
known in advance that is used by some completely separate
in terms of what they require from the noisy channel.
entity.Rather,itevolvesthroughtimeandisusedtocontrolthe
very process being encoded. This introduces two interesting
In Part II, the necessity and suf(cid:2)ciency results are general-
issues. First, causality is strictly enforced. The encoder and
izedtothecaseofmultivariablecontrolsystemsonanunstable
controller must act in real time and so taking the limit of
eigenvalue by eigenvalue basis. The role of anytime capacity
large delays must be interpreted very carefully. Second, it
isplayedbyarateregioncorrespondingtoavectorofanytime
is unclear what the status of the controlled process is. If
reliabilities.Ifthereisnoexplicitchanneloutputfeedback,the
the controller succeeds in stabilizing the process, it is no
intrinsic delay of the control system’s input-output behavior
longerunstable.AsexploredinSectionII-D,apurelyexternal
plays an important role. It shows that two systems with the
non-interactive observer could treat the question of encoding
same unstable eigenvalues can still have potentially different
the controlled closed-loop system state using classical tools
channelrequirements.Theseresultsestablishthatininteractive
for the encoding and communication of a stationary ergodic
settings, a single (cid:147)application(cid:148) can fundamentally require
different senses of reliability for its data streams. No single
5The unstable open-loop processes discussed here are (cid:2)rst-order nonsta-
tionary autoregressive processes [21], of which an important special case is numbercanadequatelysummarizethechannelandanylayered
theWienerprocessconsideredbyBerger[22]. communicationarchitectureshouldallowapplicationstoadjust
6In[23],weshowhowtheclassicalpartoftheinformationdeterminesthe
reliabilities on bitstreams.
shape of the rate-distortion curve, while the unstable core is responsible for
ashiftofthiscurvealongtherateaxis.
7How to communicate such unstable processes over noisy channels had There are many results in this paper. In order not to burden
been an open problem since Berger had (cid:2)rst developed a source-coding the reader with repetitive details and unnecessarily lengthen
theorem for the Wiener process [24]. Berger had conjectured that it was this paper, we have adopted a discursive style in some of the
impossible to transport such processes over generic noisy channels with
proofs. The reader should not have any dif(cid:2)culty in (cid:2)lling in
asymptotically(cid:2)niteend-to-enddistortionusingtraditionalmeans.
8Reference[26]raisedthepossibilityofsuchaconnectionearlyon. the omitted details.
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 3
Wt(cid:0)1 Thisde(cid:2)nitionrequirestheprobabilityofalargestatevalue
PossibleChannelFeedback to be appropriately bounded. A looser sense of stability is
? (cid:27)
-’SSycsatleamr $- DOebssiegrnveedr givDene(cid:2)bnyit:ion 2.2: A closed-loop dynamic system with state
Xt O 1Step Xt is (cid:17)-stable if there exists a constant K s.t. E[Xt (cid:17)] K
j j (cid:20)
Ut(cid:0)1 6 Delay for all t(cid:21)0.
&PossibleCo%ntrolKnowledge ? 6
’Noisy $ In both de(cid:2)nitions, the bound is required to hold for all
Channel possible sequences of bounded disturbances W that satisfy
t
f g
thegivenbound(cid:10).Wedonotassumeanyspeci(cid:2)cprobability
1Step model governing the disturbances. Rather than having to
Delay &%
specify a speci(cid:2)c target for the tail probability f, holding the
6 Ut CDoenstirgonleledr(cid:27) ? (cid:17)-moment within bounds is a way of keeping large deviations
ControlSignals C rare. The larger (cid:17) is, the more strongly very large deviations
are penalized. The advantage of (cid:17)-stability is that it allows
constant factors to be ignored while making sharp asymptotic
Fig. 2. Control over a noisy communication channel. The unstable scalar
statements. Furthermore, Section III-C shows that for generic
systemispersistentlydisturbedbyWtandmustbekeptstableinclosed-loop
throughtheactionsofO;C. DMCs, no sense stronger than (cid:17)-stability is feasible.
The goal in this paper is to (cid:2)nd necessary and suf(cid:2)cient
conditions on the noisy channel for there to exist an observer
II. PROBLEMDEFINITIONANDBASICCHALLENGES and controller so that the closed loop system shown
O C
in Figure 2 is stable in the sense of de(cid:2)nitions 2.1 or 2.2.
Section II-A formally introduces the control problem of
Theproblemisconsideredunderdifferentinformationpatterns
stabilizing an unstable scalar linear system driven by both
correspondingtodifferentassumptionsaboutwhatinformation
a control signal and a bounded disturbance. In Section II-
is available at the observer . The controller is always
B, classical notions of capacity are reviewed along with how O
assumed to just have access to the entire past history10 of
to stabilize an unstable system with a (cid:2)nite rate noiseless
channel outputs.
channel. In Section II-C, it is shown by example that the
For discrete-time linear systems, the intrinsic rate of infor-
classicalconceptsareinadequatewhenitcomestoevaluatinga
mationproduction(inunitsofbitspertime)equalsthesumof
noisychannelforcontrolpurposes.Shannon’sregularcapacity
the logarithms (base 2) of the unstable eigenvalues [9]. In the
is too optimistic and zero-error capacity is too pessimistic.
scalar case studied here, this is just log (cid:21). This means that
Finally, Section II-D shows that the core issue of interactivity 2
it is generically11 impossible to stabilize the system in any
is different than merely requiring the encoders and decoders
reasonable sense if the feedback channel’s Shannon classical
to be delay-free.
capacity C <log (cid:21).
2
A. The control problem
B. Classical notions of channels and capacity
De(cid:2)nition 2.3: A discrete time channel is a probabilistic
Xt+1 =(cid:21)Xt+Ut+Wt; t 0 (1) system with an input. At every time step t, it takes an input
(cid:21)
a and produces an output b with probability12
wcporhonectrreeoslsfXps.rtto.gceWisssaaIRnd-v(cid:10)af.lWuTetdhgisstiabstoeuapnrdboociuesnssad.sesfduUmntgeodisiseto/adihIsRotul-drvbaawluniecthde pse(tqBu2tejnaAct1e;bat1(cid:0)1;1a)2;w:h:e:r;eatt.hIengneontaetriaoln,tthae2t1ciuBsrreshnotrcthhaannndelfoorutpthuet
j tj (cid:20) 2 is allowed to depend on all inputs so far as well as on past
certainty. For convenience, we also assume a known initial
outputs.
condition X =0.
0 The channel is memoryless if conditioned on a , B is
To make things interesting, consider (cid:21) > 1 so the open- t t
independent of any other random variable in the system that
loop system is exponentially unstable. The distributed nature
occurs at time t or earlier. All that needs to be speci(cid:2)ed is
of the problem (shown in Figure 2) comes from having
p(B a ).
t t
a noisy communication channel in the feedback path. The j
observer/encoder system observes Xt and generates inputs The maximum rate achievable for a given sense of reliable
O
at to the channel. It may or may not have access to the communication is called the associated capacity. Shannon’s
control signals U or past channel outputs B as well. The
t t 1
decoder/controller9 system observeschannel(cid:0)outputsB and 10InSectionIII-C.3,itisshownthatanythinglessthanthatcannotwork
t
C ingeneral.
generates control signals Ut. Both O;C are allowed to have 11There are pathological cases where it is possible to stabilize a system
unbounded memory and to be nonlinear in general. with less rate. These occur when the driving disturbance is particularly
De(cid:2)nition 2.1: A closed-loop dynamic system with state structuredinsteadofjustbeingunknownbutbounded.Anexampleiswhen
the disturbance only takes on values (cid:6)1 while (cid:21)=4. Clearly only one bit
X is f-stable if (X >m)<f(m) for all t 0.
t P j tj (cid:21) perunittimeisrequiredeventhoughlog2(cid:21)=2.
12ThisisaprobabilitymassfunctioninthecaseofdiscretealphabetsB,but
9Because the decoder and controller are both on the same side of the ismoregenerallyanappropriateprobabilitymeasureovertheoutputalphabet
communicationchannel,theycanbelumpedtogetherintoasinglebox. B.
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 4
classical reliability requires that after a suitably large end-to- diminished since the computational burden is not superlinear
end delay13 n that the average probability of error on each bit in the block-length.
is below a speci(cid:2)ed (cid:15). Shannon classical capacity C can also For memoryless channels, the presence or absence of feed-
be calculated in the case of memoryless channels by solving back does not alter the classical Shannon capacity [1]. More
an optimization problem: surprisingly, for symmetric DMCs, the (cid:2)xed block coding
reliability functions also do not change with feedback, at
C = sup I(A;B)
least in the high rate regime [29]. From a control perspective,
(A)
P this is the (cid:2)rst indication that neither Shannon’s capacity nor
where the maximization is over the input probability distribu-
block-codingreliabilityfunctionsaretheperfect(cid:2)tforcontrol
tion and I(A;B) represents the mutual information through
applications.
the channel [1]. This is referred to as a single letter character-
ization of channel capacity for memoryless channels. Similar C. Counterexampleshowingclassicalconceptsareinadequate
formulae exist using limits in cases of channels with memory.
We use erasure channels to construct a counterexample
Thereisanothersenseofreliabilityanditsassociatedcapacity
showing the inadequacy of the Shannon classical capacity in
C called zero-error capacity which requires the probability
0 characterizing channels for control. While both erasure and
of error to be exactly zero with suf(cid:2)ciently large n. It does
AWGNchannelsareeasytodealwith,itturnsoutthatAWGN
not have a simple single-letter characterization [25].
channels can not be used for a counterexample since they
Example 2.1: Consider a system (1) with (cid:10) = 1 and (cid:21) =
can be treated in the classical LQG framework [15]. The
3. Suppose that the memoryless communication channel is a
2 deeper reason for why AWGN channels do not provide a
noiseless one bit channel. So = = 0;1 and p(B =
A B f g t counterexample is given in Section III-C.4.
1a = 1) = p(B = 0a = 0) = 1 while p(B = 1a =
j t t j t t j t 1) Erasure channels: The packet erasure channel models
0)=p(B =0a =1)=0. This channel has C =C =1>
t j t 0 situationswhereerrorscanbereliablydetectedatthereceiver.
log 3.
2 2 In the model, sometimes the packet being sent does not make
Use a memoryless observer
itthroughwithprobability(cid:14),butotherwiseitmakesitthrough
0 if x 0 correctly. Explicitly:
O(x)= 1 if x>(cid:20)0 De(cid:2)nition 2.4: TheL-bitpacketerasurechannelisamem-
(cid:26) oryless channel with = 0;1 L, = 0;1 L and
and memoryless controller A f g B f g [f;g
p(xx)=1 (cid:14) while p( x)=(cid:14).
j (cid:0) ;j
+3 if B =0
(B)= 2 It is well known that the Shannon capacity of the packet
C 3 if B =1
(cid:26) (cid:0)2 erasure channel is (1 (cid:14))L bits per channel use regardless
(cid:0)
Assume that the closed loop system state is within the of whether the encoder has feedback or not [1]. Furthermore,
interval [ 2;+2]. If it is positive, then it is in the interval because a long string of erasures is always possible, the zero-
(cid:0)
[0;+2]. At the next time, 3X +W would be in the interval errorcapacityC ofthischannelis0.Therearealsovariable-
2 0
[ 1;7]. The applied control of 3 shifts the state back to length packet erasure channels where the packet-length is
(cid:0)2 2 (cid:0)2
within the interval [ 2;+2]. The same argument holds by something the encoder can choose. See [30] for a discussion
(cid:0)
symmetryonthenegativeside.Sinceitstartsat0,byinduction of such channels.
it will stay within [ 2;+2] forever. As a consequence, the To construct a simple counterexample, consider a further
(cid:0)
second moment will stay less than 4 for all time, and all the abstraction:
other moments will be similarly bounded. De(cid:2)nition 2.5: The real packet erasure channel has =
A
=IR and p(xx)=1 (cid:14) while p(0x)=(cid:14).
In addition to the Shannon and zero-error senses of reli- B j (cid:0) j
ability, information theory has various reliability functions. ThismodelhasalsobeenexploredinthecontextofKalman
Suchreliabilityfunctions(orerrorexponents)aretraditionally (cid:2)lteringwithlossyobservations[31],[32].Ithasin(cid:2)niteclas-
considered an internal matter for channel coding and were sical capacity since a single real number can carry arbitrarily
viewed as mathematically tractable proxies for the issue of many bits within its binary expansion, while the zero-error
implementation complexity [1]. Reliability functions study capacity remains 0.
how fast the probability of error goes to zero as the relevant 2) The inadequacy of Shannon capacity: Consider the
system parameter is increased. Thus, the reliability functions problem from example 2.1, except over the real erasure
for block-codes are given in terms of the block length, channel instead of the one bit noiseless channel. The goal
reliability functions for convolutional codes in terms of the is for the second moment to be bounded ((cid:17) = 2) and recall
constraint length[27], and reliability functions for variable- that (cid:21) = 3. Let (cid:14) = 1 so that there is a 50% chance of any
2 2
length codes in terms of the expected block length [28]. With real number being erased. Assume the bounded disturbance
the rise of sparse code constructions and iterative decoding, Wt, assume that it is zero-mean and iid with variance (cid:27)2. By
the prominence of error exponents in channel coding has assuminganexplicitprobabilitymodelforthedisturbance,the
problem is only made easier as compared to the arbitrarily-
13Traditionally, thecommunityhas usedblock-lengthforablockcode as varying but bounded model introduced earlier.
thefundamentalquantityratherthandelay.Itiseasytoseethatdoingencoding
In this case, the optimal control is obvious (cid:151) set a =
and decoding in blocks of size n corresponds to a delay of between n and t
2nontheindividualbitsbeingcommunicated. Xt as the channel input and use Ut = (cid:21)Bt as the control.
(cid:0)
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 5
Wt(cid:0)1
At (cid:2)rst glance, it certainly appears that the communication
? situations are symmetric. If anything, the internal observer is
?
-#SSycastleamr -DOebssiegrnveedr EPnacsosidveer better off since it also has access to the control signals while
Xt O Ep the passive observer is denied access to them.
Suppose that the closed-loop process (1) had already been
Ut(cid:0)1 "Controlkn!owledgeato6bserver ? ? stabilizedbytheobserverandcontrollersystemof2.1,sothat
#CNhaonisnyel #CNhaonisnyel nthoeissyeccohnandnmelofmaceinntgEth[eXpt2a]ss(cid:20)iveKenfcoordaelrlits.thSeuprpeaolse1-tehraatsuthree
2
channel of the previous section. It is interesting to consider
1Step
(cid:3)Delay (cid:0) ? how well the passive observer does at estimating this process.
"!"!
6 Ut CDoenstirgonleledr(cid:27) EPstaismsiavteor The optimal encoding rule is clear, set at = Xt. It is
(cid:2) (cid:1) ControlSignals C Dp certainly feasible to use Xt = Bt itself as the estimator for
the process. This passive observation system clearly achieves
?
Xt Eis[(1X. Tt(cid:0)heXcta)u2s]a(cid:20)l dKe2co<diKngbsriunlceeisthaebplerotboaabcilhitiyevoefano(cid:2)nn-ieteraesnudre-
2
b to-enbd squared error distortion over this noisy channel in a
Fig.3. Thecontrolsystemwithanadditionalpassivejointsource-channel
encoder Ep watching the closed loop state Xt and communicating it to a causal and memoryless way.
passiveestimatorDp.ThecontrollerCimplicitlyneedsagoodcausalestimate Thisexamplemakesitclearthatthechallengehereisarising
forXt andthepassiveestimatorDp explicitlyneedsthesamething.Which from interactivity, not simply being forced to be delay-free.
requiresthebetterchannel?
The passive external encoder and decoder do not have to face
the unstable nature of the source while the internal observer
and controller do. An error made while estimating X by the
With every successful reception, the system state is reset to t
the initial condition of zero. For an arbitrary time t, the time passive decoder has no consequence for the next state Xt+1
sinceitwaslastresetisdistributedlikeageometric-1 random while a similar error by the controller does. b
2
variable. Thus the second moment is:
III. ANYTIMECAPACITYANDITSNECESSITY
t i Anytime reliability is introduced and related to classical
1 1 3
E[Xt+1 2] > ( )iE[( ( )jWt j)2] notions of reliability in [23]. Here, the focus is on the maxi-
j j 2 2 2 (cid:0)
i=0 j=0 mumrateachievableforagivensenseofreliabilityratherthan
X X
t i i the maximum reliability possible at a given rate. The two are
1 1 3
= ( )i ( )j+kE[Wt jWt k] of course related since fundamentally there is an underlying
2 2 2 (cid:0) (cid:0)
i=0 j=0k=0 region of feasible rate/reliability pairs.
X XX
t 1 i 9 Since the open-loop system state has the potential to grow
= ( )i+1 ( )j(cid:27)2 exponentially, the controller’s knowledge of the past must
2 4
i=0 j=0 become certain at a fast rate in order to prevent a bad
X X
4(cid:27)2 t 9 1 decision made in the past from continuing to corrupt the
= ( )i+1 ( )i+1
future.Whenviewedinthecontextofreliablycommunicating
5 8 (cid:0) 2
Xi=0(cid:18) (cid:19) bits from an encoder to a decoder, this suggests that the
This diverges as t since 9 >1. estimates of the bits at the decoder must become increasingly
!1 8
Notice that the root of the problem is that (3)2(1) > 1. reliable with time. The sense of anytime reliability is made
2 2
Intuitively, the system is exploding faster than the noisy precise in Section III-A. Section III-B then establishes the
channel is able to give reliability. This causes the second key result of this paper relating the problem of stabilization
moment to diverge. In contrast, the (cid:2)rst moment E[X ] is to the reliable communication of messages in the anytime
t
j j
bounded for all t since (3)(1)<1. sense. Finally, some consequences of this connection are
2 2
The adequacy of the channel depends on which moment is studied in Section III-C. Among these consequences is a
required to be bounded. Thus no single-number characteriza- sequential generalization of the Schalkwijk/Kailath scheme
tionlikeclassicalcapacitycangivethe(cid:2)gure-of-meritneeded for communication over an AWGN channel that achieves a
to evaluate a channel for control applications. doubly-exponential convergence to zero of the probability of
bit error universally over all delays simultaneously.
D. Non-interactive observation of a closed-loop process
A. Anytime reliability and capacity
Consider the system shown in Figure 3. In this, there is an
additional passive joint source-channel encoder p watching The entire message is not assumed to be known ahead of
E
the closed loop state X and communicating it to a passive time. Rather, it is made available gradually as time evolves.
t
estimator p through a second independent noisy channel. For simplicity of notation, let M be the R bit message that
i
D
Both the passive and internal observers have access to the thechannelencoder getsattimei.Atthechannel decoder,no
same plant state and we can also require the passive encoder target delay is assumed (cid:151) i.e. the channel decoder does not
and decoder to be causal (cid:151) no end-to-end delay is permitted. necessarily know when the message i will be needed by the
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 6
(cid:27) ChannelFeedback
Communicationsystemsthatachieveanytimereliabilityare
Anytime
calledanytimecodesandsimilarlyforuniformanytimecodes.
Inputmessages - Channel
Mt Encoder
Wecouldalternativelyhaveboundedtheprobabilityoferror
E
by2(cid:0)(cid:11)(d(cid:0)log2K)andinterpretedlog2K astheminimumdelay
6 imposed by the communication system.
De(cid:2)nition 3.2: The(cid:11)-anytimecapacityCany((cid:11))ofachan-
? nel is the least upper bound of the rates R (in bits) at which
SharedRandomness ’Noisy $ the channel can be used to construct a rate R communication
Channel system that achieves uniform anytime reliability (cid:11).
Feedback anytime capacity is used to refer to the anytime
&% capacitywhentheencoderhasaccesstonoiselessfeedbackof
?
the channel outputs with unit delay.
Anytime
CurrentEstim(cid:27)ates Channel (cid:27) ? The requirement for exponential decay in the probability of
M1t(t) Decoder error with delay is reminiscent of the block-coding reliability
D functionsE(R)ofachannelgivenin[1].Thereisonecrucial
c
difference. With standard error exponents, both the encoder
and decoder vary with blocklength or delay n. Here, the
Fig. 4. The problem of communicating messages in an anytime fashion.
Both the encoder E and decoder D are causal maps and the decoder in encoding is required to be (cid:2)xed and the decoder in principle
principle provides updated estimates for all past messages. These estimates has to work at all delays since it must produce updated
mustconvergetothetruemessagevaluesappropriatelyrapidlywithincreasing estimates of the message M at all times t>i.
delay. i
This additional requirement is why it is called (cid:147)anytime(cid:148)
capacity. The decoding process can be queried for a given
bit at any time and the answer is required to be increasingly
application. A past message may even be needed more than
accuratethelongerwewait.Theanytimereliability(cid:11)speci(cid:2)es
once by the application. Consequently, the anytime decoder
produces estimates M (t) which are the best estimates for the exponential rate at which the quality of the answers
i
messageiattimetbasedonallthechanneloutputsreceivedso must improve. The anytime sense of reliable transmission
far.Iftheapplicationcisusingthepastmessageswithadelayd, lies between that represented by classical zero-error capacity
therelevantprobabilityoferrorisP(M1t(cid:0)d(t)6=M1t(cid:0)d).This dCe0la(yp)roanbdabcilliatsysicoaflecrarpoarcbiteycoCm(epsrozbearboilaittyaoflaerrgreorbbuetco(cid:2)mnietes
corresponds to an uncorrected error anywhere in the distant
past (ie on messages M1;M2;:::;Mct d) beyond d channel something small at a large but (cid:2)nite delay). It is clear that
uses ago. (cid:0) (cid:11);C0 Cany((cid:11)) C.
8 (cid:20) (cid:20)
Byusingarandomcodingargumentoverin(cid:2)nitetreecodes,
De(cid:2)nition 3.1: As illustrated in (cid:2)gure 4, a rate R commu-
it is possible to show the existence of anytime codes without
nication system over a noisy channel is an encoder and
E using feedback between the encoder and decoder for all rates
decoder pair such that:
D less than the Shannon capacity. This shows:
R-bit message M enters14 the encoder at discrete time i
i
(cid:15)
The encoder produces a channel input at integer times Cany(Er(R)) R
(cid:15) (cid:21)
based on all information that it has seen so far. For
where E (R) is Gallager’s random coding error exponent
encoders with access to feedback with delay 1+(cid:18), this r
also includes the past channel outputs B1t(cid:0)1(cid:0)(cid:18). cfeaelcdublaactkedpilnaybsasaen2eassnedntRialisrothlee rinatecoinntbroitls, [i3t3t]u,rn[2s3o].uStitnhcaet
The decoder produces updated channel estimates M (t)
i
(cid:15) we are interested in the anytime capacity with feedback. It is
for all i t based on all channel outputs observed till
(cid:20) interesting to note that in many cases for which the block-
time t. c
coding error exponents are not increased with feedback, the
A rate R sequential communication system achieves any-
anytime reliabilities are increased considerably [6].
time reliability (cid:11) if there exists a constant K such that:
P(M1i(t)6=M1i)(cid:20)K2(cid:0)(cid:11)(t(cid:0)i) (2) B. Necessity of anytime capacity
Anytime reliability and capacity are de(cid:2)ned in terms of
holds for every i;t. The probability is taken over the chan-
c
digital messages that must be reliably communicated from
nel noise, the R bit messages M , and all of the common
i
point to point. Stability is a notion involving the analog value
randomness available in the system.
of the state of a plant in interaction with a controller over a
If (2) holds for every possible realization of the messages
noisy feedback channel. At (cid:2)rst glance, these two problems
M, then the system is said to achieve uniform anytime
appear to have nothing in common except the noisy channel.
reliability (cid:11).
Even on that point there is a difference. The observer/encoder
14In what follows, messages are considered to be composed of bits for O in the control system may have no explicit access to the
simplicity of exposition. The i-th bit arrives at the encoder at time i and noisy output of the channel. It can appear to be using the
R
thusMi iscomposedofthebitsSbb(iiR(cid:0)c1)Rc+1. noisy channel without feedback. Despite this, it turns out that
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 7
ttohethreelnevoaisnyt cdhigaintnalelcwomithmnuoniiscealteiossncphraonbnleeml feiendvboalvcekscoacmciensgs SimPlualnatted Ut CoPnltaronltler(cid:27) Ch.annelFeedback
C
?
back to the message encoder.
Wt #Simulated
Theorem 3.3: For a given noisy channel and (cid:17) > 0, if Source
there exists an observer and controller for the unstable ? Xt
bscoaulnadresdysdterimvinthgatnoacisheiejvWeOstjE(cid:20)[jX(cid:10)2t,jt(cid:17)h]e<n tKhefcoChraanlnlesle’qsufeenecdebsacokf #SiSmoXu(cid:20)ulrtacteed "e -!+i? Xt-ObPslOearnvter (cid:3)1DeSltaeyp(cid:0)
anytime capacity Cany((cid:17)log2(cid:21)) (cid:21) log2(cid:21) bits per channel 6
use. "! ? (cid:2) (cid:1)
#Noisy
The proof of this spans the next few sections. Assume that Channel
there is an observer/controller pair ( ; ) that can (cid:17)-stabilize
O C
an unstable system with a particular (cid:21) and are robust to all
"!
btooucnodnesdtrudcitstaurrbaatnecRes<oflosigz2e(cid:21)(cid:10)a.nTyhtiemegoeanlciosdetor aunsed tdheecopdaeirr Xt(cid:27) #SiSmouulractee(cid:27)d Ut CoPnltCaronltler.(cid:27) ?
for the channel with noiseless feedback, thereby reducing15
e
the problem of anytime communication to a problem of
"!
stabilization.
Fig.5. Theconstructionofafeedbackanytimecodefromacontrolsystem.
The heart of the construction is illustrated in (cid:2)gure 5. The The messages are used to generate the fWtg inputs which are causally
(cid:147)black-box(cid:148)observerandcontrollerarewrappedaroundasim- combined to generate fX(cid:20)tg within the encoder. The channel outputs are
used to generate control signals at both the encoder and decoder. Since the
ulated plant mimicking (1). Since the fUtg must be generated simulatedplantisstable,(cid:0)XandX(cid:20)areclosetoeachother.Thepastmessage
by the black-box controller and the (cid:21) is prespeci(cid:2)ed, the bitsareestimatedfromtheX atthedecoder.
C
disturbances Wt mustbeusedtocarrythemessage.So,the e
f g
encoder must embed the messages M into an appropriate e
t
f g
sequence W , taking care to stay within the (cid:10) size limit.
t
f g
While both the observer and controller can be simulated 1) Encodingdataintothestate: Aslongasthebound(cid:10)is
at the encoder thanks to the noiseless channel output feed- satis(cid:2)ed,theencoderisfreetochooseanydisturbance17forthe
back, at the decoder only the channel outputs are available. simulatedplant.Thechoicewillbedeterminedbythedatarate
Consequently, these channel outputs are connected to a copy R and the speci(cid:2)c messages to be sent. Rather than working
of the black-box controller , thereby giving access to the with general messages M , consider a bitstream S with bit i
i i
controls U at the decoderC. To extract the messages from becomingavailableattime i.Everythinggeneralizesnaturally
f tg R
these control signals, they are (cid:2)rst causally preprocessed to non-binary alphabets for the messages, but the notation is
throughasimulatedcopyoftheunstableplant,exceptwithno cleaner in the binary case with S = 1.
i
(cid:6)
disturbance input. All past messages are then estimated from X(cid:20) is the part of X driven only by the W .
t t t
the current state of this simulated plant. f g
The key is to think of the simulated plant state as the sum
of the states of two different unstable LTI systems. The (cid:2)rst,
with state denoted X , is driven entirely by the controls and
t
starts in state 0. X(cid:20) = (cid:21)X(cid:20) +W
t t 1 t 1
eXt+1 =(cid:21)Xt+Ut (3) t 1 (cid:0) (cid:0)
(cid:0)
X is available at both the decoder and the encoder due to = (cid:21)iWt 1 i
e e (cid:0) (cid:0)
the presence of noiseless feedback.16 The other, with state i=0
X
deenoted X(cid:20)t, is driven entirely by a simulated driving noise = (cid:21)t 1t(cid:0)1(cid:21) jW
that is generated from the data stream to be communicated. (cid:0) (cid:0) j
j=0
X(cid:20) =(cid:21)X(cid:20) +W (4) X
t+1 t t
The sum X =(X +X(cid:20) ) behaves exactly like it was coming
t t t
from (1) and is fed to the observer which uses it to generate
This looks like the representation of a fractional number in
inputs for the noiesy channel. base (cid:21) which is then multiplied by (cid:21)t 1. This is exploited in
(cid:0)
The fact that the original observer/controller pair stabilized
the encoding by choosing the bounded disturbance sequence
the original system implies that X = X(cid:20) ( X ) is small
t t
and hence X stays close to X(cid:20)j . j j (cid:0) (cid:0) j
t t
(cid:0)
e
15In traditional rate-distortion theory, this (cid:147)necessity(cid:148) direction is shown
e
by going through the mutual information characterizations of both the rate- 17In [23], a similar strategy is followed assuming a speci(cid:2)c density
distortionfunctionandthechannelcapacityfunction.Inthecaseofstabiliza- for the iid disturbance Wt. In that context, it is important to choose a
tion,mutualinformationisnotdiscriminatingenoughandsothereductionof simulated disturbance sequence that behaves stochastically like Wt. This is
anytimereliablecommunicationtostabilizationmustbedonedirectly. accomplishedbyusingcommonrandomnesssharedbetweentheencoderand
16If the controller is randomized, then the randomness is required to be decoder to dither the kind of disturbances produced here into ones with the
commonandsharedbetweentheencoderanddecoder. desireddensity.
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 8
so that:18 2) Extracting data bits from the state estimate:
Rt Lemma 3.1: Givenachannel withaccesstonoiselessfeed-
b c
X(cid:20)t =(cid:13)(cid:21)t (2+(cid:15)1)(cid:0)kSk (5) back, for any rate R < log2(cid:21), it is possible to encode bits
k=0 intothesimulatedscalarplantsothattheuncontrolledprocess
X
behaves like (5) by using disturbances given in (6) and the
where S is the k-th bit19 of data that the anytime encoder
k formulas (7) and (8). At the output end of the noisy channel,
has to send and Rt is just the total number of bits that are
available by timebt. (cid:13)c;(cid:15) are constants to be speci(cid:2)ed. it is possible to extract estimates Si(t) for the i-th bit sent for
1 which the error event
To see that (5) is always possible to achieve by appropriate
b (cid:13)(cid:15)
choice of W, use induction. (5) clearly holds for t=0. Now ! i j;Si(t)=Si(t) ! Xt (cid:21)t(cid:0)Rj 1
f j9 (cid:20) 6 g(cid:18)f jj j(cid:21) 1+(cid:15) g
assume that it holds for time t and consider time t+1: (cid:18) 1(cid:19)
(9)
b
X(cid:20) = (cid:21)X(cid:20) +W and thus:
t+1 t t
(cid:13)(cid:15)
= (cid:13)(cid:21)t+1(bRtc(2+(cid:15)1)(cid:0)kSk)+Wt ProoPf:(SH1je(rte)!6=iSs1ju(ts)e)d(cid:20)toPd(ejnXottje(cid:21)m(cid:21)emt(cid:0)bRjer(cid:18)s 1of+th1(cid:15)e1(cid:19)un)derly(1in0g)
kX=0 samplebspace.20
So setting The decoder has X = X(cid:20) X which is close to X(cid:20)
t t t
(cid:0) (cid:0)
since X is small. To see how to extract bits from X , (cid:2)rst
t t
Wt =(cid:13)(cid:21)t+1 bR(t+1)c(2+(cid:15)1)(cid:0)kSk (6) conSstiadretirnghowwitthothreecu(cid:2)rrssitevebliyt,enxottriaccettthhaotstehebistsetfroofmal(cid:0)Xl(cid:20)pteo.ssible
k=bXRtc+1 X(cid:20)t thathaveS0 =+1isseparatedfromthesetofallpossible
gives the desired result. Manipulate (6) to get Wt = X(cid:20)t that have S0 =(cid:0)1 by a gap of
Rt Rt
R(t+1) Rt b c b c
b c(cid:0)b c (cid:13)(cid:21)t (1 (2+(cid:15) ) k) ( 1+ (2+(cid:15) ) k)
(cid:13)(cid:21)t+1(2+(cid:15)1)(cid:0)bRtc (2+(cid:15)1)(cid:0)jS Rt +j 0 (cid:0) 1 (cid:0) (cid:0) (cid:0) 1 (cid:0) 1
b c k=1 k=1
j=1 X X
X @ A
= (cid:13)(cid:21)(cid:21)(2(cid:0)t+(1(cid:0)(cid:15)1R)Rlotgl(cid:0)2o(g(22b+R(cid:21)(cid:15)t1c))) bR(t+Xj1=)c1(cid:0)bRtc(2+(cid:15)1)(cid:0)jSbRtc+j => (cid:13)(cid:13)(cid:21)(cid:21)tt22((11(cid:0)kX1=11(2+) (cid:15)1)(cid:0)k)
To keep this bounded, choose (cid:0) 1+(cid:15)1
2(cid:15) (cid:13)
(cid:15)1 =2logR2(cid:21) (cid:0)2 (7) = (cid:21)t(cid:18)1+1(cid:15)1(cid:19)
which is strictly positive if R < log (cid:21). Applying that
2
substitution gives W =
t
j j
R(t+1) Rt
b c(cid:0)b c
(cid:13)(cid:21)(2+(cid:15) )Rt ( Rt ) (2+(cid:15) ) jS
1 (cid:0) b c 1 (cid:0) Rt +j
j b c j
j=1
X
< (cid:13)(cid:21)(2+(cid:15) ) Fig.6. Thedatabitsareusedtosequentiallyre(cid:2)neapointonaCantorset.
1
j j Its natural tree structure allows bits to be encoded sequentially. The Cantor
= (cid:13)(cid:21)1+R1 setalsohas(cid:2)nitegapsbetweenallpointscorrespondingtobitsequencesthat
j j (cid:2)rstdifferinaparticularbitposition.Thesegapsallowtheuniformlyreliable
extractionofbitvaluesfromnoisyobservations.
So by choosing
(cid:13) = (cid:10) (8) Noticethatthisworst-casegap21 isapositivenumberthatis
2(cid:21)1+R1 growing exponentially in t. If the (cid:2)rst i 1 bits are the same,
the simulated disturbance is guaranteed to stay within the thenbothsidescanbescaledby(2+(cid:15)1)i(cid:0)=(cid:21)Ri togetthesame
expressions above and so by induction, it quickly follows that
speci(cid:2)ed bounds.
the minimum gap between the encoded state corresponding to
two sequences of bits that (cid:2)rst differ in bit position i is given
18Foraroughunderstanding,ignorethe(cid:15)1 andsupposethatthemessage
wereencodedinbinary.ItisintuitivethatanygoodestimateoftheX(cid:20)t state by gapi(t)=
is going to agree with X(cid:20)t in all the high order bits. Since the system is
ounn.stSabolen,oabllittheerroenrccooduelddpbeitrssiesvtefnotrutaolloylboencgoamnedhsitgilhl-koeredperthbeitsesatsimtiamteecglooeses inf X(cid:20)t(S) X(cid:20)t(S(cid:22)) > (cid:21)t(cid:0)Ri 12+(cid:13)(cid:15)(cid:15)11 if i(cid:20)bRtc
to X(cid:20)t. The (cid:15)1 in the encoding is a technical device to make this reasoning S(cid:22):S(cid:22)i6=Sij (cid:0) j ( 0 (cid:16) (cid:17) otherwise
hold uniformly for all bit strings, rather than merely (cid:147)typical(cid:148) ones. This is (11)
importantsinceweareaimingforexponentiallysmallboundsandsocannot
neglectrareevents. 20If the bits to be sent are deterministic, this is the sample space giving
19For the next section, it is convenient to have the disturbances balanced channelnoiserealizations.
around zero and so we choose to represent the bit Si as +1 or (cid:0)1 rather 21Thetypicalgapislargerandsotheprobabilityoferrorisactuallylower
thantheusual1or0. thanthisboundsaysitis.
IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.??,NO.??,MONTH??2005 9
Because the gaps are all positive, (11) shows that it is always Theorem 3.5: For a given noisy channel and decreasing
possible to perfectly extract the data bitsfrom X(cid:20) by using an function f(m), if there exists an observer and controller
t
O
iterative procedure.22 To extract bit information from an input fortheunstablescalarsystemthatachieves (X >m)<
t
C P j j
I : f(m) for all sequences of bounded driving noise W (cid:10),
t j tj (cid:20) 2
1) Initialize threshold T0 =0 and counter i=0. then Cg-any(g) (cid:21) log2(cid:21) for the noisy channel considered
2) Compare input I to T . If I T , set S (t) = +1. If withtheencoderhavingaccesstonoiselessfeedbackandg(d)
t i t i i
I <T , set S (t)= 1. (cid:21) having the form g(d)=f(K(cid:21)d) for some constant K.
t i i
(cid:0)
3) Increment counter i and update threbshold Ti = Proof: For any rate R<log2(cid:21),
4) G(cid:13)(cid:21)otto sikt(cid:0)e=p10(22ab+s l(cid:15)o1n)(cid:0)gkaSski Rt (Si(t)=Si(t)) (X (cid:21)t(cid:0)Ri(cid:13)(cid:15)1)
P (cid:20)b c P 1 6 1 (cid:20) P j tj(cid:21) 1+(cid:15)
Since the gaps given bby (11) are always positive, the 1
(cid:13)(cid:15)
procedure works perfectly if applied to input It = X(cid:20)t. At b = f(1+1(cid:15) (cid:21)t(cid:0)Ri)
the decoder, apply the procedure to I = X instead. 1
t t
Withthis,(9)iseasytoverifybylooking(cid:0)atthecomplemen- Since the delay d=t i, the theorem is proved. (cid:3)
tary event ! X < (cid:21)t(cid:0)Rj(cid:13)(cid:15)1 . The bounde(11) thus implies (cid:0) R
f jj tj 1+(cid:15)1 g
that we are less than halfway across the minimum gap for
bit j at time t. Consequently, there is no error in the step 2 C. Implications
comparison of the procedure at iterations i j. (cid:3)
(cid:20) Atthispoint,itisinterestingtoconsiderafewimplications
of Theorem 3.5.
3) Probability of error for bounded moment and other
1) Weaker senses of stability than (cid:17)-moment: There are
senses of stability: Proof of Theorem 3.3: Using Markov’s
senses of stability weaker than specifying a speci(cid:2)c (cid:17)-th
inequality:
moment or a speci(cid:2)c tail decay target f(m). An example
P(jXtj>m) = PE([jXXttj(cid:17)(cid:17)]m>(cid:0)m(cid:17)(cid:17)) iusnigfoivremnlybfyortahleltr.eTquhiirsecmaennbteliemxpmlo!re1dPby(jtXaktijng>thmel)im=ito0f
(cid:20) j j
< Km (cid:17) Cany((cid:11)) as (cid:11) 0. We have shown elsewhere[33], [23] that:
(cid:0) #
Combining with Lemma 3.1, gives: limCany((cid:11))=C
(cid:11) 0
(cid:13)(cid:15) #
P(S1i(t)6=S1i(t)) (cid:20) P(jXtj(cid:21)(cid:21)t(cid:0)Ri 1+1(cid:15) ) where C is the Shannon classical capacity. This holds for all
(cid:18) 1(cid:19) discrete memoryless channels since the (cid:11)-anytime reliability
1 1
b < K( + )(cid:17)(cid:21)(cid:0)(cid:17)(t(cid:0)Ri) goes to zero at Shannon capacity but is > 0 for all lower
(cid:13) (cid:13)(cid:15)
1 rates even without feedback being available at the encoder.
1 1
= (K( + )(cid:17))2(cid:0)((cid:17)log2(cid:21))(t(cid:0)Ri) Thus, classical Shannon capacity is the natural candidate for
(cid:13) (cid:13)(cid:15)
1 the relevant (cid:2)gure of merit.
Since t i represents the delay between the time that bit i To see why Shannon capacity can not be beaten, it is
(cid:0) R
was ready to be sent and the decoding time, the theorem is usefultoconsideranevenmorelaxsenseofstability.Suppose
proved. (cid:3) the requirement were only that limm (Xt > m) =
10 5 > 0 uniformly for all t. This i!m1poPsesj thej constraint
(cid:0)
All that was needed from the bounded moment sense of that the probability of a large state stays below 10 5 for all
(cid:0)
stability was some bound on the probability that X took on time. Theorem 3.5 would thus only requires the probability
t
large values. Thus, the proof above immediately generalizes of decoding error to be less than 10 5. However, Wolfowitz’
(cid:0)
to other senses of stochastic stability if we suitably generalize strong converse to the coding theorem[1] implies that since
thesenseofanytimecapacitytoallowforotherboundsonthe the block-length in this case is effectively going to in(cid:2)nity,
probability of error with delay. the Shannon capacity of the noisy channel still must satisfy
De(cid:2)nition 3.4: A rate R communication system achieves C log (cid:21). Adding a (cid:2)nite tolerance for unboundedly large
(cid:21) 2
g anytime reliability given by a function g(d) if statesdoesnotgetaroundtheneedtobeabletocommunicate
(cid:0)
log (cid:21) bits reliably.
P(M1t(cid:0)d(t)6=M1t(cid:0)d(t))<g(d) 22) Stronger senses of stability than (cid:17)-moment: Having f
g(d) is assumed to be 1 for all negative values of d. decreaseonlyasapowerlawmightnotbesuitableforcertain
c
The g anytime capacity Cg-any(g) of a noisy channel is applications. Unfortunately, this is all that can be hoped for
(cid:0)
the least upper bound of the rates R at which the channel can in generic situations. Consider a DMC with no zero entries
be used to construct a sequential communication system that in its transition matrix. De(cid:2)ne (cid:26) = min p(i;j). For such a
i;j
achieves g anytime reliability given by the function g(d). channel,withorwithoutfeedback,theprobabilityoferrorafter
(cid:0)
d time steps is lower bounded by (cid:26)d since that lower bounds
Noticethatfor(cid:11)-anytimecapacity,g(d)=K2 (cid:11)dforsome
(cid:0) the probability of all channel output sequences of length d.
K.
This implies that the probability of error can drop no more
22ThisisaminortwistontheprocedurefollowedbyserialA/Dconverters. than exponentially in d for such DMCs. Tighter upper-bounds