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1356 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 3, MARCH 2017

Anytime Capacity of a Class of Markov ChannelsPaolo Minero, Member, IEEE , and Massimo Franceschetti, Senior Member, IEEE

Abstract—Several new expressions for the anytime ca-pacity of Sahai and Mitter are presented for a time-varyingrate-limited channel with noiseless output feedback. Thesefollow from a novel parametric characterization obtained inthe case of Markov time-varying rate channels, and includean explicit formula for the r-bit Markov erasure channel, aswell as formulas for memoryless rate processes includingBinomial, Poisson, and Geometric distributions. Besidethe memoryless erasure channel and the additive whiteGaussian noise channel with input power constraint, theseare the only cases where the anytime capacity has beencomputed. At the basis of these results is the study of thethreshold function for m-th moment stabilization of a scalarlinear system controlled over a Markov time-varying digitalfeedback channel that depends on m and on the channel’sparameters. This threshold is shown to be a continuous andstrictly decreasing function of m and to have as extremevalues the Shannon capacity and the zero-error capacity asm tends to zero and infinity, respectively. Its operationalinterpretation is that of achievable communication rate,subject to a reliability constraint.

Index Terms—Anytime capacity, entropy power inequal-ity, Markov channels, Markov jump linear systems, quan-tized control.

I. INTRODUCTION

W E consider the problem of moment stabilization of adynamical system where the estimated state is trans-

mitted for control over a time-varying, rate-limited communi-cation channel. This set-up has been studied extensively in thecontext of networked control systems and discussed in severalspecial issue journals dedicated to the topic [1]–[3]. Recently,it gained renewed attention due to its relevance for the designof cyberphysical systems [4]. A tutorial review with extensivereferences appears in [5], see also [6].

The notion of Shannon capacity is in general not sufficient tocharacterize the trade-off between the entropy rate productionof the system, expressed by the growth of the state spacespanned in open loop, and the communication rate required forits stabilization. A large Shannon capacity is useless for stabi-

Manuscript received October 27, 2015; revised May 20, 2016;accepted June 13, 2016. Date of publication June 27, 2016; date ofcurrent version February 24, 2017. This work was partially supportedby NSF award CPS 1446891. Recommended by Associate EditorS. Yuksel.

P. Minero is with the Department of Electrical Engineering, Universityof Notre Dame, Notre Dame, IN 46556 USA and also with QualcommInc., San Diego, CA 92121 USA (e-mail: [email protected]).

M. Franceschetti is with the Department of Electrical and ComputerEngineering, University of California at San Diego, La Jolla, CA 92093-0407 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2016.2585354

Fig. 1. Feedback loop model. The estimated state is quantized, en-coded, and sent to a decoder over a digital channel of state Rk thatevolves according to a Markov process of discrete time step k.

lization if it cannot be used in time for control. For the controlsignal to be effective, it must be appropriate to the current stateof the system. Since decoding the wrong codeword impliesapplying a wrong signal and driving the system away fromstability, applying an effective control signal depends on thehistory of whether previous codewords were decoded correctlyor not. In essence, the stabilization problem is an example ofinteractive communication [7], where two-way communicationoccurs through the feedback loop between the plant and thecontroller. For this reason, usage of alternative capacity notionswith stronger reliability constraints than a vanishing probabilityof error has been proposed, including the zero-error capacity[8], originally introduced by Shannon [9], and the anytimecapacity of Sahai and Mitter [10], [11]–[14].

Within this general framework, we focus on the m-th mo-ment stabilization of an unstable scalar system whose stateis communicated over a rate-limited channel capable of sup-porting Rk bits at each time step k and evolving randomly ina Markovian fashion, see Fig. 1. The rate process is knowncausally to both encoder and decoder. A motivating example isgiven by sensors and actuators communicating over a channelwhose quality varies over time due to a random fading process.For digital communication, this reflects in a time variation ofthe rate supported by the channel. If the channel variations areslow enough, transmitter and receiver can estimate the qualityof the link by sending a training sequence to acquire channelstate information, and adapt their communication scheme to thechannel’s condition.

Many variations of this “bit-pipe” model have been studied inthe literature [15]–[35], including the case of fixed rate channel;the erasure channel where the rate process can take value zero;and the packet loss channel, where the rate process can oscillaterandomly between the two values of zero and infinity, allowinga real number with infinite precision to be transported across thechannel in one time step. Connections between the rate limitedand the packet loss channel have been pointed out in [28] and[29], showing that results for the latter model can be recoveredby appropriate limiting arguments. Our Markovian channelmodel includes all of these models as special cases and our the-orems recover the corresponding previous stabilization results.

Beside the bit-pipe model, stabilization over the additivewhite Gaussian channel has also been considered in [36]–[40]

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MINERO AND FRANCESCHETTI: ANYTIME CAPACITY OF A CLASS OF MARKOV CHANNELS 1357

and in this case the Shannon capacity has been shown to besufficient to express the rate needed for stabilization. Exten-sions to the additive colored Gaussian channel [41] show thatthe maximum “tolerable instability”—expressed by the sum ofthe logarithms of the unstable eigenvalues of the system thatcan be stabilized by a linear controller with a given power con-straint over a stationary Gaussian channel—corresponds to theShannon feedback capacity [42], that assumes the presence of anoiseless feedback link between the output and the input of thechannel and that is subject to the same power constraint. Thissuggests a duality between the problems of control and com-munication in the presence of feedback, and indeed it has beenshown that efficient feedback communication schemes can beobtained by solving a corresponding control problem [43], [44].

The novel contribution of the present paper is in the contextof the bit-pipe model, and is in the introduction of a stabil-ity threshold function of the channel’s parameters and of themoment stability number m that converges to the Shannoncapacity for m → 0, to the zero-error capacity for m → ∞,and it provides a parametric characterization of the anytimecapacity for the remaining values of m. This function yieldsa novel anytime capacity formula in the special case of ther-bit Markov erasure channel, as well as novel formulas formemoryless rate processes including Binomial, Poisson, andGeometric distributions. It also provides a general strategy tocompute the anytime capacity of arbitrary memoryless channelswith given rate distributions. To prove our results, we alsorequire some extensions of the theory of Markov Jump LinearSystems (MJLS), that can be of independent value.

Coding and decoding schemes for our channel model are notan issue, since no error protection or redundancy is needed. Thisis always the case for erasure channels and limited rate channelswith causal feedback and no noise, and it represents the maindifficulty in the extension of our results to noisy channels besideerasures, and to channels with partial channel state information.On the other hand, we do not see any difficulty to extend resultsto vector systems by exploiting known bit-allocation techniquesdescribed in [28] and [29]. Additional extensions consideringdisturbances of unbounded support can also be obtained usingstandard adaptive schemes [21], [28], [29]. We refrain frompresenting these additional results to keep the length of thepaper under control, and also because they would not introduceany new methods.

The rest of the paper is organized as follows. Some pre-liminary results on Markov Jump Linear Systems, necessaryfor our derivations are presented in Section II. Section IIIdescribes the system and channel model and introduces thestability threshold function, illustrating some of its properties.Section IV describes relationships with the anytime capacity,and provides some representative examples. Section V providesthe formula for the anytime capacity of the Markov r-bit erasurechannel. Section VI considers the case of memoryless bit-pipechannels of arbitrary rate distribution.

Throughout the paper, the following notation is used. Log-arithms are assumed to be in base two; random variables aredenoted with uppercase letters, while their realizations withlowercase letters; matrices are also denoted in uppercase letters,but we denote them with the special typeset A.

II. MARKOV JUMP LINEAR SYSTEMS

Consider the scalar non-homogeneous MJLS [45] withdynamics

zk+1 = akzk + wk (1)

where zk denotes the state, wk is the realization of a stochasticdisturbance Wk, and ak is the realization of {Ak}k≥0, a Markovrate process defined on a finite set of reals

A = {a1, . . . , an} (2)

and with one-step transition probability matrix P having entries

pij = P{Ak = aj |Ak−1 = ai} (3)

for every i, j ∈ {1, . . . , n}.We assume that the initial condition is a realization of a

stochastic variable Z0 that is mutually independent of thedisturbance process {Wk}k, and both have uniformly boundedm-th moment, i.e., supk E(|Wk|m) ≤ ∞ and E(|Z0|m) ≤ ∞.The system (1) is said to be weakly m-th moment stable if

supk

E (|Zk|m) < ∞. (4)

Let A ∈ Zn×n+ be a diagonal matrix with diagonal entries

|a1|m, . . . , |an|m, i.e.,

A = diag (|a1|m, . . . , |an|m) . (5)

The following theorem states the necessary and sufficient con-dition for m-th moment stability of the system (1) in terms ofthe the spectral radius ρ(PTA), that is the maximum among theabsolute values of the eigenvalues of PTA.

Theorem 1: For any m ∈ R+, the MJLS (1) is weakly m-th

moment stable if and only if

ρ(PTA) � 1. (6)

The symbol “�” indicates that while the necessary conditionholds with a weak inequality, the sufficient condition requires astrong inequality.

Proof: The sufficient condition is obtained by subsam-pling the original MJLS at a sampling rate of τ samples andthen showing the weak stability of the subsampled systemYk = Zkτ , k = 0, 1, . . .. The weak stability of {Yk}k is suf-ficient to ensure that (4) holds, because the m-th moment ofthe system in fact cannot grow unbounded in any finite timehorizon due the assumptions that |a1|, . . . , |an| are finite andthat supk E(|Wk|m) ≤ ∞.

By iterating (1) τ times, it can be seen that the subsampledprocess is a non-homogeneous MJLS with dynamics

yk+1 = bkyk + ck (7)

where for every k ≥ 0 we define Bk =∏(k+1)τ−1

j=kτ Ak as therandom expansion of the open-loop system during the k-thsampling interval and Ck =

∑τ−1j=0(

∏τ−1i=j Akτ+i)Wkτ+j as the

total disturbance entering the system in the k-th sampling


interval. Notice that supk E(|Ck|m) < ∞, since by assumptionthe disturbance process {Wk}k has bounded m-th moment.

By taking the m-th power in both sides of (7) and by applyingthe inequality (x+ y)m ≤ 2m(xm + ym), which holds for anym > 0, it follows that:

E (|Yk+1|m) ≤ 2mE (|Bk|m|Yk|m) + const (8)

where const represents a uniform bound on the m-th momentof {Ck}k. Let vk,i = E(|Yk+1|m1{Ak=ai}), i = 1, . . . , n. Byrepeatedly applying the law of total probability on both sides of(8), it can be easily seen that the vector vk = (vk,1, . . . , vk,n)

T

satisfies the following component-wise vector inequality:

vk+1 ≤ 2m(PTA)τvk + c (9)

where c is again a constant that only depends on the statisticsof the disturbance process. If (6) holds we can choose the sub-sampling rate τ large enough to ensure that ρ(2m(PTA)

τ) < 1

and, thus, that the above recursion remains bounded. SinceE(|Yk|m) =

∑ni=1 vk,i it follows that ρ(PTA) < 1 is sufficient

to ensure that supk E(|Yk|m) < ∞ as desired.The proof of the necessary condition relies on the fact that

the non-homogeneous MJLS {Zk}k is weakly stable only ifthe homogeneous MJLS obtained by setting wk = 0 in (1) isweakly stable. Therefore, we focus on the homogeneous settingwherein wk = 0 for all k and let vk,i = E(|Zk+1|m1{Ak=ai})for i = 1, . . . , n. By (1) and the law of total probability, thevector vk = (vk,1, . . . , vk,n)

T evolves over time according tothe linear system

vk+1 = PTAvk. (10)

Since E(|Zk|m) =∑n

i=1 vk,i, we conclude that (4) can holdonly if the above linear system is stable and therefore only ifρ(PTA) ≤ 1, as claimed. �

Theorem 1 extends the well-known conditions for secondmoment stability given in [45] to m-th moment stability. Asimilar result appears in [46, Th. 3.2], limited to the special caseof a homogeneous MJLS driven by an i.i.d. rate process.

III. MOMENT STABILIZATION OVER BIT-PIPE

MARKOV CHANNELS

Results on the stability of MJLS are used to characterize thestability of linear dynamical systems where the estimated stateis sent to the controller over a digital communication link whosestate is described by a Markov process, as depicted in Fig. 1.

A. System Model

Consider the scalar dynamical system

xk+1 = λxk + uk + vk, yk = xk + wk (11)

where k ∈ N, and |λ| ≥ 1. The variable xk represents the stateof the system, uk the control input, vk is the realization of an ad-ditive stochastic disturbance Vk, yk is the sensor measurement,and wk is the realization of the random measurement noise Wk.The initial condition x0 is the realization of a stochastic variable

X0. Both disturbance and noise are independent of each otherand of the initial condition. They are also independent ofthe channel state, as defined below. X0, Vk, and Wk, k ≥ 0,all have uniformly bounded mth moment, i.e., E(|X0|m) ≤∞, supk E(|Vk|m) ≤ ∞ and supk E(|Wk|m) ≤ ∞. The distur-bance has a continuous probability density function f of finitedifferential entropy h(Vk) := −

∫f(x) log2 f(x)dx, so there

exists a constant β > 0 such that e2h(Vk) > β for all k.

B. Channel Model

The state observer is connected to the actuator through anoiseless digital communication link that at each time k al-lows transmission without errors of rk bits. The rate process{Rk}k≥0 is modeled as a homogeneous positive-recurrentMarkov process defined on the finite set

R = {r1, . . . , rn} (12)

for some integer numbers 0 ≤ r1 < · · · < rn, and with one-step transition probability matrix P having entries

pij = P{Rk = rj |Rk−1 = ri} (13)

for every i, j ∈ {1, . . . , n}. In the sequel, we define R ∈ Zn×n+

as the diagonal matrix with diagonal entries r1, . . . , rn, i.e.,

R = diag(r1, . . . , rn). (14)

Encoder and decoder are supposed to have causal knowledge ofthe rate process. More precisely, for every k ≥ 0, the coder isa function sk = sk(y0, . . . , yk) that maps all past and presentmeasurements into the set {1, . . . , 2rk}, the digital link is theidentity function on the set {1, . . . , 2rk}, and the control uk is afunction xk(s0, . . . , sk) that maps all past and present symbolssent over the digital link into R.

This channel model corresponds to a finite state channelwhere the Markov state is available causally at both the encoderand the decoder. A channel with state is defined by a triple(F × S, p(g|f, s),G) consisting of an input set F , a state setS , an output set G, and a conditional probability assignmentp(g|f, s) for every f ∈ F , s ∈ S , and g ∈ G. This channel ismemoryless if the output gk at time k is conditionally inde-pendent of everything else given (fk, sk). The state sequenceis Markov if S0, S1, . . . forms a Markov chain. According tothese definitions, our channel model is a discrete-memorylesschannel with Markov state (F × S, p(g|f, s),G) with F =G = {1, . . . , rn}, S = {r1, . . . , rn}

p(g|f, s) ={1 f = g and f ≤ s

0 otherwise(15)

and state transition probabilities

p(sk+1 = rj |sk = ri) = pij . (16)

The Shannon capacity of this channel is [47]

C =n∑

i=1

πiri (17)


where (π1, . . . , πn) denotes the unique stationary distrib-ution of P .

The zero-error capacity of this channel is [9]

C0 = r1. (18)

Next, we show that the capacities in (17) and (18) are thelimiting values of a stability threshold function indicating thechannel’s rate-reliability constraint required to achieve a givenlevel of stabilization. As m → ∞ and the system is highly sta-ble, then the stability threshold function tends to the zero-errorcapacity that has a hard reliability constraint of providing nodecoding error. Conversely, as m → 0 and the system’s stabilitylevel decreases, then the stability threshold function tends tothe Shannon capacity that has a soft reliability constraint ofvanishing probability of error.

C. Stability Threshold Function

The system (11) is m-th moment stable if

supk

E [|Xk|m] < ∞ (19)

where the expectation is taken with respect to the randominitial condition x0, the additive disturbance vk, and the rateprocess Rk.

The following Theorem establishes the equivalence betweenthe m-th moment stability of (11) and the weak moment stabil-ity of a suitably defined MJLS.

Theorem 2: It is possible to stabilize the scalar system (11)in m-th momemt sense over the rate-limited Markov channel ifand only if the scalar non-homogeneous MJLS with dynamics

zk+1 =|λ|2mrk

zk + const (20)

for some suitably defined constant const is weakly m-th mo-ment stable, i.e., if and only if

log |λ| � − 1

mlog ρ(PT2−mR) � R(m). (21)

The proof is given in the appendix assuming state feedback.In the case of output feedback the proof is more technical butthe extension is immediate from standard techniques [29]. Wealso assume the disturbance has bounded support. This assump-tion is made to compare results to the ones on the anytimecapacity that only apply to plants with bounded disturbance[10]. The extension to unbounded disturbance can be easilyobtained using standard adaptive encoding schemes describedin [21], [28], and [29].

We mention several properties of the threshold functionR(m), whose proofs are given in the appendix.

Preposition 1: The following holds:

1) Monotonicity: R(m) is continuous and strictly decreas-ing for m > 0.

2) Convergence to the Shannon capacity

limm→0

R(m) =

n∑i=1

πiri = C. (22)

3) Convergence to the Zero Error capacity

R(m) ∼ r1 −1

mlog p11, as m → ∞ (23)

and hence

limm→∞

R(m) = r1 = C0. (24)

4) The function mR(m) is nonnegative, strictly increasing,and strictly concave. If r1 �= 0, then mR(m) grows un-bounded as m → ∞. If instead r1 = 0, then

limm→∞

mR(m) = − log p11. (25)

IV. ANYTIME CAPACITY OF BIT-PIPE

MARKOV CHANNELS

We now relate the stability threshold function R(m) to theanytime capacity. For the given Markov channel, our resultsshow that R(m) provides a parametric representation of theanytime capacity in terms of system’s stability level m.

The anytime capacity is defined in the following context[12]. Consider a system for information transmission over anoisy channel that evolves in discrete time steps, allows thedecoding time to be infinite, and improves the reliability ofthe estimated message as time progresses. More precisely, ateach step k a new message mk of r bits is generated thatmust be sent over the channel. The coder and decoder at thetwo ends of the communication link are equipped with infinitememory and keep track of the messages and their correspondingestimates, respectively. At each time step, a single bit is sent bythe encoder to the decoder over the channel. Upon reception ofthe new bit, the decoder updates the estimates for all messagesup to time k. It follows that at time k the estimates:

m0|k, m1|k, . . . , mk|k

are constructed, given all the bits received up to time k. Theestimate for any message mi is then continuously refined forall k ≥ i. A reliability level α is achieved if the probabilitythat there exists at least a message in the past whose estimateis incorrect decreases α-exponentially with the number of bitsreceived, namely, for all d ≤ k

P{(M0|k, . . . , Md|k) �= (M0, . . . ,Md)

}= O

(2−α(k−d)

)(26)

as k → ∞. It follows that the described communication systemis characterized by a rate-reliability pair (r, α). The work in [12]has shown a correspondence between the ability to constructsuch a communication system over a given channel, and theability to stabilize a scalar linear system of the type (11) overthe same channel.

To precisely state this result in the context of our Markovchannel, we define the α-anytime capacity CA(α) to be thesupremum of the rate r that can be achieved with reliability α.Sahai and Mitter have proven the following result:

Theorem 3 (Sahai, Mitter [12]): The necessary and sufficientcondition for m-th moment stabilization of a scalar system with


bounded disturbances and in the presence of channel outputfeedback over a Markov channel is

log |λ| � CA (m log |λ|) . (27)

The anytime capacity is an intermediate notion between thezero-error capacity and the Shannon capacity. The zero-errorcapacity requires decoding without error. The Shannon capacityallows the decoding error to tend to zero as the length of thecode used for transmission increases. While only a critical valueof the zero-error capacity can guarantee the almost sure stabilityof the system [8], Sahai and Mitter show that for scalar systemssubject to bounded disturbances a critical value of the anytimecapacity can guarantee the ability to stabilize the system in theweaker m-th moment sense.

By combining Theorem 2 and Theorem 3, we obtain thefollowing result on the equivalence of the reliability functionR(m) and the anytime capacity CA(α) on the boundary ofthe stability region, where α = mR(m). With this choice of α,both the reliability function and the anytime capacity can be ar-bitrarily close to log |λ| and still guarantee stability. This showsthat the operational definition of anytime capacity in terms of ahypothetical communication system satisfying the exponentialconstraint (26) is not needed to characterize stability in thiscase. The simpler notion of reliability function is enough forour purposes. On the other hand, by carefully choosing m asa function of α, the reliability function can also be used tocompute the corresponding value of the anytime capacity.

Theorem 4: For Markov bit-pipe channels as defined inSection III-B, the following holds:

1) Parametric characterization of the anytime capacity: Forevery m > 0, the anytime capacity CA satisfies

CA (mR(m)) = R(m) (28)

i.e., for every α ≥ 0, there exists a unique m(α) such that

m(α)R (m(α)) = α (29)

CA(α) = R (m(α)) =α

m(α). (30)

2) CA(α) is a strictly decreasing function function of α > 0.3) Convergence to the Shannon capacity

limα→0

CA(α) =

n∑i=1

πiri = C. (31)

4) Convergence to the Zero Error capacity: If r1 = 0, thenfor every α ≥ log(1/p11)

CA(α) = 0 = C0. (32)

Conversely, if r1 �= 0, then CA(α) has unbounded sup-port and

CA(α) ∼ r1α

α− log(1/p11), as α → ∞ (33)

hence

limα→∞

CA(α) = r1 = C0. (34)

Proof: By Theorem 2 and Theorem 3, at the boundariesof the stability region we must have that

log |λ| = R(m) (35)

log |λ| = C (m log |λ|) . (36)

Therefore, (28) follows by combining (35) and (36). Next, byProposition 1, the function φ(m) = mR(m) is increasing andstrictly concave, thus invertible. It follows that for every α ≥ 0,there exists a unique m := m(α) such that:

mR(m) = α. (37)

Substituting this equality into (28), it follows that CA(α) =R(m(α)). as claimed. The remaining properties are immediateconsequences of Proposition 1. By definition, CA(α) is nonincreasing in α. Since both mR(m) and R(m) are bothmonotonic function of m, however, it follows that CA(α)must be strictly decreasing in α, thus establishing 2). Property3) follows from (28) combined with the fact that mR(m) → 0and R(m) → C as m → 0. Similarly, property 4) followsdirectly by combining (28) with properties 3) and 5) inProposition 1. �

We now give some representative examples of the stabilitythreshold function, visually showing its extremal properties andits relationship to the anytime capacity.

Example IV.1: Let n = 4, R = {1, 3, 4, 5} and P be is a4×4 circulant matrix with first row equal to (1/16)(1, 13, 1, 1),namely

P =1

16

⎡⎢⎢⎣1 13 1 113 1 1 11 1 1 131 1 13 1

⎤⎥⎥⎦ . (38)

In this case it is easy to compute C = (1/4)(1 + 3 + 4 + 5) =13/4 and C0 = 1. Fig. 2 plots the stability threshold functionR(m) (together with its asymptotic approximation) and theanytime capacity CA(α). Both curves have the same shape andthey are in fact related by an affine transformation as m grows.Furthermore, both curves have unbounded support and tend toone at infinity. There is a change of convexity for small values ofm and α, as indication that R(m) and CA(α) are generally notconvex functions. In contrast, the function φ(m) = mR(m),reported in red in the top plot of Fig. 2, is strictly convex andincreasing.

Example IV.2: Let R = {0, 3, 4, 5} and P is as in (38).The only difference with the previous example is that r1 isnow 0 instead than 1. In this case, it is easy to computeC = (1/4)(0 + 3 + 4 + 5) = 3 and C0 = 0. Fig. 3 plots thestability threshold function R(m) (together with its asymptoticapproximation) and the anytime capacity CA(α). In this case,while R(m) has unbounded support, CA(α) is zero for allα ≥ − log p11 = log 16 = 4. This occurs because the functionφ(m) = mR(m) saturates as m → ∞, tending to the limitingvalue − log p11 = 4.

When viewed together, the two examples above show thatfor some channels, like the one in Example IV.2, a commu-nication system with an arbitrary rate-reliability pair (r, α)


Fig. 2. Stability threshold function and anytime capacity forExample IV.1.

cannot be constructed, because the anytime capacity may havebounded support and tend abruptly to zero. However, in orderto achieve m-th moment stabilization it is sufficient to considerthe parametric function R(m) = CA(mR(m)), and constructa communication system whose reliability level depends onthe desired stabilization level. It follows that we do not needto compute the whole anytime capacity if we are interestedonly in moment stabilization, and we may be content withdetermining the threshold function R(m) corresponding to itsparametric representation. The extremal properties of R(m)determine the support of the anytime capacity correspondingto the achievable reliability level α. If R(m) = O(1/m) thenthe anytime capacity has support bounded by the pre-constantof the asymptotic order. On the other hand, if R(m) decreasesat most sub-linearly to zero, or it tends to a constant zero-errorcapacity, then the anytime capacity has unbounded support andany reliability level α can be achieved.

V. THE MARKOV r-BIT ERASURE CHANNEL

We use the stability threshold function to characterize theanytime capacity of the Markov r-bit erasure channel.

The Markov r-bit erasure channel corresponds to a two-state Markov process where n = 2 R = {0, r}, p12 = q, and

Fig. 3. Stability threshold function and anytime capacity forExample IV.2.

p21 = p, where 0 < p, q < 1. In this case

PT2−mR =

((1− q) 1

2mr pq 1

2mr (1− p)

)(39)

and we have the following result.Theorem 5: The anytime capacity of the Markov r-bit

erasure channel is

CA(α) =αr

α+ log2

(1−p−2α(1−p−q)

1−(1−q)2α

) (40)

if 0 ≤ α < − log2(1− q), and 0 otherwise.Proof: By (29) and the definition of R(m), for every α

there exists an m(α) such that

ρ(PT2−m(α)R

)= 2−α. (41)

In the case of a Markov erasure channel, a simple calculationshows that

ρ(PT2−mR) =tr

2+

1

2

√tr2 −4 det (42)

where tr and det denote the trace and determinant of (39),respectively. By combining (41) and (42) and squaring both


sides of the resulting equation, it follows that m(α) must satisfy:

2−αtr− 4det = 2−α+1 (43)

where tr = (1− q) + 2−m(α)r(1− p) and det = 2−m(α)r(1−p− q). Solving (43) yields

m(α) =1

r

(α+ log2

(1− p− 2α(1− p− q)

1− (1− q)2α

))(44)

for 0 ≤ α < −log2(1− q). Finally, substituting (44) into (30)yields (40). �

A. Special Cases

Several special cases recover previous results in the litera-ture. By (40) it follows that the anytime capacity of the binaryerasure channel (BEC) with Markov erasures and with noiselesschannel output feedback is:

CA(α) =α

α+ log2

(1−p−2α(1−p−q)

1−(1−q)2α

) . (45)

By letting q = 1− p, the erasure process becomes i.i.d. andwe recover the anytime capacity of the memoryless BEC witherasure probability p derived by Sahai [10, p. 129] (in paramet-ric form) and by Xu [14, Th. 1.3] (in non-parametric form)

CA(α) =α

α+ log2

(1−p

1−p2α

) . (46)

By (40), letting α → 0, we have that

limα→0

CA(α) =q

p+ qr = C. (47)

This recovers the Shannon capacity of an r-bit erasure chan-nel with Markov erasures and with noiseless channel outputfeedback.

In the case n = 2, r1 = 0, r2 = r, and an i.i.d. rate processwith P{Rk = 0} = p1 and P{Rk = r} = p2 for all k, then thestability condition becomes

|λ|m(p1 + p22

−mr)< 1

which provides a converse to the achievable scheme of Yükseland Meyn [48, Th. 3.3].

If we further let r → ∞, then the stability conditionp1 > 1/|λ|m only depends on the erasure rate of the channel.In this case, our condition generalizes the packet loss modelresult in [30].

VI. MEMORYLESS BIT-PIPE CHANNELS

We show that the stability threshold function can be usedto determine the anytime capacity of an arbitrary memorylessbit-pipe channel of given rate distribution and provide threerelevant examples of this computation. Beside the memorylesserasure channel and the additive white Gaussian noise channelwith input power constraint, these are the only cases where theanytime capacity has been computed.

Consider the special case of an i.i.d. rate process Rk whereRk ∼ R has probability mass function pi = P{R = ri}, ri ∈R. For t real, let MR(t) = E(etR) denote the moment gener-ating function of R and let M−1

R (y) denotes the inverse of theMR(t), if it exists, i.e., M−1

R (y) = t if and only if MR(t) = y.We have the following result.Theorem 6: The anytime capacity of a memoryless bit-pipe

channel with rate distribution R is

CA(α) =ln 2−α

M−1R (2−α)

(48)

for α < − log p11 if r0 = 0, or for any α > 0 if r0 �= 0.Proof: In the special case of an i.i.d. rate process

PT2−mR = (p1, . . . , pn)T (2−mr1 , . . . , 2−mrn)

is a rank-one matrix whose only nonzero eigenvalue is∑ni=1 pi2

−mri = E(2−mR). Therefore, in this case

R(m) = − 1

mlogE(2−mR)

= − 1

mlogMR(−m ln 2). (49)

By combining (29) and (49), we find that α =−logMR(−m(α) ln 2) and so

m(α) = − 1

ln 2M−1

R (2−α) (50)

where M−1R (y) exists in light of property 5) in Proposition 1.

Thus, by (30), (48) follows. �Theorem 6 shows that in the case the channel is memoryless,

the anytime capacity can be evaluated by computing the inverseof the moment generating function of R, as illustrated in thenext three examples.

Example VI.1: Suppose that R is a binomial random variablewith parameters k and 1− p. Then

MR(t) =(p+ (1− p)et

)k(51)

M−1R (y) = ln

y1k − p

1− p, y < pk (52)

and thus by (48)

CA(α) =α

α/k + log2

(1−p

1−p2α/k

) (53)

for α < −k log p. Notice that (53) recovers (46) in the specialcase k = 1 in which R is Bernoulli with parameter 1− p.

Example VI.2: Suppose that R is a Poisson random variablewith parameter λ. Then

MR(t) = eλ(et−1) (54)

M−1R (y) = ln(1 + 1/λ ln y), y > 0 (55)

and thus by (48)

CA(α) = − α

log(1− α/λ ln 2)(56)

for α < −λ/ ln 2.


Fig. 4. Comparison of the anytime capacity for different memorylesschannels. For the uniform distribution the plot is obtained numerically.

Notice in both examples r1 = 0, so the anytime capacityhas bounded support. In the next example r1 = 1 and thus theanytime capacity is defined for all α > 0.

Example VI.3: Suppose that R is a geometric randomvariable with parameter p. Then

MR(t) =pet

1− (1− p)et(57)

for t < −ln(1− t) and

M−1R (y) = ln

y

p+ y(1− p), y > 0 (58)

for y > 0, and thus

CA(α) =α

log ((1− p) + p2α), α > 0. (59)

Plots of the anytime capacity for different memoryless bit-pipe channels are given in Fig. 4. Plots for the Binomial,Poisson, and Geometric distributions correspond to the resultsin Examples VI.1, VI.2, and VI.3. The plot for the uniformdistribution is obtained by numerically inverting the momentgenerating function of the rate process.

VII. CONCLUSION

We presented a parametric characterization of the anytimecapacity in terms of the threshold function for m-th momentstabilization of a scalar linear system controlled over a Markovtime-varying digital feedback channel that depends on m andon the channel’s parameters. This parametrization allows an ex-plicit computation of the anytime capacity of the r-bit Markoverasure channel, and of Binomial, Poisson, and Geometricmemoryless rate processes. It also provides a general strategyto compute the anytime capacity of arbitrary memoryless bit-pipe channels of given rate distribution, based on the inver-sion of the moment generating function of the process. Theoperational interpretation of the stability threshold function is

that of achievable communication rate, subject to a reliabilityconstraint, and it has as extreme values the Shannon capacity,corresponding to a soft reliability constraint, and the zero-errorcapacity, corresponding to hard reliability. It follows that amore stringent stability requirement for the system, expressedin terms of higher moment, translates in a harder reliabilityconstraint for communication.

Our derivation relies on some extensions of the theory ofMJLS. We provide the generalization of the necessary andsufficient conditions for second moment stability of MJLS tom-th moment stability. A commonly encountered notion ofstochastic stability in the MJLS literature [45] is the one ofstrong moment stability, according to which the MJLS (1) isstable if there exists a finite αm ∈ R such that E[|Z|m] →αm, as m → ∞. Clearly, if an MJLS is strongly stable, thenit is also weakly stable according to our definition. It can beverified that our proof of the necessary condition in Theorem 1extends to the case of strong moment stability, thereby estab-lishing the necessity of (6) for the strong moment stability of(1). In contrast, the subsampling technique used in the directpart of Theorem 1 is not suitable to prove strong momentstability because it cannot prevent undesirable fluctuations inthe system dynamics between any two sampling periods. In [49]we presented a different proof technique based on the binomialexpansion of (akzk + wk)

m which can be used to prove that (6)is a sufficient condition for the strong moment stability of (1).This approach however requires m to be an integer number, asopposed to the subsampling technique which can be applied toany m > 0.

Finally, we see no difficulty, sic et simpliciter, but at theexpense of a more technical treatment, to extend our resultsto vector systems using bit-allocation techniques outlined in[28] and [29], and to disturbances of unbounded support usingstandard adaptive schemes [21], [28], [29].

APPENDIX AAUXILIARY RESULTS

We state a series of auxiliary results that are needed in the restof the paper. We begin with a maximum entropy result whichgeneralizes the well-known result that the second moment of acontinuous random variable X is lower bounded by its entropypower e2h(X) to the case of the Lm-norm of a continuousrandom vector.

Lemma 1: Let X be a continuous n-dimensional vector-valued random variable with E[‖X‖mm] < ∞, where m is apositive real number. Let ‖X‖m = (

∑i |Xi|m)1/m denote the

Lm-norm of X . Then

E [‖X‖mm] ≥ n

cme

mn h(X) (60)

where cm=2mm1−me(Γ(1/m))m and Γ(x)=∫∞0 e−ttx−1dt.

Equality holds if and only if X1, . . . , Xn are i.i.d ∼ X , where

fX(x) =exp

(− |x|m

mμm

)2m(1−m)/mμΓ

(1m

) , x ∈ R. (61)


Proof: First, notice that by the maximum entropy theo-rem [50, Th. 12.1.1], the unique density fX(x) that maximizesthe differential entropy h(X) over all probability densities fhaving m-th moment equal to E(|X|m) is given in (61). No-tice that if X ∼ fX(x), then h(X) = (1/m) log[cmE[|X|m]].Next, by using the fact that conditioning reduces the entropy

h(X) ≤n∑

i=1

h(Xi)

≤n∑

i=1

1

mlog [cmE [|Xi|m]]

≤ n

mlog

[cmn

n∑i=1

E [|Xi|m]

]

=n

mlog[cmn

E [‖X‖mm]]

where the second inequality follows from the maximum en-tropy theorem applied to each random variable Xi, while thelast inequality follows from Jensen’s inequality. �

It should be remarked that inequality (60) with n = 1 is aspecial case of a result proved in [51] relating the m-th momentand the Renyi entropy of a random variable.

Next, we state a known result on the log-convexity of thespectral radius of a nonnegative matrix which is needed to provesome of the properties of R(m).

Lemma 2 (Friedland, Theorem 4.2 [52]): Let Dn be the setof n× n real-valued diagonal matrices. Let A be a fixed n× nnon-negative matrix having a positive spectral radius. Defineφ : Dn → R by φ(D) := log ρ(eDA). Then, φ(D) is a convexfunctional on Dn. Specifically: for every D1, D2 ∈ Dn, andα ∈ (0, 1)

φ (αD1 + (1− α)D2) ≤ αφ(D1) + (1− α)φ(D2). (62)

Moreover, if A is irreducible and the diagonal entries of A arepositive (or A is fully indecomposable) then equality holds in(62) if and only if

D1 −D2 = cI (63)

for some c ∈ R, where I is the identity matrix.Finally, we show a result on the asymptotic behavior of

PT2−mR as m → ∞.Lemma 3: The following equality holds:

limm→0

(PT2−mR)1m= lim

m→0

⎛⎜⎝⎡⎢⎣π12

−mr1 · · · π12−mrn

......

...πn2

−mr1 · · · πn2−mrn

⎤⎥⎦⎞⎟⎠

1m

.

Proof: Let 1/m = k. By the monotonicity propertyof R(m), it is sufficient to prove the claim for k integer.Therefore, let

A :=

⎡⎢⎣p112

−r1/k · · · p1n2−r1/k

......

...pn12

−rn/k · · · pnn2−rn/k

⎤⎥⎦ = (PT2−1/kR)T

B :=

⎡⎢⎣π12

−r1/k · · · πn2−r1/k

......

...π12

−rn/k · · · πn2−rn/k

⎤⎥⎦ .

To establish the claim it suffices to show that

limk→∞

Ak = limk→∞

Bk.

To do so, we prove that

limk→∞

[Ak]i,j = limk→∞

[Bk]i,j .

Note that

[Ak]i,j =∑

l1,...,lk−1

(pil12

−i/k)· · ·(plk−1j2

−lk−1/k)

=∑

l1,...,lk−1

(pil1 · · · plk−1j

)2−(i+···+lk−1)/k

[Bk]i,j =∑

l1,...,lk−1

(πl12

−i/k)· · ·(πlj2

−lk−1/k)

=∑

l1,...,lk−1

(πl1 · · ·πlj

)2−(i+···+lk−1)/k.

Then, we denote the stationary distribution of P as Π =limk→∞ P k

limk→∞

([Ak]i,j − [Bk]i,j

)= lim

k→∞

∑l1,...,lk−1

(pil1 · · · plk−1j − πl1 · · ·πlj

)2−(i+···+lk−1)/k

≤ limk→∞

∑l1,...,lk−1

(pil1 · · · plk−1j − πl1 · · ·πlj

)

=

⎛⎝⎛⎝ lim

k→∞

∑l1,...,lk−1

pil1 · · · plk−1j

⎞⎠

−

⎛⎝ lim

k→∞

∑l1,...,lk−1

πl1 · · ·πlj

⎞⎠⎞⎠

=

(limk→∞

[P k]i,j − limk→∞

[Πk]i,j

)

= 0.

�

APPENDIX BPROOF OF THEOREM 2

Necessity. To establish the necessary condition, we prove thatfor every k = 0, 1, . . .

cm E [|Xk|m] ≥ E [|Zk|m] (64)

where cm is a constant defined in Lemma 1 and {zk} is ahomogeneous MJLS with dynamics

zk+1 =|λ|2rk

zk (65)

and z0 = eh(X0).


Let Sk = {S0, . . . , Sk} denote the symbols transmitted overthe digital link up to time k. By the law of total expectation andLemma 1

E (|Xk+1|m) ≥ 1

cmESk

(emh(Xk+1|Sk=sk)

). (66)

It follows that the m-th moment of the state is lower boundedby the average entropy power of Xk conditional on Sk. Fromthe translation invariance property of the differential entropy,the conditional version of entropy power inequality [53], andAssumption of finite differential entropy of the disturbance, itfollows that:

ESk

(emh(Xk+1|Sk=sk)

)= ESk

(emh(λXk+x(sk)+Vk|Sk=sk)

)

≥ ESk

((e2h(λXk |Sk=sk) + e2h(Vk)

)m2

)

≥ ESk

(emh(λXk |Sk=sk)

)= |λ|mESk

(emh(Xk |Sk=sk)

). (67)

We can further lower bound (67) as in Lemma 1 in [28], andobtain that for every k ≥ 0

ESk |Sk−1,Rk

(emh(Xk |Sk=sk)

)≥ 1

2mRkemh(Xk |Sk−1=sk−1)

(68)with S−1 = ∅. By the tower rule of conditional expectation, itthen follows that:

ESk

(emh(Xk |Sk=sk)

)≥ESk−1,Rk

(1

2mRkemh(Xk |Sk−1=sk−1)

).

(69)

Combining (67) and (69) gives

ESk

(emh(Xk+1|Sk=sk)

)

≥ ERk

(|λ|m2mRk

ESk−1|Rk

(emh(Xk |Sk−1=sk−1)

)). (70)

Following similar steps and using the Markov chain Sk−1 →(Sk−2, Rk−1) → Rk, we obtain:

ESk−1|Rk

(emh(Xk |Sk−1=sk−1)

)≥ |λ|mESk−1|Rk

(emh(Xk−1|Sk−1=sk−1)

)

≥ ESk−2,Rk−1|Rk

(|λ|m

2mRk−1emh(Xk−1|Sk−2=sk−2)

)

= ERk−1|Rk

(|λ|m22Rk−1

ESk−2|Rk−1,Rk

[e2h(Xk−1|Sk−2=sk−2)

]).

(71)

Substituting (71) into (70) and re-iterating k times, it followsthat ESk(emh(Xk+1|Sk=sk)) is lower bounded by:

ERk−1,Rk

(|λ|m

2m(Rk−1+Rk)ESk−2|Rk−1,Rk

×(emh(Xk−1|Sk−2=sk−2)

))

≥ ER1,...,Rk

(|λ|mk

2m(R1+···+Rk)ES1|R1,...,Rk

(emh(X1|S0=s0)

))(72)

= E(

|λ|m(k+1)

2m(R1+···+Rk)

)emh(X0) (73)

where (72) uses the fact that the initial condition of the statex0 is independent of the rate process Rk. Let {Zk} be a non-homogeneous MJLS with dynamics

zk+1 = |λ|/2rkzk (74)

with z0 = eh(X0). By taking the expectation on both sidesof (74) and iterating k times, it is easy to see that the righthand side of (73) is equal to the mth moment of Zk+1.Hence, combining (66)–(73) we conclude that E(|Xk|m) >(1/cm)E(|Zk|m), which is the claim.

Sufficiency. Let ω denote a uniform bound on the measureof the support of the initial condition, the noise, and the distur-bance. Then, we claim that the plant dynamics can be boundedas follows:

|xk| ≤ zk, k = 0, 1, . . . (75)

where Zk is a homogeneous MJLS with dynamics

zk+1 =|λ|2Rk

zk + ω, k = 0, 1, . . .

and z0 = ω. To see this, consider the following inductive proof.By assumption, |x0| ≤ ω = z0. Assume that the claim holdsfor all times up to k, so |xk| ≤ zk. Suppose that at time k theuncertainty set [−zk, zk] is quantized using a rk-bit uniformquantizer, and that the encoder communicates to the decoder theinterval containing the state. Then, the decoder approximatesthe state by the centroid xk of this interval and sends to the plantthe control input uk = −λxk. By construction |xk − xk| ≤zk/2

rk , thus

|xk+1| = |λ(xk − xk) + vk|

≤ |λ||xk − xk|+ ω

≤ |λ|2rk

zk + ω

= zk+1 (76)

i.e., the claim holds at time k + 1 as well. It follows that xk ism-th moment stable if the homogeneous MJLS Zk is weaklym-th moment stable, i.e., if and only if (21) holds.


APPENDIX CPROOF OF PROPOSITION 1

The monotonicity property is an immediate consequenceof the log-convexity of the spectral radius of a nonnegativematrix stated in Lemma 2. Let D1 = −mR ln 2, D2 = 0n×n

and α = n/m. Notice that 2−mR = eD1 , log ρ(PTeD2) = 0,and PT2−mR = PTeαD1 . Then, for every n < m, by Lemma 2

−nR(m) =n

mlog ρ(PT2−mR)

= α log ρ(PTeD1) + (1− α) log ρ(PTeD2)

> log ρ(PTeαD1+(1−α)D2

)= log ρ(PT2−nR)

= −nR(n). (77)

By dividing both sides by −n, it then follows thatR(n) > R(m), establishing that R(m) is a strictly decreasingfunction of m.

To establish the convergence of R(m) to the Shannon capac-ity as m → 0, observe that

limm→0

R(m)

= limm→0

log ρ((PT2−mR)

1m

)= log ρ

(limm→0

(PT2−mR)1m

)

(a)= log ρ

⎛⎜⎜⎝ lim

m→0

⎛⎜⎝⎡⎢⎣π12

−mr1 · · · π12−mrn

......

...πn2

−mr1 · · · πn2−mrn

⎤⎥⎦⎞⎟⎠

1m

⎞⎟⎟⎠

= limm→0

1

mlog ρ

⎛⎜⎝⎡⎢⎣π12

−mr1 · · · π12−mrn

......

...πn2

−mr1 · · · πn2−mrn

⎤⎥⎦⎞⎟⎠

= limm→0

1

mlog

(∑i

πi2−mri

)

(b)= lim

m→0

∑i πiri2

−mri∑i πi2−mri

=∑i

πiri

where (a) follows from Lemma 3 and (b) follows froml’Hôpital’s rule.

The convergence of R(m) to the zero-error capacity as m →∞ can be proved using the fact that, as m → ∞

R(m) = − 1

mlog ρ(PT2−mR)

= − log ρ((PT2−mR)

1m

)

= r1 − log ρ

⎛⎜⎜⎝⎡⎢⎣p11 · · · pn12

−m(rn−r1)

......

...p1n · · · pnn2

−m(rn−r1)

⎤⎥⎦

1m

⎞⎟⎟⎠

∼ r1 −1

mlog p11

where the last equation follows from the fact that for anycolumn vector v = (v1, . . . , vn)

T, ρ([v 0n×(n−1)

]) = v1.

From the above asymptotic approximation it immediately fol-lows that if r1 = 0, then:

limm→∞

mR(m) = − log p11. (78)

Next, the monotonicity property of −mR(m) =log ρ(PT2−mR) follows from the fact that all the entriesof the matrix PT2−mR are monotonically decreasing in m.Finally, the strict concavity of mR(m) is again a directconsequence of the log-convexity of the spectral radius of anonnegative matrix stated in Lemma 2.

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Paolo Minero (M’11) received the Laurea de-gree (with highest honors) in electrical engineer-ing from the Politecnico di Torino, Torino, Italy, in2003, the M.S. degree in electrical engineeringfrom the University of California at Berkeley in2006, and the Ph.D. degree in electrical engi-neering from the University of California at SanDiego in 2010. He is currently a Staff Engi-neer in the modem system engineering groupat Qualcomm Inc, San Diego, CA, where he isengaged in the design and development of both

4 G and 5 G wireless technology. Prior to joining Qualcomm, he wasan Assistant Professor in the Department of Electrical Engineering atthe University of Notre Dame, IN, conducting research on informationtheory and control. Dr. Minero received the U.S. Vodafone Fellowship in2004 and 2005 and the Shannon Memorial Fellowship in 2008.

Massimo Franceschetti (M’98–SM’11) re-ceived the Laurea degree (with highest honors)in computer engineering from the University ofNaples, Naples, Italy, in 1997, the M.S. andPh.D. degrees in electrical engineering fromthe California Institute of Technology, Pasadena,CA, in 1999, and 2003, respectively. He is Pro-fessor of Electrical and Computer Engineeringat the University of California at San Diego(UCSD). Before joining UCSD, he was a post-doctoral scholar at the University of California at

Berkeley for two years. He has held visiting positions at the Vrije Uni-versiteit Amsterdam, the École Polytechnique Fédérale de Lausanne,and the University of Trento. His research interests are in physical andinformation-based foundations of communication and control systems.He is co-author of the book “Random Networks for Communication”published by Cambridge University Press. Dr. Franceschetti served asAssociate Editor for Communication Networks of the IEEE TRANSAC-TIONS ON INFORMATION THEORY (2009–2012) and as Guest AssociateEditor of the IEEE Journal on Selected Areas in Communications(2008, 2009). He is currently serving as Associate Editor of the IEEETRANSACTIONS ON CONTROL OF NETWORK SYSTEMS and the IEEETRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING. He wasawarded the C. H. Wilts Prize in 2003 for best doctoral thesis in electricalengineering at Caltech; the S.A. Schelkunoff Award in 2005 for bestpaper in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, aNational Science Foundation (NSF) CAREER award in 2006, an Officeof Naval Research (ONR) Young Investigator Award in 2007, the IEEECommunications Society Best Tutorial Paper Award in 2010, and theIEEE Control theory society Ruberti young researcher award in 2012.

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