Problem statement

Chapter II: Causal frequency-constrained sampling

2.2 Problem statement

Consider the single-sampler single-estimator system in Fig.2.1. A source outputs a real-valued continuous-time stochastic process{X_t}^T_t=0 with state space(R,B_R), whereB_Ris the Borelσ-algebra onR.

sampler noiseless

channel estimator

Figure 2.1: System Model. Sampling timesτi,i = 1,2, . . . are determined by the sampling policies.

A sampler tracks the source process{X_t}^T_t=0and causally decides to sample it at a sequence of stopping times

0≤τ₁ ≤τ₂ ≤ · · · ≤τ_N ≤T (2.1) that are decided by a causal sampling policy. Thus, the total number of time stamps N can be random. The time horizon T can either be finite or infinite. At time τ_i, the sampler passes sampleX_τi to the estimator without delay through a noiseless channel. At time t, t ∈ [τ_i, τ_i+1), the estimator estimates the source process X_t, yieldingX¯_t, based on all the received samples and the sampling time stamps, i.e., (X_τj, τ_j), j = 1,2, . . . , i. Note that the sampler and the estimator can leverage

the timing information for free due to the clock synchronization and the zero-delay channel.

We formally define causal sampling and estimating policies.

Definition 1((F, d, T) causal frequency-constrained code). A time horizon-T causal frequency-constrained code for the stochastic process {X_t}^T_t=0 is a pair of causal sampling and estimating policies:

1. The causalsamplingpolicy is a collection of stopping times τ₁, τ₂, . . . (2.1) adapted to the filtration{F_t}^T_t=0at which samples are generated.

2. Given a causal sampling policy, the real-valued samples{Xτj}ⁱ_j=1and sam- pling time stampsτⁱ, the MMSE estimating policy is

X¯_t≜E[X_t|{X_τj}ⁱ_j=1, τⁱ, t < τ_i+1], t∈[τ_i, τ_i+1). (2.2) In an(F, d, T)code, the average sampling frequency must satisfy

E[N]

T ≤F (samples per sec), (T < ∞), (2.3a) lim sup

T→∞

E[N]

T ≤F (samples per sec), (T =∞), (2.3b) whereN is the total number of stopping times in(2.1), while the MSE must satisfy

1 TE

Z T 0

(X_t−X¯_t)²

≤d, (T <∞), (2.4a) lim sup

T→∞

1 TE

Z T 0

(X_t−X¯_t)²

≤d, (T =∞). (2.4b)

Allowing more freedom in designing the estimating policy will not lead to a lower MSE, since (2.2) is the MMSE estimator.

We present the assumptions on the source process and on the causal sampling policies below. Throughout, we impose the following assumptions on the source process{Xt}^T_t=0. Let{Ft}^T_t=0be the filtration generated by{Xt}^T_t=0.

(P.1) (Strong Markov property){X_t}^T_t=0satisfies the strong Markov property: For all almost surely finite stopping timesτ ∈ [0, T]and allt ∈[0, T −τ],X_t+τ is conditionally independent ofFτ givenXτ.

(P.2) (Continuous paths){X_t}^T_t=0 has continuous paths: X_t is almost surely con- tinuous int.

(P.3) (Mean-square residual error properties)For all almost surely finite stopping times τ ∈ [0, T] and all t ∈ [τ, T], the mean-square residual error X˜_t = X_t−E[X_t|X_τ, τ]satisfies:

(P.3-a) X˜_t is independent ofF_τ and X˜_t has the Markov property, i.e., for all r∈[τ, t],X˜_tis conditionally independent ofF_r givenX˜_r.

(P.3-b) X˜_tcan be expressed as

X˜t=qt(s) ˜Xs+Rt(s, τ), (2.5) where s ∈ [τ, t], q_t(s) is a deterministic function of (t, s), and R_t(s, τ) is a random process with continuous paths, i.e., R_t(s, τ) is almost surely continuous int. Furthermore, the random variableRt(s, τ)has an even and quasi-concave pdf, andq_t(t) = 1,R_t(t, τ) = 0.

We assume that the initial stateX0 = 0at timeτ0 ≜0is known both at the sampler and the estimator. For example, the Wiener process satisfies (P.1)–(P.3), whose definition is given below.

Definition 2(Wiener process, e.g., [68]). A Wiener process{W_t}_t≥0is a stochastic process characterized by the following three properties:

• For all non-negativesandt, W_s andW_s+t−W_t have the same distribution (W₀ = 0);

• The increments W_ti −W_si (i ≥ 1) are independent whenever the intervals (s_i, t_i]are disjoint;

• The random variableW_tfollows the Gaussian distributionN(0, t).

Any stochastic process of the form X_t = g₁(t)W_g2(t)+g₃(t) satisfies (P.1)–(P.3), whereg₁, g₂, g₃are continuous deterministic functions of the timet, andg₂is positive and non-decreasing int. The parameters in (2.5) for this example process areq_t(s) =

g1(t)

g1(s)andR_t(s, τ) =g₁(t)W_g2(t)−g₂(s). The Wiener process, the Ornstein-Uhlenbeck (OU) process, and the continuous Lévy processes are special cases of this form.

These processes are widely used in financial mathematics and physics. There are also other stochastic processes satisfying (P.1)–(P.3), e.g.,X_t=W_t+c1+c₂W_t, where c1, c2 ∈R, which is expressed by (2.5) withqt(s) = 1,Rt(s, τ) = (1 +c1)Wt−s.

Definition 3(time-homogeneous process). We say that a stochastic process{X_t}^T_t=0 is time-homogeneous, if for a stopping timeτ ∈[0, T]and a constants∈[0, T−τ], X_s+τ −E[X_s+τ|X_τ]follows a distribution that only depends ons.

We focus on causal sampling policies that satisfy the following assumptions.

(S.1) The sampling interval between any two consecutive stopping times,τi+1−τi, satisfies

E[τ_i+1−τ_i]<∞, i = 0,1, . . . , (2.6) and the MSE within each interval satisfies

E

Z τi+1

τi

(Xt−X¯t)²dt

<∞, i = 0,1, . . . (2.7) (S.2) The Markov chainτ_i+1−τ_i− {X_t}^τ_t=0ⁱ holds for alli= 0,1, . . .

(S.3) For alli= 0,1, . . ., the conditional pdfsf_τi+1|τ_i exist.

Note that (2.6) holds trivially if T < ∞. Sun et al. [8] and Ornee and Sun [9]

also assumed (2.6) in their analyses of the infinite time horizon problems for the Wiener [8] and the OU [9] processes. We use (2.7) to obtain a simplified form of the distortion-frequency tradeoff for time-homogeneous processes (see (2.13) below). Furthermore, (2.7) allows us to prove that the optimal sampling intervals τ_i+1 −τ_i form an i.i.d. process (see (2.12) below). We use (S.2), (S.3) to show that the optimal sampling policy is a symmetric threshold sampling policy in the frequency-constrained setting. See Appendix A.1 for a sufficient condition on the stochastic process for the optimal sampling policy to satisfy (S.2). For example, in the infinite time horizon, stochastic processes of the formX_t = cW_at +btsatisfy the sufficient condition. Assumption (S.2) implies that the stopping times form a Markov chain. In contrast, the sampling intervals of causal sampling policies are assumed to form a regenerative process in [8][9].

To quantify the tradeoffs between the sampling frequency (2.3) and the MSE (2.4), we introduce the distortion-frequency function.

Definition 4(Distortion-frequency function (DFF)). The DFF for causal frequency- constrained sampling of the process {X_t}^T_t=0 is the minimum MSE achievable by causal frequency-constrained codes,

D(F)≜inf{d:∃(F, d, T)causal frequency-constrained code

satisfying (S.1), (S.2), (S.3)}. (2.8)

In the causal frequency-constrained sampling scenario, we say that a causal sampling policy isoptimalif, when succeeded by the MMSE estimating policy (2.2), it forms an(F, d, T)code withd=D(F).