Rate-constrained sampling over imperfect channels

Chapter III: Causal rate-constrained sampling

3.7 Rate-constrained sampling over imperfect channels

We replace the perfect channel in Section3.2 by a channel with a fixed delay and a binary erasure channel (BEC), respectively. For a channel with a fixed delay, we show that the SOI code for a continuous Lévy process remains optimal. For a BEC, we show that appending the SOI compressor (3.6) to the optimal causal sampling policy for a packet-drop channel in Theorem4gives a code that attains the optimal distortion-rate tradeoff.

Channel with delay

We re-consider the communication scenario in Section2.5 with the sampling fre- quency constraint (2.3b) in (2.33) replaced by the communication rate constraint (3.3b). We denote byΠandC the set of all causal sampling policies and the set of

all causal compressing policies in the infinite horizon, respectively. The DRF for a channel with fixed delayδis defined as

D^ch(R) = inf

π∈Π, {ft}^∞_t=0∈C:

(3.3b)

lim sup

T→∞

1 TE

" _N X

i=0

Z τi+1+δ τi+δ

(X_t−Xˆ_t^ch)²dt

#

, (3.45)

whereXˆ_t^ch is the MMSE decoding policy of the continuous Lévy process (2.17) Xˆ_t^ch ≜E[X_t|Uⁱ, τⁱ], t∈[τ_i +δ, τ_i+1+δ). (3.46)

We present the optimal causal code that achievesD^ch(R)(3.45).

Proposition 3. In causal rate-constrained sampling of the continuous Lévy process (2.17) with a fixed channel delay δ and the decoding policy (3.46), the optimal causal code remains the SOI code in Theorem5and achieves

D^ch(R) = ac²

6R +ac²δ. (3.47)

Proof. Converse: We show that the DRF for a channel with fixed delayδ(3.45) is lower bounded by the DFF for a channel with fixed delayδ(2.33) atF =R, i.e.,

D^ch(R)≥D^ch(R). (3.48)

We denote by X¯_t^′ ≜ E[X_t|{X_t}^τ_t=0ⁱ ], t ∈ [τ_i +δ, τ_i+1 +δ) the MMSE estimator given all the past process by timeτ_i. Sinceσ(Uⁱ, τ_i)⊆ σ({X_t}^τ_t=0ⁱ ), the DRF with channel delayD^ch(R)is lower bounded by the right side of (3.45) withXˆ_t^ch ←X¯_t^′, (3.3b)←(2.3b), andF ←R. Since by the strong Markov propertyX¯_t^′ = ¯X_t^ch(2.34), the lower bound reduces toD^ch(R).

Achievability: We show that the SOI code achieves the converse bound (3.48).

Since the decoder can noiselessly recover the samples from the1-bit codewords, the MMSE decoding policyXˆ_t^ch in (3.46) is equal to X¯_t^ch in (2.34). Since the length ℓ(U_i) = 1, the rate constraint (3.3b) is equal to the frequency constraint (2.3b).

The first term in (3.47) is the DRF for a delay-free channel and the second term in (3.47) is the penalty due to the channel delayδ. Proposition3demonstrates that our SOI code is resilient to channel delay.

Binary erasure channel

We consider the communication scenario in Fig. 3.5, which is similar to that in Section 2.5 except that the packet-drop channel is replaced by a BEC and the sampling frequency constraint is replaced by a rate constraint. Here, we slightly abuse the notations to denote by 0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τn (2.41) a sequence of bit-generating times and to denote byU_i ∈ {0,1}a bit generated at timeτ_i.

encoder BEC decoder

Figure 3.5: System model for causal rate-constrained sampling over a BEC with feedback.

In Fig. 3.5, the encoder transmits bit U_i over a BEC(p), whose channel transition probability is given byP_V|U: {0,1} → {0,1, e},

P_V|U(u|u) = 1−p, ∀u∈ {0,1}, (3.49a) P_V|U(e|u) =p, ∀u∈ {0,1}. (3.49b) Upon receiving the channel output V_i at each sampling time τ_i, i = 1,2, . . ., the decoder sends a1-bit feedbackB_τi ∈ {0,1}to inform the encoder whether bitU_iis erased or not. If B_τi = 1, the bit is not erased, i.e., V_i = U_i; otherwise, the bit is erased, i.e.,V_i =e. Since both the encoder and the decoder knowX₀ = 0atτ₀ = 0, it holds thatB_τ0 ≜1.

We define an⟨n, d, T⟩causal code for a BEC that transmitsnbits of a source process {X_t}^∞_t=0 within an expected time horizonT at an MSE less than or equal tod. Definition 16(An⟨n, d, T⟩causal code for a BEC). Fix a source process{X_t}^∞_t=0 and fix a BEC with a single-letter transition probabilityP_V|U: {0,1} → {0,1, e}.

An ⟨n, d, T⟩ causal code for a BEC is a pair of encoding and decoding policies.

The encoding policy consists of a causal sampling policy and a causal compressing policy.

1. The causalsamplingpolicy is a collection ofn stopping times(2.41)defined in Definition6-1;

2. The causal compressing policy, characterized by the {0,1}-valued process {f_t}_t≥0adapted to the filtration generated by the source process{X_t}^∞_t=0and the feedback bit process{B_τi}ⁿ_i=1, decides the bit U_i to transmit at time τ_i, U_i =f_τi,i= 1,2, . . . , n.

Given channel outputs{V_j}ⁱ_j=1 and sampling timesτⁱ, the decoding policy is Xˆ_t^BEC≜E

h X_t

{V_j, τ_j}_j∈N(^{B_τj^}ij=1) i

, t∈[τ_i, τ_i+1), (3.50) where N {B_τj}ⁱ_j=1

is defined in (2.39), which contains all the indices of the successful transmissions by timeτi.

The expectation of then-th sampling time must satisfy(2.43), while the MSE must satisfy

1 TE

Z τn

0

(X_t−Xˆ_t^BEC)²dt

≤d. (3.51)

The causal sampling policy is assumed to satisfy (S.1) in Section 2.2 and (S.4) in Section2.5. The decoding policy in (3.50) can be suboptimal since it ignores the knowledge implied by the sampling times of the erased bits.

To quantify the tradeoffs among the number of bits n (2.41), the expected time horizon T (2.43), and the MSE (3.51), we introduce the distortion-sample-time function for a BEC.

Definition 17 (Distortion-sample-time function (DSTF) for a BEC). Fix a source process{X_t}^∞_t=0. The DSTF for a BEC is the minimum MSE (3.51)achievable by causal codes withnbits and expected horizonT:

D^BEC(n, T)≜inf{d: ∃ ⟨n, d, T⟩causal code for a BEC satisfying (S.1), (S.4)}.

(3.52) For a continuous Lévy process (2.17), we show that appending the SOI compressing policy (3.6) to the optimal causal sampling policy in Theorem4gives a causal code for a BEC that achievesD^BEC(n, T).

Theorem 8. Fix a continuous Lévy process{X_t}^∞_t=0in(2.17)and fix a BEC(p) with erasure probability p < ¹₅ (3.49). Together with the decoding policy in(3.50), the causal encoding policy that operates as:

• ifB_τi = 1, then the next sampling time is(2.47)and the encoder transmits bit U_i+1 using the SOI compressing policy(3.6);

• ifB_τi = 0, then the next sampling time isτ_i+1 =τ_iand the encoder transmits Ui+1 =Ui,i= 0,1, . . . , n−1;

achieves the DSTF for a BEC

D^BEC(n, T) = ac²T

6(1 + (n−1)(1−p)). (3.53) Proof. Converse:In AppendixB.10, we show that the DSTF for a BECD^BEC(n, T) is lower bounded by the DSTF for a packet-drop channelD^pd(n, T), i.e.,

D^BEC(n, T)≥D^pd(n, T). (3.54) Achievability: The causal encoding policy in Theorem8together with the decoding policy in (3.50) achieves the converse bound D^pd(n, T) since the decoder can noiselessly recover the samples at the sampling times using the successfully received the1-bit codewords, i.e.,Xˆ_t^BEC= ¯X_t^pd.

The optimal causal encoding policy for a BEC in Theorem8operates as follows: if the last bit is not erased, then the encoder follows the symmetric threshold sampling policy (4) and transmits a 1-bit codeword that describes the sign of the process innovation at each sampling time; otherwise, the encoder retransmits the erased bit immediately.

Considering _Tⁿ as the rate, letting R ≜ _Tⁿ, and taking n → ∞, we rewrite the sampling policy (2.47) andD^BEC(n, T)in (2.48) in terms of rateRas

τ_i+1 = inf

t≥τ_i:|X_t−Xˆ_t^BEC| ≥c

r a (1−p)R

, i= 0,1,2, . . .; (3.55) D^BEC(R) = ac²

6(1−p)R. (3.56)