Source Coding in Sensor Networks Robert L. Konsbruck School of - PowerPoint PPT Presentation

LSI Filtering �� U ( x, t ) V ( x, t ) = R 2 g ( x − ξ, t − τ ) U ( ξ, τ ) d τ d ξ g ( x, t ) Theorem If g is absolutely integrable, then V ( x, t ) is a well defined random variable. Further: � � � 2 S U (Φ , Ω) , �� S V (Φ , Ω) = g (Φ , Ω) g denotes the Fourier transform of g . where � V ( x, t ) is a homogeneous, Gaussian random field. Since � g is bounded, S V is bounded and has a compact support. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 18 / 53

LSI Filtering (cont.) Proof. ��� � � � � g ( x − ξ, t − τ ) U ( ξ, τ ) � d τ d ξ E R 2 �� �� � � � � � g ( x − ξ, t − τ ) � E � U ( ξ, τ ) � = d τ d ξ R 2 �� �� � 2 � 1 / 2 � � � � g ( x − ξ, t − τ ) � E � U ( ξ, τ ) ≤ d τ d ξ R 2 �� � � � � g ( x − ξ, t − τ ) � d τ d ξ < + ∞ = K U (0 , 0) R 2 Hence: �� � � � g ( x − ξ, t − τ ) U ( ξ, τ ) � d τ d ξ R 2 is finite with probability one. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 19 / 53

Outline Introduction 1 Sensor Network Model 2 Physical Fields & Stochastic Source Model 3 Spatio-Temporal Sampling 4 Source Coding Schemes 5 Application – Acoustic Wave Acquisition 6 Conclusions 7 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 20 / 53

Spatio-Temporal Sampling U ( x, t ) V ( x, t ) V x R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 21 / 53

Spatio-Temporal Sampling U ( x, t ) V ( x, t ) V x t R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 21 / 53

✣ ✬ ✛ ✜ ✢ ✼ ✤ ✥ ✦ ✧ ★ ✩ ✪ ✫ ✭ ✙ ✮ ✯ ✰ ✱ ✲ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✚ ✘ ✻ ✟ ❀ ✿ ✾ ✽ � ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✠ ✗ ✡ ☛ ☞ ✌ ✍ ✎ ✏ ✑ ✒ ✓ ✔ ✕ ✖ ✺ Spatio-Temporal Sampling U ( x, t ) V ( x, t ) V x independent temporal lattices t R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 21 / 53

❭ ❡ ❳ ❨ ❩ ❬ ✈ ❪ ❫ ❴ ❵ ❛ ❜ ❝ ❞ ❢ ❱ ❣ ❤ ✐ ❥ ❦ ❧ ♠ ♥ ♦ ♣ q r s ❲ ❯ ✉ ❊ ③ ② ① ✇ ❁ ❂ ❃ ❄ ❅ ❆ ❇ ❈ ❉ ❋ ❚ ● ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ P ◗ ❘ ❙ t Spatio-Temporal Sampling U ( x, t ) V ( x, t ) V x spatio-temporal lattice Λ M � Ml : l ∈ Z 2 � t R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 21 / 53

➑ ↕ ➋ ➌ ➍ ➎ ➏ ➐ ➭ ➒ ➓ ➔ → ➣ ↔ ➙ ➉ ➛ ➜ ➝ ➞ ➟ ➠ ➡ ➢ ➤ ➥ ➦ ➧ ➨ ➊ ➈ ➫ ❸ ➵ ➳ ➲ ➯ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ❶ ❷ ❹ ➇ ❺ ❻ ❼ ❽ ❾ ❿ ➀ ➁ ➂ ➃ ➄ ➅ ➆ ➩ Spatio-Temporal Sampling U ( x, t ) V ( x, t ) V x sampled field spatio-temporal � lattice Λ M V [ l ] := V ( Ml ) � Ml : l ∈ Z 2 � l ∈ Z 2 t R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 21 / 53

Spatio-Temporal Sampling (cont.) Sampling on Λ M V ( x, t ) �→ � l ∈ Z 2 . V [ l ] := V ( Ml ) , Key Questions When does Λ M allow alias-free sampling of V ( x, t )? 1 Which alias-free sampling lattice has minimal density | det M | ? How can we interpolate V ( x, t ) from its samples � V [ l ]? R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 22 / 53

Alias-Free Sampling Spectral Support of � V [ l ] If S V has a compact support S V , the support of S e V is � � � S e V = S V + k , k ∈ Λ ∗ M � 2 π M − T m : m ∈ Z 2 � where Λ ∗ M := is the spectral lattice. Alias-Free Sampling Λ M allows alias-free sampling of V ( x, t ) if � � ∀ k ∈ Λ ∗ S V ∩ S V + k = ∅ , M \{ 0 } . If S V has a compact support, M can be adjusted to avoid aliasing. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 23 / 53

Example 1 Ω Φ Support S V of S V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 24 / 53

Example 1 Ω Φ Primitive cell B of Λ ∗ M R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 24 / 53

Example 1 Ω Φ Support S e V of S e Aliasing V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 24 / 53

Critical Sampling Lower Bound on Alias-Free Sampling Density 1 | det M | ≥ µ ( S V ) . Critical Sampling Λ M allows critical sampling of V ( x, t ) if Λ M allows alias-free sampling and � � � = R 2 . S V + k k ∈ Λ ∗ M For critical sampling: 1 | det M | = µ ( S V ) . Λ M allows critical sampling of V ( x, t ) if and only if S V = B , where B is a primitive cell of Λ ∗ M . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 25 / 53

Example 1 (cont.) Ω Φ Support S V of S V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 26 / 53

Example 1 (cont.) Ω Φ Primitive cell B 1 of Λ ∗ M 1 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 26 / 53

Example 1 (cont.) Ω Φ Primitive cell B 2 of Λ ∗ M 2 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 26 / 53

Example 1 (cont.) Ω Φ R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 26 / 53

Example 1 (cont.) Ω Φ Support S e V of S e V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 26 / 53

Critical Sampling (cont.) Why Critical Sampling? Reduces the number of deployed sensor nodes. Lowers the computational burden of the sensor nodes. If S V is constant on S V , the samples � V [ l ] are uncorrelated. Illustration R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 27 / 53

Interpolation ∪ -Convex Set A set A ⊆ R 2 is said to be ∪ -convex if A can be written as the union of a finite number of convex sets. Sampling Theorem Assumptions: S V is bounded and has a compact support S V . Λ ∗ M has a ∪ -convex primitive cell B such that S V ⊆ B . � � � Then: � ( x, t ) − Ml V ( x, t ) = l . i . m . V [ l ] s , L →∞ l ∈ Q L for all ( x, t ) ∈ R 2 , and where s := | det M | F − 1 (1 B ); Q L := [ − L, L ] 2 ∩ Z 2 . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 28 / 53

Example 2 Ω Φ Example 2 Support S V of S V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 29 / 53

Interpolation (cont.) Proof. . . . The interpolation formula follows from the uniformly bounded convergence � e j � Ml , Ψ � s ( z − Ml ) = e j � z , Ψ � , lim L →∞ l ∈ Q L for Ψ ∈ B . By Saakyan, it is sufficient to prove that the 2 π M − T -periodic function g defined on B by Ψ �→ e j � z , Ψ � is of bounded harmonic variation. This follows from the ∪ -convexity of B . Proof R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 30 / 53

Outline Introduction 1 Sensor Network Model 2 Physical Fields & Stochastic Source Model 3 Spatio-Temporal Sampling 4 Source Coding Schemes 5 Application – Acoustic Wave Acquisition 6 Conclusions 7 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 31 / 53

Source Coding Schemes ˘ x 0 x K − 1 V [ k, n ] Node group size: K x V Block length: N W 0 W K − 1 Notation: � � Z N k := Z [ k, 0] , . . ., Z [ k, N − 1] Base Station Finite alphabet: A � V [ k, n ] � ˘ � � � 0 , . . . , ˘ V N V N Encoding: �→ W 0 , . . . , W K − 1 ∈ A K − 1 � � � � � 0 , . . . , � V N V N Decoding: W 0 , . . . , W K − 1 �→ K − 1 Average bit rate: R ( K,N ) := log |A| KN �� � 2 � K − 1 � � 1 Average distortion: D ( K,N ) := � ˘ − � V N V N E k k KN k =0 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 32 / 53

Source Coding Schemes (cont.) Source coding schemes: inter-node correlation taken communication into account multiterminal none spatio-temporal centralized free spatio-temporal spatially independent none temporal A rate distorion pair ( R, D ) is said to be achievable if, for every ǫ > 0, there exist K and N , and a K -terminal source code with block length N such that R ( K,N ) ≤ R + ǫ ; D ( K,N ) ≤ D + ǫ. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 33 / 53

Multiterminal Source Coding V N ˘ 0 Encoder 0 W 0 � � � 0 , . . . , � V N V N K − 1 · Decoder · · V N ˘ W K − 1 K − 1 Encoder K − 1 The nodes do not communicate while encoding their samples. The decoding at the base station is performed jointly. Rate Distortion Function Complete characterization has not been found to date. Has been determined for special cases. Inner and outer bounds for the general case have been proved. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 34 / 53

Berger-Tung Source Coding ˘ V N 0 VQ 0 SW Enc. 0 W 0 � � � 0 , . . . , � V N V N K − 1 · Decoder · · ˘ V N W K − 1 K − 1 VQ SW Enc. K − 1 K − 1 Each node first vector quantizes its samples without cooperation. The correlated outputs are then compressed via Slepian-Wolf enc.. Rate Distortion Function � � �� 1 1 1 1 + S ˘ V ( φ, ω ) R P = 2 log d φ d ω (2 π ) 2 | det M | P ( ω ) ( − π,π ] 2 �� S ˘ V ( φ, ω ) P ( ω ) 1 D P = V ( φ, ω ) + P ( ω ) d φ d ω P : ( − π,π ] → [0 , + ∞ ] (2 π ) 2 S ˘ ( − π,π ] 2 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 35 / 53

Centralized Source Coding ˘ V N 0 � � � 0 , . . . , � V N V N W K − 1 · Encoder Decoder · · ˘ V N K − 1 All the spatio-termporal samples are available at a single encoder. Rate Distortion Function � � �� 1 1 0 , 1 2 log S ˘ V ( φ, ω ) R θ = ( − π,π ] 2 max d φ d ω (2 π ) 2 | det M | θ �� � � 1 D θ = ( − π,π ] 2 min θ, S ˘ V ( φ, ω ) d φ d ω (2 π ) 2 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 36 / 53

Spatially Independent Source Coding � V N ˘ V N W 0 0 0 Encoder 0 Decoder 0 · · · · · · ˘ � V N V N W K − 1 K − 1 K − 1 Encoder K − 1 Decoder K − 1 Both the encoding at the sensors and the decoding at the base station are performed without cooperation. Rate Distortion Function � � � 1 1 0 , 1 2 log S 0 ( ω ) R θ = max d ω | det M | 2 π θ ( − π,π ] � � � D θ = 1 R min θ, S 0 ( ω ) d ω where S 0 ( ω )= 1 − ( π,π ] S ˘ V ( φ,ω ) d φ 2 π 2 π ( − π,π ] R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 37 / 53

Source Coding Schemes (Summary) Note The source coding schemes may be ordered by increasing rate distortion regions: spatially independent coding Berger-Tung coding multiterminal coding centralized coding Summary R cent ( D ) ≤ R mult ( D ) ≤ R B-T ( D ) ≤ R spin ( D ) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 38 / 53

Outline Introduction 1 Sensor Network Model 2 Physical Fields & Stochastic Source Model 3 Spatio-Temporal Sampling 4 Source Coding Schemes 5 Application – Acoustic Wave Acquisition 6 Conclusions 7 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 39 / 53

Acoustic Wave Acquisition U ( ξ, τ ) ξ U d g ( x − ξ, t − τ ) V ( x, t ) x V � Base Station V ( x, t ) Sound sources U ( ξ, τ ) on U induce an acoustic field V ( x, t ) on V . Microphones on V sample V ( x, t ), quantize the samples, and transmit the encoded samples to the base station. The base station produces an estimate � V ( x, t ). R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 40 / 53

Plenacoustic Function Green’s function: Fourier transform: � � � � �� √ � 2 − Φ 2 g (Φ , Ω) = − j d 2 + x 2 t − 1 δ 4 H ∗ � d Ω /c c 1 , 0 √ g ( x, t ) = d 2 + x 2 4 π for Ω > 0 � � � � � Φ � < � Ω � /c . Approximate passband: R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 41 / 53

Far-Field Approximation Assumptions: The sound sources are located far away from the microphones. = ⇒ The waves arriving on V appear as plane waves. There is no attenuation. Support of � g (Φ , Ω): Ω c Exact passband: Φ 1 � � � � � < � Φ � Ω � /c . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 42 / 53

Stochastic Source Model U ( x, t ): homogeneous, Gaussian random field with a bounded spectral density S U of compact support. V ( x, t ): solution of the linear PDE ∂t 2 V ( x, t ) = ∂ 2 ∂ 2 1 ∂x 2 V ( x, t ) + U ( x, t ) . c 2 Spectral relation: � � � 2 S U (Φ , Ω) . �� S V (Φ , Ω) = g (Φ , Ω) Ω 0 Ω 0 Ω Ω Ω Φ = Ω /c Φ Φ Φ ∩ = Φ 0 = Ω 0 /c � Φ 0 B q Support S U of S U Support of � g Support S V of S V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 43 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ π/ Ω 0 2Ω 0 Support S V of S V πc/ Ω 0 2Ω 0 /c � � V ( x, t ) is bandlimited to the frequencies Ω 0 /c, Ω 0 . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ π/ Ω 0 2Ω 0 Primitive cell B r of Λ ∗ Rectangular lattice Λ M r πc/ Ω 0 2Ω 0 /c M r Λ M r allows alias-free sampling if inter-microphone spacing X 0 = πc/ Ω 0 ; sampling period T 0 = π/ Ω 0 . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ Support S e V r of S e Rectangular lattice Λ M r V r Λ M r does not allow critical sampling. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ Subsample by 2 Each microphone records only every other sample. Adjacent microphones operate with an offset of T 0 . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ Primitive cell B q of Λ ∗ Quincunx lattice Λ M q M q Λ M q and Λ M ∗ q are rhombic lattices. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ Quincunx lattice Λ M q Copies of S V generated by the rectangular lattice. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ Support S e V q of S e Quincunx lattice Λ M q V q Λ M q allows critical sampling. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ Quincunx lattice Λ M q Spectral support of the interpolating function s q s q := | det M q | F − 1 (1 B q ) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Spatio-Temporal Sampling Spatio-temporal plane Frequency plane t Ω x Φ V ( x, t ) is recovered by interpolation. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 44 / 53

Source Coding Schemes Multiterminal coding: Microphones do not communicate with each other. R mult ( D ) is unknown. Centralized coding: Performance is not affected by the sampling lattice. R cent ( D ) is a lower bound for R mult ( D ). Spatially independent coding: Performance depends on how strongly the samples are correlated in space. R spin ( D ) is an upper bound for R mult ( D ). Summary R cent ( D ) ≤ R mult ( D ) ≤ R spin ( D ) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 45 / 53

Rate Distortion Functions for a White Sound Field Assumption: S V (Φ , Ω) = σ 2 V 1 B q (Φ , Ω) Centralized coding: � σ 2 � R cent ( D ) = Ω 2 V Ω 2 0 0 4 π 2 c log 2 π 2 cD Spatially independent coding, rectangular sampling lattice: � � � �� R spin,r ( D ) = Ω 2 σ 2 V Ω 2 1 − 2 π 2 cD 1 0 0 2 π 2 c log 1 + eπ 2 cD σ 2 V Ω 2 2 0 � � � + Ω 2 1 − 2 π 2 cD 0 1 − σ 2 V Ω 2 2 π 2 c 0 Spatially independent coding, quincunx sampling lattice: � σ 2 � R spin,q ( D ) = Ω 2 V Ω 2 0 0 4 π 2 c log 2 π 2 cD R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 46 / 53

Rate Distortion Functions for a White Sound Field (cont.) Corollary R cent ( D ) = R mult ( D ) = R spin,q ( D ) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 47 / 53

Rate Distortion Functions for a White Sound Field (cont.) 400 centralized coding spat. ind. coding, rectangular lattice 350 spat. ind. coding, quincunx lattice 300 Rate (kb/(m ⋅ s)) 250 200 150 100 50 0 0 50 100 150 200 250 Distortion (MSE/(m ⋅ s)) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 48 / 53

Rate Distortion Functions for a White Source � � � 2 1 B q (Φ , Ω) �� Assumption: S V (Φ , Ω) = σ 2 g (Φ , Ω) U 400 centralized coding spat. ind. coding, rectangular lattice 350 spat. ind. coding, quincunx lattice 300 Rate (kb/(m ⋅ s)) 250 200 150 100 50 0 0 50 100 150 200 250 Distortion (MSE/(m ⋅ s)) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 49 / 53

Outline Introduction 1 Sensor Network Model 2 Physical Fields & Stochastic Source Model 3 Spatio-Temporal Sampling 4 Source Coding Schemes 5 Application – Acoustic Wave Acquisition 6 Conclusions 7 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 50 / 53

Summary Interplay between physics and the source coding performance of sensor networks. Stochastic model for physical fields that is amenable to the tools of information theory. Efficient spatio-temporal sampling geometries and sampling theorem for bandlimited random fields. Applications: Wave acquisition: quincunx sampling exempts from multiterminal source coding. Temperature monitoring: some multiterminal binning is necessary. Random walk: simple scalar algorithm outperforms vector quantization. Take-Home Message Tailor your communication scheme to the physics of the problem. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 51 / 53

Future Research Sampling and compression of non-bandlimited fields: Real physical fields are not perfectly bandlimited. No spatial low-pass filtering can be applied to avoid aliasing. Sensitivity of the results to uncertainties in the sensors’ locations: Regular sampling lattices require a supervised sensor deployment. In practice, sensor nodes may be dropped from a plane. Trade-off between the spatial and temporal sampling densities: Increasing the spatial sampling density is often a lot more expensive than increasing the temporal sampling rate. Rate-accuracy trade-offs for estimating a functional of the physical field (e.g., average temperature); the source field U ( x, t ). R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 52 / 53

Thanks for your attention R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 53 / 53

Appendix R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 54 / 53

Outline Application – Temperature Monitoring 8 Application – Random Walks 9 10 Critical Sampling – Illustration 11 Spatio-Temporal Sampling – Example 2 12 Sampling Theorem – Proof 13 Aliasing in Digital Photography 14 Far-Field Approximation 15 Rate Distortion Functions Without Shaping Gain R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 55 / 53

Outline Application – Temperature Monitoring 8 Application – Random Walks 9 10 Critical Sampling – Illustration 11 Spatio-Temporal Sampling – Example 2 12 Sampling Theorem – Proof 13 Aliasing in Digital Photography 14 Far-Field Approximation 15 Rate Distortion Functions Without Shaping Gain R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 56 / 53

Temperature Monitoring U ( x, t ) V ( x, t ) x δ � Base Station V ( x, t ) Heat sources U ( x, t ) generate a temperature field V ( x, t ) inside a homogeneous rod. Sensor nodes sample V ( x, t ), quantize the samples, and transmit the encoded samples to the base station. The base station produces an estimate � V ( x, t ). ∂tV ( x, t ) = ∂ 2 Heat equation: 1 ∂ ∂x 2 V ( x, t ) − µV ( x, t ) + 1 κU ( x, t ) α R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 57 / 53

Green’s Function Green’s function: Fourier transform: 8 1 1 > e − αµt e − x 2 / (4 αt ) < √ if t > 0 g (Φ , Ω) = b α Φ 2 + j Ω + αµ 4 παt g ( x, t ) = > : 0 if t ≤ 0 g is absolutely integrable on R 2 . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 58 / 53

Stochastic Source Model and Spatio-Temporal Sampling U ( x, t ): homogeneous Gaussian random field with a bounded spectral density S U of support � � � � S U = − Φ 0 , Φ 0 × − Ω 0 , Ω 0 . �� Define V ( x, t ) := α R 2 g ( ξ, τ ) U ( x − ξ, t − τ ) d ξ d τ. κ Spectral relation: S V (Φ , Ω) = α 2 � � � 2 S U (Φ , Ω) . �� g (Φ , Ω) κ 2 � � V ( x, t ) is bandlimited to the frequencies Φ 0 , Ω 0 . The rectangular lattice Λ M allows alias-free sampling if inter-node spacing X 0 = π/ Φ 0 ; sampling period T 0 = π/ Ω 0 . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 59 / 53

Source Coding Schemes Multiterminal coding. Centralized coding. Spatially independent coding. Berger-Tung coding: Based on separate vector quantization followed by Slepian-Wolf compression. Provides an upper bound to the multiterminal R-D function. Predictive quantization: Uses the samples’ spatial correlation through prediction at the base station. Based on scalar quantization. Relies on the existence of ideal feedback channels. Provides an upper bound to the multiterminal R-D function. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 60 / 53

Predictive Quantization � E [ − ( K − 1) / 2 , n ] E [ − ( K − 1) / 2 , n ] ˘ � V [ − ( K − 1) / 2 , n ] V [ − ( K − 1) / 2 , n ] + + Encoder 0 Decoder 0 − ¯ V [ − ( K − 1) / 2 , n ] · · Predictor · · · · ¯ V [( K − 1) / 2 , n ] − + + Encoder K − 1 Decoder K − 1 ˘ � V [( K − 1) / 2 , n ] V [( K − 1) / 2 , n ] � E [( K − 1) / 2 , n ] E [( K − 1) / 2 , n ] Rate Distortion Function » – » – 2 log σ 2 2 log σ 2 Ω 0 0 , 1 ≤ R pq ( D ) ≤ Ω 0 0 , 1 + Ω 0 1 2 log πe E E X 0 π max X 0 π max 6 D D X 0 π Z „ 1 Z « E = 1 σ 2 exp log S ˘ V ( φ, ω ) d ω d φ 2 π 2 π ( − π,π ] ( − π,π ] R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 61 / 53

Rate Distortion Functions Assumption: S U (Φ , Ω) = σ 2 U 1 S U (Φ , Ω) 0.45 Berger−Tung coding centralized coding 0.4 spatially independent coding predictive quantization 0.35 0.3 Rate (bits/(m ⋅ s)) 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 Distortion (MSE/(m ⋅ s)) R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 62 / 53

Outline Application – Temperature Monitoring 8 Application – Random Walks 9 10 Critical Sampling – Illustration 11 Spatio-Temporal Sampling – Example 2 12 Sampling Theorem – Proof 13 Aliasing in Digital Photography 14 Far-Field Approximation 15 Rate Distortion Functions Without Shaping Gain R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 63 / 53

Random Walk on a Ring p i,i − 1 2 U [ n ] 1 p i,i +1 BS 0 � V [ k, n ] M − 1 V [ k, n ] Alice visits pubs located on Ring Street. (At time nT 0 , she is at pub U [ n ].) After leaving a pub, she randomly chooses one of the neighboring pubs as her next destination. At each pub, a sensor detects the presence of Alice, encodes the information and transmits it to the base station. ( V [ k, n ] ∈ { 0 , 1 } .) At the base station, Bob reconstructs the sequence of Alice’s visits. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 64 / 53

Stochastic Source Model p i,i − 1 � � 2 P U [ n ] = m = 1 /M, U [ n ] 1 ∀ n ∈ Z , ∀ m ∈ { 0 , . . . , M − 1 } p i,i +1 � 1 if j − i ≡ ± 1 (mod M ) 2 p i,j = BS 0 0 otherwise � V [ k, n ] V [ k, n ] M − 1 Alice spends a fixed amount of time T 0 at each pub. Traveling time is negligible. Alice’s decisions are independent of any past visits. = ⇒ U [ n ] is a stationary Markov chain. � Define 1 if U [ n ] = k, V [ k, n ] := 0 otherwise . R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 65 / 53

Source Compression Schemes Lossless Source Compression Problem Determine the minimal sum rate R at which the probability of error at the base station can be made arbitrarily small. Distributed compression: Sensors do not communicate with each other. Slepian-Wolf: achieves the same rate as centralized compression. Centralized compression: A single encoder has access to all V [ k, n ], and hence to U [ n ]. AEP: R c = 1 bit/time unit. Spatially independent compression: Sensor k encodes V k [ n ] := V [ k, n ], ignoring spatial correlation. R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 66 / 53

Minimal Sum Rates 3.5 3 2.5 Rate (bits/time unit) 2 1.5 1 0.5 centralized compression distributed compression spatially independent compression 0 5 10 15 20 25 30 35 40 Number of states M R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 67 / 53

Zero-Delay Distributed Compression Algorithm Assumptions: M is a multiple of 4. MAC of sum capacity C = 1 bit/time unit: Y [ n ] = � k X k [ n ] mod 2 X k [ n ] Sensors Base Station X k +1 [ n ] Algorithm: At time nT 0 , all sensors � = U [ n ] transmit 0. U [ n − 1] even U [ n − 1] odd U [ n ] ≡ 1 (mod 4) U [ n ] ≡ 3 (mod 4) U [ n ] ≡ 0 (mod 4) U [ n ] ≡ 2 (mod 4) Sensor U [ n ] sends 0 1 0 1 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 68 / 53

Outline Application – Temperature Monitoring 8 Application – Random Walks 9 10 Critical Sampling – Illustration 11 Spatio-Temporal Sampling – Example 2 12 Sampling Theorem – Proof 13 Aliasing in Digital Photography 14 Far-Field Approximation 15 Rate Distortion Functions Without Shaping Gain R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 69 / 53

Illustration Spectral lattice Λ ∗ Lattice Λ M M z 2 Ψ 2 z 1 Ψ 1 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 70 / 53

Illustration Spectral lattice Λ ∗ Lattice Λ M M z 2 Ψ 2 z 1 Ψ 1 Density halved. Density doubled. Return R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 70 / 53

Outline Application – Temperature Monitoring 8 Application – Random Walks 9 10 Critical Sampling – Illustration 11 Spatio-Temporal Sampling – Example 2 12 Sampling Theorem – Proof 13 Aliasing in Digital Photography 14 Far-Field Approximation 15 Rate Distortion Functions Without Shaping Gain R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 71 / 53

Example 2 Ω Φ Support S V of S V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53

Example 2 Ω Φ Primitive cell B 1 of Λ ∗ M 1 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53

Example 2 Ω Φ R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53

Example 2 Ω Φ R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53

Example 2 Ω Φ Primitive cell B 2 of Λ ∗ M 2 R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53

Example 2 Ω Φ R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53

Example 2 Ω Φ Support S e V of S e V R. L. Konsbruck Source Coding in Sensor Networks November 3, 2009 72 / 53