Source Coding in Sensor Networks Robert L. Konsbruck School of - - PowerPoint PPT Presentation
COLE POLYTECHNIQUE FDRALE DE LAUSANNE Source Coding in Sensor Networks Robert L. Konsbruck School of Computer and Communication Sciences (I&C) Ecole Polytechnique F ed erale de Lausanne (EPFL) Joint Work with Prof. Emre
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
Robert L. Konsbruck
School of Computer and Communication Sciences (I&C) ´ Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL) Joint Work with
Research Seminar Luxembourg, November 3, 2009
Source Coding in Sensor Networks November 3, 2009 1 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
Source Coding in Sensor Networks November 3, 2009 2 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
Source Coding in Sensor Networks November 3, 2009 3 / 53
BS
Large number of small embedded devices monitor a physical field (temperature, sound, . . .). The sensor nodes
collect samples; − → sampling process these samples; − → source coding transmit the data to a base station. − → channel coding
The base station produces an estimate of the field. Goal: minimize the used resources for a given accuracy of the
→ rate distortion function
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Sensor nodes are spatially localized objects. = ⇒ inherent spatial sampling Data results from sampling a physical phenomenon. = ⇒ spatio-temporal correlation determined by the laws of physics Sensor nodes have limited energy resources. = ⇒ no transmission of redundant information Wireless communication is the dominant factor in the energy budget. = ⇒ distributed processing Wireless sensor nodes rely on a shared communication medium. = ⇒ efficient utilization of the available bandwidth
Source Coding in Sensor Networks November 3, 2009 5 / 53
Spatio-Temporal Sampling
How to choose the samples to be collected by the sensor nodes?
Source Coding
How to efficiently represent the data gathered by the sensor nodes?
Channel Coding
How to reliably transmit this data to the base station? No separation theorem for general sensor networks. Joint source-channel codes may significantly outperform separation-based schemes.
Source Coding in Sensor Networks November 3, 2009 6 / 53
Spatio-Temporal Sampling
How to choose the samples to be collected by the sensor nodes?
Source Coding
How to efficiently represent the data gathered by the sensor nodes?
Channel Coding
How to reliably transmit this data to the base station? No separation theorem for general sensor networks. Joint source-channel codes may significantly outperform separation-based schemes. Scaling-law separation theorem for “expanding” sensor networks.
Source Coding in Sensor Networks November 3, 2009 6 / 53
Spatio-Temporal Sampling
How to choose the samples to be collected by the sensor nodes?
Source Coding
How to efficiently represent the data gathered by the sensor nodes?
Channel Coding
How to reliably transmit this data to the base station? No separation theorem for general sensor networks. Joint source-channel codes may significantly outperform separation-based schemes. Scaling-law separation theorem for “expanding” sensor networks. Digital architectures prevail in off-the-shelf sensor networks.
Source Coding in Sensor Networks November 3, 2009 6 / 53
Spatio-Temporal Sampling
How to choose the samples to be collected by the sensor nodes?
Source Coding
How to efficiently represent the data gathered by the sensor nodes?
Channel Coding
How to reliably transmit this data to the base station? No separation theorem for general sensor networks. Joint source-channel codes may significantly outperform separation-based schemes. Scaling-law separation theorem for “expanding” sensor networks. Digital architectures prevail in off-the-shelf sensor networks.
Source Coding in Sensor Networks November 3, 2009 6 / 53
Source Coding in Sensor Networks
Determine the optimal trade-off between: the quality of the base station’s estimate; the amount of resources used by the sensor nodes; in “expanding” sensor network setups. Study the effect of the physics of the observed field upon:
the efficiency of spatio-temporal sampling schemes; the performance of source coding strategies.
Provide a mathematical basis for studying source-coding-related issues in sensor networks.
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Multiterminal Source Coding
Considers distributed encoding and joint decoding of a fixed number
Determines inner and outer bounds to the rate distortion region (Berger-Tung). Solves the two-terminal Gaussian/MSE problem (Wagner et al.).
“Refining” Sensor Networks
Considers densely scattered sensor nodes monitoring a temporally uncorrelated phenomenon with relatively few degrees of freedom. Shows the feasibility of distributed encoding of the samples at a rate that stays only a constant away from the minimal rate required by joint encoding (Kashyap et al., Neuhoff et al.).
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MAC-based Sensor Networks
Considers joint source-channel coding of temporally uncorrelated sources over multiple-access channels. Indicates that joint source-channel communication may lead to significant energy savings with respect to separation-based schemes (Gastpar et al.). Uses the spatial averaging operation inherent in the MAC for projecting sparse/compressible fields onto appropriate basis functions (“compressed sensing”, Bajwa et al.).
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1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
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Base Station
V V (xk, tk,n) V (x, t)
Physical field V (x, t) monitored in a region V. “Expanding” network: fixed average sensor density. Sensors communicate through digital channels. = ⇒
sampling: V (x, t) → V (xk, tk,n) quantization:
Base station produces an estimate V (x, t).
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Base Station
V V (xk, tk,n) V (x, t)
The field V (x, t) is bandlimited. = ⇒ #(degrees of freedom) grows linearly with the dimensions of V. Channels are perfect bit pipes. = ⇒ Bits are transmitted without error. Channels are parallel. = ⇒ No interference between neighboring nodes’ transmissions.
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Spatio-Temporal Sampling
When is a discrete-space and -time representation of the analog field sufficient? How efficient is such a representation? How can we reconstruct the analog field from its samples?
Source Coding
What amount of information is lost by a discrete-amplitude representation of the analog field? What is the minimal average bit rate for a given average MSE distortion?
Source Coding in Sensor Networks November 3, 2009 13 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
Source Coding in Sensor Networks November 3, 2009 14 / 53
U V U(ξ, τ) g(x − ξ, t − τ) V (x, t)
Assumption: The observed field V (x, t) is generated by a source field U(x, t) according to a linear PDE: L V = U Examples:
Heat diffusion: L = ∂ ∂t − ∆x Wave propagation: L = ∂2 ∂t2 − ∆x
Under appropriate symmetry and regularity conditions: V (x, t) =
g(x − ξ, t − τ) U(ξ, τ) dτdξ g: Green’s function
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g(x − ξ, t − τ) U(ξ, τ) V (x, t) x ξ V U
Assumption: U and V are one-dimensional domains mapped to R. By appropriately redefining U, V and g: V (x, t) =
Note
The PDE acts as a linear, shift-invariant (LSI) filter with impulse response g.
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The source field U(x, t) is a
real-valued, zero-mean, homogeneous (stationary), Gaussian
random field with covariance function KU. Assumptions:
U(x, t) is separable and measurable. U(x, t) admits a spectral density SU: KU(ξ, τ) = 1 (2π)2
SU is bounded and has a compact support.
Source Coding in Sensor Networks November 3, 2009 17 / 53
g(x, t) U(x, t) V (x, t) =
Theorem
If g is absolutely integrable, then V (x, t) is a well defined random variable. Further: SV (Φ, Ω) =
where g denotes the Fourier transform of g. V (x, t) is a homogeneous, Gaussian random field. Since g is bounded, SV is bounded and has a compact support.
Source Coding in Sensor Networks November 3, 2009 18 / 53
Proof.
E
≤
dτdξ =
Hence:
is finite with probability one.
Source Coding in Sensor Networks November 3, 2009 19 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
Source Coding in Sensor Networks November 3, 2009 20 / 53
U(x, t) V (x, t) V x
Source Coding in Sensor Networks November 3, 2009 21 / 53
U(x, t) V (x, t) V x t
Source Coding in Sensor Networks November 3, 2009 21 / 53
U(x, t) V (x, t) V x t
independent temporal lattices
Source Coding in Sensor Networks November 3, 2009 21 / 53
U(x, t) V (x, t) V x t
❁ ❂ ❃ ❄ ❅ ❆ ❇ ❈ ❉ ❊ ❋spatio-temporal lattice ΛM
Source Coding in Sensor Networks November 3, 2009 21 / 53
U(x, t) V (x, t) V x t
④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸ ❹ ❺ ❻ ❼ ❽ ❾ ❿ ➀ ➁ ➂ ➃ ➄ ➅ ➆ ➇ ➈ ➉ ➊ ➋ ➌ ➍ ➎ ➏ ➐ ➑ ➒ ➓ ➔ → ➣ ↔ ↕ ➙ ➛ ➜ ➝ ➞ ➟ ➠ ➡ ➢ ➤ ➥ ➦ ➧ ➨ ➩ ➫ ➭ ➯ ➲ ➳ ➵spatio-temporal lattice ΛM
sampled field
l ∈ Z2
Source Coding in Sensor Networks November 3, 2009 21 / 53
Sampling on ΛM
V (x, t) → V [l] := V (Ml), l ∈ Z2.
Key Questions
When does ΛM allow alias-free sampling of V (x, t)? Which alias-free sampling lattice has minimal density
1 | det M|?
How can we interpolate V (x, t) from its samples V [l]?
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Spectral Support of V [l]
If SV has a compact support SV , the support of S e
V is
Se
V =
M
where Λ∗
M :=
is the spectral lattice.
Alias-Free Sampling
ΛM allows alias-free sampling of V (x, t) if SV ∩
∀ k ∈ Λ∗
M\{0}.
If SV has a compact support, M can be adjusted to avoid aliasing.
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Φ Ω Support SV of SV
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Φ Ω Primitive cell B of Λ∗
M
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Φ Ω Support Se
V of Se V
Aliasing
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Lower Bound on Alias-Free Sampling Density
1 | det M| ≥ µ(SV ).
Critical Sampling
ΛM allows critical sampling of V (x, t) if ΛM allows alias-free sampling and
M
For critical sampling: 1 | det M| = µ(SV ). ΛM allows critical sampling of V (x, t) if and only if SV = B, where B is a primitive cell of Λ∗
M.
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Φ Ω Support SV of SV
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Φ Ω Primitive cell B1 of Λ∗
M1
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Φ Ω Primitive cell B2 of Λ∗
M2
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Φ Ω
Source Coding in Sensor Networks November 3, 2009 26 / 53
Φ Ω Support Se
V of Se V
Source Coding in Sensor Networks November 3, 2009 26 / 53
Why Critical Sampling?
Reduces the number of deployed sensor nodes. Lowers the computational burden of the sensor nodes. If SV is constant on SV , the samples V [l] are uncorrelated.
Illustration
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∪-Convex Set
A set A ⊆ R2 is said to be ∪-convex if A can be written as the union of a finite number of convex sets.
Sampling Theorem
Assumptions:
SV is bounded and has a compact support SV . Λ∗
M has a ∪-convex primitive cell B such that SV ⊆ B.
Then: V (x, t) = l.i.m.
L→∞
for all (x, t) ∈ R2, and where
s := | det M| F−1(1B); QL := [−L, L]2 ∩ Z2.
Source Coding in Sensor Networks November 3, 2009 28 / 53
Φ Ω Support SV of SV
Example 2
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Proof.
. . . The interpolation formula follows from the uniformly bounded convergence lim
L→∞
ejMl,Ψ s(z − Ml) = ejz,Ψ, for Ψ ∈ B. By Saakyan, it is sufficient to prove that the 2πM −T-periodic function g defined on B by Ψ → ejz,Ψ is of bounded harmonic variation. This follows from the ∪-convexity of B.
Proof
Source Coding in Sensor Networks November 3, 2009 30 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
Source Coding in Sensor Networks November 3, 2009 31 / 53
Base Station
˘ V [k, n] x V x0 xK−1 W0 WK−1 Node group size: K Block length: N Notation: ZN
k :=
Encoding: ˘ V N
0 , . . . , ˘
V N
K−1
Decoding:
V N
0 , . . . ,
V N
K−1
KN Average distortion: D(K,N) := 1 KN
K−1
E
V N
k
− V N
k
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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.
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· · · Decoder Encoder 0 Encoder K−1 V N
0 , . . . ,
V N
K−1
W0 ˘ V N
K−1
˘ V N
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.
Source Coding in Sensor Networks November 3, 2009 34 / 53
· · · VQ K−1 Decoder V N
0 , . . . ,
V N
K−1
V N
K−1
˘ V N SW Enc. 0 SW Enc. K−1 VQ 0 W0 WK−1
Each node first vector quantizes its samples without cooperation. The correlated outputs are then compressed via Slepian-Wolf enc..
Rate Distortion Function
RP = 1 | det M| 1 (2π)2
1 2 log
V (φ, ω)
P(ω)
DP = 1 (2π)2
S ˘
V (φ, ω)P(ω)
S ˘
V (φ, ω) + P(ω) dφdω P : (−π,π] → [0,+∞]
Source Coding in Sensor Networks November 3, 2009 35 / 53
· · · Decoder Encoder W V N
0 , . . . ,
V N
K−1
V N ˘ V N
K−1
All the spatio-termporal samples are available at a single encoder.
Rate Distortion Function
Rθ = 1 | det M| 1 (2π)2
2 log S ˘
V (φ, ω)
θ
Dθ = 1 (2π)2
V (φ, ω)
Source Coding in Sensor Networks November 3, 2009 36 / 53
WK−1 Encoder K−1 Decoder K−1
K−1
˘ V N
K−1
· · · · · · Encoder 0 Decoder 0 W0
˘ V N
Both the encoding at the sensors and the decoding at the base station are performed without cooperation.
Rate Distortion Function
Rθ = 1 | det M| 1 2π
max
2 log S0(ω) θ
Dθ = 1 2π
min
where S0(ω)= 1
2π
R
−(π,π] S ˘ V (φ,ω) dφ
Source Coding in Sensor Networks November 3, 2009 37 / 53
Note
The source coding schemes may be ordered by increasing rate distortion regions: spatially independent coding Berger-Tung coding multiterminal coding centralized coding
Summary
Rcent(D) ≤ Rmult(D) ≤ RB-T(D) ≤ Rspin(D)
Source Coding in Sensor Networks November 3, 2009 38 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
Source Coding in Sensor Networks November 3, 2009 39 / 53
g(x − ξ, t − τ) U(ξ, τ) V (x, t) x ξ V U d Base Station
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).
Source Coding in Sensor Networks November 3, 2009 40 / 53
Green’s function: g(x, t) = δ
c
√ d2 + x2
√ d2 + x2 Fourier transform:
4 H∗
1,0
2 − Φ2
Approximate passband:
Source Coding in Sensor Networks November 3, 2009 41 / 53
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):
Φ Ω 1 c Exact passband:
Source Coding in Sensor Networks November 3, 2009 42 / 53
U(x, t): homogeneous, Gaussian random field with a bounded spectral density SU of compact support. V (x, t): solution of the linear PDE 1 c2 ∂2 ∂t2 V (x, t) = ∂2 ∂x2 V (x, t) + U(x, t). Spectral relation: SV (Φ, Ω) =
Φ Ω
Ω0 Support SU of SU Φ Ω Φ = Ω/c Support of g Φ Ω Φ0 = Ω0/c Ω0 Support SV of SV Bq
Source Coding in Sensor Networks November 3, 2009 43 / 53
Spatio-temporal plane x t πc/Ω0 π/Ω0 Frequency plane Φ Ω 2Ω0/c 2Ω0 Support SV of SV V (x, t) is bandlimited to the frequencies
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t πc/Ω0 π/Ω0 Rectangular lattice ΛMr Frequency plane Φ Ω 2Ω0/c 2Ω0 Primitive cell Br of Λ∗
Mr
ΛMr allows alias-free sampling if
inter-microphone spacing X0 = πc/Ω0; sampling period T0 = π/Ω0.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t Rectangular lattice ΛMr Frequency plane Φ Ω Support Se
Vr of Se Vr
ΛMr does not allow critical sampling.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t Subsample by 2 Frequency plane Φ Ω Each microphone records only every other sample. Adjacent microphones operate with an offset of T0.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t Quincunx lattice ΛMq Frequency plane Φ Ω Primitive cell Bq of Λ∗
Mq
ΛMq and ΛM∗
q are rhombic lattices.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t Quincunx lattice ΛMq Frequency plane Φ Ω Copies of SV generated by the rectangular lattice.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t Quincunx lattice ΛMq Frequency plane Φ Ω Support Se
Vq of Se Vq
ΛMq allows critical sampling.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Spatio-temporal plane x t Quincunx lattice ΛMq Frequency plane Φ Ω Spectral support of the interpolating function sq sq := | det Mq| F−1(1Bq)
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Spatio-temporal plane x t Frequency plane Φ Ω V (x, t) is recovered by interpolation.
Source Coding in Sensor Networks November 3, 2009 44 / 53
Multiterminal coding:
Microphones do not communicate with each other. Rmult(D) is unknown.
Centralized coding:
Performance is not affected by the sampling lattice. Rcent(D) is a lower bound for Rmult(D).
Spatially independent coding:
Performance depends on how strongly the samples are correlated in space. Rspin(D) is an upper bound for Rmult(D).
Summary
Rcent(D) ≤ Rmult(D) ≤ Rspin(D)
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Assumption: SV (Φ, Ω) = σ2
V 1Bq(Φ, Ω)
Centralized coding: Rcent(D) = Ω2 4π2c log σ2
V Ω2
2π2cD
Rspin,r(D) = Ω2 2π2c log
V Ω2
eπ2cD 1 2
σ2
V Ω2
2π2c
σ2
V Ω2
Rspin,q(D) = Ω2 4π2c log σ2
V Ω2
2π2cD
Source Coding in Sensor Networks November 3, 2009 46 / 53
Corollary
Rcent(D) = Rmult(D) = Rspin,q(D)
Source Coding in Sensor Networks November 3, 2009 47 / 53
50 100 150 200 250 50 100 150 200 250 300 350 400
Distortion (MSE/(m⋅s)) Rate (kb/(m⋅s))
centralized coding
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Assumption: SV (Φ, Ω) = σ2
U
50 100 150 200 250 50 100 150 200 250 300 350 400
Distortion (MSE/(m⋅s)) Rate (kb/(m⋅s))
centralized coding
Source Coding in Sensor Networks November 3, 2009 49 / 53
1
Introduction
2
Sensor Network Model
3
Physical Fields & Stochastic Source Model
4
Spatio-Temporal Sampling
5
Source Coding Schemes
6
Application – Acoustic Wave Acquisition
7
Conclusions
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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.
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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).
Source Coding in Sensor Networks November 3, 2009 52 / 53
Source Coding in Sensor Networks November 3, 2009 53 / 53
Source Coding in Sensor Networks November 3, 2009 54 / 53
8
Application – Temperature Monitoring
9
Application – Random Walks
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
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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U(x, t) V (x, t) x δ Base Station
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). Heat equation: 1
α ∂ ∂tV (x, t) = ∂2 ∂x2 V (x, t) − µV (x, t) + 1 κU(x, t)
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Green’s function:
g(x, t) = 8 > < > : 1 √ 4παt e−αµt e−x2/(4αt) if t > 0 if t ≤ 0
Fourier transform:
b g(Φ, Ω) = 1 αΦ2 + jΩ + αµ
g is absolutely integrable on R2.
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U(x, t): homogeneous Gaussian random field with a bounded spectral density SU of support SU =
Define V (x, t) := α κ
Spectral relation: SV (Φ, Ω) = α2 κ2
V (x, t) is bandlimited to the frequencies
The rectangular lattice ΛM allows alias-free sampling if
inter-node spacing X0 = π/Φ0; sampling period T0 = π/Ω0.
Source Coding in Sensor Networks November 3, 2009 59 / 53
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.
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+ + + + · · · · · · Encoder 0 Decoder 0 Predictor − − Decoder K−1 Encoder K−1 E[−(K−1)/2, n]
E[(K−1)/2, n]
¯ V [(K−1)/2, n]
¯ V [−(K−1)/2, n] ˘ V [−(K−1)/2, n] ˘ V [(K−1)/2, n]
Rate Distortion Function
Ω0 X0π max » 0, 1 2 log σ2
E
D – ≤ Rpq(D) ≤ Ω0 X0π max » 0, 1 2 log σ2
E
D – + Ω0 X0π 1 2 log πe 6 σ2
E = 1
2π Z
(−π,π]
exp „ 1 2π Z
(−π,π]
log S ˘
V (φ, ω) dω
« dφ
Source Coding in Sensor Networks November 3, 2009 61 / 53
Assumption: SU(Φ, Ω) = σ2
U 1SU (Φ, Ω)
1 2 3 4 5 6 7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Distortion (MSE/(m⋅s)) Rate (bits/(m⋅s))
Berger−Tung coding centralized coding spatially independent coding predictive quantization
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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U[n] V [k, n] BS
pi,i−1 pi,i+1 1 2 M − 1
Alice visits pubs located on Ring Street. (At time nT0, 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.
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U[n] V [k, n] BS
pi,i−1 pi,i+1 1 2 M − 1
P
∀ n ∈ Z, ∀ m ∈ {0, . . . , M − 1} pi,j =
2
if j − i ≡ ±1 (mod M)
Alice spends a fixed amount of time T0 at each pub. Traveling time is negligible. Alice’s decisions are independent of any past visits. = ⇒ U[n] is a stationary Markov chain. Define V [k, n] :=
if U[n] = k,
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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: Rc = 1 bit/time unit.
Spatially independent compression:
Sensor k encodes Vk[n] := V [k, n], ignoring spatial correlation.
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5 10 15 20 25 30 35 40 0.5 1 1.5 2 2.5 3 3.5
Number of states M Rate (bits/time unit)
centralized compression distributed compression spatially independent compression
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Assumptions:
M is a multiple of 4. MAC of sum capacity C = 1 bit/time unit:
Sensors Base Station Xk[n] Xk+1[n] Y [n] =
k Xk[n] mod 2
Algorithm:
At time nT0, all sensors = U[n] transmit 0.
1 1 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
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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Lattice ΛM z1 z2 Spectral lattice Λ∗
M
Ψ1 Ψ2
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Lattice ΛM z1 z2 Density halved. Spectral lattice Λ∗
M
Ψ1 Ψ2 Density doubled.
Return
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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Φ Ω Support SV of SV
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Φ Ω Primitive cell B1 of Λ∗
M1
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Φ Ω
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Φ Ω
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Φ Ω Primitive cell B2 of Λ∗
M2
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Φ Ω
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Φ Ω Support Se
V of Se V
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Φ Ω Spectral support of the interpolating function s
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Φ Ω
Return
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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Proof.
. . . The interpolation formula follows from the uniformly bounded convergence lim
L→∞
ejMl,Ψ s(z − Ml) = ejz,Ψ, for Ψ ∈ B. By Saakyan, it is sufficient to prove that the 2πM −T-periodic function g defined on B by Ψ → ejz,Ψ is of bounded harmonic variation. This follows from the ∪-convexity of B.
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Define g on B. Ψ1 Ψ2 Primitive cell B of Λ∗
M
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Consider the spectral lattice Λ∗
M.
Ψ1 Ψ2 Spectral lattice Λ∗
M
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Extend g by periodicity. Ψ1 Ψ2
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Verify that g is of bounded harmonic variation. Ψ1 Ψ2 Rectangular primitive cell of Λ∗
M
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Return
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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Without aliasing
Return
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With aliasing
Return
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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Assumptions:
The sound sources are located far away from the microphones. = ⇒ The waves arriving on V appear as plane waves. There is no attenuation.
For a single angle of arrival α:
x V α Support of
Φ Ω Slope: −
c cos α
For all possible angles of arrival α ∈ (0, π):
Support of
Φ Ω 1 c Exact passband:
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8
Application – Temperature Monitoring
9
Application – Random Walks
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
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Assumption: SU(Φ, Ω) = σ2
U 1SU (Φ, Ω)
1 2 3 4 5 6 7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Distortion (MSE/(m⋅s)) Rate (bits/(m⋅s))
Berger−Tung coding centralized coding spatially independent coding predictive quantization
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