http://nesl.ee.ucla.edu/document/edit/324
Data loss in wireless sensing applications is inevitable and while there have been many attempts at coping with this issue, recent developments in the area of Compressive Sensing (CS) provide a new and attractive perspective. Since many physical signals of interest are known to be sparse or compressible, employing CS, not only compresses the data and reduces effective transmission rate, but also improves the robustness of the system to channel erasures. This is possible because reconstruction algorithms for compressively sampled signals are not hampered by the stochastic nature of wireless link disturbances, which has traditionally plagued attempts at proactively handling the effects of these errors. In this paper, we propose that if CS is employed for source compression, then CS can further be exploited as an application layer erasure coding strategy for recovering missing data. We show that CS erasure encoding (CSEC) with random sampling is efficient for handling missing data in erasure channels, paralleling the performance of BCH codes, with the added benefit of graceful degradation of the reconstruction error even when the amount of missing data far exceeds the designed redundancy. Further, since CSEC is equivalent to nominal oversampling in the incoherent measurement basis, it is computationally cheaper than conventional erasure coding. We support our proposal through extensive performance studies.
Compressive Oversampling for Robust Data Transmission in Sensor Networks - Presented at INFOCOM 2010
1. Recovering Lost Sensor
Data through Compressed
Sensing
Zainul Charbiwala
Collaborators:Younghun Kim, Sadaf Zahedi, Supriyo Chakraborty,
Ting He (IBM), Chatschik Bisdikian (IBM), Mani Srivastava
2. The Big Picture
Lossy Communication Link
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
3. The Big Picture
Lossy Communication Link
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
4. The Big Picture
Lossy Communication Link
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
5. The Big Picture
Lossy Communication Link
How do we recover from this loss?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
6. The Big Picture
Lossy Communication Link
How do we recover from this loss?
• Retransmit the lost packets
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
7. The Big Picture
Lossy Communication Link
How do we recover from this loss?
• Retransmit the lost packets
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
8. The Big Picture
Generate Error
Correction Bits
Lossy Communication Link
How do we recover from this loss?
• Retransmit the lost packets
• Proactively encode the data with some protection bits
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
9. The Big Picture
Generate Error
Correction Bits
Lossy Communication Link
How do we recover from this loss?
• Retransmit the lost packets
• Proactively encode the data with some protection bits
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
10. The Big Picture
Generate Error
Correction Bits
Lossy Communication Link
How do we recover from this loss?
• Retransmit the lost packets
• Proactively encode the data with some protection bits
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
11. The Big Picture
Generate Error
Correction Bits
Lossy Communication Link
How do we recover from this loss?
• Retransmit the lost packets
• Proactively encode the data with some protection bits
• Can we do something better ?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 2
12. The Big Picture - Using Compressed Sensing
Lossy Communication Link
CSEC
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
13. The Big Picture - Using Compressed Sensing
Lossy Communication Link
CSEC
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
14. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
15. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
16. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
17. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
Recover from
Received Compressed
Measurements
How does this work ?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
18. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
Recover from
Received Compressed
Measurements
How does this work ?
• Use knowledge of signal model and channel
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
19. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
Recover from
Received Compressed
Measurements
How does this work ?
• Use knowledge of signal model and channel
• CS uses randomized sampling/projections
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
20. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
Recover from
Received Compressed
Measurements
How does this work ?
• Use knowledge of signal model and channel
• CS uses randomized sampling/projections
• Random losses look like additional randomness !
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
21. The Big Picture - Using Compressed Sensing
Generate Compressed
Measurements
Lossy Communication Link
CSEC
Recover from
Received Compressed
Measurements
How does this work ?
• Use knowledge of signal model and channel
• CS uses randomized sampling/projections
• Random losses look like additional randomness !
Rest of this talk focuses on describing
“How” and “How Well” this works
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 3
22. Talk Outline
‣ A Quick Intro to Compressed Sensing
‣ CS Erasure Coding for Recovering Lost Sensor Data
‣ Evaluating CSEC’s cost and performance
‣ Concluding Remarks
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 4
23. Why Compressed Sensing ?
Physical Sampling Compression Communication Application
Signal
Computationally
expensive
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 5
24. Why Compressed Sensing ?
Physical Sampling Compression Communication Application
Signal
Computationally
expensive
Physical Compressive
Communication Decoding Application
Signal Sampling
Shifts computation to
a capable server
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 5
25. Compressed Sensing - Some Intuition
Bandwidth Frequency
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 6
26. Compressed Sensing - Some Intuition
Bandwidth Frequency
How do you acquire this signal?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 6
27. Compressed Sensing - Some Intuition
Bandwidth Frequency
How do you acquire this signal?
• Nyquist rate - twice the bandwidth
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 6
28. Compressed Sensing - Some Intuition
Bandwidth Frequency
How do you acquire this signal?
• Nyquist rate - twice the bandwidth
• But what if you knew more about the signal?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 6
29. Compressed Sensing - Some Intuition
Bandwidth Frequency
How do you acquire this signal?
• Nyquist rate - twice the bandwidth
• But what if you knew more about the signal?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 6
30. Compressed Sensing - Some Intuition
Bandwidth Frequency
How do you acquire this signal?
• Nyquist rate - twice the bandwidth
• But what if you knew more about the signal?
• CS enables signal acquisition based on information content
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 6
32. Transform Domain Analysis
‣ We usually acquire signals in the time or spatial domain
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 7
33. Transform Domain Analysis
‣ We usually acquire signals in the time or spatial domain
‣ By looking at the signal in another domain, the signal may be
represented more compactly
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 7
34. Transform Domain Analysis
‣ We usually acquire signals in the time or spatial domain
‣ By looking at the signal in another domain, the signal may be
represented more compactly
‣ Eg: a sine wave can be expressed by
3 parameters: frequency, amplitude
and phase.
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 7
35. Transform Domain Analysis
‣ We usually acquire signals in the time or spatial domain
‣ By looking at the signal in another domain, the signal may be
represented more compactly
‣ Eg: a sine wave can be expressed by
3 parameters: frequency, amplitude
and phase.
‣ Or, in this case, by the index of the
FFT coefficient and its complex
value
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 7
36. Transform Domain Analysis
‣ We usually acquire signals in the time or spatial domain
‣ By looking at the signal in another domain, the signal may be
represented more compactly
‣ Eg: a sine wave can be expressed by
3 parameters: frequency, amplitude
and phase.
‣ Or, in this case, by the index of the
FFT coefficient and its complex
value
‣ Sine wave is sparse in frequency
domain
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 7
37. Acquiring a Sine Wave
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 8
38. Acquiring a Sine Wave
‣ Assume we’re interesting in acquiring a single sine wave x(t) in a
noiseless environment
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 8
39. Acquiring a Sine Wave
‣ Assume we’re interesting in acquiring a single sine wave x(t) in a
noiseless environment
‣ An infinite duration sine wave can be expressed using three
parameters: frequency f, amplitude a and phase φ.
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 8
40. Acquiring a Sine Wave
‣ Assume we’re interesting in acquiring a single sine wave x(t) in a
noiseless environment
‣ An infinite duration sine wave can be expressed using three
parameters: frequency f, amplitude a and phase φ.
‣ Question: What’s the best way to find the parameters ?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 8
41. Acquiring a Sine Wave
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 9
42. Acquiring a Sine Wave
‣ Technically, to estimate three parameters one needs three good
measurements
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 9
43. Acquiring a Sine Wave
‣ Technically, to estimate three parameters one needs three good
measurements
‣ Questions:
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 9
44. Acquiring a Sine Wave
‣ Technically, to estimate three parameters one needs three good
measurements
‣ Questions:
‣ What are “good” measurements ?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 9
45. Acquiring a Sine Wave
‣ Technically, to estimate three parameters one needs three good
measurements
‣ Questions:
‣ What are “good” measurements ?
‣ How do you estimate f, a, φ from three measurements ?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 9
47. Compressed Sensing
‣ With three samples: z1, z2, z3 of the sine wave at times t1, t2, t3
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 10
48. Compressed Sensing
‣ With three samples: z1, z2, z3 of the sine wave at times t1, t2, t3
‣ We know that any solution of f, a and φ must meet the three
constraints and spans a 3D space:
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 10
49. Compressed Sensing
‣ With three samples: z1, z2, z3 of the sine wave at times t1, t2, t3
‣ We know that any solution of f, a and φ must meet the three
constraints and spans a 3D space:
z i = x(t i ) = a sin(2π ft i + φ )
∀i ∈{1, 2, 3}
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 10
50. Compressed Sensing
‣ With three samples: z1, z2, z3 of the sine wave at times t1, t2, t3
‣ We know that any solution of f, a and φ must meet the three
constraints and spans a 3D space:
z i = x(t i ) = a sin(2π ft i + φ )
∀i ∈{1, 2, 3}
φ
‣ Feasible solution space is much smaller
a
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 10
51. Compressed Sensing
‣ With three samples: z1, z2, z3 of the sine wave at times t1, t2, t3
‣ We know that any solution of f, a and φ must meet the three
constraints and spans a 3D space:
z i = x(t i ) = a sin(2π ft i + φ )
∀i ∈{1, 2, 3}
φ
‣ Feasible solution space is much smaller
‣ As the number of constraints grows from
more measurements, the feasible solution
a
space shrinks
‣ Exhaustive search over this space reveals the
right answer knowing presence of one sine
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 10
52. Formulating the Problem
‣ We could also represent f, a and φ as a very long, but mostly
empty FFT coefficient vector.
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 11
53. Formulating the Problem
‣ We could also represent f, a and φ as a very long, but mostly
empty FFT coefficient vector.
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 11
54. Formulating the Problem
‣ We could also represent f, a and φ as a very long, but mostly
empty FFT coefficient vector. Sine wave.
Amplitude
represented by
x color
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 11
55. Formulating the Problem
‣ We could also represent f, a and φ as a very long, but mostly
empty FFT coefficient vector. Sine wave.
Amplitude
represented by
Ψ (Fourier Transform) x color
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 11
56. Formulating the Problem
‣ We could also represent f, a and φ as a very long, but mostly
empty FFT coefficient vector. Sine wave.
Amplitude
represented by
y = Ψ (Fourier Transform) x color
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 11
57. Formulating the Problem
‣ We could also represent f, a and φ as a very long, but mostly
empty FFT coefficient vector. Sine wave.
Amplitude
represented by
y = Ψ (Fourier Transform) x color
− j2 π ft + φ
ae
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 11
58. Sampling Matrix
‣ We could also write out the sampling process in matrix form
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
59. Sampling Matrix
‣ We could also write out the sampling process in matrix form
x
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
60. Sampling Matrix
‣ We could also write out the sampling process in matrix form
Φ x
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
61. Sampling Matrix
‣ We could also write out the sampling process in matrix form
z = Φ x
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
62. Sampling Matrix
‣ We could also write out the sampling process in matrix form
z = Φ x
Three non-zero entries at some
“good” locations
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
63. Sampling Matrix
‣ We could also write out the sampling process in matrix form
Three measurements
z = Φ x
Three non-zero entries at some
“good” locations
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
64. Sampling Matrix
‣ We could also write out the sampling process in matrix form
Three measurements
z = Φ x
k
n
Three non-zero entries at some
“good” locations
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 12
65. Exhaustive Search
‣ Objective of exhaustive search:
‣ Find an estimate of the vector y that meets the constraints and is the most
compact representation of x (also called the sparsest representation)
‣ Our search is now guided by the fact that y is a sparse vector
‣ Rewriting constraints:
z = Φx
y = Ψx
−1
z = ΦΨ y
Constraints from
measurements
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 13
66. Exhaustive Search
‣ Objective of exhaustive search:
‣ Find an estimate of the vector y that meets the constraints and is the most
compact representation of x (also called the sparsest representation)
‣ Our search is now guided by the fact that y is a sparse vector
‣ Rewriting constraints:
ˆ %
y = arg min y l 0
%
y
z = Φx
y = Ψx %
s.t. z = ΦΨ y −1
z = ΦΨ y −1 y l0
@ {i : yi ≠ 0}
Constraints from
measurements
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 13
67. Exhaustive Search
‣ Objective of exhaustive search:
‣ Find an estimate of the vector y that meets the constraints and is the most
compact representation of x (also called the sparsest representation)
‣ Our search is now guided by the fact that y is a sparse vector
‣ Rewriting constraints:
ˆ %
y = arg min y l 0
%
y
z = Φx
y = Ψx %
s.t. z = ΦΨ y −1
z = ΦΨ y −1 y l0
@ {i : yi ≠ 0}
Constraints from This optimization problem
measurements is NP-Hard !
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 13
68. l1 Minimization
‣ Approximate the l0 norm to an l1 norm
ˆ %
y = arg min y l 1
%
y y = ∑ yi
l1
% −1 i
s.t. z = ΦΨ y
‣ This problem can now be solved efficiently using linear
programming techniques
‣ This approximation was not new
‣ The big leap in Compressed Sensing was a theorem that showed
that under the right conditions, this approximation is exact!
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 14
69. Some CS Results
‣ Theorem: If k samples of a length n signal are acquired uniformly
randomly (if each sample is equiprobable) and reconstruction is
performed in the Fourier basis:
k
[Rudelson06] s≤C · 4 ′
w.h.p.
log (n)
‣ Where s is the sparsity of the signal
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 15
70. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 16
71. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
Missing
Communication
samples
When communication channel is lossy:
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 16
72. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
Missing
Communication
samples
When communication channel is lossy:
• Use retransmissions to recover lost data
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 16
73. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
Missing
Communication
samples
When communication channel is lossy:
• Use retransmissions to recover lost data
• Or, use error (erasure) correcting codes
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 16
74. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
Missing
Communication
samples
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 17
75. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
Recovered
Channel Channel
Missing
Communication compressed
Coding Decoding
samples domain samples
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 17
76. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz
nxn
Recovered
Channel Channel
Missing
Communication compressed
Coding Decoding
samples domain samples
+
w = Ωy wl = Cw y = ( CΩ ) wl
ˆ
mxn
m>n
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 17
77. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz Done at
nxn application layer
Recovered
Channel Channel
Missing
Communication compressed
Coding Decoding
samples domain samples
+
w = Ωy wl = Cw y = ( CΩ ) wl
ˆ
mxn
m>n
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 17
78. Handling Missing Data - Traditional Approach
Physical Compressed
Sampling Compression
Signal domain samples
n
x ∈° z = In x y = Ψz Done at
nxn application layer
Recovered
Channel Channel
Missing
Communication compressed
Coding Decoding
samples domain samples
+
w = Ωy wl = Cw y = ( CΩ ) wl
ˆ
mxn
m>n Done at physical layer
Can’t exploit signal
characteristics
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 17
79. CS Erasure Coding Approach
Physical Compressive Compressed
Communication Decoding
Signal Sampling domain samples
n
x ∈° z = Φx zl = Cz %
y = arg min y l 1
kxn %
y
k<n
%
s.t. zl = CΦΨ y −1
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 18
80. CS Erasure Coding Approach
Physical Compressive Compressed
Communication Decoding
Signal Sampling domain samples
n
x ∈° z = Φx zl = Cz %
y = arg min y l 1
kxn %
y
k<n
%
s.t. zl = CΦΨ y −1
Physical Compressive Compressed
Communication Decoding
Signal Sampling domain samples
n
x ∈° z = Φx zl = Cz %
y = arg min y l 1
mxn %
y
k<m<n
%
s.t. zl = CΦΨ y −1
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 18
81. CS Erasure Coding Approach
Physical Compressive Compressed
Communication Decoding
Signal Sampling domain samples
n
x ∈° z = Φx zl = Cz %
y = arg min y l 1
kxn %
y
k<n
%
s.t. zl = CΦΨ y −1
Over-sampling in CS is
Erasure Coding !
Physical Compressive Compressed
Communication Decoding
Signal Sampling domain samples
n
x ∈° z = Φx zl = Cz %
y = arg min y l 1
mxn %
y
k<m<n
%
s.t. zl = CΦΨ y −1
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 18
82. Effects of Missing Samples on CS
z = Φ x
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
83. Effects of Missing Samples on CS
z = Φ x
Missing
samples at the
receiver
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
84. Effects of Missing Samples on CS
z = Φ x
Missing
samples at the
Same as missing
receiver
rows in the
sampling matrix
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
85. Effects of Missing Samples on CS
z = Φ x
What happens if we over-sample?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
86. Effects of Missing Samples on CS
z = Φ x
What happens if we over-sample?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
87. Effects of Missing Samples on CS
z = Φ x
What happens if we over-sample?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
88. Effects of Missing Samples on CS
z = Φ x
What happens if we over-sample?
• Can we recover the lost data?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
89. Effects of Missing Samples on CS
z = Φ x
What happens if we over-sample?
• Can we recover the lost data?
• How much over-sampling is needed?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 19
90. Extending CS Results
‣ Claim: When m>k samples are acquired uniformly randomly and
communicated through a memoryless binary erasure channel that
drops m-k samples, the received k samples are still equiprobable.
‣ Implies that bound on sparsity condition should hold.
‣ If bound is tight, over-sampling rate (m-k) is same as loss rate
[This paper]
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 20
91. Features of CS Erasure Coding
‣ No need of additional channel coding block
‣ Redundancy achieved by oversampling
‣ Recovery is resilient to incorrect channel estimates
‣ Traditional channel coding fails if redundancy is inadequate
‣ Decoding is free if CS was used for compression anyway
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 21
92. Features of CS Erasure Coding
‣ No need of additional channel coding block
‣ Redundancy achieved by oversampling
‣ Recovery is resilient to incorrect channel estimates
‣ Traditional channel coding fails if redundancy is inadequate
‣ Decoding is free if CS was used for compression anyway
‣ Intuition:
‣ Channel Coding spreads information out over measurements
‣ Compression (Source Coding) - compact information in few measurements
‣ CSEC - spreads information while compacting !
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 21
93. Signal Recovery Performance Evaluation
Create CS Lossy CS Reconstruction
Signal Sampling Channel Recovery Error?
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 22
94. In Memoryless Channels
Baseline performance - No Loss
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 23
95. In Memoryless Channels
Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 23
96. In Memoryless Channels
Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
20 % Oversampling -
complete recovery
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 23
97. In Memoryless Channels
Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
20 % Oversampling -
complete recovery
Less than 20 %
Oversampling -
recovery does not fail
completely
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 23
98. In Bursty Channels
Baseline performance - No Loss
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 24
99. In Bursty Channels
Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
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100. In Bursty Channels
Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
20 % Oversampling -
doesn’t recover
completely
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101. In Bursty Channels
Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
Oversampling + Interleaving
- Still incomplete recovery
20 % Oversampling -
doesn’t recover
completely
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102. In Bursty Channels
Worse than baseline Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
Oversampling + Interleaving
- Still incomplete recovery
20 % Oversampling -
doesn’t recover
completely
Better than baseline
‣ Recovery incomplete because of low interleaving depth
‣ Recovery better at high sparsity because bursty channels deliver
bigger packets on average, but with higher variance
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 24
103. In Bursty Channels
Worse than baseline Baseline performance - No Loss
20 % Loss - Drop in
recovery probability
Oversampling + Interleaving
- Still incomplete recovery
20 % Oversampling -
doesn’t recover
completely
Better than baseline
‣ Recovery incomplete because of low interleaving depth
‣ Recovery better at high sparsity because bursty channels deliver
bigger packets on average, but with higher variance
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 24
104. In Real 802.15.4 Channel
Baseline performance - No Loss
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 25
105. In Real 802.15.4 Channel
Baseline performance - No Loss
15 % Loss - Drop in
recovery probability
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106. In Real 802.15.4 Channel
Baseline performance - No Loss
15 % Loss - Drop in
recovery probability
15 % Oversampling -
complete recovery
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107. In Real 802.15.4 Channel
Baseline performance - No Loss
15 % Loss - Drop in
recovery probability
15 % Oversampling -
complete recovery
Less than 15 %
Oversampling -
recovery does not fail
completely
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 25
108. Cost of CSEC
5
Rnd ADC FFT Radio TX RS
4
Energy/block (mJ)
3
2
1
0
m=256 S-n-S m=10 C-n-S m=64 CS k=320 S-n-S+RS k=16 C-n-S+RS k=80 CSEC
Sense Sense, CS Sense Sense, CSEC
and Compress and and Compress and
Send (FFT) Send Send and Send
and (1/4th with Send
Send rate) Reed with
Solomon RS
No robustness
guarantees
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109. Cost of CSEC
5
Rnd ADC FFT Radio TX RS
4
Energy/block (mJ)
3
2
1
0
m=256 S-n-S m=10 C-n-S m=64 CS k=320 S-n-S+RS k=16 C-n-S+RS k=80 CSEC
Sense Sense, CS Sense Sense, CSEC
and Compress and and Compress and
Send (FFT) Send Send and Send
and (1/4th with Send
Send rate) Reed with
Solomon RS
No robustness All options equally
guarantees robust (w.h.p.)
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 26
110. Cost of CSEC
5
Rnd ADC FFT Radio TX RS
4
2.5x
Energy/block (mJ)
lower
3
2 energy
1
0
m=256 S-n-S m=10 C-n-S m=64 CS k=320 S-n-S+RS k=16 C-n-S+RS k=80 CSEC
Sense Sense, CS Sense Sense, CSEC
and Compress and and Compress and
Send (FFT) Send Send and Send
and (1/4th with Send
Send rate) Reed with
Solomon RS
No robustness All options equally
guarantees robust (w.h.p.)
zainul@ee.ucla.edu - CSEC - Infocom - March 2010 26
111. Summary
‣ Oversampling is a valid erasure coding strategy for compressive
reconstruction
‣ For binary erasure channels, an oversampling rate equal to loss
rate is sufficient
‣ CS erasure coding can be rate-less like fountain codes
‣ Allows adaptation to varying channel conditions
‣ Can be computationally more efficient on transmit side than
traditional erasure codes
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112. Closing Remarks
‣ CSEC spreads information out while compacting
‣ No free lunch syndrome: Data rate requirement is higher than if using good
source and channel coding independently
‣ But, then, computation cost is higher too
‣ CSEC requires knowledge of signal model
‣ If signal is non-stationary, model needs to be updated during recovery
‣ This can be done using over-sampling too
‣ CSEC requires knowledge of channel conditions
‣ Can use CS streaming with feedback
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