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A User‘s Guide to Compressed Sensing for Communications Systems
Kazunori Hayashi Graduate School of Informatics
Kyoto University [email protected]
Wireless Signal Processing & Networking Workshop: Emerging Wireless Technologies
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K. Hayashi, M. Nagahara and T. Tanaka, “A User’s Guide to Compressed Sensing for Communications Systems,” IEICE Trans. Commun. , Vol. E96-B, No.3, pp.685-712, Mar. 2013.
Survey paper on compressed sensing:
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• A Brief Introduction to Compressed Sensing • Warming-up • Linear Equations Review • Problem Setting • Algorithms
• Applications to Communications Systems • Channel Estimation • Wireless Sensor Network • Network Tomography • Spectrum Sensing • RFID
Agenda
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A Brief Introduction to Compressed Sensing
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Warming-up Exercise(1)
-Linear simultaneous equations: x + y + z = 3
x − z = 02x − y + z = 2
x = 1y = 1z = 1
Unique solution
（# of Eqs. = # of unknowns）
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Warming-up Exercise(2)
No solution ?
x + y + z = 3x − z = 0
2x − y + z = 2x + y = 1
-Linear simultaneous equations:
（# of Eqs. > # of unknowns）
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Warming-up Exercise(3)
x + y + z = 3x − z = 0
2x − y + z = 2x + y = 2
x = 1y = 1z = 1
Unique solution
-Linear simultaneous equations:
（# of Eqs. > # of unknowns）
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Warming-up Exercise(4)
x + y + z = 3x − z = 0
Infinitely many solutions
x = t
y = −2t + 3z = t
-Linear simultaneous equations:
（# of Eqs. < # of unknowns）
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Linear Equations in Engineering Problems
y = Ax
-Answering “No solution” or “Non-unique solutions” is not enough
A ∈ Rm×n
x ∈ Rn
y ∈ Rm
Sensing matrix： Unknown vector： Measurement vector：
-Measurement may include noise y = Ax + v
noise
10Linear Equations Review: Conventional approach ( , noise-free) n = m
-Unique Solution (if is not singular):
x = A−1y
A
Corresponds to warming-up (1)
y = AxA ∈ R
m×n
x ∈ Rn
y ∈ Rm
Sensing matrix： Unknown vector： Measurement vector：
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n < m
x = (ATA)−1ATy
-Unique Solution (if is of full column rank): A
Corresponds to warming-up (3)
Linear Equations Review: Conventional approach ( , noise-free)
y = AxA ∈ R
m×n
x ∈ Rn
y ∈ Rm
Sensing matrix： Unknown vector： Measurement vector：
12Linear Equations Review: Conventional approach ( , noise-free) n > m
y = AxA ∈ R
m×n
x ∈ Rn
y ∈ Rm
Sensing matrix： Unknown vector： Measurement vector：
-Non-unique solutions
Corresponds to warming-up (4)
-Minimum-norm solution： xMN = argmin
x||x||22 subject to Ax = y
= AT(AAT)−1y
Select , which has minimum -norm x �2
13Linear Equations Review: Conventional approach ( , noisy) n < m
Select , which minimize squared error between and , as a solution Ax y
x
y = AxA ∈ R
m×n
x ∈ Rn
y ∈ Rm
Sensing matrix： Unknown vector： Measurement vector：
-No solution in general （ is not included in Img. of due to noise） Ay
Corresponds to warming-up (2)
-Least-square (LS) solution： xLS = argmin
x||Ax− y||22 = (ATA)−1ATy
14Linear Equations Review: Conventional approach ( , noisy)
-Non-unique solutions
n > m
y = AxA ∈ R
m×n
x ∈ Rn
y ∈ Rm
Sensing matrix： Unknown vector： Measurement vector：
-Regularized LS solution:
：regularization parameter λ
xrLS = argminx
(||Ax− y||22 + λ||x||22)
= (λI+ATA)−1ATy
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Problem Setting of Compressed Sensing
y = AxA ∈ R
m×n
x ∈ Rn
y ∈ Rm
Sensing matrix (non-adaptive): Unknown vector： Measurement vector：
Linear equations of with a prior information that the true solution is sparse
n > m
Underdetermined: Sparse unknown vector:
n > m||x||0 � n
||x||0 : # of nonzero elements of x
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Signal Recovery via Optimization
-Natural and straightforward approach:
x�0 = arg minx
||x||0 subject to Ax = y
- if , almost always true solution is obtained - NP hard in general
m > ||x||0
�0
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Signal Recovery via Optimization Replace with
x�1 = arg minx
||x||1 subject to Ax = y
�1
-It can be posed as linear programing (LP) problem -True solution can be obtained even when under some conditions
n > m
||x||0 ||x||1 =n∑
i=1
|xi|
If satisfies RIP (restricted isometry property), only measurement will be enough for exact recovery
O(k log(n/k))A
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x�1 = arg minx
||x||1 subject to Ax = y
Intuitive Illustrations of Sparse Signal Recovery
�1 recovery：
MN solution： xMN = arg minx
||x||22 subject to Ax = y
x
y = Ax y = Ax
x
00
19Reconstruction with Noisy Measurement（1/2）
x�1 = arg minx
||x||1 subject to ||Ax − y||2 ≤ ε
Constrained reconstruction: �1
- Natural constraint for bounded noise - A certain guarantee of signal recovery can
be given for Gaussian noise
(ε > 0)
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optimization： �1-�2x�1-�2 = arg min
x
(12||Ax − y||22 + λ||x||1
)
- Equivalent to constrained reconstruction for a certain choice of
- Analogy to regularized LS problem: λ
�1
xrLS = arg minx
(||Ax − y||22 + λ||x||22)
- LARS (least angle regression) algorithm
Reconstruction with Noisy Measurement（2/2）
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Equivalent approaches to optimization
-Maximum a posteriori (MAP) estimator for AWGN with Laplacian prior ∝ exp(−λ||x||1)
�1-�2-Lasso (Least absolute shrinkage and selection operator)： xLasso = arg min
x||Ax − y||22 subject to ||x||1 ≤ t
(t > 0)
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Algorithms for Compressed Sensing
http://www.msys.sys.i.kyoto-u.ac.jp/csdemo.html
Sample program (Scilab):
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Applications to Communications Systems
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Key issues in applications
• Sparsity of signals of interest � Is it natural to assume the sparsity of them? � In which domain?
• Linear measurement of the signals � Can we formulate with linear equation? � Is it happy to reduce the number of equations?
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Channel Estimation
Receiver
Multipath channel
Estimation of channel impulse response using Rx. signals
Transmitter
Convolution channel Tx. signal Rx. signal Noise
yn
vn
sn h0, . . . , hM
yn =M∑
m=0
hmsn−m + vn
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s = [s1, . . . , sP ]T ∈ RPKnown pilot signal：
-Received signal vector：
Channel matrix：
White noise：
y = [y1, . . . , yP−M ]T
= Hs + v
H =
⎡⎢⎢⎢⎢⎣
hM . . . h0 0 . . . 0
0 hM h0. . .
......
. . . . . . . . . 00 . . . 0 hM . . . h0
⎤⎥⎥⎥⎥⎦
(P − M) × P
v = [v1, . . . , vP−M ]T
Channel Estimation
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y = Sh + v
S =
⎡⎢⎢⎢⎣
sM+1 sM . . . s1
sM+2 sM+1 . . . s2
......
...sP sP−1 . . . sP−M
⎤⎥⎥⎥⎦
h = [h0, . . . , hM ]T ∈ RM+1
-LS solution ： h = (STS)−1STy(P − M ≥ M + 1)
(P − M) × (M + 1)
-MN solution ： h = ST(SST)−1y(P − M < M + 1)
Channel Estimation -Received signal vector (rewritten)：
-Conventional approaches：
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-Estimation via compressed sensing：
hS
h = arg minh
||h||1 subject to ||Sh − y||2 ≤ ε
：Sensing matrix（Random, Toeplitz） ：Unknown sparse channel impulse response
Merit: Reduction of pilot signals (Improvement of frequency efficiency)
-Sparsity of wireless channel impulse response: Wireless channel is discrete in nature, and the feature appears as the bandwidth increases．
Channel Estimation
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Wireless Sensor Network (Single-Hop)
yi =n∑
j=1
Ai,jsj + vi
-Rx. signal@time @ fusion center
y = [y1, . . . , ym]T
= As + v
i
sj :measurement data Ai,j :random coefficients
(generated from node id)
-Tx. signal@time @ -th node: Ai,jsj
i j
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-Sparsity of measurement data：
Merit: Reduction of data collection time
-Data recovery via compressed sensing：
：sensing matrix ：unknown sparse vector
x = arg minx
||x||1 subject to ||AΦx − y||2 ≤ ε
AΦx
Wireless Sensor Network (Single-Hop)
（ : invertible transformation matrix, typically denoting DFT, DCT, wavelet, etc.）
s = Φx
might be sparse in a certain representation with basis （due to spatial correlation of data）
Φ
s Φ
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yi =∑j∈Pi
Ai,jsj
Pi
Ai,j
-Rx. signal@sink node in the -th multi-hop communication：
i
: index set of nodes included in the -th multi-pop communication i: random coefficient of the -th node in the -th multi-hop communication ( if )
ij
Ai,j = 0 j /∈ Pi
y = AΦxSensing matrix depends on routing algorithm
Merit: Reduction of # of communications (power consumption)
Wireless Sensor Network (Multi-Hop)
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Network Tomography
Network Terminal (source)
Terminal (destination)
Router Link
Path
-Link-level parameter estimation based on end-end measurements -End-End performance evaluation based on link-level measurements
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xj-delay @ link ： ej
y = Ax A: Routing Matrix
yi =∑j∈Pi
Ai,jxj
Pi
-total delay in the -th path: i
: set of link index included in the -th path
Ai,j =
{1 j ∈ Pi
0 otherwise
i
Delay Tomography (Link Delay Estimation)
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-Sparsity of link delay: Only limited number of links have large delay, due to node failure or some other reasons in typical network
Merit: Reduction of # of prove packets
x = arg minx
||x||1 subject to Ax = y-Estimation via compressed sensing:
：sensing matrix (routing matrix) ：(approximately) sparse delay vector x
A
Delay Tomography (Link Delay Estimation)
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-packet loss probability @ link :
Loss Tomography (Link Loss Probability Estimation)
pjej
1 − qi =∏
j∈Pi
(1 − pj)-packet success probability @ -th path: i
-packet loss probability @ -th path: i qi
− log(1 − qi) = − log
⎛⎝ ∏
j∈Pi
(1 − pj)
⎞⎠
= −∑j∈Pi
log(1 − pj)
xj = − log(1 − pj)
y = Axyi =∑j∈Pi
Ai,jxj
yi = − log(1 − qi)
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-Rx. signal (continuous time):
Spectrum Sensing
r(t) t ∈ [0, nTs]
Ts：inverse of Nyquist rate
rt = [r(Ts), . . . , r(nTs)]T-Rx. signal with Nyquist rate sampling：
yt = Ψrt
Ψ
-Rx. signal with (discrete time) linear measurement：
m × n： sensing matrix
( sub-Nyquist rate sampling） m < n
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rf = Drt
-Sparsity of occupied spectrum： Primary system uses
D：DFT matrix
yt = ΨDHrf
-Spectrum sensing via compressed sensing：
：Sensing matrix ：Sparse spectrum vector
ΨDH
rf
Merit: Reduction of sampling rate
Spectrum Sensing
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zs = ΓDΦsrt
Sparsity of spectrum edge
-Edge spectrum
Φs: smoothing function
Γ =
⎡⎢⎢⎢⎢⎢⎢⎢⎣
1 0 . . . . . . 0
−1 1. . .
...
0. . . . . . . . .
......
. . . . . . . . . 00 . . . 0 −1 1
⎤⎥⎥⎥⎥⎥⎥⎥⎦
yt = Ψ(ΓDΦs)−1zs
Spectrum Sensing
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RFID
Reader Tag
-Each tag sends its own ID by reflecting wireless signal from reader -Conventionally, collision avoiding protocols are used
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RFID Tag Detection
-ID of tag : j
-Received signal @ reader：
y =∑j∈P
aj
= [a1, . . . ,an]x= Ax
P：Index set of tags in reader’s range
xj =
{1 j ∈ P0 otherwise
aj (j = 1, . . . , n)
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-Sparsity of RFID tags： Number of tags in the reader’s range is much smaller than that of whole ID space (similar to CS-based MUD in the previous talk)
Merit: Reduction of detection time
-Tag detection via compressed sensing： x = arg min
x||x||1 subject to Ax = y
：Sensing matrix (ID matrix) ：Unknown sparse vector x
A
RFID Tag Detection
