Slide 1Wireless Communications
Acknowledgements: D. Angelosante, J.-A. Bazerque, H. Zhu; and NSF grants CCF
0830480, CON 014658; ARL grant no. DAAD19-01-2-0011
Georgios B. Giannakis Dept. of ECE, Univ. of Minnesota
http://spincom.ece.umn.edu
(a2) H can be fat; satisfies restricted isometry property
Compressive sampling [Chen-Donoho-Saunders’98], [Candès et al’04-06]
Given y and H, unknown s can be found with high probability
Least-absolute shrinkage selection operator (Lasso)
Ex. (Scalar case) Closed-form solution
Sparse regression [Tibshirani ’96], [Tipping’01]
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Outline Context
Sparsity-aware estimation of CDMA parameters Channels, timing, and user activity
Simulated tests
Simulated tests
Conclusions and ongoing research
Sparsity-cognizant sensing for cognitive radios (CRs) If time allows … more on Monday
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constraints, e.g., [Verdu’98]; but not for sparsity!
Our focus: Exploit sparsity in CDMA parameter estimation and MUD
Sparsity-agnostic estimation of CDMA parameters
LS, subspace, blind; e.g., [Madhow’97], [Bensley-Aazhang’98], [Buzzi-Poor’03]
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Rx waveform in AWGN
Training symbol: ; Chip waveform: ; k-th user energy:
, if k-th user inactive
Symbol interval: ; Chip interval:
Max. number of users: K; Length of training seq.: P
k-th user channel of duration ;
and delay (unknowns)
Composite per-user channel:
: max. user delay; : Tx filter duration; : ISI length
Model applies to symbol-periodic as well as long spreading codes
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Composite Channel is Sparse
(Group) Lasso can estimate sparse complex parameter vector h
norm in
Asynchronism: delays
Asynchronous Under-determined CDMA
K =5 (all users active), N =15, M =1, random long codes, rectangular chip waveforms, , power controlled
Lasso outperforms sparsity-agnostic and interference-limited LS
P =4, S is fat
W =3 paths
=75 < KQ=80
Detecting Inactive Users
K =20 (total users), 5 active users, N =15, M =1, random long codes, rectangular chip waveforms,
SNR=20 dB
Energy detector
Joint Exploitation of Sparsity and FA
CDMA system with K users and spreading gain N>K
Each user active with probability
K ×1 symbol block
if user k is inactive
if user k is active
N ×1 received chip samples
Access point (AP) unaware of positions and number of zero entries in
Low activity factor implies that vector is sparse
channel matrix H available at AP (via training or pathloss model)
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Sparsity-Exploiting MUD
Augmented alphabet
Sparse (S-) MAP detector to minimize error prob. (b non-equiprobable)
Prior probability:
penalty
Relaxed S-MAP detectors
convex penalty combinatorially complex
Efficient solver via quadratic programming (QP)/coordinate descent (CD)
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S-MAP with Lattice Search
Decision-directed detector (DDD)
Sparse Sphere Decoder (SSD)
(near-) optimal at cubic avg. complexity, e.g., [Giannakis et al’07]
Smart enumeration of all the constellation points [Damen et al’03]
upper triangular
Simulated Comparisons
Independent Rayleigh fading channels between AP and users
Performance: LS < RD < LD < DDD < SSD
BPSK 4-PAM
Exploit (In)activity Across Symbols
If activity independent across slots, use k = 1
If , then < 0
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Simulation: Activity Tracking
Activity model: two-state Markov chain
SER performance similar to independent case
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Simulation: Under-determined CDMA
RD loses identifiability once N<K
DDD exhibits graceful performance with moderate N
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Sparsity for Spectrum Sensing
Benefits spatial diversity gain mitigates multipath fading and shadowing reduced sensing time and local processing increase of reliability and ability to detect hidden terminals
Multiple rasios jointly detect the spectrum [Ghasemi-Sousa’07,Ganesan-Li’06]
Major limitation: occupancy is space-time invariant
Idea: collaborate to form a spatial map of the spectrum
Given the PSD at position , find
Approach: basis expansion of
Modeling
Transmitters
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Superimposed Tx spectra measured at CR r
Average path-loss
Frequency bases
As a byproduct, Lasso localizes all sources via variable selection
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Tracking Capabilities
Normalized error
batch solutions one per time-slot
path of distributed online updates
time-slot t
Concluding Summary
user inactivity augmented alphabet with non-equiprobable symbols
Optimal S-MAP detector exploits a priori sparsity information
Optimality loss but simplicity w/ relaxation; lattice search
Ongoing research
Performance analysis (at least bounds) for sparse sphere
decoding of under-determined CDMA systems
Thank You!
Can detect user activity, and enhance use capacity (reduced training)
Group-Lasso can be efficiently implemented via coordinate descent
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S-MAP Detection viz. CS/Lasso
For , and any
S-MAP detection can be viewed as CS (or Lasso) under FA constraints
Cost is convex for >0
LS regularized by the norm mitigates overfitting [Tibshirani ’96]
penalty
(Sub-) optimal alternatives available for MUD; see e.g., [Verdu ’98]
Q: Can we develop sparsity-aware MUD schemes to solve S-MAP efficiently?
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Relaxed S-MAP: Ridge MUD
Simple (as linear MMSE) and works even for ill-posed problems
where
Error rate
Simulated Performance
Independent Rayleigh fading channels between AP and users
5 10 15 20 25
10 -3
10 -2
SNR (dB)
S E
Relaxed S-MAP: Lasso-based MUD
Quadratic programming (QP): polynomial complexity
Coordinate-descent possible: closed-form solution per coordinate
Lasso detector (LD): p = 1
Degree of sparsity depends on the activity factor
sparsity of
S-MAP with Lattice Search
Cost decomposes into scalar sub-problems, each solvable in closed form
S-MAP
Sparsity Sphere Decoder (SSD) for (near-) ML at cubic avg. complexity
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Sparse Decision-Directed MUD
Prone to error propagation; without error propagation (M=2)
Decision-directed detector (DDD): optimal if R diagonal
where
Under-determined CDMA systems
RD
Identifiability possible for general constellations and detectors
with
Lassoing Block Activity
Step 1: Relax and find nonzero rows of B
Group Lasso approach
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Simulation: Group Lassoing
K =20 users, N =32, 16, 8, and Ns = 20, 10, 1
N affects diversity order, while Ns influences accuracy of recovery
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Seek a sparse to capture the spectrum measured at
Lasso
Similar to Akaike’s Information Criterion,
it penalizes the number of parameters
Variable selection + estimation
Sparse Regression