Exploiting Sparsity for Wireless Communications

Transcription

1 Exploiting Sparsity for Wireless Communications Georgios B. Giannakis Dept. of ECE, Univ. of Minnesota Acknowledgements: D. Angelosante, J.-A. Bazerque, H. Zhu; and NSF grants 1 CCF , CON ; ARL grant no. DAAD

2 Ground-Breaking Recent Advances (a1) s is sparse (nonzero entries unknown) (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 Sparse regression [Tibshirani 96], [Tipping 01] Least-absolute shrinkage selection operator (Lasso) Ex. (Scalar case) Closed-form solution SPiNCOM University of Minnesota

3 Context Outline Sparsity-aware estimation of CDMA parameters Channels, timing, and user activity Simulated tests Sparsity-exploiting multiuser detection Optimum and sub-optimum detectors Simulated tests Conclusions and ongoing research Sparsity-cognizant sensing for cognitive radios (CRs) If time allows more on Monday SPiNCOM University of Minnesota 3

4 Context Sparsity-agnostic estimation of CDMA parameters LS, subspace, blind; e.g., [Madhow 97], [Bensley-Aazhang 98], [Buzzi-Poor 03] Multiuser detection (MUD) for CDMA accounts for finite-alphabet (FA) constraints, e.g., [Verdu 98]; but not for sparsity! Our focus: Exploit sparsity in CDMA parameter estimation and MUD SPiNCOM University of Minnesota 4

5 Setup for estimating CDMA parameters Rx waveform in AWGN Signature waveform: ; Processing gain N Training symbol: ; Chip waveform: ; k-th user energy: Max. number of users: K; Length of training seq.: P Symbol interval: ; Chip interval: k-th user channel of duration ; and delay (unknowns), if k-th user inactive SPiNCOM University of Minnesota

6 Basic CDMA Model LPF r(t) through (duration ); chip-rate oversampling by M Composite per-user channel: entries depend on and sampled composite channel of k-th user : max. user delay; : Tx filter duration; : ISI length Model applies to symbol-periodic as well as long spreading codes SPiNCOM University of Minnesota

7 Composite Channel is Sparse Sparsity due to propagation Multipath: few nonzero taps over Asynchronism: delays Sparsity due to user inactivity (Group) Lasso can estimate sparse complex parameter vector h norm in Can detect inactive users and reduce training length SPiNCOM University of Minnesota 7

8 Asynchronous Under-determined CDMA K =5 (all users active), N =15, M =1, random long codes, rectangular chip waveforms,, power controlled P =4, S is fat =75 < KQ=80 W =3 paths Lasso outperforms sparsity-agnostic and interference-limited LS SPiNCOM University of Minnesota 8

9 Detecting Inactive Users K =20 (total users), 5 active users, N =15, M =1, random long codes, rectangular chip waveforms, SNR=20 db versus Energy detector Lasso outperforms LS and can detect (in)active users SPiNCOM University of Minnesota 9

10 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 Low activity factor implies that vector is sparse Access point (AP) unaware of positions and number of zero entries in channel matrix H available at AP (via training or pathloss model) SPiNCOM University of Minnesota 10

11 Sparsity-Exploiting MUD Augmented alphabet Goal: Given at the AP, find under sparsity constraints Sparse (S-) MAP detector to minimize error prob. (b non-equiprobable) takes values in w.p. Prior probability: penalty SPiNCOM University of Minnesota 11

12 With Relaxed S-MAP detectors, (S-MAP) is equivalent to combinatorially complex Just relax! Ridge detector (RD) with p = 2 convex penalty Linear MUD with tractable performance analysis Lasso detector (LD) with p = 1 Efficient solver via quadratic programming (QP)/coordinate descent (CD) SPiNCOM University of Minnesota 12

13 S-MAP with Lattice Search With H full rank, QR decomposition yields H=QR Decision-directed detector (DDD) upper triangular Sequential detection prone to error propagation at low SNR 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] SPiNCOM University of Minnesota 13

14 K =20 users, N =32, Simulated Comparisons Independent Rayleigh fading channels between AP and users BPSK 4-PAM Performance: LS < RD < LD < DDD < SSD SPiNCOM University of Minnesota 14

15 Exploit (In)activity Across Symbols Adapt prior via recursive least-squares (RLS): 0 < k <1 If activity independent across slots, use k = 1 If, then < 0 RD/LD not applicable; DDD/SSD no problem SPiNCOM University of Minnesota 15

16 Simulation: Activity Tracking K =20 users, N =32 spreading Activity model: two-state Markov chain SER performance similar to independent case SPiNCOM University of Minnesota 16

17 Simulation: Under-determined CDMA K =20 users, N =32, 16, 8 RD loses identifiability once N<K DDD exhibits graceful performance with moderate N SPiNCOM University of Minnesota 17

18 Sparsity for Spectrum Sensing Multiple rasios jointly detect the spectrum [Ghasemi-Sousa 07,Ganesan-Li 06] Benefits spatial diversity gain mitigates multipath fading and shadowing reduced sensing time and local processing increase of reliability and ability to detect hidden terminals 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 SPiNCOM University of Minnesota 18

19 Transmitters Modeling Sensing CRs Frequency bases Sensed frequencies Sparsity present in space and frequency SPiNCOM University of Minnesota 19

20 Space-Frequency Basis Expansion Superimposed Tx spectra measured at CR r Average path-loss Frequency bases Linear model in and SPiNCOM University of Minnesota 20

21 Power Spectrum Cartography sources candidate locations, cognitive radios NNLS Lasso As a byproduct, Lasso localizes all sources via variable selection SPiNCOM University of Minnesota 21

22 Normalized error Tracking Capabilities batch solutions one per time-slot path of distributed online updates time-slot t Non-stationarity: one Tx exits at time-slot t=650 SPiNCOM University of Minnesota 22

23 Concluding Summary Sparse regression for estimating CDMA system parameters Lasso outperforms sparsity-agnostic techniques Can detect user activity, and enhance use capacity (reduced training) Group-Lasso can be efficiently implemented via coordinate descent Low activity factor motivates MUD of sparse symbol vectors 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 Sparsity-aware tracking of slowly-varying channels Performance analysis (at least bounds) for sparse sphere decoding of under-determined CDMA systems Thank You! SPiNCOM University of Minnesota 23

24 S-MAP Detection viz. CS/Lasso For, and any S-MAP detection can be viewed as CS (or Lasso) under FA constraints penalty Cost is convex for >0 LS regularized by the norm mitigates overfitting [Tibshirani 96] FA constraint renders S-MAP combinatorially complex (Sub-) optimal alternatives available for MUD; see e.g., [Verdu 98] Q: Can we develop sparsity-aware MUD schemes to solve S-MAP efficiently? SPiNCOM University of Minnesota 24

25 Relaxed S-MAP: Ridge MUD S-MAP Just relax FA constraint - Ridge detector (RD): p = 2 Simple (as linear MMSE) and works even for ill-posed problems where Performance treat interference as noise Error rate Optimal SPiNCOM University of Minnesota 25

26 K =20 users, N =32, Simulated Performance Independent Rayleigh fading channels between AP and users 10-2 Optimal = 0.5 = 0.35 = 0.65 Pe SER SNR (db) Threshold = 0.5 is near optimal SPiNCOM University of Minnesota 26

27 Relaxed S-MAP: Lasso-based MUD Lasso detector (LD): p = 1 Degree of sparsity depends on the activity factor sparsity of Closed-form solution impossible for general H Quadratic programming (QP): polynomial complexity Coordinate-descent possible: closed-form solution per coordinate SPiNCOM University of Minnesota 27

28 S-MAP with Lattice Search S-MAP With H full rank, QR decomposition yields H=QR Cost decomposes into scalar sub-problems, each solvable in closed form Sparsity Sphere Decoder (SSD) for (near-) ML at cubic avg. complexity SPiNCOM University of Minnesota 28

29 Sparse Decision-Directed MUD At k-th stage, only depends on Closed form for general M-ary constellations where Decision-directed detector (DDD): optimal if R diagonal Prone to error propagation; without error propagation (M=2) SPiNCOM University of Minnesota 29

30 Under-determined CDMA systems Fat H matrix (N<K) RD S-MAP with lattice search over with Identifiability possible for general constellations and detectors SPiNCOM University of Minnesota 30

31 Lassoing Block Activity User k (in)active per block of N s symbols Group Lasso approach Step 1: Relax and find nonzero rows of B Step 2: Standard MUD on active set of users SPiNCOM University of Minnesota 31

32 Simulation: Group Lassoing K =20 users, N =32, 16, 8, and N s = 20, 10, 1 N affects diversity order, while N s influences accuracy of recovery SPiNCOM University of Minnesota 32

33 Sparse Regression Seek a sparse to capture the spectrum measured at Lasso Soft threshold shrinks noisy estimates to zero Similar to Akaike s Information Criterion, it penalizes the number of parameters Variable selection + estimation Spectrum is non-negative add non-negativity constraints Distributed solution possible via consensus [Schizas-Giannakis 08] SPiNCOM University of Minnesota 33