#### Transcript CS-CC - Department of Systems and Computer Engineering

Wideband and Cooperative Spectrum Sensing in Cognitive Radios Amir H. Banihashemi (joint work with Zeinab Zeinalkhani and Ebrahim Karami) BCWS Centre Dept. of Systems and Computer Engineering Carleton University Feb 17, 2011 Compressive Sensing [In part courtesy of R. Baraniuk, Rice University] The Digital Universe Size: 281 billion gigabytes generated in 2007 digital bits > stars in the universe growing by a factor of 10 every 5 years Growth fueled by multimedia data audio, images, video, surveillance cameras, sensor networks, … In 2007 digital data generated > total storage Solution: Compression 4 Why Compressive Sensing? Problem: Today’s multimedia sensor systems acquire massive amounts of multimedia data only to throw much/most of it away. Solution: compressive sensing - enables the design of radically new sensors and systems such as new cameras, imagers, ADCs, …, - beyond sensing to inference on massive data sets 5 Digital Data Acquisition Foundation: Shannon sampling theorem “if you sample densely enough (at the Nyquist rate), you can perfectly reconstruct the original analog data” Time Space 6 Sense by Sampling sample 7 Sense by Sampling sample too much data! 7 Sense (Sample) then Compress sample compress JPEG JPEG2000 … decompress 8 Sparsity / Compressibility (Transform Coding) pixels large wavelet coefficients (blue = 0) 9 Sparsity / Compressibility (Transform Coding) large wavelet coefficients pixels wideband signal samples frequency (blue = 0) large Gabor (TF) coefficients time 9 What’s Wrong with this Picture? • Why go to all the work to acquire N samples only to discard all but K pieces of data? sample compress decompress 10 Compressive Sensing • Directly acquire “compressed” data • Replace samples by more general “measurements” compressive sensing recover 11 Sampling • Signal is -sparse in basis/dictionary – WLOG assume sparse in the canonical domain sparse signal nonzero entries 12 Sampling • Signal is -sparse in basis/dictionary – WLOG assume sparse in the canonical domain • Sampling (matrix multiplication) samples sparse signal nonzero entries 13 Compressive Sampling • When data is sparse/compressible, one can directly acquire a condensed representation with no/little information loss through linear dimensionality reduction measurements sparse signal nonzero entries 14 How Can It Work? • In general, loss of information • Ex. Infinitely many ’s map to the same 15 How Can It Work? columns • In general, loss of information • But we are only interested in sparse vectors 16 How Can It Work? columns • In general, loss of information • But we are only interested in sparse vectors • is effectively M x K 16 How Can It Work? columns • In general, loss of information • But we are only interested in sparse vectors • Design so that each of its M x K submatrices are full rank (ideally orthobases) 16 How Can It Work? Goal: Design so that for any K-sparse vector norm of x is “close” to norm of . [Restricted Isometry Property (RIP)] x x, 17 How Can It Work? Goal: Design so that for any K-sparse vector norm of x is “close” to norm of . [Restricted Isometry Property (RIP)] x x, Unfortunately, this is NP-hard. 17 How Can It Work? Goal: Design so that for any K-sparse vector norm of x is “close” to norm of . [Restricted Isometry Property (RIP)] x x, Unfortunately, this is NP-hard. Good news: Draw at random, e.g., i.i.d. Gaussian or i.i.d. 1 Bernoulli. Then has the RIP with high probability provided 17 Compressive Data Acquisition • Measurements = random linear combinations of the entries of • No information loss for sparse vectors probability measurements with high sparse signal nonzero entries 18 CS Signal Recovery • Goal: Recover signal from measurements 19 CS Signal Recovery • Goal: Recover signal from measurements • Problem: ill-posed inverse problem 19 CS Signal Recovery • Goal: Recover signal from measurements • Problem: ill-posed inverse problem • Solution: Exploit the sparse/compressible nature of acquired signal 19 CS Signal Recovery • Random projection • Recovery problem: given find • Null space • So search in null space for the “best” according to some criterion – ex: least squares (N-M)-dim hyperplane at random angle20 CS Signal Recovery • Recovery: (ill-posed inverse problem) • given find (sparse) fast pseudoinverse 21 CS Signal Recovery • Recovery: (ill-posed inverse problem) • given find (sparse) fast, wrong pseudoinverse 21 CS Signal Recovery • Reconstruction/decoding: (ill-posed inverse problem) • given find fast, wrong • number of nonzero entries “find sparsest in translated nullspace” 21 CS Signal Recovery • Reconstruction/decoding: (ill-posed inverse problem) • fast, wrong • correct given find 21 CS Signal Recovery • Reconstruction/decoding: (ill-posed inverse problem) • fast, wrong • correct given find slow: NP-hard 21 CS Signal Recovery • Recovery: (ill-posed inverse problem) given find • fast, wrong • correct, slow • correct, efficient mild oversampling [Candes, Romberg, Tao; Donoho] (sparse) linear program number of measurements required 21 “Single-Pixel” CS Camera scene single photon detector DMD image reconstruction or processing DMD random pattern on DMD array w/ Kevin Kelly “Single-Pixel” CS Camera scene single photon detector image reconstruction or processing DMD DMD random pattern on DMD array … • Flip mirror array M times to acquire M measurements • Sparsity-based (linear programming) recovery 23 Wideband Spectrum Sensing Based on Compressive Sensing Spectrum Sensing in Cognitive Radio • Motivation: To detect and use spectrum holes efficiently • The frequency range of interest is often wide (wideband spectrum sensing) • Sampling at Nyquist rate is very challenging • Solution: Compressive Spectrum Sensing 32 Compressive Spectrum Sensing • Sub-Nyquist sampling: Analog to Information Conversion (AIC) y(t) y[m] M/N Nyquist Rate p(t) Random at Nyquist Rate 38 Compressive Spectrum Sensing Signal X is in time domain and is sparse in frequency domain y mn x s Fnn x y mnFm1ns mns We solve the following optimization problem to find s min s l1 s.t. y mns Application of compressive sensing to spectrum sensing: Tian and Giannakis, 2007 39 Background • Application of compressive sensing to spectrum sensing ( l1 optimization): Tian and Giannakis, ICCASP 2007. • Improving the performance for a block sparse signal ( l1 l2 optimization): Stojnic, Parvaresh, and Hassibi, IEEE Trans. Signal Proc., Aug. 2009. • Application of l1 l2 optimization in spectrum sensing: Liu and Wan, Arxiv., 2010. • Improving the performance of l1 optimization (Iterative Support Detection (ISD)): Wang and Yin, SIAM Journal on Imaging Sciences, Aug. 2010. • Our contribution: Application of ISD- l1 optimization and ISD- l1 l2 optimization to spectrum sensing 40 l1 l2 minimization • The received signal is block sparse • For a licensed user (PU), its operating spectrum is in a certain band • Uses the priori information of the spectrum boundaries between PUs min s1 s l2 s2 l2 sK l2 s.t. y mns • Can be formulated as a convex optimization 41 Iterative Support Detection (ISD) • Reduced requirement on the number of measurements compared to the classical l1 minimization • Recovers the signal iteratively in a small number of iterations 42 Simulations • n=256 , m=140, sparsity ratio = 30% (k = 77) m= =1 14 40 0 )) (( m 1 1 n p u P S D III n np pu u ttt P PS SD D 0.9 0.9 ISD l1 - l1 / l2 l1 / l2 0.8 0.8 0.7 0.7 PSD 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0 50 50 100 100 150 150 n n 200 200 250 250 Simulations • n=256 , m=120, sparsity ratio = 30% (k = 77) (m = 120) 1 II n np pu u tt P PS SD D 0.9 l1 / l2 ISD - l1 / l2 0.8 0.7 PSD 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 n 200 250 Simulations • n=256 , m=85, sparsity ratio = 30% (k = 77) (m = 85) 1 II n np pu u tt P PS SD D 0.9 l1 / l2 ISD - l1 / l2 0.8 0.7 PSD 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 n 200 250 MSE Comparison • n=256, sparsity ratio = 30% (k = 77) 1.4 ISD - l1 / l2 l1 / l2 1.2 ISD - l1 l1 1 MSE 0.8 0.6 0.4 0.2 0 70 80 90 100 110 120 130 140 # of m eas urem ent s (m ) 46 Cooperative Spectrum Sensing Introduction • Probability of detection (PD) and probability of false alarm (PF) are two important parameters to measure the performance of spectrum sensing algorithms • The interference induced from the secondary user (SU) on the primary user (PU) is proportional to (1-PD) • The throughput of the SU is proportional to 1-PF • Goal: Minimize PF for a given PD. 48 Introduction • Spectrum sensing can be performed as either distributed or cooperative. • Cooperative spectrum sensing - Advantage: improves PF for a given PD - Disadvantage: requires extra bandwidth for communication between SUs • Goal: To optimize the cluster size for maximum effective throughput R Pd tot m N 1 Pf tot Pd tot , N , , N , 1 Ts - E. Karami and A. H. Banihashemi, ``Cluster size Optimization in Cooperative Spectrum Sensing,” 9th Conference on Communication Networks and Services Research (CSNR 2011), May 2011. 49 Simulations Normalized achievable throughput for m=0.05 Ts 1 AND rule OR rule 0.9 0.8 SNR=5dB Normalized throughput 0.7 0.6 0.5 0.4 0.3 SNR=0dB 0.2 SNR=-5dB 0.1 0 2 4 6 8 10 12 Cooperation cluster size 14 16 18 20 50 Simulations Normalized achievable throughput for m=0.2 Ts 1 0.9 AND rule OR rule SNR=5dB 0.8 Normalized throughput 0.7 0.6 0.5 SNR=0dB 0.4 0.3 SNR=-5dB 0.2 0.1 0 2 4 6 8 10 12 Cooperation cluster size 14 16 18 20 51 Future Research • Low complexity wideband compressive spectrum sensing: Sparse measurement matrices, Adaptive algorithms • Cooperative spectrum sensing: optimal fusion rules, distributed fusion schemes • Cooperative wideband spectrum sensing 52 Compressive Sensing with Sparse Graphs Graphical Model of CS x signal xi y j ji xi Measurements y x ji i y • RIP: Graph is dense 33 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! • Measurements M O( K log(N / K )) 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! • Measurements M O( K log(N / K )) Low (good)! 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! • Measurements M O( K log(N / K )) Low (good)! • What if we use sparse graphs? 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! • Measurements M O( K log(N / K )) Low (good)! • What if we use sparse graphs? • Why? 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! • Measurements M O( K log(N / K )) Low (good)! • What if we use sparse graphs? • Why? - Fast measurement 34 Revisiting CS with Dense Graph 3 • Recovery complexity using LP O( N ) High! • Measurements M O( K log(N / K )) Low (good)! • What if we use sparse graphs? • Why? - Fast measurement - Fast recovery (iterative) 34 CS with Sparse Graphs • Graph structure (degree distribution)? 35 CS with Sparse Graphs • Graph structure (degree distribution)? • Edge weights? 35 CS with Sparse Graphs • Graph structure (degree distribution)? • Edge weights? • Recovery algorithms? 35 CS with Sparse Graphs • Graph structure (degree distribution)? • Edge weights? • Recovery algorithms? • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) • Edge weights? • Recovery algorithms? • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? • Recovery algorithms? • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? 0 and 1 • Recovery algorithms? • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? 0 and 1 [similar to a Tanner graph] • Recovery algorithms? • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? 0 and 1 [similar to a Tanner graph] • Recovery algorithms? Verification-based • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? 0 and 1 [similar to a Tanner graph] • Recovery algorithms? Verification-based [similar to iterative decoding algorithms over BEC] • Performance analysis? 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? 0 and 1 [similar to a Tanner graph] • Recovery algorithms? Verification-based [similar to iterative decoding algorithms over BEC] • Performance analysis? Asymptotic ( N ) 35 CS with Sparse Graphs • Graph structure (degree distribution)? Random regular (dv , dc ) [similar to a regular Tanner graph] • Edge weights? 0 and 1 [similar to a Tanner graph] • Recovery algorithms? Verification-based [similar to iterative decoding algorithms over BEC] • Performance analysis? Asymptotic ( N ) [same general techniques as in Density Evolution] 35 Example: Graph x1 N 8, M 6, d v 3, d c 4 x2 y1 x3 y2 x4 y3 x5 y4 x6 y5 x7 y6 x3 x4 x6 x7 x8 36 Example: Recovery Algorithm 37 Example: Recovery Algorithm Measurements 38 Example: Recovery Algorithm Random Integer Signal: 0 w.p. 1-α Measurements 39 Example: Recovery Algorithm Integer Signal Measurements 40 Example: Recovery Algorithm Integer Signal Measurements 41 Example: Recovery Algorithm Integer Signal Measurements 42 Example: Recovery Algorithm Integer Signal Measurements 43 Example: Recovery Algorithm Integer Signal Measurements 44 Example: Recovery Algorithm Integer Signal Measurements 45 Example: Recovery Algorithm Integer Signal Measurements 46 Example: Recovery Algorithm Integer Signal Measurements 47 Example: Recovery Algorithm Integer Signal Measurements 48 Example: Recovery Algorithm Integer Signal Measurements 49 Example: Recovery Algorithm Integer Signal Measurements 50 Example: Recovery Algorithm Integer Signal Measurements 51 Example: Recovery Algorithm Integer Signal Measurements 52 Example: Recovery Algorithm Integer Signal Measurements 53 Example: Recovery Algorithm Integer Signal Measurements 54 Example: Recovery Algorithm Integer Signal Measurements 55 Example: Recovery Algorithm Integer Signal Measurements 55 Example: Recovery Algorithm Integer Signal Measurements 56 Example: Recovery Algorithm Integer Signal Measurements 57 Example: Recovery Algorithm Integer Signal Measurements 58 The Evolution of the Ratio of Unresolved Variables with Iterations Above and Below the Threshold (5,6) Graphs Original SBB Algorithm 0.45 0.4 Simulation Below Threshold Simulation Above Threshold Theoretical Below Threshold Theoretical Above Threshold 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Iteration Number 61 Sparse vs. Dense Graphs Encoding Complexity: sparse ? dense 62 Sparse vs. Dense Graphs Encoding Complexity: sparse dense 62 Sparse vs. Dense Graphs Encoding Complexity: sparse dense Recovery Complexity: sparse ? dense 62 Sparse vs. Dense Graphs Encoding Complexity: sparse dense Recovery Complexity: sparse dense 62 Sparse vs. Dense Graphs Encoding Complexity: sparse dense Recovery Complexity: sparse dense Measurement Size: sparse ? dense 62 Sparse vs. Dense Graphs Encoding Complexity: sparse dense Recovery Complexity: sparse dense Measurement Size: sparse dense 62