Single Sensor Estimation of Radio Activity via Blind Block- - PowerPoint PPT Presentation

Single Sensor Estimation of Radio Activity via Blind Block- Partitioned Tensor Decomposition Christopher Mueller-Smith chris.muellersmith@rutgers.edu

Problem Formulation Sensor

Problem Formulation Transmissions are non-orthogonal in time and frequency

Problem Formulation Transmissions are non-orthogonal in time and frequency

Problem Formulation Signal 1 Signal 2 Signal 3

Problem Formulation ? FHSS OFDM ? Sensor PSK ?

Problem Formulation ? FHSS OFDM Separate Receivers: OFDM ? PSK Sensor FHSS -?- PSK ?

Problem Formulation ? FHSS OFDM Separate Receivers: OFDM ? PSK Sensor FHSS -?- PSK ?

System Model M transmitters • R m ∞ X X Linear/Linear Approx. Modulation: x m ( t ) = a m,r,k p m,r ( t − kT m,r ) • r =1 k = −∞ Packetized (non-continuous in-time transmissions) • Overlapping in time and frequency • M X Received at sensor: y ( t ) = h m ( t ) ∗ x m ( t ) + n ( t ) • m =1

Preprocessing Group statistically similar receptions together into I cl groups. 1 2 3 4 1 2 3 4 1 4 3 2 1 Frequency Time [1] G. Ivkovic, P. Spasojevic, and I.Seskar,“Mean shift based segmentation for time frequency analysis of packet based radio signals,” in 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers, Nov 2010, pp. 1526–1530.

Trispectrum Slice Estimate anti-diagonal slice of the trispectrum for each group: 04 ( f, v ) = S x 4 ( f, v, − v ) S x = E {| X ( f ) X ( v ) | 2 } − | E { X ( f ) X ∗ ( v ) }| 2 − | E { X ( f ) X ( v ) }| 2 − E {| X ( f ) | 2 } E {| X ( v ) | 2 }

Trispectrum Slice Components I cl trispectrum slices: M X | H m ( f ) | 2 | H m ( v ) | 2 S x m 04 ( f, v ) c im + S n S 04 ( f, v, i ) = 04 ( f, v ) m =1 R m S a mr 04 ( f, v ) X S x m | P mr ( f ) | 2 | P mr ( v ) | 2 04 ( f, v ) = T mr Pulse shape r =1 Symbol sequence trispectrum S a mr 04 ( f, v ) = cum 4 ( a mrk , a mrk , a ∗ mrk ) mrk , a ∗ = k mr T mr T mr

Trispectrum Slice Components I cl trispectrum slices: M X | H m ( f ) | 2 | H m ( v ) | 2 S x m 04 ( f, v ) c im + S n S 04 ( f, v, i ) = 04 ( f, v ) Signal Noise m =1 Channel Response trispectrum trispectrum Signal activity R m S a mr 04 ( f, v ) X S x m | P mr ( f ) | 2 | P mr ( v ) | 2 04 ( f, v ) = T mr Pulse shape r =1 Symbol sequence trispectrum S a mr 04 ( f, v ) = cum 4 ( a mrk , a mrk , a ∗ mrk ) mrk , a ∗ = k mr T mr T mr

Trispectrum Slice Components I cl trispectrum slices: M X | H m ( f ) | 2 | H m ( v ) | 2 S x m 04 ( f, v ) c im + S n S 04 ( f, v, i ) = 04 ( f, v ) Signal Noise m =1 Channel Response trispectrum trispectrum Signal activity R m S a mr 04 ( f, v ) X S x m | P mr ( f ) | 2 | P mr ( v ) | 2 04 ( f, v ) = T mr Pulse shape r =1 Symbol sequence trispectrum S a mr 04 ( f, v ) = cum 4 ( a mrk , a mrk , a ∗ mrk ) mrk , a ∗ = k mr T mr T mr

Trispectrum Slice Components I cl trispectrum slices: M X | H m ( f ) | 2 | H m ( v ) | 2 S x m 04 ( f, v ) c im + S n S 04 ( f, v, i ) = 04 ( f, v ) Signal Noise m =1 Channel Response trispectrum trispectrum Signal activity R m S a mr 04 ( f, v ) X S x m | P mr ( f ) | 2 | P mr ( v ) | 2 04 ( f, v ) = T mr Pulse shape r =1 Symbol sequence trispectrum S a mr 04 ( f, v ) = cum 4 ( a mrk , a mrk , a ∗ mrk ) mrk , a ∗ = k mr T mr T mr

Discretize Trispectrum Y Z j ∆ f Z n ∆ f S y y jni = 04 ( f, v, i ) d f dv I cl ( j − 1) ∆ f ( n − 1) ∆ f M R m X X = c im k rm f jrm f nrm m =1 r =1 N fb Z n ∆ f | H rm ( f ) | 2 | P rm ( f ) | 2 d f nrm = f ( n − 1) ∆ f N fb

CP Tensor Model R X X = J U , V , W K = u r � v r � w r r =1 • Explored fitting tensor to Canonical Decomposition/ Parallel Factors (CP) Tensor Model in [3] • Good performance for linear modulations • Poor performance for non-linear modulations (decomposition not unique) [3] C. Mueller-Smith and P. Spasojevic, “Single Sensor Blind Time-Frequency Activity Estimation of a Mixture of Radio Signals via CP Tensor Decomposition,” presented at the Military Communications Conference (MILCOM), 2014 IEEE, 2014, pp. 617–622.

Rank-(R m ,R m ,1) Block Partitioned Tensor The 3-way trispectrum array can be modeled as a Rank-(R m ,R m ,1) Block Partitioned Tensor (BPT): ⇥ ⇤ A 1 | A 2 | . . . | A M A = ⇥ ⇤ B 1 | B 2 | . . . | B M B = ⇥ ⇤ C = c 1 c 2 c M . . . M X ( A m · B T T = m ) � c m m =1 M R m X X t jni = c im a jrm b nrm m =1 r =1 [2] L. De Lathauwer, “Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness,” SIAM. J. Matrix Anal. & Appl. , vol. 30, no. 3, pp. 1033–1066, Jan. 2008.

Rank-(R m ,R m ,1) Block Partitioned Tensor M R m X X t jni = c im a jrm b nrm General BPT: m =1 r =1 M R m X X y jni = c im k rm f jrm f nrm Trispectrum BPT: m =1 r =1

BPT Formulation for Trispectrum Slice ⇥ F 1 | F 2 | . . . | F M ⇤ F = ⇥ c 1 ⇤ c 2 c M C = . . . ⇤ T ⇥ k T 1 | k T 2 | . . . | k T k = M M X F m · diag( k m ) · F T Y = m � c m m =1 Note in our case we have repeated factor matrix e.g. A = B .

BPT Formulation for Trispectrum Slice M X F m · diag( k m ) · F T Y = m � c m m =1

Current BPT Decomposition Algorithms • Alternating Least Squares (ALS) • Non-blind (partitioning known) • A ≠ B - ALS not intended for decompositions with repeated factor matrices

Reformulate BPT Reformulate BPT to look like a CANDECOMP/PARAFAC (CP) tensor model: R X X = J U , V , W K = u r � v r � w r r =1 Y = J F , F , CP · diag( k ) K = J F , F , CW K

Reformulate BPT Reformulate BPT to look like a CANDECOMP/PARAFAC (CP) tensor model: Y = J F , F , CP · diag( k ) K = J F , F , CW K  1 R 1 0 R − R 1  0 R 1 1 R 2 0 R − R 1 − R 2     . . ...   . . . .   P =   0 P m 1 R m 0 R − P m i =1 R i − R 1 i =1 R i     . . ...   . . . .   0 P M 1 R M i =1 R i − R 1

Reformulate BPT Reformulate BPT to look like a CANDECOMP/PARAFAC (CP) tensor model: Y = J F , F , CP · diag( k ) K = J F , F , CW K R-sparse R-sparse  1 R 1 0 R − R 1  0 R 1 1 R 2 0 R − R 1 − R 2     . . ...   . . . .   P =   0 P m 1 R m 0 R − P m i =1 R i − R 1 i =1 R i     . . ...   . . . .   0 P M 1 R M i =1 R i − R 1

Estimation Strategy 1 2 k Y � J F , F , CW K k 2 + λ k W k 1 [ F , C , W ] = arg min F , C , W • Under certain conditions the BPT decomposition of Y is unique (up to permutation and scaling of columns) • Output of algorithms is ‣ F : estimates of signals’ power spectra ‣ C : estimates of signals’ on/off activity (in time) ‣ W : estimate of signal grouping

Simulation Power Spectrum 0 • 3 overlapping signal − 20 groups − 40 • Each signal group Power (dB) − 60 experiences different − 80 Rayleigh fading − 100 channels − 120 • Varying SNR − 140 − 1 − 0.5 0 0.5 1 Frequency (radians)

Preliminary Results Probability of Correct Activity Sequence Estimation Normalized Mean Square Error of F − 5 1 N f b=32, M=3, R m ={3,2,1} − 6 0.95 N fb =64, M=3, R m ={3,2,2} Probability of correct C − 7 0.9 − 8 NMSE (dB) 0.85 − 9 0.8 − 10 0.75 − 11 0.7 N f b=32, M=3, R m ={3,2,1} N fb =64, M=3, R m ={3,2,2} − 12 0.65 − 13 0.6 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 SNR (dB) SNR (dB) [3] C. Mueller-Smith and P. Spasojevic, “Single Sensor Blind Time-Frequency Activity Estimation of a Mixture of Radio Signals via CP Tensor Decomposition,” presented at the Military Communications Conference (MILCOM), 2014 IEEE, 2014, pp. 617–622.

Estimate of PSD Power Spectrum 15 PSD m=1 PSD m=2 Est. PSD m=1 10 Est. PSD m=2 5 0 Power (dB) − 5 − 10 − 15 − 20 − 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 Frequency

Estimate of PSD Power Spectrum 0 − 2 − 4 − 6 − 8 Power (dB) − 10 − 12 − 14 − 16 − 18 − 20 − 1 − 0.9 − 0.8 − 0.7 − 0.6 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 0 0.1 Frequency ( π radians)

Continuing Work • How to choose appropriate L1 regularization coefficient. • Alternative algorithms that avoid tuning of L1 regularization coefficient. • Alternative algorithms that do not require estimating appropriate values for M and R. • Characterization of algorithm performance in varying conditions (SNR, signal orthogonality in freq. & time). • System implementation in software defined radio.

Thank You