Subspace intersection tracking using the Signed URV algorithm

Transcription

1 Subspace intersection tracking using the Signed URV algorithm Mu Zhou and Alle-Jan van der Veen TU Delft, The Netherlands 1

2 Outline Part I: Application 1. AIS ship transponder signal separation 2. Algorithm based on Generalized SVD (GSVD) Part II: Subspace tracking 1. Signed (hyperbolic) URV to approximate the GSVD 2. Updating the SURV 2

3 AIS signal separation Automatic Idenfication of Ships (AIS) A default AIS message is a binary sequence of 256 bits GMSK modulated, kbps, MHz Short data packets in a TDMA system (225 time slots = 1 minute) Data includes ID, GPS location, course, speed Used for ship-ship (anti-collision) and ship-shore (tracking) 3

4 AIS signal separation Idea: use LEO satellites for ship tracking On surface: 5 km range; from satellite: 5 km range: many packet collisions (also only partial synchronization) Significant Doppler shifts (only partial frequency overlap) Many partially overlapping signals, no user codes: need blind source separation 4

5 AIS signal separation ISIS AIS satellite prototype (Triton-1 mission) Launched December 213 5

6 AIS signal separation Global ship distribution and a satellite field of view. The red dots denote ships within the FoV. 6

7 AIS signal separation TU Delft experimental AIS 4-channel receiver 7

8 AIS signal separation AIS overlapping signals Example of a measurement 5 x antenna measurements 4 Amplitude Samples 3 2 one of the signals (after separation) Amplitude Samples 8

9 AIS signal separation Proposed multi-user receiver Blind beamforming stage 1: asynchronous interference suppression stage 2: synchronous interference cancellation (block constant modulus algorithm) Demodulator Bank of standard single-channel GMSK receivers 9

10 Data model Received signal Assume antennas, stack received signals into columns : : tall, full column rank; columns normalized to targets interference Analysis window 1 # "!

11 Data model The signals can be considered zero constant modulus. Constant modulus algorithms cannot directly be applied because part of the signal is zero. We will derive a blind separation algorithm for the structure zero/non-zero. That will suppress the asynchronous interference. The targets can be further separated using constant modulus algorithms (e.g. ACMA). Data model The noise is considered white with power. targets interference Analysis window 11

12 Data model Covariance model Assume that The signal covariance matrices are We assume these are diagonal matrices; the diagonal entries contain the signal powers. The signals are considered independent. targets interference Analysis window 12 and contain stationary data (they don t):

13 where Data model Covariance model The distinction between target signals and interfering signals is defined by I.e., target signals are stronger (more samples present) in the first data block than in the second data block. (This can be generalized to.) Objective Compute a separating beamforming matrix of size, such that is any full rank matrix (residual mixing of the target signals). 13

14 Tools from linear algebra Generalized SVD For two matrices GSVD, (both, wide ), the GSVD is is an invertible matrix, and are square positive diagonal matrices, are semi-unitary matrices of size Columns of are scaled to norm 1. This definition is transposed compared to the Matlab definition. Also the scaling is different: Matlab has. 14

15 contains the common column span, i.e.,, partition correspondingly as but not in is the subspace of columns that are in Tools from linear algebra Generalized SVD (cont d). Given some tolerance and as and is the subspace of columns that are in but not in,, is a common left null space. Thus, the GSVD provides subspace intersection. 15

16 , is invertible and Unclear if the decomposition exists if Tools from linear algebra Generalized Eigenvalue Decomposition (GEV) Squaring the GSVD, we obtain (for positive definite matrices GEV ) where are diagonal and positive ( ). and indefinite ( and may become complex). Can partition in the same way as for the GSVD. 16

17 Source separation Noise-free case Recall the data model: The GEV of is For a small threshold, partition,,as and moreover, sort s.t. 17

18 Using Now, Source separation Comparing the sorted GEV with the data model, we immediately obtain, we can construct a separating beamformer as or, alternatively Case with white noise with known covariance! from GEV changes (unlike EVD of a single matrix in white noise which will shift eigenvalues but not change the eigenvectors). Could compute GEV ; but risk that matrices become indefinite. First need to remove the noise subspace. Single matrix: If the noisefree decomposition is, then with noise 18

19 Source separation Algorithm using SVD and GEV 1. Preprocessing to remove noise subspace: compute the SVD: Then apply a rank and dimension reduction: 2. Compute the rank-reduced covariance matrices 3. Compute the GEV of the noise-shifted rank-reduced covariance matrices, GEV 4. Sort the entries of and correspondingly partition. The term should be absent as the noise subspace has been removed. 5. The separating beamformer is 19

20 Source separation Separation performance: SINR as function of SIR for SNR = 15 db Packets with random time offsets (2 targets, 3 interferers), antennas. 2

21 Source separation Extensive simulation Carrier frequency Channel bandwidth Satellite altitude Satellite speed Orbit period Radius of FoV Ship visible time MHz 25 khz (modulation 9.6 kbps GMSK, 6 km m/s s nautical miles 74 s per sat. pass Ship emission power 12.5 W(Class A)/2 W(Class B) Ship transmit antenna Sat. receive antenna Sat. antenna spacing Array spinning speed Max. SNR at the receiver Cell size Half-wave dipole Array of directional elements Half wavelength 1 round/3 s 25 db (square) Num. of Cells in FoV 5184 Ship report interval 6 s 21

22 Source separation Sip detection probability GSVD-T+ACMA GSVD-SI+ACMA ACMA ESPRIT+Capon Ship detection probability Uniform ship distribution System time period = 74 s Sat. altitude = 6 km Number of ships in FoV = 5, Number of ship IDs = 12,747 Ship report interval = 6 s Number of sent messages = 296,32 Avg. number of messages per slot = Number of antennas 22

23 Source separation Tracking The analysis window slides over the data. This allows to receive new messages as targets. Need updating and downdating. Analysis window 23

24 Source separation Tracking The analysis window slides over the data. This allows to receive new messages as targets. Need updating and downdating. Analysis window 24

25 Source separation Tracking The analysis window slides over the data. This allows to receive new messages as targets. Need updating and downdating. Analysis window 25

26 Towards part II The source separation algorithm works nice, but... Uses both SVD and GEV, thus not suitable for tracking (sliding window operation); The noise shifting is awkward. We propose to use a new tool, the Schur subspace estimator (SSE), which can replace the SVD and GSVD, and is easily updated allowing sliding window tracking of subspaces. Recall, the Schur algorithm establishes the stability of a polynomial (roots inside unit circle) without explicitly computing the roots. Likewise, the SSE partitions the space into a dominant and a minor subspace w.r.t. a threshold, without computing the SVD. 26

27 !!! " " " Intermezzo Elementary rotations ladder lattice Consider a rotation: Conservation of energy: ( ) 27! "

28 Intermezzo Schur recursion Such elementary rotations are used in the familiar Schur recursion: the analysis filter consists of hyperbolic rotations which create zeros in the input vectors, the synthesis filter of Givens rotations. The are the reflection coeffients. e stable allpass filter e e Synthesis e Analysis 28

29 is Intermezzo Properties of elementary hyperbolic rotations With we have conservation of energy in the -inner product: Define With it follows that -unitary: Note also that and. This generalizes to larger -unitary matrices. The case where is problematic and should be avoided. 29

30 is a perbolic QR factorization, where the role of is played by Replacing GEV by SSE Schur subspace estimator (SSE) We show how the SSE partitions the space into a positive and negative subspace, without computing the SVD. For two given matrices and, compute (not unique) such that SSE [ ] [ ] where is square and -unitary matrix: decomposes into a series of hyperbolic rotations, so this looks like a hy-. 3

31 Replacing GEV by SSE If we square the data, we obtain and capture the positive and negative part of using factors of minimal dimensions. In our application, we had the asymptotic data model: (Note that the noise covariance is cancelled in the difference.) We can show there exists a such that (asymptotically) In particular,,. For finite, these become good approximations. Thus, the SSE gives directly the required subspaces. But how is it computed? 31

32 The Schur Subspace Estimator Subspace estimation is related to the following problem: Problem For a given matrix and tolerance level, find all approximants such that where is equal to the number of singular values of that are larger than. ( denotes the matrix 2-norm.) The usual solution goes via a truncated SVD: TSVD is expensive to compute, especially for on-line applications 32

33 The Schur Subspace Estimator TSVD OTHER SOLUTION + Results There are many other approximants that do not set singular values to zero. They are still optimal in 2-norm, not in the Frobenius norm. A generalized Schur algorithm provides a parametrization of all solutions without computing SVDs, but rather a Hyperbolic QR (actually Hyperbolic URV) 33

34 and The Schur Subspace Estimator Schur subspace estimator (SSE) For two given matrices, compute such that is a where has full column rank and -unitary matrix: If we square the data, we obtain and capture the positive and negative part of using factors of minimal dimensions. The decomposition always exists but, and are not unique. 34

35 HURV Hyperbolic URV decomposition An example is given by the signed Cholesky factorization, where is lower or upper triangular. This corresponds to a hyperbolic QR factorization. However, this decomposition doesn t always exist, the triangular shape is too restrictive. This motivates to introduce a QR-factorization of : where is unitary and is lower (or upper) triangular. The result is a two-sided decomposition ( hyperbolic URV ) 35

36 Assume that Hyperbolic URV decomposition Low-rank approximation Consider, where is a threshold, and introduce the SVD of as where has singular values larger than ; none equal to. We compute the SSE has is parametrized as where (inertia) has columns and columns. 1 Theorem parametrize all rank- approximants such that (matrix 2-norm) In particular, the column span of any such with with 36

37 Hyperbolic URV decomposition Example A valid rank- approximant is Indication of proof: Rank because has columns The norm property follows from } {{ } } {{ } 37

38 Hyperbolic URV decomposition Subspace estimation All subspace estimates are given by We could choose and simply use as an estimate for the principal column span of In particular we will use (SSE-2), but there are other choices.. The TSVD is a special case of such an approximant, corresponding to a decomposition with and a specific. 38

39 Hyperbolic URV decomposition Pre-whitened low-rank approximation More in general, consider. Then all low-rank approximants with such that such that have a column span parametrized by. In applications, could be a data matrix (including noise), and could be an imitation of the noise process, e.g., from a nearby frequency. 39

40 Hyperbolic URV decomposition Relation to GSVD We can show that the GSVD is a special case of the SSE: The GSVD of two matrices is where the sorting and partitioning is such that, (for simplicity of notation, assume there is no common null space: is missing). Compare this to the SSE 4

41 particular, In, Hyperbolic URV decomposition Squaring the GSVD, we have the GEV partitioned such that,. Then Squaring the SSE gives We can show there exists a such that. 41

42 SURV updating The signed URV (SURV) is a stable algorithm to compute and update the HURV. The decomposition is not unique, and we will subsequently place an additional constraint that leads to favorable properties. Elementary rotations Let be an (unsorted) signature matrix, and similar for. A matrix is an elementary rotation if it satisfies,. Given and input signature. We can determine such that The output signature follows from sign of and inertia. 42

43 well-defined: or or or where where SURV updating Elementary rotations such that 1. If (Hyperbolic rotation), and, : ; ;. 2. If (Hyperbolic rotation), and, : ; 3. If (Givens rotation) where :, ; ;. Case 1 or 2 (hyperbolic rotation): If, then is unbounded but the result is 43 (sign reversal);

44 SURV updating Suppose we have already computed the decomposition where is square, lower triangular and sorted according to signature. To update, let us say that we want to find a new factorization where either (downdate), or (update). It suffices to find [ ] [ ] where (signature ), or (signature );. Denote the signature of by. The rank of the principal subspace before the update is, after the update. 44 and such that

45 -th column of SURV updating Zeroing schemes for GCR: Givens Column Rotations Apply only if 1. Compute Givens rotation 2. Apply to the : such that and (no sign change). GCR 45

46 -th column of SURV updating Zeroing schemes for HCR: Hyperbolic Column Rotations Apply if 1. Set 2. Apply : to the, and compute and and ; update signatures following (possible sign change). HCR Try to avoid this operation as can be very large (unbounded if ). 46 such that

47 SURV updating Zeroing schemes for GRCR: Givens Row and Column Rotations Apply only if 1. Compute Givens row rotation 2. Apply to rows 3. Compute Givens column rotation 4. Apply to columns of of : such that ; apply such that [ ] [ to columns (no sign change). of ] ; ; ; 47

48 SURV updating Zeroing schemes for GRR: Givens Row Rotations to zero Apply only if : 1. Compute Givens row rotation 2. Apply to rows of such that ; apply [ ] to columns of [ ] ; GRR This is used as a clean-up operation after is zeroed. 48 ".

49 Case ( Case ( SURV updating Updating sequence for GCR GRCR HCR or ): no sign change, no rank change;. Done. ): sign reversal, rank decrease;. Continue: 49

50 SURV updating Signature sorting steps GRR swap 5

51 SURV updating Updating sequence for GRCR swap Tentative rank increase. Continue as in step () for, before. 51 GRCR

52 SURV updating At most a single hyperbolic rotation is used (corresponding to a single rank change decision). It involves and. If then is unbounded but the result is well defined, and this unbounded acts only on columns for which the other entries are already. Thus, will remain bounded. This is one of the keys to show numerical stability, despite the use of hyperbolic rotations. Computational complexity: per update. Only and are tracked/stored. 52

53 SURV updating SSE-2 definition and properties The HURV decomposition is not unique, and we can place additional constraints to reach desired properties. All valid subspace estimates have the form such that where is a contractive matrix that parametrizes all solutions. Given a specific it is always possible to transform, using additional ro- tations to a new, i.e., the same subspace is obtained using and a new parameter. 53

54 SURV updating The Schur Subspace Estimate SSE-2 [2] is obtained for where This is interesting because of the following: Theorem 2 Given an HURV decomposition, and consider =. Then. This shows that the estimator is unbiased and bounded by the input data. The SSE-2 is still not unique. The SVD subspace estimate is a special case of an SSE-2. 54

55 SURV updating The SURV algorithm provides an SSE-2 decomposition Idea: use the available freedom on to add constraints that ensure. Theorem 3 For given matrices,, such that and, there exist matrices [ ] [ ] [ ] where is unitary, is an invertible matrix (actually, is ). -unitary, is lower triangular, and Let. Then is an SSE-2 subspace estimate. 55

56 SURV updating Corollary 1 For this decomposition, is bounded if is nonsingular. In any case we have of the decomposition are bounded by the inputs, even if Thus, the results may be unbounded. Also the corresponding subspaces are well-defined. The norm properties could be key to a formal proof on numerical stability of this algorithm.. Theorem 4 The SURV algorithm presented before provides the required decomposition (without explicitly computing or storing and ). 56

57 Conclusions GSVD is a nice tool for separating partially overlapping data packets. SURV is a nice tool to replace the GSVD in subspace tracking applications. Similar algorithms are applicable for separating airplane signals (SSR system) and RFID signals, and for suppressing Bluetooth interference from WiFi signals. 57

58 Background material References [1] J. Götze and A.-J. van der Veen, On-line subspace estimation using a Schurtype method, IEEE Trans. Signal Process., vol. 44, no. 6, pp , Jun [2] A.-J. van der Veen, A Schur method for low-rank matrix approximation, SIAM J. Matrix Anal. Appl., vol. 17, no. 1, pp , [3] M. Zhou and A.-J. van der Veen, Stable subspace tracking algorithm based on a signed URV decomposition, IEEE Trans. Signal Process., vol. 6, no. 6, pp , Jun [4] Mu Zhou and A.J. van der Veen, Blind Beamforming Techniques for Automatic Identification System using GSVD and Tracking, in Proc. Int. Conf. Acoustics, Speech, Signal Proc. (ICASSP 214), Florence (Italy), May