Subspace intersection tracking using the Signed URV algorithm

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Subspace intersection tracking using the

Signed URV algorithm

Mu Zhou and Alle-Jan van der Veen

TU Delft, The Netherlands

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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

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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 (2250 time slots = 1 minute)

Data includes ID, GPS location, course, speed

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AIS signal separation

Idea: use LEO satellites for ship tracking

On surface: 50 km range; from satellite: 500 km range: many packet colli-sions (also only partial synchronization)

Significant Doppler shifts (only partial frequency overlap)

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AIS signal separation

ISIS AIS satellite prototype (Triton-1 mission)

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AIS signal separation

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

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AIS signal separation

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AIS signal separation

AIS overlapping signals

Example of a measurement 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 4 5x 10 −3 Samples Amplitude 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 −1 0 1 2 3 Samples Amplitude

one of the signals (after separation) 4-antenna measurements

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AIS signal separation

Proposed multi-user receiver

Blind beamforming

stage 1: asynchronous interference suppression

stage 2: synchronous interference cancellation (block constant modulus algorithm)

Demodulator

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Data model

Received signal

Assume antennas, stack received signals

into columns :

: tall, full column rank; columns normalized to

   targets ! " interference # Analysis window

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Data model

The signals can be considered “zero constant modulus”. Constant modulus al-gorithms 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 sepa-rated using constant modulus algorithms (e.g. ACMA).

Data model

The noise is considered white with power

. targets interference Analysis window

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Data model

Covariance model

“Assume” that

and

contain stationary data (they don’t):

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

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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 where is any

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Tools from linear algebra

Generalized SVD

For two matrices , (both , ’wide’), the GSVD is GSVD   

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

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Tools from linear algebra

Generalized SVD (cont’d)

. Given some tolerance

, partition and as                                 and correspondingly as

contains the common column span, i.e.,

is the subspace of columns that are in

but not in ,

is the subspace of columns that are in

but not in ,

is a common left null space.

Thus, the GSVD provides subspace intersection.

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Tools from linear algebra

Generalized Eigenvalue Decomposition (GEV)

Squaring the GSVD, we obtain (for positive definite matrices

) GEV   

where is invertible and , are diagonal and positive (

).

Unclear if the decomposition exists if

and indefinite ( and may become complex). Can partition

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Source separation

Noise-free case

Recall the data model:

                The GEV of is

For a small threshold

, partition , , as                    

and moreover, sort s.t.

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Source separation

Comparing the “sorted” GEV with the data model, we immediately obtain

Using , we can construct a separating beamformer as

or, alternatively

Case with white noise with known covariance

!

Now, 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

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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

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Source separation

Separation performance: SINR as function of SIR for SNR = 15 dB

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Source separation

Extensive simulation

Carrier frequency 162.025 MHz

Channel bandwidth 25 kHz (modulation 9.6 kbps GMSK,

Satellite altitude 600 km Satellite speed 7561.65 m/s Orbit period 5792.52 s

Radius of FoV 1396.25 nautical miles Ship visible time 704 s per sat. pass

Ship emission power 12.5 W(Class A)/2 W(Class B) Ship transmit antenna Half-wave dipole

Sat. receive antenna Array of directional elements Sat. antenna spacing Half wavelength

Array spinning speed 1 round/30 s Max. SNR at the receiver 25 dB

Cell size

(square)

Num. of Cells in FoV 5184 Ship report interval 6 s

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Source separation

Sip detection probability

1 2 4 8 16 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of antennas S h ip d et ec ti o n p ro b a b il it

y Uniform ship distribution_{System time period = 704 s}

Sat. altitude = 600 km

Number of ships in FoV = 5,000 Number of ship IDs = 12,747 Ship report interval = 6 s

Number of sent messages = 296,320

Avg. number of messages per slot = 11.1111 GSVD-T+ACMA

GSVD-SI+ACMA ACMA

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Source separation

Tracking

The analysis window slides over the data. This allows to receive new messages as targets. Need “updating” and “downdating”.

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Source separation

Tracking

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Source separation

Tracking

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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.

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Intermezzo

Elementary rotations lattice ladder ! ! " ! ! " " " Consider a rotation:               Conservation of energy:

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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 Synthesis e e Analysis

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Intermezzo

Properties of elementary hyperbolic rotations

With   

 we have conservation of energy in the -inner product:

        Define         With it follows that is -unitary:

Note also that

and .

This generalizes to larger -unitary matrices.

The case where

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Replacing GEV by SSE

Schur subspace estimator (SSE)

We show how the SSE partitions the space into a “positive” and “negative” sub-space, without computing the SVD.

For two given matrices

and

, compute

(not unique) such that

SSE h i h i where is square and is a -unitary matrix:    

decomposes into a series of

hyperbolic rotations, so this looks like a

“hy-perbolic QR” factorization, where the role of “ ” is played by

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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

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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:

       

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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)

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The Schur Subspace Estimator

Schur subspace estimator (SSE)

For two given matrices and , compute

such that where

has full column rank and is a

-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

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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”)

HURV

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Hyperbolic URV decomposition

Low-rank approximation

Consider , where is a threshold, and introduce the SVD of

as where

Assume that has singular values larger than ; none equal to .

We compute the SSE

where (inertia) has columns and has

columns. Theorem 1

parametrize all

approximants such that

(matrix 2-norm)

In particular, the column span of any such is parametrized as

with with

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Hyperbolic URV decomposition

Example

A valid rank- approximant is

Indication of proof:

Rank because has columns

The norm property follows from

    | {z } | {z }

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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 , but there are other choices.

In particular we will use (SSE-2)

   .

The TSVD is a special case of such an approximant, corresponding to a decom-position with and a specific .

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Hyperbolic URV decomposition

“Pre-whitened” low-rank approximation

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

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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

In particular, , .

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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.

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SURV updating

Elementary rotations such that

1. If or , and : (Hyperbolic rotation)     where , q ; ; . 2. If or , and : (Hyperbolic rotation)     where , q ; (sign reversal); 3. If or : (Givens rotation)     where , q ; ; p .

Case 1 or 2 (hyperbolic rotation): If

, then is unbounded but the result is

well-defined:

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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 and such that h i h i where (signature ), or (signature ); .

Denote the signature of by

.

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SURV updating – Zeroing schemes for

GCR: Givens Column Rotations

Apply only if

:

1. Compute Givens rotation such that

2. Apply to the

-th column of and (no sign change).

GCR

0

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SURV updating – Zeroing schemes for

HCR: Hyperbolic Column Rotations

Apply if : 1. Set

, and compute and

such that 2. Apply to the

-th column of and ; update signatures

following

(possible sign change).

HCR

0

Try to avoid this operation as can be very large (unbounded if

).

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SURV updating – Zeroing schemes for

GRCR: Givens Row and Column Rotations

Apply only if

:

1. Compute Givens row rotation such that ; 2. Apply to rows of ; apply to columns of ; 3. Compute Givens column rotation such that

; 4. Apply to columns of

(no sign change).

0

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SURV updating – Zeroing schemes for

GRR: Givens Row Rotations to zero

" Apply only if :

1. Compute Givens row rotation such that ; 2. Apply to rows of ; apply to columns of .

GRR

0

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SURV updating

Updating sequence for

GRCR HCR _or 0 0 0 0 0 0 0 0 0 0 0 0 0 GCR 0 0 0 0 0 0 0

Case ( ): no sign change, no rank change;

. Done.

Case ( ): sign reversal, rank decrease;

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SURV updating

Signature sorting steps

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 GRR swap 0 0

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SURV updating

Updating sequence for

GRCR swap 0 0 0 0 GRCR 0 0

Tentative rank increase.

Continue as in step ( ) for

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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:

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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

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 such that

, i.e., the same subspace is obtained using and a new parameter

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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

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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

and

, there exist matrices

, , such that h i h i h i     where is unitary, is -unitary,

is lower triangular, and is an invertible matrix (actually,

). Let . Then

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SURV updating

Corollary 1 For this decomposition, is bounded if

is nonsingular. In any case we have

Thus, the results

of the decomposition are bounded by the inputs, even if

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 decom-position (without explicitly computing or storing and ).

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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)

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Background material

References

[1] J. Götze and A.-J. van der Veen, “On-line subspace estimation using a

Schur-type method,” IEEE Trans. Signal Process., vol. 44, no. 6, pp. 1585–1589, Jun. 1996.

[2] A.-J. van der Veen, “A Schur method for low-rank matrix approximation,” SIAM

J. Matrix Anal. Appl., vol. 17, no. 1, pp. 139–160, 1996.

[3] M. Zhou and A.-J. van der Veen, “Stable subspace tracking algorithm based

on a signed URV decomposition,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 3036–3051, Jun. 2012.

[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,