RobustRange-rateEstimationofPassiveNarrowbandSourcesin …people.ee.duke.edu/~jk/SAM Group_files/Hailiang... · 2010. 10. 26. · The SWellEx-96 Environment Typical shallow water

Robust Range-rate Estimation of Passive Narrowband Sources inShallow Water

Hailiang Tao and Jeffrey Krolik

Department of Electrical and Computer Engineering

Duke University

Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water – p. 1/23

Outline

• Introduction• The Independent Mode Range-rate Estimator• Comparison with Cramer-Rao Bound• Application to SWellEx-96 data• Conclusions.


Range-Rate Discrimination in Passive Sonar

Motivation:• Range-rate is a more robust searching dimension with regard to

wavenumber mismatch in Matched Field Processing (MFP) comparedwith absolute range.

• The discrimination capability in range-rate is helpful to detect targets inthe presence of interferences with similar bearings but different relativerange-rate.

Background:

• Normal mode theory for a narrowband moving source derived byHawker (JASA, 1979). Projection of target Doppler onto different modesinduces signal fluctuations which are target range-rate dependent.

• Direct application of MUSIC (Song, 1990) does not take advantage ofavailable environmental information and requires a long observationtime.

• Direct extension of Matched Field Processing (Zala, 1992) with range-rateis computational intensive due to parameter coupling.


Acoustic Normal Mode Theory For a Moving Source

The velocity potential emitted by a narrowband, horizontally uniform movingpoint source, in a range-independent stratified oceanic waveguide: (Hawker,1979)

ψ(t) ≈ CM

X

m=1

Um(z)Um(zs)√kmR0

exp

»

jωmt− jkmR0„

1 − vrvGm

«–

(1)

where Um(z) and Um(zs) are mode eigenfunctions at receiver depth z andsource depth zs. km is the horizontal wavenumber. R0 the initial range.

ωm = ω0 − kmvr(1 −vrvGm

) ≈ ω0 − kmvr

is the Doppler frequency, in which ω0 is the intrinsic frequency, vr therange-rate of the source and vGm the group velocity of mode m.


Differential Doppler for a Moving Source

Differential Doppler for moving vs. stationary source evident from slope offrequency distribution as a function of modal wavenumber for a single tonalsource. Note that range-rate dependence does not require relative phasebetween modal components.

0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22

47.5

48

48.5

49

49.5

50

50.5

Horizontal wavenumber (1/m)

Fre

quen

cy (

Hz)

f−k diagram for stationary and moving source

v=10m/sv=0


The Time-Frequency Data Snapshot Vector

Consider snapshot of narrowband data which consists of a concatenation of Nconventional frequency domain snapshots over time.

Note range-rate processing can be performed before or after conventionalbeamforming.


The Signal Model (1)

Model the (pN × 1) space-time snapshot r as:

r = sX

m

am(dm ⊗ um) + n (2)

Where s is the signal amplitude, unknown nonrandom. am is a zero-meancomplex random variable representing mode m’s amplitude. dm is a timeharmonic vector for mode m:

dm = [1 e−jkmvrTd e−jkmvr2Td · · · e−jkmvr(N−1)Td ]T

⊗ is the Kronecker product. um represents mth mode eigenfunctionsevaluated at the depths of each array element:

um = [Um(z1) Um(z2) . . . Um(zp)]T

and n ∼ CN(0, σ2nI) represents complex white Gaussian noise.


The Signal Model (2)

In matrix formr = sDa + n (3)

whereD = [d1 ⊗ u1 d2 ⊗ u2 . . . dM ⊗ uM ]

anda = [a1 a2 . . . aM ]

T

The variance of mode amplitude is

σ2m = E(amaHm) = Um(zs)

2/km (4)

In this model:• Assume am and an are uncorrelated for m 6= n.• The model is independent of absolute range.


The Independent Mode Range-rate Estimator (IMRE)

The covariance matrix is:

R = E(rrH) = PsDSDH + σ2nI (5)

where Ps = |s|2 and S = E(aaH) = diag[σ21 σ22 . . . σ2M ].Given the environment and target depth zs, the signal covariance matrixRs = DSD

H could be computed for each hypothesized range-rate vr .Suppose the dominant eigenvector of Rs is h1 (normalized), then a Bartletttype estimator can be expressed as:

P (vr) =1

λ1h

H1 R̂h1 (6)

which maximizes output SNR in white noise. R̂ is the sample covariancematrix and λ1 is the maximum eigenvalue of R̂.


The SWellEx-96 Environment

Typical shallow water environmental profiles of SWellEx-96 are used insimulations:

1485 1490 1495 1500 1505 1510 1515 1520 1525

0

20

40

60

80

100

120

140

160

180

200

216.5

Sound Velocity (m/s)

Dep

th (

m)

S5S59

SWellEx-96 Water Column SoundVelocity Profile

SWellEx-96 Sea Bottom Properties


Typical IMRE Output

−6 −4 −2 0 2 4 6−35

−30

−25

−20

−15

−10

−5

0

Range−rate (m/s)

IMR

E o

utpu

t pow

er (

dB)

Swellex96−S5:ds=51.0m, f

s=50Hz, v

s=2.50m/s, d

a=70.0m, n

snap=20, T

d=1.0s, nSample=200, SNR=20dB

The sidelobe is 13dB down from the mainlobe.


Typical Eigenspectrum of Covariance Matrix R

Same settings as previous slide without noise. Source is moving at 2.5m/s.

Eigenvalue number Eigenvalues (dB) Normalized eigenvalues1 −18.3 0.97022 −33.4 0.02973 −59.6 0.7231 × 10−44 −85.2 0.1998 × 10−65 −111.7 0.4465 × 10−96 −140.8 0.5507 × 10−127 - 08 - 0...

......

20 - 0

The dB difference between 1st and 2nd eigenvalues corresponds tomainlobe/sidelobe difference in previous slide.


Range-rate Aliasing

Range-rate difference between grating lobes:

vgr ≈2π

k̄Td(7)

where k̄ is the mean value of km. Td is the snapshot delay.

−20 −15 −10 −5 0 5 10 15 20−14

−12

−10

−8

−6

−4

−2

0

Range−rate (m/s)

IMR

E o

utpu

t pow

er (

dB)

Swellex96−s5:ds=51.0m, f

s=50Hz, v

s=2.50m/s, d

a=70.0m, n

snap=20, nSample=200, SNR=0dB

Td = 2s

Td = 0.5s

Grating Lobe Distance vrg


Insensitivity to Target Depth

Range−rate (m/s)

Dep

th (

m)

Swellex96−s5:ds=51.0m, f

s=50Hz, v

s=2.50m/s, d

a=70.0m, n

snap=20, T

d=1.0s, nSample=200, SNR=0dB

−6 −4 −2 0 2 4 6

0

20

40

60

80

100

120

140

160

180

200

−12

−10

−8

−6

−4

−2

IMRE search in both depth and range-rate with one sensor


Cramer-Rao Low Bound (1)

• The unknown parameters are: Θ = (vr, Ps, σ2n)T .• each element of Fisher Information Matrix (FIM) could be represented by:

Jij = Nstr

»

R−1 ∂R

∂θiR

−1 ∂R

∂θj

–

(8)

where Ns is the number of sampled r.To compute Jij , recall

R = PsDSDH + σ2nI

So∂R

∂Ps= DSDH (9)

∂R

∂σ2n= I (10)


Cramer-Rao Low Bound (2)

Let c = [0 1 2 . . . (N − 1)]T and k = [k1 k2 . . . kM ]T .Define matrix G:

G = −jTdckT

andH = G ⊗ h

where vector h is a p (array length) by 1 vector with all its elements being 1.Thus

∂R

∂vr= Ps(

∂D

∂vrSD

H + DS∂DH

∂vr)

= Ps((H � D)SDH + DS(H � D)H) (11)

where � is element-by-element Hadamard product.


Comparison of IMRE with CRB

−20 −15 −10 −5 0 5−50

−40

−30

−20

−10

0

10

20

SNR(dB)

Ran

ge−

rate

MS

E (

dB)

Swellex96−S5: ds=51.0m, f

s=100Hz, v

s=2.50m/s, d

a=70.0m, n

snap=20, nSample=200, NMC=200

Monte CarloCRLB

Between SNR −15 ∼ 0dB, the mean square error of IMRE achieves CRLB. Thethreshold happens at −16dB.


High SNR Behaviour Analysis

Express the dominant eigenvector of the sample covariance matrix as:

b̂1 = b1 + η (12)

The estimator error vector η has the following asymptotic correlation function:

E[ηηH ] =λ1Ns

NX

k=2

λk(λ′1 − λ′k)2

bkbHk

where λ′s are the eigenvalues of PsDSDH . Looking at λ1λk(λ′1−λ′

k)2

, we have:

• When the noise variance σ2n is far smaller than the second eigenvalue λ′2of PsDSDH , the performance of the algorithm will not improve withhigher SNR, i.e., not stick to the CRB.

• The smallest MSE achieveable is determined by λ′

2

λ′1

.


High SNR Performance at Different Frequencies

0 5 10 15 20 25 30 35 40 45−55

−50

−45

−40

−35

Ran

ge−

rate

MS

E (

dB) Swellex96:ds=51.0m, fs=20Hz, vs=2.50m/s, da=70.0m, nsnap=20, nSample=200, NMC=200

0 5 10 15 20 25 30 35 40 45−55

−50

−45

−40

−35

Ran

ge−

rate

MS

E (

dB)

Swellex96:ds=51.0m, f

s=50Hz, v

s=2.50m/s, d

a=70.0m, n


0 5 10 15 20 25 30 35 40 45−55

−50

−45

−40

−35

SNR(dB)

Ran

ge−

rate

MS

E (

dB)

Swellex96:ds=51.0m, f

s=100Hz, v

s=2.50m/s, d

a=70.0m, n


λ’2/λ’

1 = 0.0018

λ’2/λ’

1=0.0306

λ’2/λ’

1=0.1096


Source Tracks in the SWellEx-96 Experiment

Source: UCSD Marine Physical Laboratory SWellEx-96 Website.


Range-rate Track of S5 and S59 Events in SWellEx96

Time (m)

Ran

ge−

rate

(m

/s)

SwellEx96:s59 79Hz, zs=54.0m, z

a=94.1m, nSnap=20, t

FFT=1.0s

10 20 30 40 50 60−6

−4

−2

0

2

4

6

−40

−30

−20

−10

0

Time (m)

Ran

ge−

rate

(m

/s)

SwellEx96:s5 79Hz, zs=54.0m, z

a=94.1m, nSnap=20, t

FFT=1.0s

10 20 30 40 50 60 70−6

−4

−2

0

2

4

6

−40

−30

−20

−10

0


Summary and Comments

• A narrowband range-rate estimator, IMRE, is proposed. The methodexploits existing environmental information to obtain a more robust andaccurate estimation of range-rate.

• The aliasing and robustness of IMRE are discussed.• The performance of IMRE is compared favorably with Cramer-Rao

Lower Bound. The high SNR behavior is analysized.• Application of IMRE to the SWellEx-96 data set illustrates the practical

usage of the algorithm. Note in application of IMRE to real data, thedemodulation of FFT will introduce a bias, which will be addressed infuture research.


Thank You!


OutlineRange-Rate Discrimination in Passive SonarAcoustic Normal Mode Theory For a Moving SourceDifferential Doppler for a Moving SourceThe Time-Frequency Data Snapshot VectorThe Signal Model (1)The Signal Model (2)The Independent Mode Range-rate Estimator (IMRE)The SWellEx-96 EnvironmentTypical IMRE OutputTypical Eigenspectrum of Covariance Matrix $R$Range-rate AliasingInsensitivity to Target DepthCramer-Rao Low Bound (1)Cramer-Rao Low Bound (2)Comparison of IMRE with CRBHigh SNR Behaviour AnalysisHigh SNR Performance at Different FrequenciesSource Tracks in the SWellEx-96 ExperimentRange-rate Track of S5 and S59 Events in SWellEx96Summary and Comments

Documents

RobustRange-rateEstimationofPassiveNarrowbandSourcesin …people.ee.duke.edu/~jk/SAM Group_files/Hailiang... · 2010. 10. 26. · The SWellEx-96 Environment Typical shallow water