Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water

Transcription

1 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 1/23 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water Hailiang Tao and Jeffrey Krolik Department of Electrical and Computer Engineering Duke University

2 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 2/23 Outline Introduction The Independent Mode Range-rate Estimator Comparison with Cramer-Rao Bound Application to SWellEx-96 data Conclusions.

3 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 3/23 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) compared with absolute range. The discrimination capability in range-rate is helpful to detect targets in the presence of interferences with similar bearings but different relative range-rate. Background: Normal mode theory for a narrowband moving source derived by Hawker (JASA, 1979). Projection of target Doppler onto different modes induces signal fluctuations which are target range-rate dependent. Direct application of MUSIC (Song, 1990) does not take advantage of available environmental information and requires a long observation time. Direct extension of Matched Field Processing (Zala, 1992) with range-rate is computational intensive due to parameter coupling.

4 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 4/23 Acoustic Normal Mode Theory For a Moving Source The velocity potential emitted by a narrowband, horizontally uniform moving point source, in a range-independent stratified oceanic waveguide: (Hawker, 1979) ψ(t) C MX m=1 U m (z)u m (z s ) km R 0 exp»jω m t jk m R 0 1 v «r vm G (1) where U m (z) and U m (z s ) are mode eigenfunctions at receiver depth z and source depth z s. k m is the horizontal wavenumber. R 0 the initial range. ω m = ω 0 k m v r (1 v r ) ω vm G 0 k m v r is the Doppler frequency, in which ω 0 is the intrinsic frequency, v r the range-rate of the source and v G m the group velocity of mode m.

5 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 5/23 Differential Doppler for a Moving Source Differential Doppler for moving vs. stationary source evident from slope of frequency distribution as a function of modal wavenumber for a single tonal source. Note that range-rate dependence does not require relative phase between modal components f k diagram for stationary and moving source v=10m/s v= Frequency (Hz) Horizontal wavenumber (1/m)

6 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 6/23 The Time-Frequency Data Snapshot Vector Consider snapshot of narrowband data which consists of a concatenation of N conventional frequency domain snapshots over time. Note range-rate processing can be performed before or after conventional beamforming.

7 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 7/23 The Signal Model (1) Model the (pn 1) space-time snapshot r as: r = s X m a m (d m u m ) + n (2) Where s is the signal amplitude, unknown nonrandom. a m is a zero-mean complex random variable representing mode m s amplitude. d m is a time harmonic vector for mode m: d m = [1 e jk mv r T d e jk mv r 2Td e jk mv r (N 1)T d ] T is the Kronecker product. u m represents mth mode eigenfunctions evaluated at the depths of each array element: u m = [U m (z 1 ) U m (z 2 )... U m (z p )] T and n CN(0, σ 2 ni) represents complex white Gaussian noise.

8 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 8/23 The Signal Model (2) In matrix form where and r = sda + n (3) D = [d 1 u 1 d 2 u 2... d M u M ] a = [a 1 a 2... a M ] T The variance of mode amplitude is σ 2 m = E(a m a H m) = U m (z s ) 2 /k m (4) In this model: Assume am and a n are uncorrelated for m n. The model is independent of absolute range.

9 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 9/23 The Independent Mode Range-rate Estimator (IMRE) The covariance matrix is: R = E(rr H ) = P s DSD H + σ 2 ni (5) where P s = s 2 and S = E(aa H ) = diag[σ 2 1 σ σ 2 M]. Given the environment and target depth z s, the signal covariance matrix R s = DSD H could be computed for each hypothesized range-rate v r. Suppose the dominant eigenvector of R s is h 1 (normalized), then a Bartlett type estimator can be expressed as: P (v r ) = 1 λ 1 h H 1 ˆRh 1 (6) which maximizes output SNR in white noise. ˆR is the sample covariance matrix and λ 1 is the maximum eigenvalue of ˆR.

10 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 10/23 The SWellEx-96 Environment Typical shallow water environmental profiles of SWellEx-96 are used in simulations: Depth (m) S5 S SWellEx-96 Sea Bottom Properties Sound Velocity (m/s) SWellEx-96 Water Column Sound Velocity Profile

11 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 11/23 Typical IMRE Output Swellex96 S5:d s =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 IMRE output power (db) Range rate (m/s) The sidelobe is 13dB down from the mainlobe.

12 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 12/23 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 eigenvalues The db difference between 1st and 2nd eigenvalues corresponds to mainlobe/sidelobe difference in previous slide.

13 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 13/23 Range-rate Aliasing Range-rate difference between grating lobes: v g r 2π kt d (7) where k is the mean value of k m. T d is the snapshot delay. 0 Swellex96 s5:d s =51.0m, f s =50Hz, v s =2.50m/s, d a =70.0m, n snap =20, nsample=200, SNR=0dB T d = 2s T d = 0.5s 2 4 Grating Lobe Distance v r g IMRE output power (db) Range rate (m/s)

14 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 14/23 Insensitivity to Target Depth Swellex96 s5:d s =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 Depth (m) Range rate (m/s) IMRE search in both depth and range-rate with one sensor

15 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 15/23 Cramer-Rao Low Bound (1) The unknown parameters are: Θ = (vr, P s, σ 2 n) T. each element of Fisher Information Matrix (FIM) could be represented by: J ij = N s tr» R 1 R R 1 R θ i θ j (8) where N s is the number of sampled r. To compute J ij, recall R = P s DSD H + σ 2 ni So R P s = DSD H (9) R σ 2 n = I (10)

16 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 16/23 Cramer-Rao Low Bound (2) Let c = [ (N 1)] T and k = [k 1 k 2... k M ] T. Define matrix G: G = jt d ck T and H = G h where vector h is a p (array length) by 1 vector with all its elements being 1. Thus R = P s ( D SD H + DS DH ) v r v r v r = P s ((H D)SD H + DS(H D) H ) (11) where is element-by-element Hadamard product.

17 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 17/23 Comparison of IMRE with CRB Swellex96 S5: d s =51.0m, f s =100Hz, v s =2.50m/s, d a =70.0m, n snap =20, nsample=200, NMC=200 Monte Carlo CRLB 0 Range rate MSE (db) SNR(dB) Between SNR 15 0dB, the mean square error of IMRE achieves CRLB. The threshold happens at 16dB.

18 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 18/23 High SNR Behaviour Analysis Express the dominant eigenvector of the sample covariance matrix as: ˆb1 = b 1 + η (12) The estimator error vector η has the following asymptotic correlation function: E[ηη H ] = λ 1 N s NX k=2 λ k (λ 1 λ k )2 b kb H k where λ s are the eigenvalues of P s DSD H. Looking at λ 1 λ k (λ 1 λ k )2, we have: When the noise variance σ 2 n is far smaller than the second eigenvalue λ 2 of P s DSD H, the performance of the algorithm will not improve with higher SNR, i.e., not stick to the CRB. λ The smallest MSE achieveable is determined by 2. λ 1

19 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 19/23 High SNR Performance at Different Frequencies Range rate MSE (db) Range rate MSE (db) Range rate MSE (db) Swellex96:d s =51.0m, f s =20Hz, v s =2.50m/s, d a =70.0m, n snap =20, nsample=200, NMC= Swellex96:d s =51.0m, f s =50Hz, v s =2.50m/s, d a =70.0m, n snap =20, nsample=200, NMC= λ 2 /λ 1 = Swellex96:d =51.0m, f =100Hz, v =2.50m/s, d =70.0m, n =20, nsample=200, NMC=200 s s s a snap λ 2 /λ 1 = λ 2 /λ 1 = SNR(dB)

20 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 20/23 Source Tracks in the SWellEx-96 Experiment Source: UCSD Marine Physical Laboratory SWellEx-96 Website.

21 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 21/23 Range-rate Track of S5 and S59 Events in SWellEx96 6 SwellEx96:s5 79Hz, z s =54.0m, z a =94.1m, nsnap=20, t FFT =1.0s 0 Range rate (m/s) Time (m) SwellEx96:s59 79Hz, z s =54.0m, z a =94.1m, nsnap=20, t FFT =1.0s 40 0 Range rate (m/s) Time (m) 40

22 Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 22/23 Summary and Comments A narrowband range-rate estimator, IMRE, is proposed. The method exploits existing environmental information to obtain a more robust and accurate 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, the demodulation of FFT will introduce a bias, which will be addressed in future research.

23 Thank You! Robust Range-rate Estimation of Passive Narrowband Sources in Shallow Water p. 23/23