ESPRIT Approach

8/6/2019 ESPRIT Approach

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A Report on

Direction-of-Arrival Measurementusing

ESPRIT Approach



ESPRIT (estimation of signal parameters via rotational invariance techniques) is a highly popular

eigenstructure (subspace) source localization method. Eigenstructure (subspace) source localization methods

are so labeled because they decompose the column space of the data correlation matrix into a signal subspace

and a noise subspace. Eigenstructure (subspace) methods are increasingly popular vis-a-vis the maximum-

likelihood (ML) approach because eigenstructure methods: 1) demand less computation; 2) produce at

moderate signal-to-noise ratio (SNR) estimation performance comparable to the optimal ML methods; and 3)

require only the second-order statistics of the additive noise, but not of the sources’ joint spectral densities.

ESPRIT requires a certain invariance structure in the sampled data set, and this invariance structure has

generally been realized in most ESPRIT-based algorithms as a spatial invariance dependent on some known

translational displacement (Δ) between two identical subarrays of sensors. This spatial invariance relates the

two subarray’s responses to the k th impinging source through an invariant phase-factor exp( j2π ( Δ / λ)u(θ k ,φk ))

where λ denotes the signal wavelength and u(θ k ,φk ) represents the k th source’s direction-cosine parameterized

by the elevation angle θk and the azimuth angle φk . Thus, estimation of this invariant phase factor would yield

an estimate of the k th source’s direction cosine (and hence its direction-of-arrival). However, this spatial-

invariance array structure: 1) imposes a nontrivial burden in array calibration to ensure that the two subarrays

are in fact exactly identical and are translationally (and not rotationally) separated by the nominal distance and2) requires precise a priori knowledge of the impinging signals’ wavelengths.

In contrast, this proposed algorithm advances a non-spatial realization of ESPRIT’s invariance structure, such

that the invariance factors would depend only on the impinging signals’ direction cosines and not on any array

parameters. As a consequence, 1) array elements may become arbitrarily located in a three-dimensional (3-D)

region, (thereby allowing irregular array geometries, for example, when multiple identical and similarly

oriented sonobuoys are scattered over a wide area, or when multiple sensors are mounted on the surface (of

arbitrary configuration) of an autonomous underwater vehicle (AUV), or to extend array aperture by spacing

array elements over the Nyquist half-wavelength maximum); 2) no knowledge whatsoever is needed of any

array sensor’s location; and 3) no knowledge whatsoever is needed of the signal frequencies. This resulting

algorithm is also highly parallel in structure, rendering it suitable for concurrent processing in real-time

applications. All these innovations are realized by the creative use of vector hydrophones as explained below.

The use of vector hydrophones is pivotal to the success of the present algorithm. Each vector hydrophone

consists of two (or alternately three) identical, co-located, but orthogonally oriented velocity hydrophones plus

another co-located pressure hydrophone. Many different types of velocity hydrophones are available and have

been constructed using a variety of technologies, with designs ranging from mechanically based to optically-

based to derivative-based. The DIFAR array exemplifies the construction of such a vector-hydrophone array.

Each velocity hydrophone has intrinsic directional response to the incident sonar wave field, measuring only

one Cartesian component of the 3-D particle velocity vector of the incident sonar wave field. Two (or three)

spatially co-located but orthogonally oriented velocity hydrophones plus another co-located pressure

hydrophone are formed into a three-component (or a four-component) vector hydrophone in a point-like

geometry. This array of vector hydrophones separately measures two of the three (or all three) Cartesian

components of the sonar velocity field plus the overall acoustic pressure. Normalization of the three Cartesian

velocity-field components gives the three Cartesian direction cosines (and thus the azimuth angles and the

elevation angles) of the impinging sonar signals.

In this nonspatial realization of ESPRIT’s invariance structure (denoting the number of vector hydrophones inthe array as L ), the subarray made up of the L pressure hydrophones would be related by the x-axis direction



cosine u(θ k , φk ) to the subarray composed of the L velocity hydrophone aligned along the x-axis. Similarly, the

subarray made up of the L pressure hydrophones would be related by the y-axis direction cosine v(θ k , φk ) to

the subarray composed of the L velocity hydrophones aligned along the y-axis. And the subarray made up of

the L pressure hydrophones would be related by the z-axis direction cosine w(θ k ,) to the subarray composed of

the L velocity hydrophones aligned along the z axis. Note that none of these three invariant factors involve

any array variables; thus, this realization of ESPRIT’s invariance structure does not require any information

on array geometry.

The present source localization approach recognizes the vector-field nature (i.e., the velocity field) of the

sonar wavefield, thereby distinguishing itself from more customary source localization methods deploying

only pressure hydrophones and treating the sonar wavefield merely as a scalar wavefield (i.e., intensity field).

From another perspective, the vector hydrophone array exploits the impinging signals’ directional diversity

differently from the discrete-time spatio-temporal FIR filtering approach where the signals’ directional

diversity is encapsulated in the phase factors between the phased array’s spatially displaced sensors. In the

present case, no such phase factor exists among the components of a vector hydrophone or among the

subarrays, because all intravector hydrophone components, and thus all subarrays, are spatially co-located.

Here, the sources’ directional diversity directly influences the individual scalar response of each constituenthydrophone that makes up the vector hydrophone.

Because this ESPRIT-based source localization method places no geometric constraint on the spatial location

of the array elements, the present algorithm can provide blind and closed-form estimates for irregularly or

randomly spaced array configurations, of which no a priori knowledge is assumed. This possibility of closed-

form solutions for irregular and unknown array geometries stands in stark contrast with the more cumbersome

multiple signal estimation (MUSIC) algorithm — the customary technique used to handle irregular array

geometries, which are presumed to be known precisely. MUSIC requires a D – dimensional search for K

minima of a highly nonlinear cost function for K sources each having D signal parameters. Whether MUSIC

converges to the global optimum and how fast MUSIC converges depend very much on the availability of

good initial parameter estimates to start off its computationally expensive iterative optimization search.

However, without a priori information on the incident sources, initial estimates are generally unobtainable.

In contrast, ESPRIT-based source localization algorithms are closed-form in nature. Thus, ESPRIT-based

methods are: 1) often computationally more efficient because iterative searches are avoided; 2) more

autonomous in not requiring any a priori initial parameter estimates; 3) more versatile in not needing any

detailed knowledge (and thus no computer memory storage) of the array manifold function, provided this

array manifold is calibrated to contain within it certain invariance structure; and 4) more reliable in avoiding

the problem of converging to local minima, However, ESPRIT is normally inapplicable to irregularly spaced

arrays. Nonetheless, the nonspatial realization of ESPRIT’s invariance structure through the use of vector

hydrophones would render ESPRIT applicable to irregular and even unknown array geometries. Thus, all

aforementioned advantages of ESPRIT-based algorithms can be appropriated for irregular and unknown array

geometries.

One use of the arbitrary array geometry permitted by the present ESPRIT-based source localization algorithm

is to space the array elements sparsely so as to extend the physical aperture without adding sensors. When

uniform interelement spacing exceeds a half wavelength, the one-to-one correspondence between the kth

source’s intervector -hydrophone phase factor exp( j2π ( Δ / λ)u(θ k ,φk )) and the k th source’s direction cosine u(θ k ,φk ) no longer exists. Instead, the cyclic ambiguity in the phases of the phase factor produces a set of



ambiguous direction-cosine estimates, each of which is separated from the others by some integer multiples of

λ / Δ. However, in this present algorithm, the invariance factor exploited is no longer the intervector-

hydrophone phase factor but a non-spatial invariance factor that relate the vector hydrophone’s various

components; thus, the uniform inter-element spacing of a uniform array may extend beyond the half-

wavelength limit ordinarily allowed by the spatial version of the Nyquist Sampling Theorem.

1.1 DATA MODEL FOR ARBITRARILY SPACED VECTOR HYDROPHONES

The signal model involves K narrow-band underwater acoustic signals impinging upon an array of L vector

hydrophones (with L>K) at arbitrary and possibly unknown locations in a 3-D region with completely

arbitrary geometry. The incident sources are to be narrow-band in that their bandwidths are very small

compared to their carrier frequency. Each vector hydrophone consists of either two or three identical but

orthogonally oriented velocity hydrophones plus another pressure hydrophone, all of which are spatially co-

located in a point-like geometry. The four-component vector hydrophone (with three orthogonally oriented

velocity hydrophones and a pressure hydrophone) would produce the following 4*1 manifold with regard to

the kth source impinging from (θk , φk ):

(1)

To avoid directly dealing with the vertical component of the underwater acoustical particle motion, the vector

hydrophone may also comprise only of the two horizontally oriented velocity hydrophones plus a pressure

hydrophone, producing the following 3*1 manifold with regard to the same source:

(2)

where 0< θk < π denotes the k th source’s elevation angle measured from the vertical z-axis and 0 < φk < 2π

denotes the k th source’s azimuth angle. Whilst the locations of these vector hydrophones need not be known

in any way, all vector hydrophones need to be identically oriented.

The intervector-hydrophone spatial phase factor related to the kth source and the lth vector hydrophone at the

(possibly unknown) location ( xl ,y,l zl) is:

(3)

The JL*1 array manifold for the entire L-element vector hydrophone array is:



(4)

where J equals 3 or 4 depending if the three-component or the four-component vector hydrophones are

deployed, and symbolizes the Kronecker-product operator.

With a total of K< L co-channel signals, the entire array would yield a JL*1 vector measurement z(t) at times t :

(5)

(6)

(7)

(8)

where n(t) is the JL*1 additive zero-mean white noise vector, P k denotes the k th signal’s power, σ k(t)

represents a zero-mean unit-variance complex random process, λ refers to the signals’ wavelength, c the

propagation speed, and φ k denotes the k th signal’s uniformly distributed random carrier phase.

With a total of N (with N>K), snapshots taken at the distinct instances {tn, n = 1,..,N }the vector-hydrophone

source localization problem is to determine {θ k , φk , k=1,..,K }from the JL*N data set.

(9)

1.2 CLOSED FORM SOURCE LOCALISATION WITH ARBITRARILY SPACED VECTOR

HYDROPHONES

1.2.1 OVERVIEW.

The pivotal insight underlying the present algorithm is that the JL*1 array manifold may be divided into J

number of L*1 subarray manifolds, which are related by invariant factors dependent only on the sources’

direction cosines but not on the vector hydrophones’ spatial locations. To be precise, the JL*1 array manifold

a( θ k ,φ k )=a(J)( θ k ,φ k )ǿ q( θ k ,φ k) may be alternately expressed as:



(10)

if four-component vector hydrophones are deployed, or as

(11)

if three-component vector-hydrophones are deployed. J number of L*1 sub-array manifolds may be formed

out of the JL*1 array manifold a( θ k ,φ k ) and J number of L*K subarray data blocks may be formed out of the

JL*K matrix A:

(12)

(13)

where Jj is an L*JL sub array selection matrix

(14)

where Om,n denotes an m*n zero matrix and Im denotes an m*m identity matrix. The {A1,…,AJ} sub-blocks

are interrelated as:

(15)

(16)

if J=4 :



(17)

In other words, the matrix pencil {A1,AJ}has generalized eigenvalues equal to the x-axis direction cosines,

which form the diagonal entries of Φu, the matrix-pencil {A2,AJ} has generalized eigenvalues equal to the y-

axis direction cosines, which form the diagonal entries of Φv

and if J=4, the matrix-pencil {A3,AJ} has

generalized eigenvalues equal to the z-axis direction cosines, which form the diagonal entries of Φw

. Note

that none of { Φu

,Φv

,Φw}depends on {(xl ,yl ,zl ), l =1,…,L}. The foregoing analysis suggests that without any

constraint on nor any knowledge of the location of any of the vector hydrophones, application of ESPRIT to

the matrix pencil in 15 would yield estimates of the direction cosines along the x-axis, whereas application of

ESPRIT to the matrix pencil in (16) would yield estimates of the direction cosines along the y-axis, and

application of ESPRIT to the matrix pencil in (17) would yield estimates of the direction cosines along the z-

axis. Moreover, ESPRIT may be applied to these matrix pencils in parallel to facilitate real-timeimplementation.

1.2.2 SUBSPACE DECOMPOSITION

In Eigen structure (subspace) direction-finding methods such as ESPRIT, the overall array’s element space is

decomposed into a K-dimensional signal subspace and a (JL-K)-dimensional noise subspace. Therefore, the

first step in the proposed algorithm is to compute the K (JL*1) signal-subspace eigenvectors by eigen-

decomposing the JL * JL data correlation matrix R zz= ZZT . Let Es be the JL*K matrix composed of the K

eigenvectors corresponding to the K largest eigenvalues of R zz

(18)

where T is an unknown but nonsingular K*K coupling matrix. T is nonsingular because both Es and A are full

rank. For the case of noiselessness or the case of Gaussian additive noise with infinite number of snapshots,

the approximation becomes exact.

1.2.3 ESTIMATION OF THE X-AXIS DIRECTION COSINES.

Analogous to (15), J1 and JJ are next used to construct the signal-subspace matrix pencil {Es1, EsJ}

(19)

(20)

There exists a K*K nonsingular matrix Ψu relating the two L*K full-ranked matrices Es1 and EsJ

(21)



(22)

From these, the direction cosines along the x-axis (i.e.,{ u(θ k ,φk ), k=1,…,K } ) may be estimated :

(23)

where {[Φu]kk , k = 1,…,K} constitutes the diagonal elements of the diagonal matrix Φu and is approximated

by the eigenvalues of Ψu

. Furthermore, the eigenvector corresponding to the eigenvalue [Φu]kk constitutes

the kth column of (Tu)-1

where

(24)

where Pu

is some unknown permutation matrix whose kth column is a K*1 vector with all zeroes except one at

the ik th position and {i1… ,iK } is some permutation of {1,…,K}. This unknown permutation of the rows of T

(i.e., the permutation of the eigenvectors of Ψu

) as columns of (Tu)-1

arises in the Eigen-decomposition of —

Ψu

for any T satisfying (22), PuT would likewise satisfy (22). For the case of noiselessness or the case of

Gaussian additive noise with infinite number of snapshots, the approximation becomes exact.

1.2.4 ESTIMATION OF THE Y-AXIS DIRECTION COSINES.

Analogous to (16), J2 and JJ are to be used to construct the signal-subspace matrix pencil { Es2, EsJ } :

(25)

(26)

Similar to the x-axis case, there exists a K*K nonsingular matrix Ψvrelating the two L*K full-ranked matrices

Es2 and EsJ

(27)

(28)

where the eigenvalues of Ψv

approximate {[Φv] jj , j= 1,…,K}. That is,

(29)

and the eigenvector corresponding to the eigenvalue [Φv] jj constitutes the jth column of (T

v)-1

where

(30)

where Pv

is another unknown permutation matrix. The above computation to estimate the -axis direction

cosines may be performed in parallel with the computation to estimate the x-axis direction cosines.

Note that different indices are used to enumerate uk and v j and in general T

u

=/=T

v

even thoughΨ

u

andΨ

v

share the same set of eigenvectors. That is, the eigenvectors are ordered differently in (T

u)-1

as in (Tv)-1

. No

mismatch, however, exists between uk and its corresponding eigenvector, which is simply the kth column of



perturbation weights are found to be

(37)

Thus, the improved direction-cosine estimates are :

Improved x-axis direction-cosine estimate:

Improved y-axis direction-cosine estimates:

Improved z-axis direction-cosine estimate:

(38)

1.2.7 AZIMUTH AND ELEVATION ANGLE ESTIMATES

From the direction-cosine estimates derived above, the signal’s azimuth and elevation arrival angles can be

estimated as

(39)

Note that once the direction cosines are paired, the azimuth and elevation estimates are also automatically

matched with no additional processing.



1.3. CALIBRATION AND REMEDY FOR VECTOR HYDROPHONE MISORIENTATION

This algorithm requires no a priori knowledge of and no regularity in the intervector-hydrophone spacings;

however, the foregoing development presumes all vector hydrophones to be oriented in accordance to some

nominal identical direction. Here we will analyze the effects of misorientation or random vector hydrophone

orientation and will introduce a basic calibration method and a simplistic remedy to illustrate that the

possibility of using the foregoing algorithm in some modified form to accommodate vector-hydrophone

misorientation.

1.3.1 EFFETCS OF VECTOR HYROPHONE MISORIENTATION.

Suppose that the lth vector hydrophone deviates from its nominal orientation such that the velocity

hydrophone nominally aligned along the x-axis is actually aligned along the xl axis, and similarly for the y-

axis and the z-axis. The ( xl, yl, zl) coordinate system may be constructed from the ( x,y,z) coordinate system by:

1) a φt rotation about the z-axis and then 2) a θ t rotation about the newly constructed xl axis. (Note that the

order of these two steps is not commutable. φt , θ t } in itself does not uniquely relate ( xl, yl, zl) to ( x,y,z).) Then

any incident signal from the direction { u(θ k ,φ

k ), v(θ

k ,φ

k ), w(θ

k ,)}would become misidentified as coming from

{ u(θ k (l) ,φk

(l)), v(θ k

(l) ,φk

(l)), w(θ k

(l) ,)}where

(40)

1.3.2 CALIBRATION OF VECTOR HYDROPHONE MISORIENTATION:

Calibration algorithms of varied degrees of sophistication and capability may be devised to determine the state

of misorientation of each vector hydrophone in the array. A rather simplistic calibration algorithm will be

introduced below using three reference signals. This rather unsophisticated method is used only as an

illustration. More sophisticated alternatives, perhaps with self-calibration capabilities, are currently under

development.

Suppose three reference signals from known directions of -{( θ 1 , φ1 ), ( θ 2 , φ 2 ), ( θ 3 , φ 3 )} arrival, are incident upon

the array in use. Further suppose that the spectra of all three reference signals are non-overlapping among

themselves nor with the spectra of any other signals. Then each of the three reference signals may be easily

isolated from the sampled data by passing the sampled data through a bandpass filter with the appropriate

passband.

For the four component vector-hydrophone case (i.e.J=4) the misoriented vector-hydrophone array manifold



a(l)

( θ k ,φ k ) is related to the nominal direction cosines { u(θ k ,φk ), v(θ k ,φk ), w(θ k ,)} as

(41)

The 3*3 coordinate rotational matrix Rl may be determined with the three reference signals because { u(θ k ,φk ),

v(θ k ,φk ), w(θ k ,)} are known a priori and { a(l) ( θ k ,φ k ), k=1,2,3} may be measured as discussed above. Thus

(42)

where

(43)

Similarly, for the three-component vector-hydrophone case (i.e.,J=3 ), the misoriented vector-hydrophone

array manifold a(l)

( θ k ,φ k ) is related to the nominal direction cosines { u(θ k ,φk ), v(θ k ,φk ), w(θ k ,)} as

(44)

(45)

where

(46)

1.3.3 REMEDY FOR VECTOR-HYDROPHONE MIS-ORIENTATION

Having identified the true orientation of each vector hydrophone by the foregoing calibration algorithm, the

original source localization algorithm may be modified as follows to accommodate the vector hydrophones’

mis-orientation:

(47)



where

(48)

The original source localization algorithm can now be applied with Z^

substituted by Z. Note that under noisy

and non-asymptotic situations, R(J)

would be estimated with nonzero error, thereby adding another error source

toward the estimation error in {θ k ^ , φk,

^ , k=1,… ,K }. This is a very crude remedy; more sophisticated remedies

are under development.

ESPRIT Approach

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