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BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to processing of signals from antenna arrays
to provide direction finding for signal sources, suppression of near field
interference and noise, extension of the aperture of arbitrary antenna
arrays or providing minimum redundancy array design and calibrating
antenna arrays without calibration sources. More particularly, the
invention includes an arrangement of actual sensors in an antenna array
with a computation of higher-order statistics to provide virtual second
order statistics corresponding to virtual elements in the array;
employment of the actual and virtual elements of the array for covariance
based direction finding; and, with the addition of a separate sensor
spaced from the main array, suppression of non-Gaussian measurement noise.
This is accomplished employing cross-correlation of the virtual sensors or
alternatively calibrating the existing actual array employing
cross-correlation of the array and its virtual sensors.
2. Prior Art
The use of discrete arrays of sensors as an antenna for receiving signals
generated by multiple sources and estimating the parameters of the signals
received is well known in the art. Applications of the parameter
estimation include source direction finding (often identified as direction
of arrival (DOA) of signal wavefronts) or, reversing the known and unknown
parameters, applying known signal locations for calibration of an array of
sensors of unknown array manifold.
Conventional array processing techniques utilize only second order
statistics of received signals. Second-order statistics are sufficient
whenever the signals can be completely characterized by knowledge of the
first two moments as in the Gaussian case, however, in real applications
far field sources may emit non-Gaussian signals. Exemplary of such an
application is a communications scenario with multiple receivers. Failure
of second-order statistics to completely characterize the signal
parameters may be ameliorated by the use of higher-order statistics.
Exemplary of the prior art in array signal processing are the Ph.D. thesis
by R. O. Schmidt entitled "A Signal Subspace Approach to Multi-Emitter
Location and Spectral Estimation," Stanford University 1981 and U.S. Pat.
No. 4,750,147 to Roy, III, et al., issued Jun. 7, 1988. The Multiple
Signal Classification (MUSIC) algorithm of Schmidt and the Estimation of
Signal Parameters using Rotational Invariance Techniques (ESPRIT) of Roy
III employ second order statistics based on an array covariance matrix for
establishing the signal parameters. The MUSIC approach requires extensive
calibration of the array geometry to allow its proper characterization for
use with the algorithms employed in MUSIC. The ESPRIT system alleviates
the need for array calibration by employing sensor pairs having known
spacing and orientation while allowing variable geometry with respect to
the pairs. The ESPRIT system consequently requires doubling the number of
sensors with commensurate added cost and merely replaces the problem of
calibration of the overall array manifold with the requirement for strict
orientation of the sensor pairs in the array.
Both the MUSIC and ESPRIT systems are designated as subspace parameter
estimation algorithms. Various methods have been recommended for
increasing the capability of such subspace algorithms by the use of
higher-order statistics for noise reduction. See, e.g., Shamsunder, S. and
Giannakis, G., "Modeling of Non-Gaussian Array Data Using Cumulants", IEEE
Transactions on Signal Processing, March 1993; Pan, R. and Nikias, C. L.,
"Harmonic Decomposition Methods in Cumulant Domains", Proceedings ICASSP
'88, pp. 2356-2359 New York, N.Y., April 1988; and Chiang, H. H. and
Nikias, C. L., "The ESPRIT Algorithm With Higher-Order Statistics,"
Proceedings Vail Workshop, Higher-Order Spectral Analysis, pp. 163-168,
June 1989. Such applications of higher-order statistics provide techniques
for noise reduction/elimination, however, the approches as disclosed in
the prior art are supplementary to the conventional array processing
techniques.
The present invention overcomes the difficulties of the subspace algorithms
and provides an integrated approach to the use of higher-order statistics
for direct calculation of estimated parameters as opposed to mere signal
correction.
SUMMARY OF THE INVENTION
Estimation of signal parameters employing the present invention is
accomplished by a plurality of sensors receiving signal inputs and
transferring the sensor measurements to a virtual cross-correlation
computer which generates virtual second-order statistics by processing
higher-order statistics of available sensor measurements to provide
cross-statistics between the real sensors and a set of virtual sensors
which are not actually present at desirable locations. The virtual second
order statistics are then provided to a standard subspace calculation
system for use in parameter estimation in applications such as direction
finding (wavefront DOA) and waveform recovery.
In a second embodiment of the invention a separate sensor, remote from the
plurality of sensors forming the array, provides an additional signal
input to the virtual cross-correlation computer again allowing calculation
of virtual second-order statistics based on processing of the higher-order
statistics of the available sensors and employing the cross-statistics
between the actual sensors computed in the presence of unknown
interference wherein the higher-order statistical calculations are
insensitive to the interference whereby the resulting virtual second-order
statistics are similarly unaffected by the interference. The output of the
virtual cross-correlation computer is again provided to a standard
subspace algorithm for calculation of desired signal parameters.
A third embodiment of the invention employs as a portion of the plurality
of sensor elements a pair of elements having known relative displacement
and orientation. Application of the known vector for the displacement and
distance of the interrelated sensors allows specification of the virtual
sensors in the virtual cross-correlation computer to comprise doublets
with the remaining actual sensors having similar displacement and
orientation thereby providing "self calibration". The output of the
cross-correlation computer is then provided to a standard ESPRIT algorithm
for estimation of signal parameters.
In yet another embodiment of the invention a second separate sensor
proximate the first separate sensor displaced from the plurality of
sensors forming the array is provided with known displacement and
orientation relative to the first separate sensor. Signal measurements
from the first and second separate sensor are provided to the virtual
cross-correlation computer which, with a prior knowledge of the
displacement vector of the first and second separate sensor, calculates
virtual second order statistics eliminating noise and providing virtual
sensor locations corresponding to doublets having displacement vectors
from the actual array sensors identical to the displacement vector of the
first and second separate sensors. The virtual second order statistics
provided by the virtual cross-correlation computer are then employed by a
standard ESPRIT algorithm for generation of signal parameters wherein
non-Gaussian noise has been eliminated.
BRIEF DESCRIPTION OF THE DRAWINGS
The present invention will be more clearly understood with reference to the
following drawings and detailed description.
FIG. 1 is a block diagram of a sensor array and signal parameter estimation
system employing second order statistics of the prior art;
FIG. 2 is a block diagram of a signal parameter estimation system of the
present invention employing an array of actual sensors and a virtual
cross-correlation computer using higher-order statistics for generation of
virtual second order statistics for a virtual sensor to provide
cross-correlation data for a standard subspace parameter estimating
algorithm;
FIG. 3 is a diagram of the vector relationships of the actual sensors in
the array, a virtual sensor created by calculation of higher-order
statistics and the impinging wavefront;
FIG. 4 is a diagram of a virtual aperture extension of an actual array of
three sensors to an array employing four virtual and three actual sensors;
FIG. 5 is a block diagram of an embodiment of the virtual cross-correlation
computer of FIGS. 2 and 3;
FIG. 6 is a diagram of an embodiment including an array employing actual
and virtual sensors wherein the chosen virtual sensors form doublets with
the actual sensors for generation of second-order statistics employable in
a standard ESPRIT algorithm;
FIG. 7 is a block diagram identifying the interrelationship of the virtual
cross-correlation computer of the present invention with standard subspace
algorithms for parameter estimation including source direction, source
steering vectors and source waveforms;
FIG. 8 is a schematic block diagram of another embodiment of the invention
employing an actual sensor separate from the array and providing data to
the virtual cross-correlation computer for elimination of noise in the
calculation of virtual second-order statistics of the array;
FIG. 9 demonstrates an embodiment of the invention employing the separate
sensor as identified in FIG. 8 for a communication system application
employing a satellite sensor for noise suppression;
FIGS. 10a and 10b demonstrate the reduction in sensors available for a
circular array pattern employing the present invention to replace actual
sensors with virtual sensors;
FIG. 11 demonstrates an extension of the virtual array of FIG. 10 wherein
all virtual sensors available based on the actual sensors present are
demonstrated;
FIG. 12(a) is a diagram of a linear array having an extended aperture of
virtual sensors;
FIG. 12(b) is a diagram of a rectangular array and the corresponding
virtual array available through use of a virtual cross-correlation
computer as identified in the present invention;
FIGS. 13a and b demonstrate the elimination of redundant sensors in a
uniform linear array by use of virtual sensors calculated by the virtual
cross-correlation computer of the present invention.
FIG. 14 is a diagram demonstrating minimum redundancy array design wherein
a rectangular virtual array pattern is accomplished employing selected
actual sensors located on the border of the virtual array; and
FIG. 15 demonstrates the calculation of higher-order statistics employed in
the virtual cross-correlation computer for generating virtual second order
statistics to be used in signal parameter estimation.
DETAILED DESCRIPTION
In the prior art, second-order statistics systems for signal parameter
estimation employ covariance matrices for parameter estimation. Using the
example of direction finding, wherein the desired signal parameter is
direction of arrival of the wavefront, FIG. 1 discloses the elements of a
prior art system. Algorithms employed by such systems require the
computation of cross-correlations between all sensor elements of the
array. As shown in FIG. 1, the array comprises four sensors, 2, 4, 6 and 8
which provide sensor measurements r(t), y(t), x(t) and v(t) respectively,
arising from the arrival of signal wavefronts #1 and #2. Band-pass
filtering and base-band conversion sampling of the sensor measurements is
employed for the data fed to the cross-correlation computer. A
cross-correlation computer 10 receives the sensor measurements and
calculates the cross-correlations, e.g. c.sub.ry, for each of the sensors
wherein the cross-correlation is defined as c.sub.ry E{r*(t)y(t)}. The
cross-correlations are then provided to a high resolution direction
finding algorithm 12, which calculates the direction of arrival estimates.
Referring now to FIG. 2, the present invention is demonstrated in block
diagram form. The array receiving the signals represented by wavefront #1
and wavefront #2 now comprises three actual sensors, 2', 4' and 6', which
provide relative sensor measurements r(t), y(t) and x(t). The sensor
measurements are once again subjected to band-pass filtering, base-band
conversion and sampling. The sensor measurements are provided to a virtual
cross-correlation computer (VC.sup.3).sup.14. The virtual
cross-correlation computer provides calculation of higher-order statistics
for the three actual sensors, which in turn, allow calculation of second
order statistics for a virtual sensor 8' having an effective signal
measurement v(t).
The VC.sup.3 employs cumulants for calculation of the virtual second order
statistics as shown in the following analysis. It should be noted that for
the examples shown herein, fourth order cumulants are employed to allow a
generalized multidimensional problem. The invention disclosed herein is
not limited to fourth order cumulants and may employ lower or higher-order
cumulants as constrained by the dimensionality of the signal sources under
investigation.
To handle symmetric probability density functions from the sources of
interest, fourth order cumulants of the sensor outputs are calculated.
Fourth order (zero-lag) cumulants are defined in a balanced way for a
signal vector from an array of M sensors defined as {r.sub.k
(t).sub.k=1.sup.M as follows:
##EQU1##
This definition is consistent with the definition of cross-covariance
which can be expressed as E{r.sub.i (t)r*.sub.j (t)} and has only two
arguments. The properties of cumulants for order (n>2) can be defined as
follows:
[CP1] If {.sigma..sub.i }.sub.i=1.sup.n are constants and {x.sub.i
}.sub.i=1.sup.n are random variables, then
##EQU2##
[CP2] Cumulants are additive in their arguments,
cum(x.sub.1 +y.sub.1,x.sub.2, . . . ,x.sub.n)=cum(x.sub.1,x.sub.2, . . .
,x.sub.n)+cum(y.sub.1,x.sub.2, . . . ,x.sub.n) (3)
[CP3] If the random variables {x.sub.i }.sub.i=1.sup.n are independent of
the random variables {y.sub.i }.sub.i=1.sup.n, then
cum(x.sub.1 +y.sub.1,x.sub.2 +y.sub.2, . . . ,x.sub.n
+y.sub.n)=cum(x.sub.1,x.sub.2, . . . ,x.sub.n)+cum(y.sub.1,y.sub.2, . . .
,y.sub.n) (4)
[CP4] Cumulants suppress Gaussian noise of arbitrary covariance i.e., if
{z.sub.i }.sub.i=1.sup.n are Gaussian random variables independent of
{x.sub.i }.sub.i=1.sup.n and n>2, we have
cum(x.sub.1 +z.sub.1,x.sub.2 +z.sub.2, . . . ,x.sub.n
+z.sub.n)=cum(x.sub.1,x.sub.2, . . . ,x.sub.n) (5)
[CP5] If a subset of random variables {x.sub.i }.sub.i=1.sup.n are
independent of the rest, then
cum(x.sub.1,x.sub.2, . . . ,x.sub.n)=0 (6)
[CP6] The permutation of the random variables does not change the value of
the cumulant.
Referring to FIG. 3, the elements of the array of FIG. 2 are separately
shown. For convenience, the actual elements of the array 2', 4' and 6' are
assumed to be isotropic and the sources that illuminate the array are
assumed to be statistically independent. In this case, we can further
assume the presence of a single source having a signal s(t) without loss
of generality based on [CP3] above. The signal source has a propagation
vector k where k equals k=k.sub.x a.sub.x +k.sub.y a.sub.y (a.sub.x and
a.sub.y denote the unit vectors along the x and y axis respectively for
the propagation vector). The propagation vector k further has power
.sigma..sub.x.sup.2 and forth-order cumulant .gamma..sub.4,s
Computation of the cross-correlation (ignoring noise effects) between real
signal r(t) and virtual signal v(t), E{r*(t)v(t)}, assuming the reference
point to be the position of the sensor that records r(t), is defined for
the purposes of the present invention as a "virtual" cross-correlation
since no sensor exists for the generation of signal v(t). Because
r(t)=s(t) then,
x(t)=s(t)exp(-jk.multidot.d.sub.x)
y(t)=s(t)exp(-jk.multidot.d.sub.y)
v(t)=s(t)exp(-jk.multidot.d)
It follows that the directional information provided by the correlation
operation is embedded in the dot product, k.multidot.d, (see FIG. 3) i.e.,
E{r*(t)v(t)}=.sigma..sub.s.sup.2 exp(-jk.multidot.d) (7)
The source power .sigma..sub.s.sup.2 does not provide any directional
information. The information recovered by the cross-correlation of two
sensor outputs can therefore be represented by the vector extending from
the conjugated sensor, to the unconjugated sensor. Cross-correlation is
therefore a vector in the geometrical sense. Examining fourth-order
cumulants based on the geometrical interpretation of cross-correlation as
described above we note,
cum(r*(t),x(t),r*(t),y(t))=cum(s*(t),s(t)exp(-jk.multidot.d.sub.x),s*(t)exp
(-jk.multidot.d.sub.y)) (8)
and using [CP1] we obtain:
cum(r*(t),x(t),r*(t),y(t))=.gamma..sub.4,s
exp(-jk.multidot.d.sub.x)exp(-jk.multidot.d.sub.y)=.gamma..sub.4,s
exp(-jk.multidot.d) (9)
And by comparing Equations (7) and (9), the following may be observed:
##EQU3##
Equation (10) relates the fourth-order statistic provided by the cumulant
to a second-order statistic typically associated with a cross-correlation.
It should be noted that the cross-correlation identified on the right hand
side of Equation (10) employs the virtual signal v(t) while the left side
of Equation (10) employs only real signals. It is therefore possible to
recover the directional information of signal v(t) without requiring the
use of an actual sensor to measure the v(t). From the development of
Equation (10) above it is demonstrated that fourth-order cumulants may be
interpreted as an addition of two vectors each extending from a conjugated
channel to an unconjugated channel. Using principles [CP3] and [CP6] it
can be demonstrated that Equation (10) holds for multiple independent
sources and the presence of additive colored Gaussian noise. Further [CP6]
which claims that permutation of random variables does not change the
value of a cumulant merely restates the fact demonstrated in the present
analysis that addition of two vectors is a commutative operation.
As shown by the above analysis, the use of cumulants increases the
effective aperture of an array without any design procedure or
configuration constraints by the addition of virtual sensors. The three
elements of the array shown in FIG. 3 are redrawn in FIG. 4 in a lattice
structure wherein the three actual sensors, 2', 4', 6' are interconnected
by vectors d.sub.x,d.sub.y,d.sub.z, respectively. The three actual
sensors, as previously described, provide signals r(t), y(t) and x(t).
Employing the concept of vector addition previously described for
obtaining the positioning of a virtual sensor, the intersection of the
lines in the lattice shown in FIG. 4 determine the candidate locations for
virtual-sensors. To implement a covariance-like subspace algorithm, it is
necessary to compute the cross-correlation of all sensor outputs, actual
or virtual. Consequently, the sensors to be used must be connected with a
single vector. Through the use of fourth-order cumulants, as previously
demonstrated, this constraint is relieved by allowing use of two vectors
for connection purposes. These connecting vectors must be selected from
the set of vectors that define the lattice. The virtual sensors 20, 22,
24, and 26 shown in FIG. 4 having virtual signals v.sub.1 (t), v.sub.2
(t), v.sub.3 (t), and v.sub.4 (t), respectively, comprise the chosen
sensors for this example. Note that this choice is not a unique selection.
The four virtual and three actual sensors can communicate by two jumps
(vector additions). Computation of the cross-statistics is accomplished as
follows:
##EQU4##
Referring now to FIG. 5 the elements of the VC.sup.3 are shown. The
measurements from the actual sensors r(t), x(t), and y(t) are acquired and
stored in block 30. One measurement is chosen and conjugated (in the
example shown r(t) in block 32). Employing x(t), r*(t) and y(t) the
cumulant of Equation 11(b) is estimated in block 34. This estimate,
C.sub.4, may be derived using several approaches for example:
##EQU5##
The estimated cumulant C.sub.4 is then scaled by .sigma..sub.x.sup.2 and
.gamma..sub.4,s in block 36 to provide the virtual second-order
cross-correlation statistic:
E{r*(t)v(t)}
It should be noted that if all of the signals measured are circularly
symmetric (E{s(t)s(t)}=0) then computation of these terms in the cumulant
expression of Equation (1) are not required. Cross-correlations between
actual sensors, between actual and virtual sensors, and between virtual
sensors as identified by the representative Equations (11a, b and c) is
conducted in the VC.sup.3 as described above for the first actual to
virtual sensor cross-correlation. Construction of the matrix C to be
produced by the VC.sup.3 for use in a direction finding processor is shown
diagrammatically in FIG. 15. Upon the completion of these calculations the
entire cross-correlations matrix for the entire array of real and virtual
sensors described in FIG. 4 is available.
A covariance-based algorithm can estimate the DOA's of two sources using a
three element array, whereas the cumulant-based virtual aperture extension
described above can estimate the parameters of six sources (one less than
the number of elements (actual and virtual) that form the aperture). In
addition, the cumulant approach employed in the VC.sup.3 survives the
presence of colored noise due to [CP4]. Additional detailed examples of
different array configurations will be discussed subsequently.
As identified in Equation (11) it is possible to compute cross-correlations
among the actual sensors through the use of cumulants, because
cross-correlation is a "vector" and any vector can be expressed as the
addition of the zero vector to itself. In other words, cross-correlation
between two channels can be computed by using the two arguments of the
fourth-order cumulant as required by correlation, and then using the
remaining two arguments by repeating one of the channels twice, e.g.,
##EQU6##
The advantage of computing the cross-correlation of the signals x*(t) and
y(t) using equation 12 is that additive Gaussian noise can be suppressed
by the cumulant calculation. If the cross-correlation were computed
directly it would be severely affected by additive Gaussian noise. This
approach was previously recognized in the prior art described herein for
suppression of Gaussian noise.
Joint Array Calibration and Parameter Estimation
The virtual elements of an array created by the VC.sup.3 are employed in
one embodiment of the present invention as shown in FIG. 6. Calibration of
an array is a long-standing issue in the prior art. As previously
described, the ESPRIT algorithm employs an array with an identical copy of
the array displaced in space with a known displacement vector .DELTA.. The
capabilities of the VC.sup.3 allows the "copy" of the array for an ESPRIT
application to comprise virtual sensor elements. A single sensor in
addition to the desired array, with a known displacement vector, provides
the capability for vector calculation of the remaining elements of the
second array by the VC.sup.3 as virtual sensors. Referring to FIG. 6 the
original array is represented by sensors 60, 62 and 64. The array may be
of arbitrary size and element 64 represents the last sensor in the array.
These sensors provide measurement signals r.sub.1 (t) r.sub.2 (t) and
r.sub.M (t) respectively. An additional sensor 66 is provided at a known
displacement and orientation with regard to sensor 60 and provides a
signal v.sub.1 (t). A wavefront 68 having a vector k impinges on the
sensors in the array.
Considering the arbitrary array of FIG. 6 it is desired to employ an ESPRIT
algorithm to jointly estimate the DOA parameters of multiple sources and
the associated steering vectors. It is therefore necessary to compute the
cross-correlations between sub-arrays resulting, for example, in the
equation
E{r*.sub.1 (t)v.sub.M (t)}=.sigma..sub.s.sup.2 a*.sub.1 a.sub.M
exp(-jk.multidot.d.sub.1M)exp(-jk.multidot..DELTA.) (13)
where a.sub.M denotes the response of the Mth sensor to the wavefront from
the source. If no additional sensors other than the original array are
known it would not be possible to compute v.sub.M (t). However, it is
known that
E{r*.sub.1 (t)r.sub.M (t)}=.sigma..sub.s.sup.2 a*.sub.1 a.sub.M
exp(-jk.multidot.d.sub.1M)=E{r*.sub.1 (t)v.sub.M
(t)}exp(jk.multidot..DELTA.) (14)
is computable and related to the correlation in equation 13. If the term
e.sup.j.multidot.k.multidot..DELTA. was known, equation 14 could be solved
for the term E{r*.sub.1 (t)v.sub.M (t)}. However, the propagation vector k
is not known. If the additional element 66 is employed as shown in FIG. 6,
it can be noted that all vectors joining two sensors in separate
sub-arrays can be decomposed as the addition of two vectors, one in the
actual sub-array, the other being the displacement vector .DELTA., e.g.,
d.sup./.sub.12 =d.sub.12 +.DELTA.. The computable correlation on the
left-hand side of equation 14 lacks the common term
exp(-j.multidot.k.multidot..DELTA.). The bridge between the sub-arrays to
recover this phase term is provided by the additional sensor.
From the results previously described assuming that one doublet {r.sub.1
(t), v.sub.1 (t)} (from sensors 60 and 66) is available, the
cross-correlations between all sub-array elements can be calculated.
##EQU7##
Similarly, cross-correlation of the actual sensors can be computed by
cumulants.
##EQU8##
finally, the cross-correlations between virtual-sensors can be computed
as:
##EQU9##
Equations 15 and 16 are used to form the covariance matrix required for use
of the ESPRIT algorithm in calculating estimated parameters for direction
of arrival. The combination of the VC.sup.3 with an ESPRIT parameter
estimation system may be characterized as a virtual ESPRIT or VESPA. The
VESPA embodiment of the present invention requires only a single doublet
rather than a full copy of the array as required by the prior art ESPRIT
parameter estimation system. This results in enormous hardware reductions.
VESPA further alleviates the problems resulting from the perfect sampling
synchronization requirements of the covariance ESPRIT for the two
sub-arrays. In VESPA, synchronization must be maintained only between the
elements of the single doublet. As previously described with regard to
FIG. 5 true cumulants are replaced by consistent estimates which converge
to true values as the data length grows to infinity. Prior art indicates
that this approach provides rapid convergence to the true values at high
signal to noise ratio (SNR). See, Moulines, E. & Cardoso, J. F.,
"Direction-Finding Algorithms Using 4th-Order Statistics: Asymtotic
Performance Analysis" Proceedings ICASSP-92, Vol. 2, pp. 437-440, March
1992.
This embodiment of the invention employing 4th-Order cumulants which are
blind to Gaussian processes provides significant advantage over the prior
art as identified in Rockah, Y. & Schultheiss, P. N., "Array Shaped
Calibration Using Sources In Unknown Locations--Part 1: Far-field
Sources," IEEE Transactions in Acoustics, Speech, Signal Processing, Vol.
ASSP-35, No. 3, pp. 286-299, March 1987. The disclosed embodiment allows
multiple sources sharing the same frequency band based on [CP3] and is
applicable to arbitrary arrays. Further, the embodiment shown is
applicable to nominally linear arrays. The cumulant based approach does
not require information about noise spatial correlation, unlike a
covariance based algorithm and, employing VC.sup.3, the embodiment
provides an approach which is non-iterative and eliminates parameter
search by using ESPRIT. In addition, in the presence of white observation
noise, the present embodiment provides better estimates using a maximum
likelihood approach proposed in Weiss, A. & Friedlander, B., "Array Shaped
Calibration Using Sources in Unknown Locations-A Maximum Likelihood
Approach," IEEE Transactions in Acoustics, Speech, Signal Processing, Vol.
ASSP-37, No. 12, pp. 1958-1965, December 1989. In the presence of colored
Gaussian noise, the present embodiment may be employed using a
tri-spectral maximum likelihood approach to calibrate arbitrary arrays
without knowledge of noise color.
Recovery of waveforms associated with far-field sources is accomplished
using the disclosed embodiment of the invention by first estimating the
steering vectors by subspace rotation as defined in Roy, R. and Kailath,
T. "ESPRIT-Estimation of Signal Parameters via Rotational Invariance
Techniques," Optical Engineering, Vol. 29, No. 4, pp. 296-313, April 1990.
Referring to FIG. 6 and generalizing, the measured signals are represented
as r(t)=[r.sub.1 (t),r.sub.2 (t), . . . ,r.sub.M (t),v.sub.1 (t)].sup.T
for the actual sensors in the array. The steering matrix is augmented by
the estimated response of the added sensor 66 which provides signal
measurement v.sub.1 (t). If a.sub.1 is the (M+1).times.1 steering vector
of the signal of interest (SOI) characterized as s.sub.1 (t) with
estimated bearing .theta..sub.1 and the augmented steering matrix is
decomposed as A=[a.sub.1, A.sub.I ], two approaches may be used for signal
recovery.
A minimum-variance distortionless response beamformer (MVDR) is employed to
estimate the SOI waveform in the mean-square sense.
s.sub.1 (t)=w.sub.1.sup.H r(t)=(R.sup.-1 a.sub.1).sup.H r(t) (27)
where R=E{r(t)r.sup.H (t)}
The second approach for signal recovery employs MVDR with perfect nulling
(Null-MVDR). This beamformer estimates the SOI waveform in the mean square
sense while putting perfect nulls on the interferors.
s.sub.2 (t)=w.sub.2.sup.H r(t) (18)
where the weight vector w.sub.2 is the solution of the linear-constrained
minimum variance problem
##EQU10##
which has the solution
w.sub.2 =R.sup.-1 A(A.sup.H R.sup.-1 A).sup.-1 f (20)
It should be noted that both the MVDR and null MVDR beamformers do not
require knowledge of the measurement noise covariance matrix.
FIG. 7 shows a block diagram of the VESPA system employing the VC.sup.3 to
provide covariance matrices for direction finding and waveform recovery.
The VESPA system generally designated 72 receives the measurements as
acquired from the sensors 74. The actual measurements and virtual sensor
statistics are employed by the VC.sup.s 14 to provide two virtual
covariance matrices to the ESPRIT subspace parameter estimation system 76.
The ESPRIT system provides source direction and source steering vectors,
both of which are employed in a waveform recovery calculator 78 which
receives additional input from a correlation matrix computation 79 to
provide the source waveform employing the MVDR process previously
described.
Suppression of Non-Gaussian Noise
If the sensor array employed by the present invention comprises (1)
isotropic elements (2) and is illuminated by multiple independent
non-Gaussian sources and (3) it is assumed that the array measurements are
contaminated by non-Gaussian sensor noise which is independent from sensor
to sensor and (4) noise components can have varying power and kurtosis
over the aperture, the present invention can still estimate DOA parameters
of far-field sources by subspace techniques. Variations of noise power and
kurtosis over the sensors pose no problem for determining the signal
subspace and it is possible to achieve virtual aperture extension as
previously described. The invention achieves this capability based on the
assumptions made about the structure of the non-Gaussian noise. Since the
far-field sources are independent, their second and higher-order
statistics are spatially stationary implying that auto-correlations and
auto-cumulants of signals do not vary over the aperture. Since the noise
is independent between array elements, computation of the cross statistics
is possible regardless of noise, and computation of the auto-correlation
at a sensor in the presence of noise is accomplished employing the
following equation
##EQU11##
The left side of equation 21 is computed in a scenario where additive
non-Gaussian noise is present. The right side of equation 21 is computed
in the hypothetical (virtual) case where there is no measurement noise.
This convention will be maintained throughout the following discussion of
noise suppression. It is important to note that equation 21 is valid for
ensemble averages. With finite samples the standard deviations of the two
sides will be different.
Equation 21 can be interpreted geometrically. With cumulants the geometric
effect is motion from one sensor to another sensor, which has non-Gaussian
but independent noise, and back to the starting sensor using the same
path. Previous use of higher-order statistics for accomplishing
non-Gaussian noise sensitivity in this manner was proposed in Cardoso, J.
F., "Higher-Order Narrow Band Array Processing," Proceedings of the
International Conference on Higher-Order Statistics, pp. 121-130,
Chamrousse France Jul. 10-12, 1991. The limitation in noise reduction
employing this method relates to the assumption of sensor-to-sensor
independence of the non-Gaussian noise. Employing virtual aperture
extension of the present invention the cumulants are computed for
different sensor measurements. Consequently, the cumulants which are used
for aperture extension (the virtual sensors) are insensitive to
non-Gaussian noise which is independent from sensor-to-sensor.
Additional noise reduction capability employing the present invention is
accomplished through another embodiment of the invention as shown in FIG.
8. The actual sensors of the array 2', 4' and 6' are supplemented by a
separate sensor 80. The signals from the sensors r(t), y(t) and x(t) are
provided to the VC.sup.3 14, however the signals are corrupted by a noise
cloud 82. The signal from the separate sensor g(t) is also provided to the
VC.sup.3 corrupted by a separate noise cloud 84. For the embodiment of the
invention shown in FIG. 8, it can be assumed that the array of arbitrary
sensors is illuminated by multiple linearly-correlated non-Gaussian
sources. The noise cloud contaminates the measurements with noise of
arbitrary cross statistics. The separation of the separate sensor 80 from
the array allows the additional assumption that the contamination of the
sensor measurement g(t) by noise cloud 84 is independent of the noise
component of the other sensors. If the array comprises M sensors which
measure {x.sub.k }.sub.k=1.sup.M then because
##EQU12##
Equation 22 allows calculation of the second order statistics as if the
noise cloud for the original array were not present. Equation 22 can be
interpreted as follows: to implement E{x*.sub.j (t)x.sub.k (t)} first move
from x.sub.j (t) to the separate sensor providing the signal g(t). Then
return from the separate sensor to the sensor having the signal x.sub.k
(t). The array covariance matrix R E{x(t)x.sup.H (t)} can be constructed
as
R=A.GAMMA.A.sup.H (23)
where A is the steering matrix for M sensors and .GAMMA. is a diagonal
matrix whose kth diagonal entry is .gamma..sub.4,k .vertline.g.sub.k
.vertline..sup.2 and g.sub.k is the response of the separate sensor to the
kth source (the row vector g can be defined as the collection of
responses). Any subspace method may be applied to R whose elements are
computed using equation 22. Even in the presence of colored non-Gaussian
noise there is no need to know the response of the separate sensor to the
far-field sources although it must be non-zero. The time series recorded
by the separate sensor provides the information necessary for the
VC.sup.3.
Consider source signals s(t) correlated in the following way
S(t)=Qu(t) (24)
where Q is non-singular (but arbitrary), and components of u(t) are
independent. Let B=AQ and h=gQ; then R, computed as described in (22), for
the independent sources scenario, takes the form:
R=B.GAMMA.B.sup.H (25)
where .GAMMA. is a diagonal matrix whose kth diagonal entry is
.gamma..sub.4,k .vertline.h.sub.k .vertline..sup.2 and .gamma..sub.4,k is
the fourth-order cumulant of u.sub.k (t). Note that Q.GAMMA.Q.sup.H is
full-rank, so that R, expressed as
R=A(Q.GAMMA.Q.sup.H)A.sup.H (26)
maintains all the requirements for use of subspace algorithms like MUSIC
and ESPRIT for direction finding. The structure of FIG. 8 allows such
direction finding even in the presence of correlated sources, correlated
non-Gaussian noise and arbitrary array characteristics.
The embodiment of FIG. 8 is employable in a communications scenario as
shown in FIG. 9 wherein the array of M sensors comprises a plurality of
communications receivers 90, 92, 94 and 96 with multiple intermediate
sensors not shown wherein the signal measurements by the sensors comprise
x.sub.1 (t) through x.sub.M (t) and the separate sensor comprises a
satellite sensor 98. The applications of the embodiment of the invention
described with respect to FIG. 8 are equally applicable to the embodiment
of FIG. 9 wherein the geometrical interpretation of the equation 22 with
respect to the satellite sensor comprises vector motion from a source to
the satellite sensor which then distributes the message (i.e., returning
from the satellite sensor to a second source (x.sub.j (t) to g(t) and
returning from g(t) to x.sub.k (t)).
With the embodiment of the invention as shown in FIGS. 8 and 9, if the
original array is linear and consists of uniformly spaced sensors of
identical response estimation of the parameters of coherent sources (i.e.,
when Q is singular) is possible through a spatial-smoothing algorithm on
the covariance matrix of equation 26 as proposed in Shan, T. and Kailath,
T., "Adaptive Beam Forming for Coherent Signals and Interference" IEEE
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