Self-coherence restoring signal extraction and estimation of signal direction of arrival

A processor and method for extracting or estimating directions of arrival of signals from a received data vector x(t) which has been corrupted by interfering signals and noise is described. The processor extracts signals by forming the scalar product of x(t) and a weight vector which is chosen such that the spectral self-coherence or conjugate spectral self-coherence of the processor output is maximized. The processor estimates the directions of arrival of signals by spectral self-coherence-selective performance surfaces for maxima.

BACKGROUND OF THE INVENTION 
1. Technical Field of the Invention 
The present invention relates to signal processing and, more particularly, 
but still generally, to signal processing systems for extracting 
communication signals from environments containing uncorrelated co-channel 
interference, and for signal-selective direction finding. 
2. Description of Background and Related Art 
The problem of extracting a signal from a noisy environment is well known 
in the signal processing arts. The fundamental problem facing any receiver 
designer is how to improve the reception of a desired signal in the 
presence of unknown and undesired interfering signals, channel distortion, 
and thermal background noise. In principle, this can be accomplished by 
signal processing. For example, consider a multisensor receiver consisting 
of an array of spatially-separated antennae which is receiving a desired 
signal from a first direction and interfering signals from other 
directions. By forming the appropriate linear combination of the sensor 
outputs, the signals arriving from the desired direction will be 
accentuated while signals from other directions are attenuated. Similarly, 
in single sensor receivers, notch filters can be used to place notches at 
the frequencies of narrowband interfering signals and other filter types 
can be used to equalize linear channel distortion. In both cases, the 
desired signal reception can be significantly improved by passing the 
received signal (or signals) through a linear combiner with the proper 
combiner weights. 
The basic problem is to set the combiner weights. If all of the parameters 
of the interfering signals are known, the proper combiner weights can be 
calculated. Unfortunately, such information is not always available. In 
many practical communications problems, the dominant type of channel 
corruption may be known. However, even in those cases in which the type of 
corruption is known, it may not be possible to know the number, strength, 
or directions-of-arrival of the interfering signals at any given time. For 
example, in mobile radio applications, these interference parameters will 
depend on the time of day and the physical location of the transmitter and 
receiver. In this case, not only will the parameters be unknown at the 
beginning of the transmission, but also, they may vary during the 
transmission. 
In such environments, it is usually impossible to preset the linear 
combiner weights to significantly improve reception of the desired signal. 
Hence, an adaptive algorithm must be used to learn the correct weight 
settings and to vary these settings over the course of the transmission if 
the parameters vary. This is accomplished by exploiting some known 
characteristic of the desired signal that distinguishes it from the 
interfering signals and noise. 
Prior art methods for calculating the combiner weights have focused 
primarily on methods that optimize some measure of signal quality. A 
number of different quality measures have been used. For example, the 
Applebaum algorithm (S. P. Applebaum, "Adaptive Arrays," Syracuse 
University Research Corporation," Rep. SPLTR66-1, August 1966) maximizes 
the signal-to-noise ratio at the output to the array. The Widrow-Hoff 
least-mean-square algorithm (B. Widrow, "Adaptive Filters I: Fundamentals, 
Stanford University Electronics Laboratory, Rep. SU-SEL-66-12, Tech. Rep. 
6764-6, December 1966) minimizes the mean-square-error between the desired 
signal and the output of an adaptive array. 
More recently, exact least squares algorithms which optimize deterministic, 
time-averaged measures of output signal quality have been developed (P. E. 
Mantey, L. J. Griffiths, "Iterative Least-Squares Algorithms for Signal 
Extraction," Second Hawaii Conference on Systems Science, January 1969, 
pp. 767-770, B. Friedlander, "Lattice Filters for Adaptive Processing," 
Proc. pp. 879-867, August 1982). In directly implementing an adaptive 
processor that optimizes any such quality measure, the receiver designer 
must have accurate knowledge of the cross-correlation between the 
transmitted and received signals. In practice, this requires close 
cooperation between the receiver and the desired-signal transmitter. In 
applications where the cross-correlations are not known at the start of 
the desired-signal transmission, these statistics must be learned by the 
receiver at the start of the transmission and, on time-varying channels, 
updated over the course of the transmission. 
The earliest methods for accomplishing this in telecommunications 
applications required the transmitter to send a known signal over the 
channel. This signal was sent at the beginning of the transmission or 
intermittently in lieu of the information-bearing signal. In this manner, 
the receiver could be trained at the start of the transmission and the 
combiner weights updated during the transmission. Other embodiments of 
this type of system transmit a pilot signal along with the 
information-bearing signal. This pilot signal is used to train and 
continuously adapt the receiver. These methods are effective in those 
situations in which there is cooperation between the transmitter and 
receiver. 
In many applications, however, quality-optimizing techniques cannot be 
directly used. For example, the channel may be varying too rapidly for 
start-up or intermittent adaptation to be effective. In addition, the 
system resources (power, dynamic-range, bandwidth, etc.) may be too 
limited to allow pilot signals to be added to the information-bearing 
signal 
Alternatively, the receiver may not have the necessary control over the 
transmitter. This is the case when the transmitter is a natural source 
such as a person speaking, or when the receiver that must be adapted is 
not the intended receiver in the communication channel, e.g., in 
reconnaissance applications. 
In applications in which a known desired signal cannot be made available to 
the receiver, the designer must make use of blind adaptation techniques 
that exploit other observable properties of the desired signal or the 
environment in which the signal is transmitted. Prior art algorithms for 
accomplishing this may be divided into three classes. In the first class 
of techniques, referred to as demodulation-directed techniques, a 
reference signal is produced by demodulating and remodulating the 
processor output signal. This reference signal is then used as a training 
signal in a conventional adaptive processing algorithm. This technique is 
commonly employed in decision-directed and decision-feedback equalizers in 
telephony systems (J. G. Proakis, "Advances in Equalization of Intersymbol 
Interference," Advances in Communications Systems, ed. by A. V. 
Balakrishnan, A. J. Viterbi, N.Y. Academic Press, 1975). It has also been 
used to adapt antenna arrays in spread-spectrum communication systems (A. 
T. Compton, "An Adaptive Array in a Spread Spectrum Communications 
System," Proc. IEEE, Vol. 66, March 1978). 
The primary advantage of demodulation-directed techniques lies in their 
efficient use of system resources and in their convergence speed. These 
algorithms rely on the demodulator-remodulator loop providing a very clean 
estimate of the desired signal. In most of these techniques, this 
requirement will be met after the demodulator has locked on to the 
received signal. However, until the demodulator does lock on, the 
reference signal estimate will generally be poor. For this reason, most 
demodulation-directed techniques are employed as tracking algorithms after 
a more sophisticated technique has been used to lock onto the desired 
signal. Many demodulation directed techniques encounter additional 
problems in dynamic environments where signals are appearing and 
disappearing over the course of the desired-signal transmission. In 
addition, these techniques are expensive to implement and are inflexible 
in their implementation, since they require a built-in demodulator matched 
to a specific desired signal to operate. This drawback renders them 
inapplicable to a system in which a variety of signals are of interest, 
e.g., in satellite transponders and reconnaissance systems. 
The second class of techniques, referred to as channel-directed techniques, 
exploit known properties of the receiver channel or environment such as 
the spatial distribution of the received signals. In this class of 
techniques, knowledge of the receiver channel is used to generate and 
apply a reference signal to a conventional adaptation algorithm, or to 
estimate key statistics which are used to optimize the combiner weights. 
When applied to antenna arrays, most channel-directed methods exploit the 
discrete spatial distribution of the signals received by the array, i.e., 
the fact that the received signals impinge on the array from discrete 
directions of arrival. Examples of channel-directed techniques include the 
Griffiths P-vector algorithm (L. J. Griffiths, "A Simple Adaptive 
Algorithm for Real-Time Processing in Antenna Arrays," Proc. IEEE, Vol. 
57, October, 1969) and the Frost constrained LMS algorithm (O. L. Frost, 
"An Algorithm for Linearly-Constrained Adaptive Array Processing," Proc. 
IEEE, Vol. 60, August 1972). Both of these techniques exploit the known 
direction-of-arrival of the desired signal. Techniques have also been 
devised to deal with the situation in which the direction-of-arrival of 
the desired-signal is unknown. Examples of such techniques are the 
generalized sidelobe canceller (L. J. Griffiths, M. J. Rude, "The P-Vector 
Algorithm: A Linearly Constrained Point of View," Proc. Twentieth Asilomar 
Conf. on Signals, Systems and Computers, Pacific Grove, Calif., November 
1986) and the signal subspace techniques referred to as MUSIC (R. O. 
Schmidt, "Multiple Emitter Location and Signal Parameter Estimation," 
Proc. RADC Spectrum Estimation Workshop, October 1979. M. Wax, T. Shan, T. 
Kailath, "The Covariance Eigenstructure Approach to Detection and 
Estimation by Passive Arrays," IEEE Trans. ASSP, 1985) and ESPRIT (A. 
Paulraj, A. Roy, T. Kailath, "Estimation of Signal Parameters via 
Rotational lnvariance Techniques-ESPRIT," Proc. Nineteenth Asilomar Conf. 
on Signals, Systems and Computers, Pacific Grove, Calif., November 1985). 
These approaches can all be thought of as high-resolution spatial spectrum 
estimation techniques for locating lines in the received signal spatial 
spectrum. 
All of the channel-directed techniques suffer from the common weakness that 
they require knowledge of the sensor geometry and/or the individual sensor 
or subarray characteristics to adapt the array. In practice, this 
characterization is usually obtained by a series of experiments, referred 
to as array calibration, to determine the so-called array manifold of the 
sensor network. The cost of array calibration can be quite high, and the 
measurement procedure is, in many applications, impractical. For example, 
a 16.times.16 planar array calibrated over a sphere with a one degree 
resolution in elevation and azimuth, and using 16 bit accuracy requires 
approximately 64 megabytes of storage. This storage requirement increases 
exponentially as the number of search dimensions is increased, e.g., if 
the calibration is performed over temporal frequency or polarization in 
addition to elevation and azimuth. In addition, systems with considerably 
more sensors are desirable. Phased array communication systems have 
currently been proposed with over 10,000 elements, and current advances in 
low-cost microwave radio are pushing this figure higher. The storage 
requirements, not to mention the calibration times, for such large arrays 
renders these methods impractical. Furthermore, in certain applications, 
e.g., lightweight spaceborne arrays, airborne arrays, and towed acoustic 
arrays, the array geometry and even the sensor characteristics may be 
changing slowly with time; hence, an accurate set of calibration data may 
never be available. 
Even when calibration data is available, the computational cost of using 
this data can be prohibitive. Both the generalized sidelobe canceller and 
MUSIC techniques require a search over the set of calibration data during 
the operation of the algorithm. In addition, the computational complexity 
of the MUSIC algorithm increases as the cube of the number of sensors in 
the array. The required computational complexity can be prohibitive. Also, 
the additional classification operations required to recognize the one 
desired signal among the multiple signals extracted by the algorithms 
increase the complexity even more. 
The ESPRIT technique was proposed in an effort to overcome these 
computational and storage problems. This technique does not require 
calibration data to operate and hence, avoids many of the problems 
associated with the other channel-directed approaches. However, ESPRIT 
requires accurate knowledge of the noise covariance matrix of the received 
data, and its computational difficulty increases as the cube of the number 
of sensors in the array. The ESPRIT algorithm has a number of other 
shortcomings. It does not perform optimal signal extraction in the sense 
that the signal-to-noise-ratio is maximized, but instead, each output 
solution nulls simply all signals but one impinging on the array. In 
addition, the ESPRIT algorithm requires that the array elements be grouped 
into doublets with identical characteristics and common geometrical 
displacement. These conditions impose serious constraints on both the 
manufacture and performance of an ESPRIT array. 
In the third class of techniques, referred to as set-theoretic 
property-mapping and property-restoral techniques, the output of the 
receiver is forced to possess a set of known properties possessed by the 
transmitted signal. Here, the receiver processor is adapted to restore 
known modulation properties of the desired signal to the processor output 
signal. Modulation properties are defined here as observable properties of 
the desired signal imparted by the modulation format used at the 
desired-signal transmitter. In many cases, these properties are destroyed 
by transmission over the communication channels. For example, the constant 
modulus property shared by FM, PSK, and CPFSK is destroyed by the addition 
of noise, other signals, or multipath interference to the transmitted 
signal. The property-restoral approach adapts a receiver processor to 
optimize an objective function that measures this property in the output 
signal. 
The first use of the property-restoral concept was in Sato's algorithm (Y. 
Sato, "A Method of Self-Recovering Equalization for Multilevel Amplitude 
Modulation Systems," IEEE Trans. Comm., vol. COM-23, pp. 679-682, June 
1975) which was designed to equalize channel distortion in a BPSK 
telephony signal by minimizing the mean square error between the squared 
signal and unity. This algorithm was extended to restoral of general 
constant-modulus and QAM communication signals with the Constant-Modulus 
Algorithm (J. R. Treichler, B. C. Agee, "A New Approach to Multipath 
Correction of Constant Modulus Signals," IEEE Trans. ASSP, vol. ASSP-31, 
pp. 459-472, April 1983) and Godard's algorithm (D. N. Godard, 
"Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data 
Communication Systems," IEEE Trans. Comm., vol. COM-28, pp. 1867-1875, 
November 1980). More recently, set-theoretic property mapping (J. A. 
Cadzow, "Signal Enhancement--a Composite Property Mapping Algorithm," IEEE 
Trans ASSP, January 1988) has been advanced as a general technique for 
designing property-restoral algorithms (B. G. Agee, "The Property Restoral 
Approach to Blind Adaptive Signal Extraction," University of California, 
Davis Calif., September 1989 (Doctoral Dissertation). 
The property-restoral algorithms described above have been successfully 
applied to adaptive signal extraction in both filters and antenna arrays 
and appear to have strong advantages over both the demodulation-directed 
and channel-directed techniques. However, these techniques still have 
drawbacks. The convergence and capture characteristics of all of these 
algorithms are still not well understood. In addition, the 
Constant-Modulus Algorithms are highly nondiscriminatory, requiring 
multitarget implementations to recover all the signals present in dense 
interference environments. The former drawback limits the application of 
these algorithms in automatic (unsupervised) communication systems where 
they must operate with a minimum of attention. The latter drawback is of 
critical importance in large-aperture systems and directed search 
applications, since it requires the algorithms to extract every signal in 
the environment using a multitarget implementation and to then classify 
the signals to find the one desire signal. 
Another problem encountered in signal processing is providing high 
resolution estimation of the directions of arrival (DOA) of signals 
impinging on an antenna array. 
The conventional MUSIC algorithm for high-resolution DOA estimation 
exploits the spatial coherence properties of signal sources having a 
discrete spatial distribution by exploiting the resulting structure of the 
array autocorrelation matrix. As is known in the art, given correct 
knowledge of the noise autocorrelation matrix, the MUSIC algorithm can 
remove the noise contribution from the array autocorrelation matrix, 
leaving only the signal components. If there remain fewer such signal 
components than there are array sensors, and if no two signals are 
perfectly correlated, then the null space of the signal-only 
autocorrelation matrix is orthogonal to the direction vectors of those 
signals. Thus, using calibration data for the array, the MUSIC algorithm 
searches over all possible DOAs for the directions that maximize a 
specific measure of orthogonality. Those DOAs are taken to be the DOA 
estimates. If the interference autocorrelation matrix is known as well, 
then interfering signal components can also be removed, leaving degrees of 
freedom available for estimating DOAs of desired signals. 
The relevant limitations of the MUSIC algorithm are summarized here. (1) 
Lack of knowledge of the interference autocorrelation matrix requires 
MUSIC to estimate DOAs of all signals impinging on the array. (2) There 
must be fewer signals impinging on the array (excluding signals accounted 
for in the interference autocorrelation matrix, if any) than there are 
sensors in order to obtain useful DOA estimates. (3) With limitations on 
the amount of data processed, the DOA of a desired signal cannot be 
distinguished from that of a sufficiently closely spaced interferer if 
that interferer is not accounted for in the interference autocorrelation 
matrix. (4) Correct knowledge of the noise autocorrelation matrix is 
required to obtain useful DOA estimates. (5) No two signals may be 
perfectly correlated as can occur in multipath environments or in the 
presence of "smart" jammers. 
Because of these disadvantages, the MUSIC algorithm for high resolution DOA 
estimation, as well as most others, performs poorly or fails entirely in 
many environments. 
A further problem encountered in signal processing is that the developed 
methods for extracting signals and/or estimating signal DOA, which adapt 
to narrowband signals, may not be valid for wideband conditions. Thus, it 
is desirable to extend the method for extracting and/or estimating signal 
DOA for narrowband conditions to wideband conditions. 
Broadly, it is an object of the present invention to provide improved 
methods for adapting receivers for the estimation of directions-of-arrival 
of communications signals, and for the extraction of communications 
signals. 
It is another object of the present invention to provide an apparatus and 
method for extracting signals that do not require a knowledge of the 
desired signal waveform. 
It is yet another object of the present invention to provide an extraction 
apparatus and method that do not require a knowledge of the 
direction-of-arrival of the desired signal. 
It is a still further object of the present invention to provide an 
extraction apparatus and method that do not require a knowledge of the 
background interference environment. 
It is yet another object of the present invention to provide an extraction 
apparatus and method that do not require a knowledge of the geometry of 
the sensor array or of the individual sensor characteristics. 
It is a still further object of the present invention to provide an 
extraction apparatus and method that can be programmed to sort and 
automatically extract signals with desired statistical properties from 
dense interference environments. 
It is yet another object of the present invention to provide an extraction 
apparatus and method that require substantially less computation and 
storage than prior art competing methods. 
It is a still further object of the present invention to provide an 
extraction apparatus and method that have unambiguous and well-understood 
convergence and capture properties. 
It is another object of the present invention to provide an apparatus and 
method that are more effective for high resolution of the DOA of signals 
impinging on an antenna array. 
It is still another object of the present invention to provide an apparatus 
and method which are able to perform DOA estimation without knowledge of 
the noise and interference autocorrelation matrices. 
It is another object of the present invention to provide an apparatus and 
method able to perform DOA estimation even if an arbitrary number of 
arbitrarily closely spaced interferers are present. 
It is still another object of the present invention to provide an apparatus 
and method for DOA estimation, which operate effectively in the presence 
of perfectly correlated signals. 
It is another object of the present invention to provide an apparatus and 
method for DOA estimation wherein the only requirement is that the number 
of signals spectrally self-coherent at the chosen frequency-shift are less 
than the number of sensors. 
It is yet another object to apply the methods and apparatus of the present 
invention to wideband conditions. 
These and other objects of the present invention will be apparent to those 
skilled in the art from the following detailed description of the 
invention and the accompanying drawings. 
SUMMARY OF THE INVENTION 
The present invention comprises a processor and method for extracting a 
signal s(t) from a signal input vector x(t) in which each component 
comprises a complex measurement of s(t) together with interference from 
noise and/or other signals. The processor computes an output signal y(t) 
by forming the scalar product of x(t) and a vector w having the same 
number of components as x(t) and chosen such that the self-coherence of 
y(t) at frequency shift .alpha. and time shift .tau. is substantially 
maximized for predetermined values of .alpha. and .tau.. 
The present invention also estimates the direction of arrival of a signal 
by processing the same frequency-shifted and time-shifted data that is 
used in the process of performing signal extraction. 
The present invention also extends the method of extracting a signal and 
the method of estimating the direction of arrival of a signal to wideband 
conditions.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Least Squares SCORE and Cross-SCORE Extraction 
The present invention comprises a method and apparatus for adapting an 
antenna array to extract a communication signal-of-interest with known 
self-coherence or conjugate self-coherence properties from environments 
containing uncorrelated co-channel interference. The present invention 
does not require a knowledge of the waveform or of the 
direction-of-arrival of the signal-of-interest. Furthermore, the present 
invention accomplishes this goal without requiring a knowledge of the 
array geometry or of the individual sensor characteristics. Finally, the 
method has a relatively simple and flexible implementation compared to the 
competing present art techniques described above. 
The present invention may be most easily understood with reference to an 
array of M sensors. If the individual sensors have radio frequency 
outputs, the outputs are first downconverted to baseband or an 
intermediate frequency (IF). The converted sensor output signals are then 
converted to a complex representation either by a Hilbert transform 
operation or a separate downconversion of in-phase and quadrature 
components, to form a multisensor received signal represented by a vector 
x(t) which is defined by the following relationship 
EQU x(t)=[x.sub.k (t)]k=1/M, (1) 
where x.sub.k (t) is the complex signal from the kth sensor in the array. 
The corrected signal, i.e., the signal-of-interest, is obtained by forming 
a weighted sum of the individual sensor signals, i.e., 
EQU y(t)=.SIGMA.w.sub.k.sup.* x.sub.k (t)=wx (2) 
where y(t) is the corrected signal, and the w.sub.k are complex weight 
factors which are computed according to the present invention. Here, the 
superscripts "*" and denote the complex-conjugate and the complex 
conjugate transpose, respectively, of the quantity in question. For 
notational convenience, the w* will be interpreted as the elements of a 
vector w. 
In the present invention, w is calculated assuming that the 
signal-of-interest is either self-coherent or conjugate self-coherent. A 
waveform s(t) is said to be self-coherent at a frequency .alpha. if the 
cross-correlation of s(t) and s(t) frequency shifted by .alpha. is nonzero 
at some lag-time .tau., i.e., if 
##EQU1## 
is different from zero for some value of .tau.. Here &lt;.. &gt;.infin. denotes 
infinite time-averaging of the quantity enclosed by the brackets. A 
waveform s(t) is said to be conjugate self-coherent at frequency .alpha. 
if the cross-correlation of s(t) and the complex conjugate of s(t) 
frequency-shifted by .alpha. is nonzero at some lag-time .tau., i.e., if 
##EQU2## 
is different from zero for some value of .tau.. 
Self-coherence is commonly induced by periodic switching, gating, mixing, 
or multiplexing operations at the transmitter. For example, self-coherence 
is induced at multiples of the baud-rate in PCM signals and at multiples 
of the pilot-tone frequency in FDM-FM signals. Conjugate self-coherence is 
commonly induced by unbalanced in-phase and quadrature carrier modulation 
at the transmitter. For example, conjugate self-coherence is induced at 
twice the carrier frequency in DSB-AM, VSB-AM, BPSK, and unbalanced QPSK 
signals. 
The self-coherence (or conjugate self-coherence) of a received signal is 
degraded if it is corrupted by additive interference which does not share 
that coherence. For example, a PCM signal-of-interest is corrupted at the 
receiver if a second PCM signal modulated at a different baud-rate is 
present. The present invention adapts a receiver processor such that the 
self-coherence (or the conjugate self-coherence) of the signal-of-interest 
is restored to the receiver output signal. Consequently, the acronym SCORE 
(for Self COherence REstoral) is used to denote the invention. 
In the present invention, the weight vector w is computed so as to maximize 
the degree of self-coherence or conjugate self-coherence in the output 
signal. The goal here is to linearly combine the elements of x(t) to 
enhance the signal-of-interest and suppress the noise and interference 
received by the array. In addition, this is accomplished without using 
knowledge of the signal-of-interest waveform, signal-of-interest aperture 
vector, or interference autocorrelation matrix to calculate the linear 
combiner weights w. In the present invention, this is accomplished by 
minimizing the function, F.sub.LS, where 
EQU F.sub.LS (w,c)=&lt;.vertline.y(t)-u(t).vertline..sup.2 &gt; (5) 
or by maximizing the function, F.sub.cross, where 
##EQU3## 
where &lt;. .&gt; denotes time-averaging and u(t), which will be referred to as 
the SCORE reference signal, is defined by 
EQU u(t)=&lt;[cx(t-.tau.)](*)e.sup.j2.pi..alpha.t ]&gt; (7) 
where (*) denotes optional complex-conjugation. The control parameters 
.tau. and .alpha. are referred to as the control delay and self-coherence 
frequency, respectively; the control parameter c is referred to herein as 
the control vector of the processor. 
The control parameters (*), .tau., and .alpha. are set on the basis of the 
self-coherence properties of the signal being sought. In the preferred 
embodiment of the present invention, these parameters are set to yield a 
strong value of the self-coherence function (or conjugate self-coherence 
function) of the signal-of-interest, s(t), where the self-coherence 
function is given by 
##EQU4## 
Here, &lt;. .&gt;.infin. denotes infinite time-averaging and the optional 
conjugation *) is applied if and only if conjugate self-coherence is to be 
restored. Of these control parameters, only .alpha. and the conjugation 
control are crucial to the operation of the processor. Eq. (8) has been 
analytically evaluated for a variety of signal modulation types (see, W. 
A. Gardner, Introduction to Random Processes with Applications to Signals 
and Systems, Macmillan, New York, N.Y., 1985 and W. A. Gardner, 
Statistical Spectral Analysis: A Nonprobabilistic Theory, Prentice Hall, 
Englewood Cliffs, N.J., 1987), and for many signals of interest and 
appropriate values of .alpha. the self-coherence function has been found 
to be a slowly varying function with respect to .tau.; hence there is 
considerable flexibility in choosing .tau.. 
The control vector c is either fixed or jointly adapted with w over the 
data reception time. In the following discussion, any algorithm that 
adapts w to minimize Eq. (5) or maximize Eq. (6) for a fixed c will be 
referred to as a least squares SCORE algorithm, whereas any algorithm that 
adapts both w and c using these equations subject to any gain constraint 
on w and/or c will be referred to as a cross-SCORE algorithm. 
The values of w and c that optimize Eqs. (5) and (6) can be determined in 
terms of the estimated correlation matrices 
EQU R.sub.xx =&lt;x(t)x(t)&gt; (9) 
EQU R.sub.rr =&lt;r(t)r(t)&gt; (10) 
EQU R.sub.xr =&lt;x(t)r(t)&gt; (11) 
EQU R.sub.xu =&lt;x(t)u*(t)&gt; (11') 
where r(t) is referred to as the reference signal vector and is defined by 
EQU r(t)=[x(t-.tau.)](*)e.sup.j2.pi..alpha.t (12) 
From the above definitions, it can be shown that 
EQU R.sub.xr =R.sub.xx.sup..alpha. (*)(.tau.) (13) 
Matrices R.sub.xx and R.sub.rr are the conventional autocorrelation 
matrices of x(t) and r(t), respectively. Matrix R.sub.xx.sup..alpha. 
(*)(.tau.) is the cyclic autocorrelation matrix or the cyclic conjugate 
correlation matrix of x(t), depending on whether the conjugation (*) is 
applied in r(t). 
In terms of this notation, the w that minimizes Eq. (5) (or maximizes Eq. 
(6)) for fixed c is given by 
##EQU5## 
where g is an arbitrary scalar gain variable. Similarly, the w and c that 
jointly minimize Eq. (5) (or maximize Eq. (6)) subject to a gain 
constraint are given by the solutions of the eigenequations 
EQU R.sub.xx w=R.sub.xr R.sub.rr.sup.-1 R.sub.rx w (15) 
EQU .lambda.R.sub.rr c(*)=R.sub.rx R.sub.xx.sup.-1 R.sub.xr c(*)(16) 
corresponding to the maximum eigenvalue, .lambda., of these equations. The 
optimal w and c can also be found by setting w using Eq. (14) and then 
setting c using Eq. (16), or by obtaining the solution to the joint 
eigenequation 
##EQU6## 
corresponding to the maximum eigenvalue, .nu., of this equation. An 
apparatus according to the present invention either approximately or 
exactly solves these equations using estimates of the correlation matrices 
R.sub.xx and R.sub.xr. 
It should be noted that the other eigenvalues obtained from Eq. (17) 
provide useful information. If several signals that are self-coherent (or 
conjugate self-coherent) at .alpha. are present, multiple solutions to 
this eigen equation will be found. Each of the solutions adapts the 
processor to receive one of these signals if the self-coherence functions 
of these signals are distinct, when evaluated at the chosen control 
parameters (*), .alpha., .tau.. 
An extension of the cross-SCORE algorithm is obtained by generalizing the 
reference signal vector r(t) according to 
EQU r(t)=[h(t)x(t)](*)e.sup.j2.pi..alpha.t (18) 
where h(t) is a general scalar linear time-invariant filter impulse 
response, and denotes the convolution operation. This filter will be 
referred to as the control filter. The reference signal vector defined in 
Eq. (12) utilizes a filter that is a delay. Algorithms employing a control 
filter with an impulse response that is different from a simple time delay 
are referred to hereinafter as generalized cross-SCORE algorithms. This 
generalization does not affect the mathematical form of the cross-SCORE 
algorithm; however, it does affect the implementation of the cross-SCORE 
algorithm. If the control filter is chosen to be a delay, then R.sub.rr 
can be replaced with R.sub.xx in Eqs. (15)-(17), thereby considerably 
reducing the computational complexity of the cross-SCORE apparatus. The 
complexity of the control filter will also strongly affect the complexity 
of the overall cross-SCORE apparatus, since the filter must process the 
individual elements of x(t). However, the convergence time of the 
apparatus can be improved by matching h(t) to the self-coherence 
properties of the signal-of-interest s(t) in the environment. That is, 
h(t) should be chosen to maximize 
##EQU7## 
where 
EQU s(t)=h(t)s(t) (20) 
In summary, the method of this embodiment comprises the following steps: 
1. Using an analog or digital measurement of the complex data x(t) from a 
set of sensor elements, compute the R.sub.xx using Eq. (9). This can be 
accomplished by updating R.sub.xx after each new measurement of x(t). 
2. Form r(t) from x(t) using Eq. (12) or (18) and compute R.sub.rr and 
R.sub.xr using Eqs. (10) and (11). 
3. If the cross-SCORE algorithm is being implemented, update c.sup.(*) by 
exactly or approximately solving Eq. (16). 
4. Update w using Eq. (14), and form the array output signal using Eq. (2). 
The various correlation matrices which consist of the average values of 
products of the components of x(t) and r(t) may be calculated by 
recursively updating the matrices after each new measurement of x(t). 
Algorithms for such recursive updating are well known in the processing 
arts (See, P. E. Mantey and L. J. Griffiths, "Iterative Least-Squares 
Algorithms for Signal Extraction", Second Hawaii Conference on Systems 
Science, January 1969, pp. 767-770, or R. A. Monzingo, T. W. Miller, 
Introduction to Adaptive Arrays, John Wiley and Sons, N.Y., N.Y., 1980). 
Additional information may be obtained from the parameters obtained in the 
above described procedure. For example, the maximum eigenvalue obtained 
from the cross-SCORE procedure can be used to detect or recognize the 
presence of the signal-of-interest in the received environment. The phase 
angle of the complex inner product of the processor and control vectors, w 
and c, can be used to estimate the signal carrier/clock phase. In 
addition, the array aperture estimate, R.sub.xx w, can be used to estimate 
additional parameters such as the polarization of the signal-of-interest 
or the direction of arrival of the signal-of-interest. The direction of 
arrival information requires calibration data or a special array geometry 
for the array in question. 
Having described the method by which the values of w and c are generated, a 
more detailed description of an apparatus according to the present 
invention will now be given. The apparatus of the present invention will 
be referred to hereinafter as a SCORE processor. FIG. 1(a) illustrates a 
receiver front-end which might be used to provide the inputs, x(t), 
required by the SCORE processor when said processor is used with an 
antenna array comprised of antennae 26. To simplify FIG. 1 only two 
signals, I(t) and S(t), are shown impinging on the antenna array. S(t) is 
the signal-of-interest, and I(t) represents an interfering signal. The 
directions of incidence of these signals are indicated by arrows 22 and 
24, respectively. Each antenna 26 of the array is connected to a receiver 
circuit 28. The receiver circuits also introduce noise signals 27. 
The preferred embodiment of a receiver circuit 28 is shown in FIG. 1(b). 
The signal 29 from the antenna connected to said receiver circuit is split 
into two signals on lines 30 and 32. These signals are downconverted to 
baseband or an intermediate frequency (IF) utilizing a local oscillator 
34. The downconverted sensor output signals are converted to a complex 
representation by a separate conversion of inphase and quadrature 
components utilizing phase shifter 35 which shifts the local oscillator 
signal by 90 degrees prior to mixing by mixer 31. Mixer 33 is used to 
produce the in-phase component of the signal. Each component is passed 
through a low pass filter 40. To simplify the drawings, only one signal 
line will be shown for each component x.sub.k (t) of x(t); however, it is 
to be understood that each such signal line consists of two lines, one 
carrying the real component, R, and one carrying the imaginary component, 
J. 
FIG. 2 illustrates the preferred embodiment of a least-squares SCORE 
processor 200. The x(t) signal from the receiver front-end is fed to a 
scalar product circuit 202 which calculates the scalar product of x(t) and 
w to form the output signal y(t). To clarify FIG. 2 and the following 
drawings, those signal paths that carry a vector or matrix are shown in 
bold. It is assumed that x(t) has not been digitized, i.e., it is still in 
analog form. Scalar product circuits are known to the prior art; hence, 
the details of scalar product circuit 202 will not be given here. For the 
purposes of this discussion, it is sufficient to note that scalar product 
circuit 202 may be constructed from a plurality of operational amplifiers 
whose gains are controlled by the values of the components of w. The 
outputs of these amplifiers are combined using summing amplifiers to form 
y(t). 
The values of w are calculated by minimizing the Least Squares function 
given in Eq. (5). The steering signal, u(t), is calculated in analog form 
by circuit elements 206, 208, 210, 211, and 214 as follows. A scalar 
signal is created by computing the scalar product of x(t) and c in scalar 
product circuit 206. The output of this circuit is delayed by a time T by 
delay circuit 208. Here T is set to the desired value of .tau.. If w is to 
be updated on the basis of conjugate self-coherence, the delayed signal is 
then replaced by the complex conjugate thereof by circuit 210. If 
conjugate self-coherence is not being used, circuit 210 merely reproduces 
the input signal at its output. Finally, the output of circuit 210 is 
frequency shifted by multiplying the signal by e.sup.j2.pi..alpha.t 
utilizing a local oscillator 214 whose frequency f.sub.a has been set 
equal to .alpha.. This multiplication is performed by mixing circuit 211. 
Update circuit 204 calculates the value w that minimizes the time average 
of the squared error .vertline.y(t)-u(t).vertline..sup.2. 
The preferred embodiment of update circuit 204 is shown in FIG. 3. Update 
circuit 204 utilizes the first relationship in Eq. (14) to calculate the 
value of w that minimizes F.sub.LS. The value of w is calculated on a 
recursive basis under the control of a clock 501. During each clock cycle, 
the values of the components of x(t) are digitized by a bank of A/D 
converters 502. These values are fed to an update circuit 506 which 
updates the values in R.sub.xx.sup.-1, preferably utilizing an algorithm 
such as that of Woodward for updating an inverse of a correlation matrix. 
Circuits for updating the inverse of a correlation matrix are well known 
in the signal processing arts and hence will not be described in detail 
here. For the purposes of this discussion, it is sufficient to note that 
update circuit 506 typically comprises a bank of multiply and accumulate 
circuits. Each such circuit calculates the product of two components of 
vector x(t) and accumulates the results so as to form the appropriate time 
averages. 
Similarly, A/D converter 504 digitizes u(t) and outputs the digitized 
result to update circuit 508 which updates the average of R.sub.xu. The 
updated values calculated by update circuits 506 and 505 are then used to 
calculate a new value for w by update circuit 510 which performs the 
multiplications given in Eq. (14) above. 
Update circuit 204 is preferably a three stage pipeline processor. At each 
clock cycle, the current values of x (t) and u(t) are digitized by 
circuits 502 and 504. The values so digitized one cycle earlier are used 
to update R.sub.xx.sup.-1 and R.sub.xu. The values of R.sub.xx.sup.-1 and 
R.sub.xu calculated one cycle earlier, i.e., based on the digitized values 
of x and u two cycles earlier, are then used to update w which is 
outputted on bus 511. 
The value of the gain constant, g, shown in Eq. (14) is preferably chosen 
so as to prevent overflows or underflows in the various calculational 
elements of update circuit 204. 
SCORE processor 200 does not determine the values of the various components 
of the control vector c; hence these values must be provided. If 
calibration data is available for the antenna array and if the direction 
of incidence of the signal-of-interest is also known, then c can be chosen 
so as to beamform in the direction of incidence of the signal-of-interest. 
If such data is not available, c is preferably chosen to provide an 
isotropic response for the antenna array. 
However, if the proper values for c are not known, it is preferred that 
both w and c be simultaneously optimized, i.e., that a cross-SCORE 
processor be used. A block diagram of such a processor is shown at 300 in 
FIG. 4. In a manner similar to that described above with reference to 
SCORE processor 200, the scalar product of the vector input x(t) received 
on bus 301 and w is calculated by scalar product circuit 302 to form the 
output signal y(t). The vector input x(t) is also used to construct the 
vector r(t) given in Eq. (12) above using circuits 306, 310, 312, and 314. 
Delay circuit 306 delays each component of x(t) by a time, T, Which is set 
equal to .tau.. If w and c are being chosen using the conjugate 
self-coherence formulas, then each component of the delayed signal is 
replaced by its complex conjugate by circuit 310. Finally, each component 
of the output of circuit 310 is multiplied by e.sup.j2.pi..alpha.t by 
mixing the output with a signal from a local oscillator 314 whose 
frequency f.sub.a has been set equal to .alpha.. This mixing operation is 
performed by circuit 312. 
The resulting signal, r(t), is inputted to an update circuit 304 via bus 
315. Update circuit 304 also receives x(t) as one of its inputs. The 
preferred embodiment of update circuit 304 is shown in FIG. 5 at 600. 
Referring to FIG. 5, the measured values of x(t) are inputted to a bank of 
A/D converts 602 which digitize x(t). Similarly, the components of r(t) 
are digitized by a bank of A/D converters 604. The digitized values of the 
components of x(t) and r(t) are inputted to circuits 606-608 which are 
used to update the values of the correlation matrices given in Eqs. 
(9)-(11) or the appropriate inverses thereof. In the preferred embodiment, 
w is updated using Eq. (14) by update circuit 610. Then c is updated using 
the new value of w and Eq. (16) by update circuit 609. However, as noted 
above, there are several algorithms that may be used to find the values of 
w and c that optimize Eqs. (5) and (6). Hence, it will be obvious to those 
skilled in the art that update circuits 609 and 610 may be replaced by 
other processing means that calculate the solution to one of the above 
described equations. 
In the preferred embodiment, update circuit 600 is a four stage pipeline 
processor which is timed by clock 620. In each clock cycle, the current 
values of x(t) and r(t) are calculated by A/D banks 602 and 604. The 
values calculated by these banks in the previous clock cycle are used by 
update circuits 606-608 to update the various matrices used to calculate 
new values for w and c. The values of these matrices calculated in the 
previous cycle, i.e., based on x(t) and r(t) values digitized two cycles 
earlier, are used in conjunction with the previous value of w to update c. 
Finally, the new value of c is used in conjunction with these matrices to 
calculate a new value for w. 
Although the present invention has been described with reference to certain 
specific algorithms for optimizing Eqs. (5) and (6), it will be apparent 
to those skilled in the art that other methods of optimizing these 
equations may be utilized. For example, the signals u(t) and y(t) 
generated in FIG. 2 may be inputted to a processor that searches for the 
values of w and/or c that minimize the time average of the square of the 
absolute value of the difference of u(t) and y(t), subject to an equality 
constraint on either &lt;.vertline.u(t).vertline..sup.2 &gt; or 
&lt;.vertline.y(t).vertline..sup.2 &gt; (which can be interpreted as a 
weighted-norm constraint on either c or w, without using the matrices 
described above). Algorithms that find the constrained minimum of an 
arbitrary function of one or more variables are well known to those 
skilled in the signal processing and mathematical arts. 
In this regard, it should be noted that signal processors that calculate 
the value of w so as to maximize an objective function are known to the 
prior art. The present invention differs from such processors in the 
manner in which the reference signal, i.e., u(t) or r(t), is obtained. In 
the present invention, a reference signal that depends on the measured 
signal x(t) frequency shifted by .alpha. is used to generate the processor 
weight vector w. The processor weight vector is calculated such that the 
self-coherence (or conjugate self-coherence) of the output signal, y(t), 
is greater than the self-coherence (or conjugate self-coherence) of the 
input signals, x.sub.k (t), that makeup the input signal vector x(t). In 
the above examples, w is calculated so as to maximize the self-coherence 
(or conjugate self-coherence) of the output signal. 
In the examples discussed above, the reference signal depends on the 
measured signal delayed by .tau.. As noted above with reference to Eq. 
(18), a more generalized form of reference signal may be generated by 
first filtering x t) and then frequency shifting the filtered signal to 
produce the reference signal. The delay used in the above described 
embodiments of the present invention is merely one case of a "filter". 
FIG. 6 illustrates a more general SCORE processor 400 that utilizes such a 
filter. Generalized-SCORE processor 400 constructs the output signal y(t) 
by taking the scalar product of x(t) is which inputted on bus 401 and a 
vector w utilizing circuit 402 which operates in a manner analogous to 
that described with reference to previously described embodiments of the 
present invention. 
The processor vector w is calculated so as to optimize Eqs. (5) or (6). The 
reference signal r(t) is generated by filtering x(t) using filter bank 
406. In the preferred embodiment of the present invention, the same filter 
is applied to each component of x(t); however, embodiments in which 
different filters are applied to different components of x(t) will be 
apparent to those skilled in the art of signal processing. If conjugate 
self-coherence is being optimized, the filtered signals are conjugated by 
circuit 410, otherwise circuit 410 merely copies the filtered signals to 
mixer 412 which shifts the frequency of the filtered signals by f.sub.a 
(=.alpha.) to generate r(t) which is inputted to update circuit 404 on bus 
415. Update circuit 404 calculates new values for c and w such that 
self-coherence (or conjugate self-coherence) with a delay of zero is 
maximized. Update circuit 404 can utilize any one of the approaches 
discussed above with reference to Eqs. (9)-(17) with r(t) from Eq. (18) 
instead of the r(t) defined in Eq.(12). 
Phase SCORE Extraction 
As stated above, each of the Least-Squares SCORE algorithm and the 
Cross-SCORE algorithm maximizes the objective function F.sub.cross (Eq. 
6). Furthermore, as stated in reference to Eq. (17), each solution to the 
cross-SCORE eigenvalue equation adapts the processor to receive one of the 
signals that are self-coherent (or conjugate self-coherent) at the chosen 
.alpha.. However, as stated, such adaptation occurs only if the 
self-coherence functions of these signals are distinct when evaluated at 
the chosen control parameters (*), .alpha., and .tau.. This required 
condition does not hold when two or more signals having the same non-zero 
self-coherence at (*), .alpha. and .tau. impinge on the array, regardless 
of the time-difference of arrival (TDOA) between the signals. 
The TDOA affects the complex phase of the complex-valued self-coherence 
function 
##EQU8## 
However, although it is not obvious that F.sub.cross destroys the phase 
information of the Eq. (21), it is possible (but very difficult) to show 
that this is indeed the case. See "Spectral Self-coherence Restoral: A New 
Approach to Blind Adaptive Signal Extraction Using Antenna Arrays", B. G. 
Agee, S. V. Schell, and W. A. Gardner, to appear in Proceedings of the 
IEEE, April 1990. Some insight can be obtained by examining Eq. (15) 
wherein the matrix product R.sub.xr R.sub.rr.sup.-1 R.sub.rx is used to 
find w. Since R.sub.xr and its conjugate transpose R.sub.rx individually 
contain phase information related to Eq. (21), forming their product 
destroys that information in the same way that the product of a complex 
scalar and its conjugate contains no phase information of the original 
scalar. 
An algorithm that is similar to the cross-SCORE algorithm but that 
preserves this phase information, thereby extending applicability, is 
described here and is correspondingly referred to as the phase-SCORE 
algorithm. The phase-SCORE algorithm uses the same correlation matrices as 
the cross-SCORE algorithm but uses them differently. Each solution w to 
the phase-SCORE algorithm is given by a solution of the eigenvalue 
equation 
EQU .lambda.R.sub.xx w=R.sub.xr w (22) 
for which the eigenvalue .lambda. is non-negligible. This eigenequation can 
be obtained by replacing c.sup.(*) with w in Eq. (14). Loosely, the 
phase-SCORE algorithm (22) can be thought of as the cross-SCORE processor, 
Eq. (15), after omitting the product R.sub.rr.sup.-1 R.sub.rx from the 
full product R.sub.xr R.sub.rr.sup.-1 R.sub.rx. Since R.sub.xr contains 
phase information related to Eq. (21), and that information is not 
destroyed by multiplying R.sub.xr by its conjugate transpose, then the 
phase-SCORE algorithm is able to retain the phase information related to 
Eq. (21) and exploit it to advantage. 
If several signals that are sell-coherent at .alpha. are present, then each 
of the non-negligible eigenvalues of Eq. (22) provides useful information. 
Each of the solutions adapts the processor to receive one of these signals 
if the complex-valued self-coherence functions of these signals are 
distinct when evaluated at the chosen control parameters (*), .alpha., and 
.tau.. In practice, this condition is much more likely to hold than the 
corresponding condition for the cross-SCORE solutions. For example, a 
commercial PCM communication signal is usually bandlimited to one 
bandwidth of a small set of bandwidths agreed upon as part of a standard. 
Thus, it is quite possible that two PCM signals having the same baud rate 
also have the same bandwidth and thus have the same value of 
self-coherence. However, it is very unlikely that this condition holds and 
also that the two signals have the same TDOA or relative pulse timing, and 
thus it is very unlikely that two PCM signals have the same complex-valued 
self-coherence. 
An apparatus of the prior embodiment which is similar to the apparatus that 
solves the cross-SCORE eigenvalue Eq. (15) for multiple solutions either 
approximately or exactly solves the phase-SCORE eigenvalue equation using 
estimates of the correlation matrices R.sub.xx and R.sub.xr. 
Just as the cross-SCORE algorithm in the prior embodiment can be 
generalized by obtaining the reference signal vector r(t) as a filtered 
and frequency-shifted version of x(t) according to Eq. (18), so, too, can 
the phase-SCORE algorithm be generalized in an identical manner. Likewise 
this generalization does not affect the mathematical form of the 
phase-SCORE algorithm. Moreover, unlike the case for the cross-SCORE 
algorithm, the actual choice of filter h(t) does not affect the 
computational complexity of the phase-SCORE apparatus, except insofar as 
the filter must process the individual elements of x(t). Therefore, except 
for this last factor, no computational penalty in the phase-SCORE 
algorithm is assessed for using an optimally chosen filter h(t) instead of 
a simple delay. 
In summary, the method of this embodiment referred to as the phase-SCORE 
algorithm comprises the following steps: 
1. Using an analog or digital measurement of the complex data x(t) from a 
set of sensor elements, compute the R.sub.xx matrix using Eq. (9). This 
can be accomplished by updating R.sub.xx after each new measurement of 
x(t). 
2. Form r(t) from x(t) using Eq. (12) or (18), and compute R.sub.xr using 
Eq. (11). 
3. Update w by solving Eq. (22) and form the array output signal using Eq. 
(2). 
The formulas in the prior embodiment apply to the present embodiment. In 
the present embodiment, Eq. (22) replaces Eq. (15) to obtain each solution 
for the linear combining weight w. Thus, all the circuits in the prior 
embodiment can be used in the present embodiment, except that the present 
embodiment needs a circuit to obtain the linear combining weight w by 
solving the Eq. (22). 
FIG. 7A depicts a phase-SCORE processor including a mechanism for 
preserving phase information, according to the present invention. As such, 
the structure of proposed FIG. 7A is similar to the cross-SCORE processor 
embodiment depicted in FIG. 5. The difference between FIG. 5 and FIG. 7A, 
however, is that the R.sub.xx.sup.-1 block 606 shown in FIG. 5 is changed 
to the R.sub.xx block 706 shown in FIG. 7A. Function blocks 706, 707 and 
710 in FIG. 7A perform a preserving phase information function. 
Estimation of DOA 
The phase-SCORE algorithm discussed above performs signal selective signal 
extraction; that is, only those signals that have spectral self-coherence 
at the chosen cycle frequency, .alpha., are considered to be signals of 
interest, and all other signals are ignored or rejected. This signal 
selectivity is also useful for estimation of direction-of-arrival (DOA), 
and circumvents the major limitations in the conventional MUSIC algorithm 
for estimation of DOA, many of which are also shared by almost all other 
existing DOA estimation algorithms. 
The algorithms used in this DOA-estimation embodiment include the 
phase-SCORE algorithm, a closely related alternative algorithm called 
cyclic MUSIC, and a third related algorithm called Cyclic Least Squares. 
Although Cyclic Least Squares does not bear a resemblance to the other two 
algorithms in terms of its algorithmic implementation, its conceptual 
basis is similar. 
The Cyclic MUSIC algorithm performs signal selective DOA estimation, but 
unlike phase-SCORE it does not perform signal extraction. However, it 
possesses a slightly simpler implementation than does the phase-SCORE 
algorithm. 
The Cyclic Least Squares algorithm also performs signal selective DOA 
estimation as well as signal extraction and like the phase-SCORE and 
Cyclic MUSIC algorithms, exploits the spectral self-coherence properties 
of the signals of interest. However, it does not bear any real algorithmic 
resemblance to phase-SCORE or Cyclic MUSIC. However, a heuristic 
interpretation (to be stated later) reveals much similarity between this 
approach and the phase-SCORE and Cyclic MUSIC approaches. 
These inventions may be collectively referred to as the SCORE direction (or 
DOA) estimation inventions because they exploit the spectral 
self-coherence of the desired signals and are closely linked to the 
self-coherence-restoral eigenvalue equations. The SCORE DOA estimation 
algorithms have the following advantages: (1) only those signals having 
spectral self-coherence at the chosen value of the cycle frequency .alpha. 
are selected for DOA estimation by the algorithms; (2) knowledge of the 
noise and interference autocorrelation matrices is not required; (3) an 
arbitrary number of arbitrarily closely spaced interferers can be present; 
(4) the Cyclic Least Squares algorithm can operate effectively in the 
presence of perfectly correlated signals; and (5) the only requirement is 
that the number, d, of signals spectrally self-coherent at the chosen 
.alpha. be less than the number of sensors M. 
The phase-SCORE algorithm for DOA estimation is an extension of the 
phase-SCORE algorithm for signal extraction in the following way. The 
phase-SCORE eigenequation (Eq. 22) is solved for each of the eigenvectors 
w, having negligible eigenvalue .lambda.. That is, the phase-SCORE 
algorithm for DOA estimation uses the solutions not used by the 
phase-SCORE signal extraction algorithm. The corresponding eigenvectors 
form the columns of a matrix E.sub.N, referred to as the generalized null 
space of the matrix pair (R.sub.xr, R.sub.xx) in Eq. (22), 
EQU E.sub.N =[w.sub.d+1 w.sub.d+2 . . . w.sub.M ], (23) 
where the following equation holds for i=1, . . . , M, M being the number 
of sensors, 
EQU .lambda..sub.i R.sub.xx w.sub.i =R.sub.xr w.sub.i, (24) 
where w.sub.i is the ith eigenvector and the eigenvalues .lambda..sub.d+1, 
. . . , .lambda..sub.M are negligibly small compared with .lambda..sub.1, 
. . . ,.lambda..sub.d, d being determined from this partitioning of the 
eigenvalues. The generalized null space E.sub.N is then used to form the 
measure of orthogonality, P.sub.1 (.theta.), 
EQU P.sub.1 (.theta.)=.vertline..vertline.E.sub.N 
a(.theta.).vertline..vertline..sup.-2, (25) 
where () denotes the Hermitian or conjugate transpose operation, .theta. is 
the DOA estimation parameter, and a(.theta.) is the corresponding 
direction vector obtained from the calibration data for the array. The 
measure P(.theta.) is searched for peaks, or local maxima: the number of 
peaks is equal to the number of desired signals, d, and the location of 
each peak is the DOA of a desired signal. 
FIG. 7F depicts a phase-SCORE processor for direction of arrival ("DOA") 
estimation, including a mechanism for generating a measure of 
orthogonality, according to the present invention. The structure shown in 
FIG. 7F is similar to the structure for the phase-SCORE processor shown in 
FIG. 7A, except that FIG. 7B includes additional function blocks 762, 764, 
766 and 768. The function of these four additional blocks is to generate a 
measure of orthogonality. More specifically, function blocks 764, 766 and 
768 are details of function block 714 in FIG. 7B. 
A modified form of the phase-SCORE algorithm for DOA estimation uses both 
the solutions used above and the solutions used by the phase-SCORE signal 
extraction algorithm. The matrix E.sub.N is defined as above, and in 
addition a matrix E.sub.S is defined as containing the eigenvectors having 
non-negligible eigenvalues: 
EQU E.sub.S =[w.sub.1 w.sub.2 . . . w.sub.d ] (26) 
This signal subspace matrix is then used along with the null space matrix 
to form a different orthogonality measure P.sub.2 (.theta.), given by 
##EQU9## 
This new measure P.sub.2 (.theta.) is then searched for peaks as before. 
The new measure P.sub.2 (.theta.) is referred to as the ARMA-like 
orthogonality measure due to its similarity with the power spectral 
density of an autoregressive-moving moving average (ARMA) scalar time 
series. In contrast, the measure P.sub.1 (.theta.) is similar to the power 
spectral density of an autoregressive scalar time series. The new 
orthogonality measure (27) has the advantage that it better suppresses 
strong interference. 
The apparatus of the present invention for solving the eigenvalue Eq. (22) 
is identical to the apparatus used in the phase-SCORE signal extraction 
algorithm, except that eigenvectors corresponding to only the negligible 
eigenvalues are needed if the first measure P.sub.1 (.theta.) is used. 
Just as cross-SCORE and phase-SCORE can be generalized by obtaining the 
reference signal vector r(t) as a filtered and frequency-shifted version 
of x(t) according to Eq. (18), so too can the phase-SCORE DOA estimation 
algorithm be generalized in an identical manner without affecting the 
mathematical form. 
In summary, the method of the present invention referred to as the 
phase-SCORE DOA estimation algorithm is composed of the following steps: 
1. Using an analog or digital measurement of the complex data x(t) from a 
set of sensor elements, compute the R.sub.xx matrix using Eq. (9). This 
can be accomplished by updating R.sub.xx after each new measurement of 
x(t). 
2. Form r(t) from x(t) using Eq. (12) or (18), and compute R.sub.xr using 
Eq. (11). 
3. Solve Eq. (22) for all eigenvectors if using P.sub.2 (.theta.) or for 
the eigenvectors of the negligible eigenvalues if using P.sub.1 (.theta.); 
then form both matrices E.sub.S and E.sub.N, or form only E.sub.N, 
respectively. 
4. Search over all possible .theta. for the peaks in the function P.sub.1 
(.theta.) in Eq. (25) or in the function P.sub.2 (.theta.) in Eq. (27); 
the resulting peak locations are the DOA estimates. 
FIG. 7B depicts a phase-SCORE processor for direction of arrival ("DOA") 
estimation, including a mechanism for locating peaks, according to the 
present invention. The structure of this phase-SCORE processor for DOA 
estimation is similar to the structure of the phase-SCORE processor shown 
in FIG. 7A, except that FIG. 7B includes additional function blocks 712, 
714 and 716. These additional function blocks perform a peak locating 
function. 
FIG. 7C depicts a modified phase-SCORE processor for direction of arrival 
("DOA") estimation, including a mechanism for locating peaks, according to 
the present invention. The structure for the modified phase-SCORE 
processor for DOA estimation is similar to the structure for the 
phase-SCORE processor for DOA estimation depicted in FIG. 7B, except that 
FIG. 7C includes an additional function block 738. Function blocks 732, 
734, 736 and 738 perform a peak locating function. 
The alternative algorithm, Cyclic MUSIC, is obtained by simply deleting the 
matrix R.sub.xx in Eqs. (22) and (24) and eliminating step 1 in the 
four-step procedure, yielding the following three-step procedure: 
1. Form r(t) from x(t) using Eqs. (12) or (18), and compute R.sub.xr using 
Eq. (11). 
2. Solve the following equation for eigenvectors corresponding to the 
negligible eigenvalues, 
EQU .lambda.w=R.sub.xr w, (28) 
and form the matrix E.sub.N as in Eq. (23). 
3. Search over all possible .theta. for the peaks in the function P.sub.1 
(.theta.) in equation (25); the resulting peak locations are the DOA 
estimates. 
FIG. 7D depicts a Cyclic MUSIC processor for direction of arrival ("DOA") 
estimation, including a mechanism for locating peaks, according to the 
present invention. The structure for the Cyclic MUSIC processor for DOA 
estimation is similar to the structure for the phase-SCORE processor for 
DOA estimation as shown in FIG. 7B, except that in FIG. 7D the output from 
block 706 (R.sub.xx) is disconnected from block 710 (update w). Function 
blocks 742, 744 and 746 perform a peak locating function. 
The phase-SCORE and Cyclic MUSIC direction estimation inventions use the 
fact that the null space of the matrix pair {R.sub.xr, R.sub.xx } or of 
the matrix R.sub.xr, respectively, becomes orthogonal to the direction 
vector of each spectrally self-coherent signal after an infinite number of 
data samples has been collected. An alternative approach to exploiting the 
spectral self-coherence properties of the signals of interest is used in 
the Cyclic Least Squares (CLS) algorithm. 
The CLS algorithm finds the direction vector estimates that are closest to 
the actual direction vectors of the spectrally self-coherent signals of 
interest, where closeness is measured as follows. Given a set of possible 
direction vector estimates, an estimate of the spectrally self-coherent 
components of the measured data that are impinging on the array from those 
directions is formed. The estimates, which are linear combinations of 
frequency-shifted (by .alpha.) data, are subtracted from the measured 
data. The energy in the resultant residual is taken to be the squared 
distance between the direction vector estimates and the actual direction 
vectors. If the signal estimates are accurate, then the residual energy 
will be very small when evaluated using the correct direction vectors. 
Thus, searching for the set of direction vectors that minimizes this 
distance should yield accurate estimates of the actual direction vectors 
and, thus, of the actual directions of arrival of the spectrally self 
coherent signals. 
Note that the phase-SCORE and Cyclic MUSIC algorithms can also be thought 
of as minimizing the distance between the estimated and actual direction 
vectors, but their respective distance measures are different from the one 
used in CLS. 
Mathematically, the CLS algorithm can be expressed as 
##EQU10## 
where d is the number (assumed to be known) of transmitted signals being 
received by the processor (unknown d is considered later) and where the 
estimate of the spectrally self-coherent signal vector is given by the 
d.times.1 vector 
EQU s(t)=Fr(t), (30) 
where r(t) is the reference signal vector (12) obtained by frequency 
shifting the received data x(t). The final processor output signal y(t) is 
obtained by computing the vector product of Fx(t). Minimizing in Eq. (29) 
with respect to the d x M matrix F yields, 
EQU F=(A A).sup.-1 AR.sub.xr R.sub.rr.sup.-1, (31) 
which, when substituted back into Eq. (30), yields the CLS algorithm 
##EQU11## 
where tr{.} is the trace operator and where R.sub..alpha. is given by 
EQU R.sub..alpha. =R.sub.xr R.sub.rr.sup.-1 R.sub.xr, (33) 
the projection matrix for the space spanned by the direction vectors is 
given by 
EQU P.sub.A =A(A A).sup.-1 A, (34) 
a given set of DOA estimates, A=[a(.theta..sub.1) . . . a(.theta..sub.d)], 
where a(.theta.) is defined as in Eq. (25). 
Once the estimates of the directions of arrival .theta..sub.1, 
.theta..sub.2, . . . , .theta..sub.d have been found from Eq. (32), they 
can be substituted into Eq. (31) to perform signal extraction, which can 
be substituted into Eq. (30). 
In the case where d is unknown, it can be estimated as follows. Assume d=1 
and solve Eq. (32), obtaining the maximum value of tr{P.sub.A 
R.sub..alpha.) and referring to this value as CLS(1). Repeat this 
procedure for d=2, d=3, and so on, saving the maximum values CLS(2), 
CLS(3), and so on, until the condition CLS(d)=CLS(d+1) is satisfied 
(either exactly or within some allowable error tolerance). This resulting 
value of d should then be taken to be the number of transmitted desired 
signals being received by the processor, and thus is the number of DOA 
estimates to find. Note that these DOA estimates are found automatically 
in the process of determining d (by virtue of having to solve Eq. (32)). 
A slightly modified version of the above CLS algorithm, which requires 
slightly less computation, has been shown to provide accurate DOA 
estimates, but has not been shown to be the solution to an optimization 
problem. This modified version is simply the CLS algorithm with the matrix 
R.sub..alpha. replaced by R.sub.xr. This is the same modification that is 
made to obtain the phase-SCORE eigenequation from the cross-SCORE 
eigenequation. 
Just as the other SCORE algorithms can be generalized by obtaining the 
reference signal vector r(t) as a filtered as well as frequency-shifted 
version of x(t), so too can the CLS algorithm be generalized in an 
identical manner without affecting the mathematical form. 
In summary, the method of the present invention referred to as the Cyclic 
Least Squares algorithm is composed of the following steps: 
1. Form r(t) from x(t) using Eq. (12) or (18), and compute R.sub.rr and 
R.sub.xr using Eqs. (10) and (11). 
2. Form the matrix R.sub..alpha. using Eq. (33). 
3. Perform a multidimensional search over .theta..sub.1, . . . , 
.theta..sub.d for the maximum of Eq. (32), recomputing the projection 
matrix P.sub.A using Eq. (34) as needed. The resulting .theta..sub.1, . . 
. .theta..sub.d are the DOA estimates. 
4. If signal extraction is desired, use the DOA estimates from step 3 in 
(31), and then use (31) to form the signal estimates as in (30). 
For the special case of setting the cycle frequency parameter .alpha. equal 
to zero, both CLS algorithms reduce to an existing algorithm in the 
literature known as Maximum Likelihood by Alternating Projections. The 
paper is titled, "Maximum Likelihood Localization of Multiple Sources by 
Alternating Projection," by I. Ziskind and M. Wax in IEEE Transaction on 
Acoustics, Speech, and Signal Processing, volume 36, number 10, October 
1988, pp. 1553-1560. 
FIG. 7E depicts a cyclic least squares ("CLS") processor for direction of 
arrival ("DOA") estimation, including a mechanism for locating peaks, 
according to the present invention. Function blocks 754 and 756 perform a 
peak locating function. 
The same alternating projections technique used in that paper can be 
applied to the CLS algorithms to obtain computationally efficient 
implementations. 
Extraction and Estimation of DOA for Wideband 
The above embodiments address the problems of blind adaptive spatial 
filtering for signal extraction and signal-selective DOA estimation. 
However, those embodiments are discussed only in the context of narrowband 
signals. In the field of antenna array signal processing, a received 
signal is considered to be narrowband if its bandwidth is less than the 
reciprocal of the time required for the signal to propagate across the 
array. When this condition is violated, the array response vector is no 
longer constant across the received signal bandwidth, and thus the 
received signal model upon which the aforementioned algorithms are based 
becomes invalid. For example, signals having bandwidth equal to a 
significant fraction of their center frequency typically violate the 
narrowband assumption. Here these algorithms are extended to accommodate 
wideband signals. 
Wideband Signal Extraction 
The narrowband signal model can be represented as 
##EQU12## 
where .theta..sub.1 is the DOA of the lth signal, and s.sub.1 (t) and i(t) 
are the interference and noise, is not valid for wideband signals because 
the array response vector a(.theta..sub.1) is no longer constant over the 
bandwidth of s.sub.1 (t). A more appropriate model can be expressed as 
##EQU13## 
where a(.theta.,t) is the impulse responses of the array to a signal from 
direction .theta. and denotes convolution. For signals that are 
Fourier-transformable, this model can be reexpressed as 
##EQU14## 
where f is the frequency variable and the upper case letters denote the 
Fourier transforms of their lower case counterparts. 
Since the corresponding narrowband algorithms perform spatial filtering 
using a linear combiner, 
EQU S.sub.1 (t)=w x(t), (38) 
but do not perform spectral filtering, then it is difficult for a weight 
vector w to reject wideband interference and thus extract the desired 
signal(s). 
Three different methods for solving this problem are presented here. The 
first method is actually the application of the narrowband SCORE 
inventions to the wideband data without modification; useful signal 
extraction can still be achieved. The second method, Dual-Band SCORE, 
extracts the desired signal(s) from each narrow frequency band by 
exploiting the spectral coherence between the signal components in that 
band and in another band; these narrowband signals are then recombined to 
form an estimate of the wideband signal. The third method, Wideband SCORE, 
is a full extension of the SCORE technique to wideband signals that 
employs tapped delay lines on each sensor to perform joint spatial and 
spectral filtering. 
A fourth method, Spatial ALC, that does not exploit spectral coherence and 
is not applicable to the aforementioned wideband received signal model is 
presented. However, if the received data satisfies the narrowband 
assumption yet the desired signals have substantially greater bandwidth 
than the interference, then this method can reject the interference and 
separate the signals of interest. The method uses the basic structure of 
the cross-SCORE processor with an adaptive line canceller in place of the 
frequency-shift device. 
Application of SCORE to Wideband Signals 
As mentioned in the above, spatial filtering alone is often insufficient to 
satisfactorily reject wideband interference. However, in the special case 
of narrowband interference, the cross-SCORE and phase-SCORE processors can 
reject the interference despite the wideband nature of the desired 
signal(s). That this behavior will indeed occur can be seen by considering 
the fact that each narrowband interfering signal does satisfy the 
narrowband assumption so its contribution to the received signal 
autocorrelation and cyclic autocorrelation matrices has rank equal to one. 
Thus, any weight vector that is nearly or completely orthogonal to the 
array response vectors of the interference attenuates it or rejects it 
completely. However, if the desired signal(s) are wideband, then each 
processor output will be a linear combination of distorted (due to the 
frequency dependence of the array response vector) versions of the desired 
signals. If more than one desired signal is present, then since no 
memoryless weight vector can reject a wideband signal, each processor 
output will contain all wideband signals of interest. Nonetheless, since 
the interference has been screened out, a post-processor employing 
spectral filtering can equalize the distortion and separate the desired 
signal from each other if they are in disjoint spectral bands. 
Dual-Band SCORE 
Unlike the application of the narrowband SCORE algorithm to wideband 
signals, the method presented here, Dual-Band SCORE, explicitly recognizes 
the wideband nature of the received signals and is structured accordingly. 
In this method, the wideband data is decomposed into disjoint frequency 
bands having sufficiently small bandwidth such that the narrowband 
assumption is valid for each band. Each band can then be processed by one 
(or more, if more than one desired signal is present) weight vector to 
remove interference and separate the desired signals. The individual 
narrowband components of any given desired signal can then be recombined 
and equalized to yield a satisfactory estimate of the wideband desired 
signal. Thus this approach can be interpreted as frequency decomposition 
followed by narrowband spatial processing followed by frequency 
recomposition. 
The weight vectors are chosen by the cross-SCORE or phase-SCORE algorithm, 
in which r(t) is obtained from a different band than that from which x(t) 
is obtained. That is, the two signals to be cross correlated for any 
particular band of interest are the received signal for the band of 
interest and the received signal for a different band separated from the 
band of interest by a cycle frequency .alpha.. Because the desired signal 
components of these two bands are correlated and the interference 
components are not, only the desired signal components will remain in the 
cross correlation matrix after sufficient averaging time. Furthermore, 
since the narrowband assumption is satisfied, the cross-SCORE or 
phase-SCORE algorithms can reject the interference and separate the 
desired signals, provided the usual assumptions (discussed above) are 
satisfied. Although the reconstructed wideband estimate of a desired 
signal may be distorted, post-processing techniques such as modulus 
restoral or decision-direction may be used to correct this by finding the 
appropriate spectral equalization filter. 
Specifically, consider the wideband signal x(t) being split into two 
frequency bands x.sub.1 (t) and x.sub.2 (t) where the center frequencies 
of the two bands are separated by a cycle frequency .alpha.. Then a weight 
vector w(1) to be applied to x.sub.1 (t) can be found by applying the 
cross-SCORE or phase-SCORE algorithm to x.sub.1 (t) (instead of to x(t)) 
and replacing r(t) by x.sub.2 (t) (instead of obtaining r(t) from x.sub.1 
(t)). Similarly, a weight vector w(2) to be applied to x.sub.2 (t) can be 
found by applying the cross-SCORE o r phase-SCORE algorithm to x.sub.2 (t) 
and replacing r(t) by x.sub.1 (t). Finally, the extracted signals from the 
two bands are recombined to form the final estimate of the desired signal, 
EQU y(t)=w (1)x.sub.1 (t)+w (2)x.sub.2 (t). 
FIGS. 8A and 8B depict a dual-band SCORE processor for signal extraction, 
according to the present invention. Function blocks 802 and 844 decompose 
wideband signal x(t) into two frequency bands x.sub.1 (t) and x.sub.2 (t). 
Function blocks 806 generates scalar products w(1) and x.sub.1 (t), and 
function block 808 generates scalar product of w(2) and x.sub.2 (t). As 
described above, weight vectors w(1) and w(2) can be found using either 
the cross-SCORE or phase-SCORE algorithm. Consequently, weight vectors 
w(1) and w(2) can be obtained by using either the cross-SCORE processor 
shown in FIG. 5, or the phase-SCORE processor shown in FIG. 7A. 
Wideband SCORE 
Another method that explicitly recognizes the wideband nature of the 
desired signals is the Wideband SCORE algorithm. Unlike the Dual-Band 
SCORE which explicitly decomposes the wideband data into narrow bands that 
can be individually spatially filtered and then recombined, the Wideband 
SCORE algorithm (effectively) finds the spatial and spectral filters that 
jointly restore the spectral coherence of the processor output signal. 
This is accomplished by adding a tapped delay line of length K and tap 
spacing to each antenna sensor. A composite received signal vector x(t) 
can then be defined as the stack of the delay line outputs for the 
sensors, 
EQU x(t)=[x(t),x(t-.tau..sub.o), . . . x(t-(K-1)].sup.T, (39 ) 
where [.].sup.T denotes transpose. 
This extended vector is then processed by cross-SCORE or phase-SCORE in the 
usual way, effectively performing joint spatial and temporal processing to 
restore spectral coherence. In so doing, it can apply spectral and/or 
spatial filtering to reject narrowband and wideband interference and 
separate desired signal components. However, the distortion introduced by 
the spectral filtering would need to be equalized as discussed in regard 
to Dual-Band SCORE. 
FIG. 9 depicts a block diagram of a wideband SCORE processor for signal 
extraction, according to the present invention. Function blocks 936, 934 
and 932 delay received signal x(t) and generate (k-1) delayed signals. 
Function block 938 generates composite received signal vector based on the 
received signal and (k-1) delayed signals. Function block 942 is a 
cross-SCORE processor that extracts a desired signal, e.g. the processor 
shown in FIG. 6. 
Spatial ALC 
Unlike the other algorithms considered above, the one discussed here does 
not exploit spectral coherence nor is it considered for the case of 
wideband signals. However, it does exploit the novel structure of the 
cross-SCORE processor by replacing the frequency-shift operation in the 
reference path with an adaptive line canceller (ALC). Thus, oscillators 
214, 314 and 414 and multipliers 211, 311 and 411 of FIGS. 2, 4 and 6 may 
be replaced with respective ALCs. Furthermore, in FIGS. 4 and 6, only one 
element of the output vector of the optional complex conjugations 310 and 
410, respectively, is processed by the ALC. The spectral filter found by 
the ALC is then applied individually and separately to the remaining 
elements of the output vector of blocks 310 and 410. The resulting vector 
composed of the ALC output and the filtered versions of the remaining 
elements is the reference signal vector r(t). 
The ALC is a familiar algorithm to practitioners skilled in the art of 
signal processing. See, e.g., Adaptive Signal Processing by Widrow and 
Stearns, Prentice-Hall, Englewood Cliffs, N.J. 1985. The ALC operates on 
the data vector x(t). The ALC uses a spectral filter to reject 
interference having bandwidth substantially less than the bandwidth of the 
desired signal(s). Thus, the output of the ALC followed by the linear 
combiner with weight vector c is a distorted version of the received data 
less the narrowband interference. If all interferers present are 
sufficiently narrowband that they are cancelled by the ALC, then the ALC 
output will consist of only residual noise and components that are 
correlated with the desired relatively wideband signals. Thus, the output 
of the ALC and linear combiner serves the same purpose as the cross-SCORE 
reference path derived from frequency-shifted and beamformed data. That 
is, the cross-SCORE processor with the frequency-shift device following 
the linear combiner replaced by an ALC preceding the linear combiner will 
adapt the weight vector w so as to reject the narrowband interference by 
spatial filtering and pass the desired signals (which are not distorted by 
spectral filtering). 
One example in which this method would be applicable is described here. In 
the 100-1000 MHz band, desired signals having bandwidths less than 10% of 
their respective center frequencies (e.g., 10-100 MHz) approximately 
satisfy the narrowband assumption. Also present in that band are many 
commercial television signals which contain relatively powerful spectral 
lines (due to the horizontal synchronization component of the signal). The 
Spatial ALC can reject the TV interference and separate the desired 
signals. 
Wideband Direction Estimation 
As mentioned in the above, the narrowband signal model does not hold when 
wideband signals are received. In particular, the cyclic autocorrelation 
matrix can have rank greater than the number of signals exhibiting 
spectral coherence at the cycle frequency of interest. For example, in the 
Cyclic MUSIC algorithms, the corresponding reduction in the dimension of 
the null space of that matrix or elimination of the null space can prevent 
the algorithms from operating properly. 
FIG. 10 depicts a block diagram of a spatial adaptive line canceler ("ALC") 
processor for signal extraction, according to the present invention. 
The structure of FIG. 10 is similar to that of FIG. 6. Specifically, 
function blocks 902, 904, 906 and 910 in FIG. 10 correspond respectively 
to function blocks 402, 404, 406 and 410 in FIG. 6. Function blocks 914, 
916, 922 and 924 generate the reference vector r(t) for the cross-SCORE 
processor. 
Two methods for accommodating wideband signals are considered here. The 
first method, Dual-Band Cyclic MUSIC, finds the null space of a cross 
correlation matrix in which the two signals being correlated come from two 
disjoint frequency bands separated by the desired cycle frequency and 
having bandwidth sufficiently small that the narrowband signal model 
applies. The usual spatial spectrum .g., from Cyclic MUSIC) is then 
searched for peaks that correspond to the desired DOAs. The second method, 
Wideband Cyclic MUSIC, is a cyclic version of the conventional wideband 
MUSIC algorithm (see. M. Wax, et. al., "Spatio-Temporal Spectral Analysis 
by Eigenstructure Methods," IEEE Trans. on Acoustics, Speech, and Signal 
Processing, vol. ASSP-32, no. 4, August 1984, pp. 817-827. Using 
correlation matrices for multiple pairs of bands, Wideband Cyclic MUSIC 
simultaneously searches both in the DOA domain and in the frequency domain 
for the desired signal(s). 
Dual-Band Cyclic MUSIC 
Dual-Band Cyclic MUSIC simply cross correlates the signals from two narrow 
bands separated by the desired cycle frequency. The two bands must be 
sufficiently narrow that the narrowband assumption is valid. For example, 
the complex envelopes of the received data x(t)=x.sub.1 (t-.tau.) and 
r(t)=x.sub.2 (t-.tau.) from two bands separated by the desired cycle 
frequency .alpha. consists of desired signal components that are 
correlated and undesired interference and noise that are not: 
##EQU15## 
where f.sub.i is the center frequency of the ith frequency band, s.sub.1 
(f.sub.i,t) is the lth desired signal component from that band, s.sub.1 
(f.sub.1,t) and s.sub.1 (f.sub.2,t) are correlated, A(f.sub.i,.THETA.) is 
a matrix having a(f.sub.i,.theta..sub.1), 1-1, . . . , L as its columns, 
and i.sub.1 (t) and i.sub.2 (t) are uncorrelated. Thus, the cross 
correlation matrix with lag .tau. Rx.sub.1 x.sub.2 (.tau.)=R.sub.xr is 
given by (in the limit as the averaging time goes to infinity), 
EQU R.sub.xr =R.sub.x1x2 (.tau.)=A(f.sub.1,.THETA.)R.sub.s1s2 (.tau.)A 
(f.sub.2,.THETA.). (42) 
The right null space of this matrix is orthogonal to the array response 
vectors corresponding to the second of the two narrow bands. That is, the 
right null space of R.sub.xr is orthogonal to the columns of 
A(f.sub.2,.THETA.). After computing the right null space using Eq. (28) 
and forming the spatial spectrum using A(f.sub.2,.theta.), the usual 
search for peaks in the spatial spectrum is performed. That is, for Dual 
Band Cyclic MUSIC R.sub.x1x2 in Eq. (42) replaces R.sub.xr in Eq. (28) for 
Cyclic MUSIC. 
In addition to the advantages due to signal selectivity shared by the 
narrowband versions of Cyclic MUSIC, this algorithm exploits the spectral 
coherence of the desired signals in a wideband received-signal environment 
but requires calibration data for only one narrow frequency band. 
FIG. 11 is a block diagram of a Dual-Band Cyclic MUSIC processor for 
direction of arrival ("DOA") estimation, according to the present 
invention. In fact, the structure of function blocks is similar to that in 
FIG. 8A. Function blocks 952 and 954 decompose wideband signal x(t) into 
two frequency bands x.sub.1 (t) and x.sub.2 (t). Block 956 performs the 
function of estimating direction of arrival. Function block 956 (the 
Cyclic MUSIC processor) is shown in detail in FIG. 7D, except that x.sub.1 
(t) of FIG. 11 replaces x(t) in FIG. 7D, and x.sub.2 (t) of FIG. 11 
replaces r(t) in FIG. 7D. 
Wideband Cyclic MUSIC 
Wideband Cyclic MUSIC is essentially a cyclic version of the conventional 
Wideband MUSIC algorithm. However, the two algorithms differ significantly 
in their performance and applicability. Whereas the conventional Wideband 
MUSIC algorithm processes each frequency band independently of every other 
band, the Wideband Cyclic MUSIC algorithm exploits the correlation between 
the desired signal components in different bands. Furthermore, 
conventional Wideband MUSIC suffers from the constraint that the number of 
signals and interferers present in any frequency band must be less than 
the number of sensors, whereas Wideband Cyclic MUSIC requires only that 
the number of desired signals in any band be less than the number of 
sensors. All other advantages of Cyclic MUSIC algorithms over conventional 
algorithms also apply. 
Decompose the wideband received data into K+k narrow bands, where x.sub.i 
(t) is the received data in band i, i=1, . . . , K+k, and where the band 
centers f.sub.i and f.sub.i+k are separated by the desired cycle frequency 
.alpha., for i=1, . . . ,K. Compute K cross-correlation matrices 
R.sub.x.sbsb.i+k.sub.x.sbsb.i for i=1, . . . , K where 
EQU R.sub.x.sbsb.i+k.sub.x.sbsb.i =&lt;x.sub.i+k (t)x.sub.i (t)&gt;. (43) 
The right null space E.sub.N (i) of the ith cross-correlation matrix (in 
the limit as averaging time goes to infinity) is orthogonal to the array 
response vectors a(f.sub.i,.theta.) for the desired signals in the ith 
band, where 
EQU E.sub.N (i),=[w.sub.d+1 (i) . . . w.sub.m (i)] (44) 
and w.sub.m (i), m=d+1, . . . , M are the eigenvectors of 
R.sub.x.sbsb.i+k.sub.x.sbsb.i corresponding to negligible eigenvalues 
.lambda..sub.m (i), m=d+1, . . . , M, where the eigenvectors and 
eigenvalues are obtained by solving 
EQU .lambda..sub.m (i)w.sub.m (i)=R.sub.x.sbsb.i+k.sub.x.sbsb.i w.sub.m (i)(45) 
for i=1, . . . , K and m=1, . . . , M, and d is determined by the 
partitioning of the eigenvalues. Thus, the spatial spectra as a function 
of the angle .theta. and the frequency band i having center frequency 
f.sub.i is given by 
EQU P(.theta.,i)=.vertline..vertline.[E.sub.N 
(i)]a(f.sub.i,.theta.).vertline..vertline..sup.-2. (46) 
The locations along the G-axis of the peaks in P(.theta.,i) are the desired 
DOA estimates. 
Notice that Eqs. (40) and (41) in Dual-Band Cyclic MUSIC generalize easily 
to describe x.sub.i (t) for i=1, . . . ,K+k and that Eq. (42) generalizes 
easily to the case of R.sub.x.sbsb.i+k.sub.x.sbsb.i. 
FIG. 12 depicts a block diagram of a wideband Cyclic MUSIC processor for 
direction of arrival ("DOA") estimation, according to the present 
invention. Function blocks 961-963 decompose the wideband received data 
into K+k narrow bands. Function block 964 generates K pairs of signals, as 
indicated by equation (43). Function blocks 965-967 are Cyclic MUSIC 
processors. Function blocks 965-967 are shown in detail in FIG. 7D. As 
shown by FIG. 12, there are i cyclic MUSIC processors, where 
1.ltoreq.i.ltoreq.K. The processor for band i obtains E.sub.N (i) by 
solving equation (45) as set forth in the Specification at page 57, where 
E.sub.N (i)=[W.sub.d+1 (i) . . . W.sub.m (i)] as stated by equation (44). 
This value of E.sub.N (i) is then used in equation (46) to locate peaks 
representing desired DOA estimates (see Specification, page 58, lines 
1-3). Note that each cyclic MUSIC processor in FIG. 12 is identically 
represented by FIG. 7D, if the x(t) and r(t) inputs in FIG. 7D are 
replaced by x.sub.i+k (t) and by x.sub.i (t) respectively. 
Accordingly, there has been described herein novel signal processing 
systems. Various modifications to the present invention will become 
apparent to those skilled in the art from the foregoing description and 
accompanying drawings. Accordingly, the present invention is to be limited 
solely by the scope of the following claims.