Discrete source location by adaptive antenna techniques

A signal processor for radar systems having phased array antenna structures. The processor determines the angle of arrival of the signal from a discrete energy source. The separate elements or output ports of the antenna structure are sampled to form a matrix of signal values. These values are converted into polynomial expressions consistent with one of the spectral estimating methods of maximum entropy, maximum likelihood, and thermal noise. The roots of the polynomial expressions are found and compared to the distance of the unit circle in a pole-zero diagram. Roots sufficiently close to the unit circle are selected as representative of the source signal peaks and the angular directions to such sources are determined by the corresponding angular location of the selected roots in the pole-zero diagram.

BACKGROUND OF THE INVENTION 
This invention relates, in general, to radar systems and, more 
specifically, to apparatus and methods for processing radar signals to 
determine the direction to the source of the signal energy. 
With phased array antenna systems having many separate antenna elements or 
output ports, a large amount of signal information can be acquired for 
processing by various methods. Generally, the phase and amplitude of each 
separate antenna element or output port at different instants of time are 
used by the prior art processing techniques for obtaining the desired 
information. The desired information may be the presence of and direction 
toward two or more signal sources located within the normal bandwidth of 
the main lobe of radiation of the antenna. These sources may be targets or 
jammers, or a combination of the two. Discriminating between the sources 
in regions within the beamwidth of the antenna is known as "super 
resolution" and requires sophisticated signal processing to see signal 
sources located so close together. Basically, the process determines the 
angle or direction from the receiving antenna to the signal sources. 
Several processing methods have been used for giving super resolution 
capabilities to radar signal processors. These methods include the maximum 
entropy method (MEM), the maximum likelihood method (MLM), and the thermal 
noise method (TNM). In each case, there is the requirement that matrices 
formed from the antenna output signal components, including the in-phase 
(I) and quadrature (Q) components, be mathematically manipulated to 
produce the signal source direction. Since the size of such matrices is 
proportional to the number of antenna elements or output ports employed, 
the number of values which must be mathematically manipulated becomes very 
large and the processing speed of the processor is required to be high in 
order to achieve the results in the desired time periods. Consequently, it 
is desirable, and it is an object of this invention, to provide apparatus 
and methods for determining the location of signal sources without having 
to perform all of the mathematical steps on the signal matrices. 
SUMMARY OF THE INVENTION 
There are disclosed herein new and useful signal processing apparatus and 
methods for determining the angular direction to a source of signal energy 
received by a phased array radar antenna. The signal processing techniques 
may be used with the spectral estimating methods of maximum entropy, 
maximum likelihood, and thermal noise. 
With each of these methods, the separate elements or output ports of the 
phased array antenna are sampled to obtain values of the amplitude and 
phase of the signal at the antenna. These values are used in polynomial 
expressions which are characteristic of the specific estimating method 
being used. The roots of the expressions are found and compared to the 
location of the unit circle in the pole-zero diagram of the solution. 
Roots which are represented by zeros within a predetermined distance of 
the unit circle are considered representative of the source signals. The 
angular direction to the sources are determined by the corresponding 
angular direction of the selected roots in the pole-zero diagram. In the 
spectral estimating methods which produce more than one pole-zero diagram, 
a common arrangement or placement of the corresponding roots is also used 
in the criteria to select the roots which are representative of the source 
signal. 
The novel signal processor disclosed herein provides the approximate 
direction to the signal source without all of the computation required by 
prior art systems. The techniques disclosed give approximate angular 
directions to the sources but, nevertheless, they are of sufficiently high 
accuracy in actual practice to be very useful.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Throughout the following description, similar reference characters refer to 
similar elements or members in all of the Figures of the drawing. 
Referring now to the drawing, and to FIG. 1 in particular, there is shown a 
block diagram of a radar system constructed according to this invention. 
The phased array antenna 10 includes separate antenna elements or output 
ports, such as the antenna elements 12, 14, 16 and 18. Such antenna 
elements could be distinct individual radiating elements or individual 
radiating elements in the form of overlapped subarrays. The signal sampler 
20 obtains the phase and amplitude values of the signals captured by the 
antenna elements and transfers that data to the next stage of the system. 
The data at this point in the processing is converted into the form of a 
covariance matrix, with the elements of the matrix corresponding to the 
values of the signal samples. It is emphasized that the separate antenna 
elements could be effectively provided by a common antenna structure 
having multiple ports for connection thereto, and that the invention is 
applicable to phased array antennas with either type or with similar 
functioning antenna elements. 
The matrix elements are constructed into polynomial expressions according 
to block 22, with the exact form of the expressions being dependent upon 
the spectral estimating method being used, as more fully explained later 
herein. Regardless of the form of the polynomial expressions, the system 
of this invention finds the roots of the expressions, as shown in block 
24. Next, the location of the roots, or zeros, with respect to the unit 
circle of a Z-transform, pole-zero diagram is accomplished by the root 
locator 26. Depending upon the criteria used, the roots which satisfy the 
criteria are selected by the root selector 28 and the angular direction to 
each selected root on the pole-zero diagram is measured or calculated. 
Because of the direct correlation between direction to the root on the 
pole-zero diagram and the physical direction to the signal source sampled 
by the phased array antenna elements, the angular direction converter 30 
changes the root location to direction data 32 for further use by the 
radar system and/or operator. 
Although described thus far in terms of manipulating matrix values into 
polynomial expressions, finding the roots to the expressions, and 
selecting roots which meet previously established selection criteria, it 
is emphasized that these functions may be performed by programs in a 
general purpose computer system, or other digital or analog processing 
techniques. Actual plotting of the roots on a pole-zero diagram is not the 
preferred way of implementing this invention. Rather, computer processing 
of the signal values to perform these functions constitutes the preferred 
embodiment of the invention, and the specific programming steps to provide 
these individual functions would be obvious to those skilled in the art. 
The location of the roots near the unit circle corresponds to the angular 
location of signal sources causing electromagnetic energy to be directed 
to the antenna elements. In the case of passive targets, this energy is 
reflected signals from the radar transmitter and, in the case of radar 
jammers, this energy is usually emanating from the jamming source. The 
mathematical analysis of this invention, included later herein, explains 
in more detail the theory of the apparatus and methods taught by this 
invention. It is emphasized that the description of the embodiment herein 
is limited to a one-dimensional discussion of the determination of the 
source direction. In actual implementation, two-dimensional direction 
determination would be used wherein the teachings of this invention would 
be used for each dimension without departing from the scope of the 
invention. 
FIG. 2 is a Z-transform, pole-zero diagram illustrating the root selection 
process according to an embodiment of this invention when all of the roots 
are contained in one pole-zero diagram. Axis 34 is the real axis and axis 
36 is the imaginary axis. The unit circle 38 is between limit circles 40 
and 42 which indicate the distance from the unit circle 38 within which a 
root is considered as being representative of a zero or null of the 
transform functions. Typical roots are shown as zeros 44, 46 and 48 in 
FIG. 2. Zero 44 is outside the circle 42 and zero 46 is inside the circle 
40. Therefore, both of these zeroes are further from the unit circle than 
the deviation distance D. Thus, neither are considered as representing 
directions to sources of signal energy. However, zero 48 is sufficiently 
close to the unit circle 38 and is considered as representing a signal 
source. The direction to the signal source can be determined by use of the 
angle .theta. made with the real axis 34. 
Some of the polynomial expressions produce more than one set of roots 
associated with the polynomial expressions. FIGS. 3A, 3B and 3C illustrate 
such a case where the roots appear in three pole-zero diagrams. These 
roots are represented by zeros 50 through 66. In determining if the zeros 
represent signal sources, a two-step criterion is used. In order to be a 
zero representative of a signal source, the zero must be both close to the 
unit circle and similarly or commonly located within each of the pole-zero 
diagrams. Zeros 50, 56 and 62 meet these requirements. Zeros 52, 54, 58, 
60, 64 and 66 do not meet these requirements. As shown in FIG. 3A, zeros 
50, 56' and 62' are all close to the unit circle 68 and are commonly 
located with respect to angular position. The primed zeros in FIG. 3A 
correspond to the location of the same unprimed zeros in FIGS. 3B and 3C. 
Thus, they satisfy the closeness and commonality criteria and are 
considered to be indicative of the direction of the signal source. 
The invention disclosed herein may be better understood by a mathematical 
analysis of three methods with which the invention may be used. FIG. 4 is 
a block diagram illustrating the use of this invention with the maximum 
entropy method (MEM). With N+1 beams of the antenna 70 having uniformly 
spaced phase centers, the spatial spectrum function is: 
##EQU1## 
where w is a vector of complex weights, s is a steering vector with 
elements of the form: 
EQU s.sub.k =e.sup.-jk.phi. (2A) 
and .phi. is proportional to sin .theta., where .theta. is the spatial 
angle. 
For the thermal noise method shown in FIG. 5, the spatial spectrum function 
is: 
##EQU2## 
where the weight vector w is: 
EQU w=M.sup.-1 s* (4A) 
w.sup.+ is the conjugate transpose of w, and M is the covariance matrix (or 
sample covariance matrix) associated with the source distribution. For an 
N - beam system as illustrated by antenna 72, M is an N.times.N matrix and 
the index, k, of s.sub.k runs from zero to (N-1). 
For the maximum likelihood method shown in FIG. 6, the spatial spectrum 
function is: 
##EQU3## 
It can be seen from equations 1A and 2A that S.sub.MEM (.phi.) is the 
reciprocal of the power response of a finite impulse response (FIR) filter 
having tap weights that can be represented by the vector v, with elements 
v.sub.k, k=0, 1, 2 . . . N, where v.sub.o =1 and v.sub.k =-w.sub.k, k=1, 
2, . . . N. The peaks of S.sub.MEM, therefore, correspond to minima, or 
zeroes of the filter response: 
##EQU4## 
It should be noted that evaluation of equation 6A at uniformly spaced 
values of .phi. can be accomplished efficiently by use of the fast Fourier 
transform (FFT) algorithms, but there is no guarantee that the desired 
minima will fall on one of the sampled values. If they do not, then some 
form of subsidiary centroiding method is required, or more closely spaced 
data points must be calculated. It is more attractive to find the 
locations of the minima (or zeros) directly by the computationally simple 
method of this invention. 
Direct expansion of equation 6A produces, in general, a function of the 
form: 
##EQU5## 
where the coefficients a.sub.k and b.sub.k are functions of the complex 
vector elements v.sub.k only. The sines and cosines of the multiple angles 
k.phi. (k=2, 3 . . . N) can be further expanded as functions of cos .phi., 
sin .phi., and their integer powers up to order N. By using the 
substitutions x=cos .phi. and .sqroot.1-x.sup.2 =sin .phi., the equation 
can be further reduced to a polynomial in x. However, the existence of the 
radical in the expression for sin .phi. requires that a squaring operation 
must, in general, be employed in the reduction process. This produces a 
polynomial equation in x, of order 2N, and may introduce as many as N 
spurious roots that do not correspond to solutions of equation 7A. The 
resultant polynomial, and its first two derivatives, may then be solved to 
determine the zeros, if any, and/or the minima, of the polynomial; but the 
solutions must be substituted back into equation 7A to eliminate the 
spurious roots. According to the prior art, the computational procedure 
required by the above method is rather complex. 
The present invention results from consideration of the Z-transform of the 
FIR filter having tap weights v. It is: 
##EQU6## 
Moreover 
EQU S.sub.v (.phi.)=.vertline.H(z).vertline..sup.2 z=e.sup.j.phi.(9A) 
The behavior of S.sub.v (.phi.) is, therefore, determined completely by the 
locations, in the complex Z-plane, of the zeros of equation 8A. In 
particular, those zeros that are on or close to the unit circle produce 
zeros or minima in equation 9A, each corresponding to one of the desired 
peaks of equation 1A. 
Each point on the unit circle corresponds to a specific value of .phi., by 
means of the relationship: 
EQU z=e.sup.j.phi. (10A) 
When a particular zero, Z.sub.k, is close to, but not actually on, the unit 
circle, the corresponding minimum in equation 9A will occur at a value of 
.phi. that is close to: 
EQU .phi..sub.k =tan .sup.-1 [Im (Z.sub.k)/Re(Z.sub.k)] (11A) 
The true minimum will, however, be somewhat influenced by the presence of 
other zeros elsewhere in the Z-plane. In practice, when one is seeking 
peaks corresponding to relatively strong discrete sources in the adaptive 
antenna case, the interaction between zeros is very small, and the 
locations of the individual .phi..sub.k correspond to very good 
approximations for the source locations. The quantity: 
EQU d.sub.k =1-.vertline.Z.sub.k .vertline. (12A) 
is a measure of the closeness of Z.sub.k to the unit circle. The magnitude 
of z.sub.k is therefore an indication of the validity of the approximation 
in equation 11A. The selection of zeros that lie within a specific 
distance of the unit circle (i.e., within a ring centered on the unit 
circle as shown in FIG. 2) provides a method of choosing only those zeros 
which will provide good approximations to resolvable peaks in equations 6A 
and 1A. Note that equations 11A and 12A effectively involve a conversion 
from rectangular to polar coordinates, as shown by block 74 in FIG. 4. 
The approximate method described above has been evaluated in a three-beam 
system (N=2) and been found to give excellent results. Moreover it is 
computionally very simple in this case, since determination of the zeros 
of equation 8A involves merely the solution of a quadratic equation with 
complex coefficients, as shown by block 76. The usual formula for the 
solution of a quadratic equation applies even when the coefficients are 
complex. In the general case, the zeros of equation 8A are the roots of an 
(N-1) order polynomial in Z with complex coefficients. 
The denominator of equation 3A is equal to the sum of the squares of the 
absolute values of the N elements of the weight vector w. However, 
examination of equation 4A shows that each element of w, w.sub.k, can be 
considered as the output of an N-element FIR filter having tap weights 
equal to the element values in the k.sup.th row of M.sup.-1. The inputs to 
each of these filters are the same, namely the conjugate steering vector 
s*. The denominator of equation 3A can, therefore, be represented as the 
combined output of the N filters, where the combination comprises 
summation following square-law detection. 
FIG. 7 illustrates transversal filters arranged to give the combined output 
(w.sup.+ w). The outputs of the square law detectors 78, 80 and 82 are 
summed by the summer 84 to provide the desired output. The delay elements 
86 through 112 and the tap weight elements 114 through 138 direct the 
input signal to the summers 140, 142 and 144. The circuit of FIG. 7 is of 
interest, not only because it leads to an approximate Z-transform method 
for locating the sources directly, as will be described in more detail 
later, but also because it provides a means of computing the complete 
response by efficient means. Each of the N separate filter outputs can be 
computed, at any desired density of sample points in the spectrum, by use 
of the FFT algorithm, as was discussed previously for the single filter 
MEM case. The sum of the squares of the absolute values of the outputs of 
all N filters then provides values of the denominator of equation 3A, at 
the same set of equally spaced points in the spectrum. Minima in these sum 
values correspond to the desired peaks, within the accuracy of the sample 
spacings. 
To determine the approximate locations of the sources directly, it is noted 
that the peaks of S.sub.TN (.phi.), corresponding to minima of the 
quantity w.sup.+ w, will occur at those spatial frequencies where all the 
filters in FIG. 7 simultaneously produce low outputs. In terms of the 
Z-transform representation of the filter responses, this will occur when 
each of the filters has a zero on or close to that point on the unit 
circle that corresponds to the spatial frequency of a peak in S.sub.TN. 
For a system with N beams, M.sup.-1 will be an N.times.N matrix. Each of 
the filters in FIG. 7 will have N taps, and hence each will exhibit (N-1) 
zeros. Only if there are (N-1) discrete sources present will it be 
necessary for the zeros of each of the filters to be in the same, or 
closely the same, locations. When there are less than N-1 sources present, 
each of the filters will have one or more zeros that are not closely in 
common with those of all the other filters. 
The invention uses these considerations to develop an approximate method of 
determining the peaks of S.sub.TN (.phi.), and hence of the source 
locations. It comprises finding the zero locations of the N filters with 
tap weights corresponding to the rows of M.sup.-1, followed by a 
determination of the number and the location of those zeros which are both 
close to the unit circle and common to all the filters. 
Any zero which is determined to be closely in common between all N filters 
and, hence, to correspond to a peak in S.sub.TN, will in practice have a 
slightly different value for each of the N filters, due to statistical 
variations in the formation of the sample covariance matrix from the 
antenna output data. An average value for the common value of .phi., 
corresponding to the source location, must therefore be taken. In general, 
each of the filters can have different gain factors, as can be 
characterized, for example, by their noise power gains. Noise power gain 
is defined as the sum of the squares of the absolute values of the filter 
tap weights. A reasonable procedure is, therefore, to form a weighted 
average of the source locations as determined by the individual filters, 
using, for example, the noise power gain as weights in the averaging 
process. 
According to the maximum likelihood method shown in FIG. 6, a comparison of 
equations 3A and 5A shows that they are of the same functional form. This 
can be seen by combining equations 3A and 4A, noting that M.sup.-1 is 
Hermitian, to give: 
##EQU7## 
and to express equation 5A as: 
##EQU8## 
Thus, equations 13A and 14A differ only to the extent that the matrix 
involved in equation 14A is M.sup.-1/2 as compared with M.sup.-1 in 
equation 13A. 
The methods of analysis described for the thermal noise method of FIG. 5 
are, therefore, equally applicable to the MLM method, except that the 
covariance matrix data must be used to first generate M.sup.-1/2 in the 
MLM case, rather than M.sup.-1 for the TN case. Generation of M.sup.-1/2 
can be accomplished by first forming the spectral representation of M, 
namely: 
##EQU9## 
where the .lambda..sub.k are the eigen values of M, and the idempotents, 
E.sub.k, are: 
EQU E.sub.k =e.sub.k e.sub.k.sup.+ (16A) 
where each of the N vectors e.sub.k is an eigenvector of M, corresponding 
to the eigenvalue .lambda..sub.k. M.sup.-1/2 is then given by: 
##EQU10## 
In all of the estimating methods shown by FIGS. 4, 5 and 6, the phased 
array antennas, such as antenna 140 of FIG. 6, provide data for the 
formation of matrices, such as shown by blocks 142, 144 and 146 of FIGS. 
4, 5 and 6, respectively. The form of the matrix used is indicated by 
blocks 148, 150 and 152 for each method. According to blocks 74, 76 and 
154 through 166, shown in FIGS. 4, 5 and 6, the roots of the polynomial 
expressions are found, converted to polar coordinates, and selected as 
desired zeros when a nearness and commonality criterion is met. 
It is emphasized that numerous changes may be made in the above-described 
system within the scope of the invention. Since different embodiments of 
the invention may be made without departing from the spirit thereof, it is 
intended that all of the matter contained in the foregoing description, or 
shown in the accompanying drawing, shall be interpreted as illustrative 
rather than limiting.