US 3780372 A
The object of this invention is to provide a new nonuniformly spaced antenna array, where the most optimum positions of the array elements, and their corresponding amplitudes, are systematically determined by a rigorous synthesis technique of the given radiation pattern, and/or its requirements and specifications, in order to achieve it with the minimum possible number of array elements. This new array will be designated the Nonuniformly, Optimally Spaced Antenna Array, or in short, the NOSA Array.
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1451 Dec. 18, 1973 NONUNIFORMLY OPTIMALLY SPACED ANTENNA ARRAY  Inventor: Hillel Unz, c/o Electrical 5 Engineering Dept. University of Kansas, Lawrence, Kans. 66044 221 .Filed: Jan. 17, 1972 211 Appl.No.:21 8,495
3,553,706 l/l97l Charlton 343/844 3,262,! l5 7/1966 Macalpine 3,182,330 5/1965 Blume 343/844 Primary ExaminerEli Lieberman Attorney- Hillel Unz 5 7 ABSTRACT The object of this invention is to provide a new nonuniformly spaced antenna array, where the most optimum positions of the array elements, and their corre-  US. Cl. 343/844, 343/719 sponding amplitudes, are systematically determined by  Int. Cl. l-l0lq 21/00 a rigorous synthesis technique of the given radiation  Field of Search 343/719, 8 4, 85 pattern, and/or its requirements and specifications, in 3/854 order to achieve it with the minimum possible number of array elements. This new array will be designated  References Cited the Nonuniformly, Optimally Spaced Antenna Array,
UNITED STATES PATENTS or in short, the NOSA y- 3,l30,4l0 4/1964 Cutleber 343/844 6 Claims, 2 Drawing Figures n 0 l 2 3 4 5 b 7 8 TABLE A NONUNIFORMLY OPTIMALLY SPACED ANTENNA ARRAY The general idea of the non-uniformly spaced antenna arrays was invented by Unz in 1955, and was publicly proposed and published for the first time in-his University of California (Berkeley, I956) doctoral dissertation, where the first significant work on the subject was reported. From 1960 on many additional contributions have been made on non-uniformly spaced arrays by numerous authors, and references to most of them may be found in recent books on antennas. However, the main synthesis problem of the non-uniformly spaced arrays, namely, finding the most optimum positions of the elements of the array in order to produce a given specified radiation pattern, has not been solved rigorously to-date. This is primarily due to the great difficulties in the solution of this highly non-linear problem. The trial and error computer techniques and the other pseudo-optimum synthesis methods suggested so far are of very limited utility, and are almost impossible to employ with arrays of a large number of elements. Thus, the non-uniformly spaced arrays cannot be technically designed at the present time to their full potential advantage, if at all, and therefore are not generally used.
The object of this invention is to provide a nonuniformly spaced antenna array, where the array elements are distributed in the most optimum positions along the axis of the array, with reference to the given radiation pattern requirements and specifications. The optimal distribution of the elements along the array axis is determined so that the Nonuniformly Optimally Spaced Antenna Array (the NOSA Array) of this invention will require a smaller number of elements, than an equivalent uniformly spaced array, giving the same required approximate radiation pattern, or its specified required characteristics, such as the sidelobe level, and/or beamwidth, and/or gain, etc. Alternatively, the NOSA Array will give a better performance in its radiation pattern and its characteristics than an equivalent uniformly spaced array of an equal number of elements. A rigorous synthesis technique is provided for the systematic determination of the most optimum positions of the el' ements in the Nonuniformly Optimally Spaced Antenna Array (the NOSA Array), and their corresponding amplitudes, in order to accomplish the radiation pattern given specifications with the minimum possible number of elements in the array. Thus this invention will provide a substantial savings in the number of elements required in an antenna array for a given performance. While the rigorous synthesis technique described later is for the symmetric linear NOSA Array, the invention by no means is limited to this particular case. It could be extended quite easily to the non-symmetric NOSA Array. Furthermore, since the NOSA Array is the building stone for any number of other more sophisticated arrays, this invention covers all other arrays, where they include in whole or in part the NOSA Array. Thus this invention covers, but is not limited to the following arrays, if they include in whole or in part the NOSA Array: symmetric and non-symmetric arrays, linear, two and three dimensional arrays, arrays on curved surfaces, transmitting, receiving and retrodirective arrays, arrays of arrays, correlation arrays, steerable beam, scanning arrays and phased arrays, active and adaptive arrays, time modulated arrays, radioastronomy arrays, broadband arrays and frequency independent arrays, superdirective arrays, slot arrays, omnidirectional arrays, broadside arrays, end-fire arrays, resonant and nonresonant arrays and many others.
The invention is illustrated by the accompanying drawing in which:
Table A gives the seventeen (l7) complex Fourier series coefficients f, =f specified from the given real and symmetric radiation pattern to be approximated by the symmetric linear antenna array. FIG. 1 and Table A give the seventeen l7) element half-wavelength uniformly spaced antenna array which will give the specified Fourier coefficients f listed in Table A. The synthesis has been done in accordance with well established procedures for uniformly spaced arrays. FIG. 2 and Table A give the nine (9) element Nonuniformly Optimally Spaced Antenna Array (the NOSA Array) which will give the same specified Fourier coefi'lcients f listed in Table A. The systematic determination in FIG. 2 of the most optimum array element distribution and their corresponding amplitudes for the given Fourier coefficients in Table A, have been accomplished by using the rigorous synthesis technique described below. Thus the NOSA Array in FIG. 2 will give the same required performance as the half-wavelength uniformly spaced antenna array in FIG. 1, but with a substantial saving of almost half of the number of the elements required by the standard uniformly spaced array. A substantial saving of the number of the elements of the array for equivalent performance is of essential importance especially in arrays with a large number of elements.
The rigorous synthesis technique for the symmetric linear Non-uniformly Optimally Spaced Antenna Array (NOSA Array) for a given real and symmetric radiation pattern or its complex Fourier series coefficients f f, real specified in Table A, in order to determine the most optimum positions of the elements of the NOSA Array, and their corresponding amplitudes, in FIG. 2, is described as follows:
The radiation pattern F (0) of a linear array with nonuniform spacings of the elements is given by:
F) 2 A ikd m a where u 1r sin 0 11' 5 15 11') and x,=d,/M2 gives the distance of the array element p from the origin in terms of half wavelengths.
For the synthesis problem the given radiation pattern F (u) may be expanded in a complex Fourier series, for n integer:
For the particular case of half wavelength d p M2 uniformly spaced linear array one has x, p integer in (Lb), which then has the same form as Q a). Thus, each element of the uniformly spaced array corresponds to one complex Fourier series term A =f,,, and F(u) in general may be approximated to any degree of accuracy by a finite number of array elements, corresponding to the same number of complex Fourier series terms.
in case of a non-uniformly spaced array, x p) in (1b) has to be determined, as well as A,,, for the best approximation of F(u). Substituting (Lb) in (2.1;) one obtains after integrating:
Using the trigonometric indentity and taking sin mr O and cos mr (1 Y, one has:
sin(n x,,)rr sin mr cos x 'n' cos m-r sin x n' (i )"''sin x n' and one obtains from (3.a):
Taking the array element p of the amplitude A to be at the origin x 0, one obtains from (4.a) and (4.b):
Let us assume that the given radiation pattern F(u) is a real and symmetric function F(-u) F(u). Thus one finds from (211):
Let us further assume that the non-uniformly spaced array has an odd number of elements and is symmetric with respect to the element A at the origin x,, 0. Using (4.b) one obtains:
Substituting (6.b) in 7.11 and (7.b) one obtains after rearranging:
From (8.11) one finds thatf =f,, as in (6.a). Taking in (8.a) D, B,,x,,, N n, y X92 and using (4.b) one obtains:
We will discuss here only the solution of (9.a) for the symmetric non-uniformly spaced array.
In equation (9.0) we have two sets of unknowns, y and D,,, which have to be solved, and thus give us x, and A, from (9.1)). Writing (9.0) explicitly for s terms and multiplying by the common denominator on both sides, one obtains for N n 2 l:
where (11) for given f, is a linear equation for D, and a non-linear equation for y,,. Opening the brackets on both sides and rearranging in descending powers of N, one obtains:
---+ s (yr y2 y3 -y.. 1) (m) Since y, and D are the unknowns to be found, and thus 04,, and B defined above are also the unknowns to be found. Thus (l2.a) couldbe rearranged and rewritten as follows for)! n 2 1:
or alternatively, dividing (13.12) by N one obtains:
alternatively (l3.b) may be rewritten in the following form:
Equation (1.3.a) or (l3.b) or (13.0) represents a linear equation with 2s unknowns, namely, s unknown terms of a,,, and s unknown terms of [3 One may rewrite one of the equations (13), i.e. (13.0), 2s times for the given 2s Fourier coefficients f,, of the given radiation pattern F(u), namely for (f f f, .f where N= n 1. Thus one obtains 2; linear equations with 2s unknowns 01,, and B The method of solving Zr linear equations with 2s unknowns by the use of determinants and by well established and modified computer techniques is ye ry well known Thus if (f f ,f are given, one may solve l 3.c) for a (a,, a 11,), by determinants with the aid of a computer. As it will be shown later, there is no need to solve for [3,.
Once (a a 01,) are thus found, one should solve the s non-linear equations (l2.b) for (y,, y .,y It is well known that a polynomial equation with the solutions y y,, y y y y, could be written in the form: (yyi)(yyz)(yya)----(y-y8)= (14 11) From the left hand side of (l l) and (12.0) and from (l2.b) one finds by inspection and by multiplying and opening the brackets of (l4.a) one obtains: ya,y"+a y a y +(l)a,=0
But the coefficients ((1,, a 01,) have been solved above. Thus the positions of the elements x, along the axis of the non-uniformly spaced array could be found as the roots of the polynomial equation l4.b) y, =x,, Finding the roots of the polynomial equation (l4.b) is a well established process with the aid of a computer. The positions x thus found from the roots of the algebraic equation (l4.b) represent the optimum positions of the symmetric NOSA Array elements.
Once the s roots y have been found from (l4.b), one may substitute them in (9.a) and rewrite it s times for (f,,f ,fl). Thus one obtains s linear equations with s unknown D,,, which could be solved by using determinants and well known computer techniques. With all y, and D thus found, one may use (9.b) to obtain: x,, A, D,,/(x,, sin x qr) (15 1} and thus the position of each element x, and its amplitude A, are found for all the side elements of the symmetric non-uniformly spaced array. Once this is known, one may use (8.b) and (9.b) to find the amplitude of the center element A at x O:
and the synthesis problem for a given F (u) is solved for the non-uniformly optimally spaced array.
The details of the procedure of the rigorous synthesis technique for the determination of the most optimum positions of the elements of the NOSA Array and their corresponding amplitudes for the symmetric case in FIG. 2 are given as follows:
Assuming that 2s Fourier series coefficients of f, have to be used, one rewrites the linear equation (13.0) of 2s unknowns 01,, and B, for a total of 2s times, using the given f, for l 3 n s 2s and taking N n The 2s linear equations with 2s unknowns are solved for all a (a a 0,) by the use of well known computer techniques.
Using the 01,, coefi'icients found previously, one obtains the algebraic equation l4.b) of order s. The positive real roots of the algebraic equation (l4.b) are found by using well known computer techniques, and
they represent the optimum positions of the symmetric NOSA Array elements, where x, V'yj.
Substituting the value of y, found above in (9.0), and rewriting it 5 times for different f, for l n 5 s, taking N n one obtains s linear equations for s unknowns D,,, which can be solved by the use of known computer techniques. Once D and y, have been solved for the symmetric linear NOSA Array, obe obtains the position of the p element x,,, and its amplitude .4 by using (91;). The amplitude A,, of the center element at x, O is found by using (l5.b). Using the above rigorous synthesis technique, the symmetric linear NOSA Array in FIG. 2 was technically designed from the given complex Fourier co-efficients in Table A. The nine (9) element NOSA Array in FIG. 2 is equivalent with its required performance to the seventeen 17) element half wavelength uniformly spaced array given in FIG. 1. Both the uniformly spaced array in FIG. 1 and the nonuniformly optimally spaced array in FIG. 2 give almost the same identical radiation pattern which has been prescribed and defined by the f, complex Fourier series coefficients given in Table A.
Let us assume that in prder to approximate a given symmetric and real function radiation pattern F (u) to a certain given degree of average mean square approximation, one needs to have a symmetric linear array which will give exactly the values of the complex Fourier coefficients f,, for s n s 2M. If we chose a symmetric uniformly spaced array of half wavelength spacings, we determine in advance the element positions and will need, in accordance with equation a total of(4M l antenna elements, for the symmetric array with the center element. However, if we choose a nonuniformly spaced symmetric linear array, where both the positions of the elements and their corresponding amplitudes are to be determined for the best approximation, we have twice as many degrees of freedom, and we could approximate 2M Fourier coefficients by using only M radiation elements in accordance with (l3). Thus the nonuniformly spaced symmetric linear array with the center element would require a total of only(2M l )non-uniformly spaced antenna elements in order to achieve approximately the same radiation pattern; therefore one has for a symmetric linear array, where f for 0 s n s 2M is given:
No. of elements uniform 4M+ 1 1 N 2 No. of elements nonuniform 2M 1 2M 1 One needs therefore half the number of elements in the Non-uniformly Optimally Spaced Antenna Array (the NOSA Array) as compared to a uniformly spaced array giving the same approximate radiation pattern with all its side lobes. In this statement we assume that all the roots in the polynomial equation (l4.b) are positive and real: thus the result depends on all the coefficients a,,, and on the given Fourier Coefficientsf through the linear equation (l3.c). The results of such a numerical example for a given prescribed set of Fourier coefficients f,, are given above.
Furthermore, one should check an additional number of Fourier array coefficients f,, for n 25, which have not been approximated, but which also may be found from a given radiation pattern F(u), as well as the final resulting radiation pattern F(u) given by the NOSA Array and its elements A, at positions x, in accordance with equation (lb). In case the approximation requirements of the NOSA Array radiation pattern with respect to the given prescribed radiation pattern are not acceptable, one may repeat the process by adding one more antenna element on each side of the NOSA Array and repeat the process again, and check again for the approximation acceptability. The number of the elements in the NOSA Array will be still smaller than an equivalent uniform array, but by less than the factor of two given in (16). The final result depends on each indivudal or class of given radiation patterns.
. half wavelength uniformly spaced linear array giving the same prescribed radiation pattern with all its sidelobes.
Since the non-uniformly optimally spaced array described in the foregoing specifications could be used as the basis for building more sophisticated arrays with specific or minimax performance indices requirements, this invention covers, but is not limited to, arrays with specific requirements on the radiation patterns, impedance, bandwidth, mutual coupling, gain and directivity, polarization, noise temperature, signal to noise or interference ratio and any other specific indices, if the array includes in whole or in part certain definitive aspects of this invention as a part of its analysis or synethsis. This invention also covers this array, when the elements used in the array are of different types, including but not limited to, dipoles, slots, horns, apertures, parabolic reflectors, dishes and many others. Many of the design techniques in prior art developed for electromagnetic arrays can and has been applied with modification to acoustic arrays, seismic arrays and arrays in other fields. Thus this invention of non-uniformly optimally spaced array described in the foregoing specifications covers, but is not limited to, all such arrays or combination of them in other fields for any purpose. This invention covers non-uniformly optimally spaced acoustic arrays and sonar arrays, used with acoustic elements of any type, when used under water or above water for whatever purpose. This invention covers non-uniformly optimally spaced seismic arrays, used with seismic elements, geophones, seismometers seismographs, or other elements, when used underground for whatever purpose. Thus the foregoing invention of non-uniformly optimally spaced array covers any array of any shape and size, and used with any kind of elements for any purpose, provided that the invention described here will be used there in whole or in part, or will be used there as a part of its analysis or synthesis during the design procedure.
While in the foregoing specification, I have set forth certain details of the Nonuniformly Optimally Spaced Antenna Array (the NOSA Array) and its rigorous synthesis technique for a specified radiation pattern, for the purpose of illustrating one mode of the invention, it will be understood that such details may be varied widely by those skilled in the art without departing from the spirit of my invention, and it is therefore aimed to cover all such changes and modifications in all areas of endeavor where arrays are used, as fall within the true spirit and scope of this invention.
What I claim is l. A method for optimizing the spacing and ampli- D for n0 where f, is the n-th complex Fourier series coefficient of the given radiation pattern, N=n y x, where x,, is the distance of the p-th element of the array from the origin in terms of half wavelengths, and D, A r: sin x,,1r where A is the amplitude of the p-th element of the array, the said method comprising the selecting of the optimum positions of the individual elements from the s roots y=y,,=x,, of the following algebraic equation: 3 -04 y y -a y" .+(l) a, =0 where the coefficients 01,, are determined by the solution of the following 2s linear equations for the given f coefficientsz I if i S 1) aq. 1 S om n once the optimum positions x are determined, the optimum amplitude of each element is found from (A) by solving 5 linear equations for the unknowns D,, the amplitude of the center element A, at the origin x 0 being given by:
2. A method for optimizing the spacing and the amplitude of the individual elements in a non-uniformly spaced non-symmetric array, as compared to a conventional uniformly spaced array, in order to accomplish a given radiation pattern with the minimum number of elements, said method accomplished essentially as in claim 1, except it is based on the following relationship for a non-symmetric array:
where B, A, sin x,,1r and (E) is almost identical in form to (A) in claim 1.
3. An arrangement of a set of elements in a nonuniformly optimally spaced array, as specified in claim 1, in which each element is an electromagnetic antenna radiator or receiver, like dipole, slot, horn, aperture, dish, parabolic reflector, etc. or a combination of them.
4. An arrangement of a set of elements in a nonuniformly optimally spaced array, as specified in claim 1, in which each element is an acoustic or sonar radiator or receiver, or a combination of them, used under water or above water 5. An arrangement of a set of elements in a nonuniformly optimally spaced array, as specified in claim 1, in which each element is a seismic radiator or receiver, like geophone, seismometer, seismograph, etc. or a combination of them, used under ground or above ground.
6. An array comprising a grouping of subarrays each designed in accordance with the method set out in claim 1.