Linear array antenna employing the summation of subarrays
The invention described herein may be manufactured, used and licensed by or for the Government for governmental purposes without the payment of any royalties thereon or therefor. 
    
    
     FIELD OF THE INVENTION 
     This invention relates generally to coded linear array antennas and more particularly to a design procedure which utilizes the total flexibility of controlling both element space positions and/or amplitude levels. 
     BACKGROUND OF THE INVENTION 
     Space coded linear array antennas and methods for obtaining a desired antenna pattern therefrom are shown and described in applicant&#39;s prior U.S. Pat. No. 3,130,410, entitled, &#34;Space Coded Linear Array Antennas&#34; issuing on Apr. 21, 1964, and U.S. Pat. No. 3,605,106, entitled, &#34;Slot Fitting of Coded Linear Array Antenna&#34;, issuing on Sept. 14, 1971, which patents are furthermore incorporated herein by reference. In U.S. Pat. No. 3,130,410, there is disclosed the concept of sidelobe control of linear array antennas by amplitude and/or space coding of antenna elements and which comprises adding a second element to each existing element in order to force a zero in an antenna pattern for some specific value of space angle. In U.S. Pat. No. 3,605,106, another method of designing a coded array antenna is disclosed which is more general and involves adding h-1 additional elements to each element of an array whose vector fields are the h th  complex roots of unity. 
     SUMMARY OF THE INVENTION 
     Briefly, the subject invention is directed to a method for designing a linear array antenna whereby undesired side lobes are reduced by controlling both element amplitude and space positions using sums of antenna patterns formed by a plurality of uniform subarrays each comprising multiple elements which are spaced equidistantly from one another in each array and centered about a common axis of symmetry and with positioning of individual antenna elements being determined in accordance with the normalized equation ##EQU2## where i=1, 2, 3, . . . h, n m  is equal to the maximum number of elements in the length of the array, h is the total number of elements in the respective subarray and n is proportional to the spacing between elements of that subarray. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a diagram illustrating the geometry of one subarray of a linear array antenna and which is helpful in deriving the equations involved in the subject invention; and 
     FIG. 2 is a schematic diagram of five subarrays whose elements are positioned in accordance with the principles of the subject invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring now to the drawings and more particularly to FIG. 1, reference numeral 10 denotes a linear antenna axis which has a maximum array length L=n m  d wherein n m  is equal to the maximum number of elements h which can be positioned between the aperture of the array, as defined by the points 12 and 14, for a predetermined constant spacing d. As shown, h=4 elements 16 1 , 16 2 , 16 3  and 16 4  which are symmetrically positioned on either side of the axis of symmetry 18 such that the elements 16 1  and 16 2  lie on the left side of the axis 18 whereas the elements 16 3  and 16 4  lie on the right hand side of said axis. Moreover, the elements 16 1  . . . 16 4  have a mutually constant spacing of &#34;x&#34; with the first element 16 1  being spaced from point 12 by a distance of x 0 . The first element 16 1 , moreover, is positioned from the axis of symmetry 18 by a distance of ##EQU3## Thus the subarray is comprised of multiple elements which are respectively spaced equidistantly apart and positioned symmetrically on either side of a common axis. 
     Accordingly, a wave front 20 arriving at a space angle θ with respect to the antenna axis 10 will impinge on the antenna elements 16 1  . . . 16 4  with received radiation phases of ψ 1 , ψ 2 , ψ 3  and ψ h , respectively. The design equations for positioning of the elements are developed as follows. 
     The resultant received E t  signal for the subarray shown in FIG. 1 can be represented by the following equation: ##EQU4## where λ is equal to the wavelength of the incident radiation and θ is equal to space angle. 
     Now let ##EQU5## where d corresponds to a predetermined arbitrary constant element spacing. 
     Making the following substitution, ##EQU6## 
     yields, 
     
         ψ=2πK                                               (3) 
    
     Combining equations (1) and (2) results in: ##EQU7## where (x/d) is proportional to the element spacing. 
     Now letting (x/d)=n, equations (3) and (4) can be combined as: ##EQU8## 
     The bracketed term in equation (5) can furthermore be reduced to a (sin m x  /sin x) function in the following manner. 
     Consider the equation 
     
         E.sub.t =1+e.sup.jθ +e.sup.j2θ +. . . +e.sup.j(n-1)θ(6) 
    
     Factoring out e -j θ  yields 
     
         E.sub.t =e.sup.-jθ (e.sup.jθ +e.sup.j2θ +. . . +e.sup.jnθ)                                         (7) 
    
     
         e.sup.jθ E.sub.t =(cos θ+cos 2θ+. . . +cos nθ)+j(sin θ+sin 2θ+. . . +sin nθ) (8) 
    
     Which becomes, ##EQU9## 
     In the same manner, equation (5) may be reduced to the form: ##EQU10## 
     Combining the phase terms results in: ##EQU11## 
     Now the glometry of FIG. 1, ##EQU12## 
     Hence, equation (14) simplifies to: ##EQU13## 
     In general, a plurality of subarrays with different spacings and amplitudes positioned with their center at (n m  d/ 2) would simply result in a composite pattern given by the sum of the individual patterns, that is, ##EQU14## where the element amplitudes are given by a i , the element spacings are equal to n i  ×d and h i  is the number of elements in the i th  subarray. 
     The normalized element positions n&#39; for each h element subarray are furthermore given by, ##EQU15## 
     The above set of equations can be written more compactly as, ##EQU16## where i=1, 2, 3, . . . h 
     Referring now to FIG. 2, the present invention contemplates employing the plurality of subarrays having sin mx/sin x patterns whose elements are positioned in accordance with the foregoing design procedure. As shown, five subarrays 22, 24, 26, 28 and 30 are coupled to a signal combiner 32. The subarray 22 is comprised of antenna elements 34 1  through 34 8  which have mutual equal spacings of x 1 . With respect to the subarray 24, it is comprised of elements 36 1  through 36 6  which have an equal element spacing of x 2 . In a like manner, the subarrays 26, 28 and 30 are comprised of elements 38 1  . . . 38 4 , 40 1  . . . 40 4 , 42 1  . . . 42 4 , respectively, having respective element spacings of x 3 , x 4  and X 5 . 
     Thus what has been shown and described is a procedure for reducing the lobes of an array antenna by controlling both elements, amplitudes and space positions by using sums of antenna patterns employing uniform subarrays whose element positions are constrained within a predetermined maximum array antenna length to produce a composite antenna pattern in accordance with the design equation (18).