Abstract:
A method and apparatus for forming a set of simultaneous multiple antenna beams. The digital technique utilizes a Fermat Number Transform (FNT) processor which takes the transform of number sequences consisting of the output signals from the elements of a preferably circular, equiangularly spaced antenna array, then forms the complex linear vector product of the transformed signals and a stored beam-forming function, and finally processes the complex linear vector product through an inverse FNT network to generate a set of simultaneous multiple antenna beams. The advantages of utilizing the FNT include reduced hardware requirements, greater processing speed due to a reduction in multiplications required to be performed, and sharper output beams due to the absence in the FNT of round-off errors.

Description:
RIGHTS OF THE GOVERNMENT 
     This invention may be manufactured, used, and/or sold by or for the United States Government for governmental purposes without the payment to me of any royalty thereon. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates generally to antenna arrays and, more particularly, to beam-forming systems for such arrays. Even more particularly, the invention relates to a digital beam former capable of generating multiple output beams from an array of antenna elements. Applications for the invention include surveillance radar, instantaneous automatic direction finding, and adaptive arrays. 
     2. Description of the Prior Art 
     Techniques for performing a discrete cyclic convolution of two sequences of numbers are known. A definition for such a convolution is as follows: ##EQU1## If computed directly, this convolution requires N 2  multiplications for two input sequences x and h, each of a length N. However, as is well known to those skilled in the art of digital signal processing, there exist arithmetic transforms having the cyclic convolution property, i.e., the transform of the cyclic convolution of two sequences is equal to the product of their transforms. Thus, if the two sequences x and h are transformed by a transform having the cyclic convolution property into two new sequences, it is necessary only to find the linear vector product of the transformed sequences (which requires only N multiplications) to produce the transform of the desired result. When this result is processed through an inverse transform operation, the final result is the desired output sequence, in this case y. 
     The most well known of such transforms is the Fast Fourier Transform (FFT) which is an algorithm for computing the discrete Fourier transform (DFT) of a sequence. Another such transform found useful in digital convolution is the Fermat Number Transform (FNT). The FNT and its application to digital signal processing are described, for example, in &#34;Fast Convolution Using Fermat Number Transforms with Applications to Digital Filtering&#34; by R. C. Agarwal and C. S. Burrus, IEEE Transforms on Acoustics, Speech, and Signal Processing, April, 1974, pages 87-97. 
     The discrete Fourier transform F(k) of a sequence x(n) may be defined as ##EQU2## 
     In contrast, the Fermat Number Transform F(k) of a sequence x(n) may be defined as ##EQU3## Equations (2) and (3) apply to sequences of length N, where N is an integral power of 2. In equation (3), F t  is a Fermat number, defined as F t  =2 b  +1, b=2 t  ; and α is the Nth root of 1 (mod F t ), as will be discussed in greater detail below. 
     It is seen that the FNT resembles the DFT with α replacing exp(-2πj/N), and with all arithmetic performed modulo a Fermat number. 
     The significance of the FNT to digital signal processing lies in the fact that if α is an integral power of 2, multiplications by powers of α are accomplished by merely rotating bits in a register. In addition, the FFT algorithm may be used to compute the FNT as long as the length of the sequences is a power of two. Thus, the FNT can be implemented using only adders and bit shifters, with the only multiplications necessary being the linear vector product of the two transforms. The FFT algorithm is discussed, for example, in &#34;What is the Fast Fourier Transform?&#34; by W. T. Cochran et al, IEEE Transactions on Audio and Electroacoustics, June, 1967, at pages 45-55. 
     It is also quite common to design specialized hardware to perform Fermat arithmetic. Many such designs are based on novel digital coding schemes for the representation of numbers, such as described, for example, in &#34;Hardware Realization of a Fermat Number Transform&#34; by J. H. McClellan, IEEE Transactions on Acoustics, Speech &amp; Signal Processing, June, 1976, pages 216-225, and in &#34;Modified Circuits for Fermat Transform Implementation&#34; by H. Nussbaumer, IBM Technical Disclosure Bulletin, October, 1976, pages 1720-1. 
     Electrical networks for forming multiple beams from linear antenna arrays have also been described in the literature. See, for example, &#34;Multiple Beams from Linear Arrays,&#34; J. R. Shelton, IRE Transactions on Antennas and Propagation, March, 1961, which describes the well known Butler matrix which forms multiple beams by utilizing passive analog networks of couplers and phase shifters. 
     Digital beam forming techniques using the FFT are also known. The Butler matrix is a hardwired analog FFT that produces, at its n output spigots, n antenna beams that are mutually orthogonal sinc functions; one for each of the n antenna elements. The difficulties of analog computation are preserved when hardware realization of a Butler matrix is attempted, thus many systems use the FFT to form antenna beams. In either case, the antenna beams generated have definite drawbacks unless some amplitude weighting function is applied to the antenna element outputs, and even such an adjustment avails little in beam improvement. 
     The FFT may be regarded conceptually as producing the cross-correlation of the antenna array and a circular function whose angular progression from point to point is related to the angle of arrival of the beam being formed. Since such a cross-correlation is a constant function of its variable, each output &#34;spigot&#34; represents this cross-correlation evaluated at a single point, and each &#34;spigot&#34; or output coefficient is due to a different angle of arrival, therefore due to a different (special) frequency circular function. Although the low quality beam patterns generated by the FFT can be used in linear combination (since they form a basis) to produce any realizable beam pattern or set of beam patterns, an additional computation load is thus generated whenever beam sharpening is required. 
     OBJECTS AND SUMMARY OF THE INVENTION 
     It is therefore a primary object of the present invention to provide a novel and unique method and apparatus for generating multiple output beams from an array of antenna elements. 
     Another object of this invention is to provide a novel and unique method and apparatus for generating multiple simultaneous output beams from an antenna array having equispaced elements, and more particularly from a circular antenna array having equiangularly spaced elements. 
     Yet another object of this invention is to provide a novel and unique method and apparatus for generating multiple output beams from an antenna array which utilizes a digital processor to process the outputs of an antenna array in order to generate multiple simultaneous output beams. 
     The foregoing and other objects are attained in accordance with one aspect of the present invention through the provision of a method for forming multiple beams from an array of antenna elements which provide outputs, the method comprising the step of convolving the output of the elements with a beam forming function to produce the multiple beams. More particularly, the step of convolving utilizes a transform having the convolution property and comprises the steps of taking the transform of the antenna outputs to produce transformed antenna outputs, providing a transformed beam forming function, computing the product of the transformed antenna outputs and the transformed beam forming function, and taking the inverse transform of the product. 
     In accordance with another aspect of the present invention, the step of providing the transformed beam forming function may comprise the steps of computing the outputs of the antenna elements which are induced by reception of a plane wave, taking the transform of the plane wave induced antenna outputs, taking the inverse transform of a desired beam pattern, and computing the product of the inverse transform of the desired beam pattern with the multiplicative inverse of the transform of the plane wave induced antenna outputs. 
     In accordance with more specific aspects of the present invention, the transform comprises the Fermat Number Transform, and the transform operations are performed by separately treating the real and imaginary portions of the antenna outputs to produce transformed antenna outputs, the beam forming function to produce a complex transformed beam forming function, and the multiple beams. The product comprises a complex linear vector product of the complex transformed antenna outputs, represented by C i  +jD i , and the complex transformed beam forming function, represented by E i  +jF i , where i=0, 1, . . . , N-1, and N equals the number of antenna elements. The step of computing the product comprises the step of computing the complex linear vector product modulo a Fermat number F t , t=0, 1, . . . , of the form F t  =2 b+1 , b=2 t  and comprises the steps of replacing complex numbers of the form C i  +jD i  by numbers of the form C i  ±2 b/2  D i  and replacing complex numbers of the form E i  +jF i  by numbers of the form E i  ±2 b/2  F i . 
     In accordance with more specific aspects of the present invention, the step of computing the complex linear vector product further comprises the steps of computing the first product of C i  +2 b/2  D i  and E i  +2 b/2  F i  and computing the second product of C i  -2 b/2  D i  and E i  -2 b/2  F i . The sum of the first product and the second product are then computed, as are the difference between the first product and the second product. The sum of the products is divided by 2 to form the real portion of the desired complex product, while the difference between the products is divided by 2·2 b/2  to form the imaginary portion of the desired complex product. 
     The antenna elements are preferably equiangularly spaced about a central point, and the antenna array is, in a preferred embodiment, circular. 
     In accordance with still more specific aspects of the present invention, the step of convolving comprises the steps of converting the outputs to digital form to produce digital antenna outputs, digitally taking the Fermat Number Transform of the digital antenna outputs to produce transformed digital antenna outputs, providing the beam forming function in digital form, digitally computing the vector product of the transformed digital antenna outputs and the digital form of the beam forming function, and digitally taking the inverse Fermat Number Transform of the vector product. The digital representation of all numbers adheres to the formula: 
     B=A-1 if the decimal value of A≧1, and 
     B=A-1 if the decimal value of A≦1, wherein A denotes a one&#39;s complement binary number of p bits, and p denotes a binary number of (p+1) bits wherein the (p+1)th bit equals one if the decimal value of A is zero. 
     In accordance with still more particular aspects of the present invention, the step of computing the product of the terms C i  +2 b/2  D i  and E i  +2 b/2  F i  and the step of computing the product of the term C i  -2 b/2  D i  and E i  -2 b/2  F i  comprise the steps of inverting the bits of those of the terms whose sign bit is zero, forming a 2p bit product of the two p bit terms, subtracting the more significant p bits of the 2p bit products from the less significant p bits of the 2p bit product, the step of subtracting being performed modulo a Fermat number, and inverting the bits of the results of the subtraction step which resulted from those terms of differing sign. 
     In accordance with another aspect of the present invention, there is provided an apparatus for forming multiple beams from an array of antenna elements which provide outputs, comprising means for convolving the outputs of the elements with a beam forming function to produce the multiple beams. The convolving means includes means for using a transform having the convolution property and comprises means for taking the transform of the antenna outputs to produce transformed antenna outputs, means for providing a transformed beam forming function, means for computing the product of the transformed antenna outputs and the transformed beam forming function, and means for taking the inverse transform of the product. 
     Another aspect of the present invention specifies that the means for providing the transformed beam forming function comprises means for computing the outputs of the antenna elements which are induced by reception of a plane wave, means for taking the transform of the plane wave induced antenna outputs, means for taking the inverse transform of a desired beam pattern, and means for computing the product of the inverse transform of the desired beam pattern with the multiplicative inverse of the transform of the plane wave induced antenna outputs. 
     The transform, in a best mode, comprises the Fermat Number Transform, and the means for convolving further includes means for separately treating the real and imaginary portions of the antenna outputs, the beam forming function, and the multiple beams, which all comprise complex numbers, to respectively produce complex transformed antenna outputs, a complex transformed beam forming function and complex multiple beams. The product comprises a complex linear vector product of the complex transformed antenna outputs, which may be represented by C+jD, and the complex transformed beam forming function, which may be represented by E+jF. The means for computing the product comprises, more particularly, means for computing the complex linear vector product modulo a Fermat number F t , t=0, 1, . . . , of the form F t  =2 b+1 , b=2 t  and comprises means for representing numbers of the form C+jD by numbers of the form C±2 b/2  D and means for representing numbers of the form E+jF by numbers of the form E±2 b/2  F. 
     In accordance with more specific aspects of the present invention, the means for computing the complex linear vector product further comprises first means for computing the product of C+2 b/2  D and E+2 b/2  F and second means for computing the product of C-2 b/2  D and E-2 b/2  F. Third means are also provided for computing the sum of the output of the first means and the output of the second means, as are fourth means for computing the difference between the output of the first means and the output of the second means. First means are provided for dividing the output of the third means by 2 to form the real portion of the desired complex product, while the imaginary portion of the desired complex product is formed by second means for dividing the output of the fourth means by 2·2 b/2 . 
     In a best mode, the antenna elements are equiangularly spaced about a point, and in a preferred embodiment the array of elements is circular. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other objects, advantages, and features of this invention will become more apparent from the following detailed description of the present invention, when read in conjunction with the accompanying drawings, in which: 
     FIG. 1 is a diagram illustrating one possible placement of antenna elements in accordance with the present invention; 
     FIG. 2 is a block diagram of a preferred embodiment of this invention; 
     FIGS. 3A and 3B consist of a more detailed diagram of the components which comprise the preferred embodiment illustrated in FIG. 2; 
     FIG. 4 is a schematic diagram illustrating a preferred embodiment of the adders and subtractors of the system shown in FIGS. 3A and 3B; 
     FIG. 5 is a schematic diagram which shows a preferred design of the multipliers of the system of FIG. 3B; and 
     FIG. 6 is a table helpful in understanding the structure and function of the scalers of FIGS. 3A and 3B. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A preferred embodiment of the present invention will now be described in connection with a circular, equispaced array of antenna elements, although it is understood that the present inventive technique and apparatus may be equally applicable to other array configurations and computation situations. For example, the array elements may be placed in non-circular (e.g., ellipitical) patterns, as long as they are equally angularly spaced about a point. 
     Referring now to FIG. 1, an array of N antenna elements 10 are located on a circle 11 which has its center at 12. The elements 10 are equally angularly separated by an angle θ. The output signal from a given element is denoted by A(kθ), where θ is the angular spacing between adjacent elements and k ranges from 0 to N-1. Thus, for the elements shown, A(0) is the output of the zeroeth element (k=0), A(θ) is the output of the first element (k=1), and A(N-1)θ) is the output of the (N-1)th element (k=N-1). 
     In order to produce a set of N simultaneous multiple beams from this array according to the method of this invention, the element outputs are convolved with a suitable beam forming function. That is, the function A(kθ), representing the array outputs, must be convolved with a beam-forming function F(kθ) to produce an output B(kφ) representing the desired output beams pointing in directions kφ. 
     The discrete cyclic convolution of the two sequences A(kθ) and F(kθ) which must be evaluated can be written: ##EQU4## 
     One convenient procedure for computing the function F(kθ) according to the present invention involves the following steps: 
     1. Assume that a plane wave is exciting the antenna array from some direction ψ. Compute the outputs A(kθ) due to this plane wave; 
     2. Compute the Fermat Number Transform (FNT) of A(kθ) from Step 1; 
     3. Compute the inverse FNT of a desirable beam pattern B(kφ), peaking at kφ=ψ; and 
     4. Form the linear vector product between the result of Step 3 and the multiplicative inverse of Step 2. 
     The result is the FNT of the desired beam forming function F(kθ), or at least a first approximation thereof. The beam pattern that results from application of the function F(kθ) must be investigated for its response between values of kθ that are achieved with integral k. The function F(kθ) as derived above will be referred to hereinafter as the &#34;stored beam forming function&#34;. Note that it is the FNT of the stored beam forming function which is actually required for the convolution process. 
     Referring now to FIG. 2, a preferred embodiment of a beam forming system according to the present invention is known in block form. The preferred embodiment utilizes, for the sake of explanation, eight antenna elements 13a-h which may be physically arranged as are the elements 10 of FIG. 1, or in any other suitable configuration. Although eight elements are illustrated, any number of elements which are an integral power of 2 can be used in accordance with a preferred mode of carrying out the present invention. 
     The outputs of the elements 13a-h are fed to respective amplifiers 14a-h whose preferably low-noise high-gain characteristics establish the overall noise figure for the system. The eight signals from amplifiers 14a-h are then fed to respective quadrature detectors 15a-h, of any conventional design, whose ouputs provide a &#34;real&#34; and an &#34;imaginary&#34; component on respective channels Q and I for each array element. 
     Since the FNT is a real transform which operates only on real numbers, the Q and I channels from detectors 15a-h are transformed separately, the eight imaginary number channels I being treated as if they are real. Indicated by reference numeral 16 is a block representation of eight analog-to-digital (A/D) converters followed by an 8-point FNT for digitizing and then transforming the eight real channel components Q from the quadrature detectors 15a-h. Shown at 17 is a similar digitizer/Fermat Number Transform for processing the imaginary components I of the antenna voltages from the quadrature detectors 15a-h. 
     The sum and difference of the real and imaginary components are then formed in a Fermat adder 18 and a Fermat subtractor 19. The sum and difference process is part of a novel method of the present invention of forming the product of two complex vectors, and will be more fully described below. Bitshifters may be necessary to scale the quantities involved, and are not illustrated since, in the present hardwired pipeline convolution processor, the bit shifts may be accomplished by merely relabeling the outputs of a given register as its contents are passed to the next stage, as will be described in greater detail hereinafter. 
     Blocks 20 and 21 represent units which perform binary multiplication modulo a Fermat number. They operate to multiply the processed input from the array by the stored beam function F(kθ). The transformed beam function F(kθ) can reside in a read-only-memory (ROM) 22, or can simply be hardwired. 
     Fermat arithmetic units 23 and 24 also perform part of the novel technique of the present invention for forming a complex vector product. The outputs from units 23 and 24 are the real and imaginary components, respectively, of the transform of the desired output. Following conversion from the Fermat number domain to the time domain in inverse transform units 25 and 26, the beam voltages are ready to be utilized according to the desired application. 
     The object of the pipeline configuration set forth above is to be able to achieve an output that is as close to real time as possible (time for one pass through the processor must therefore be short) and to be able to update the outputs to reflect changes in the inputs (i.e., changes in the signals received by the antenna elements) as quickly as possible (the throughput rate of the processor must therefore be high). As is known in the art, a pipeline processor achieves both of these objectives by permitting data to enter the &#34;pipe&#34; before the previous block of data has exited. The hardware design of the present processor includes banks of latches to store intermediate results and prevent interference between one block of data and the next. Thus, many blocks of data can undergo processing simultaneously. 
     FIGS. 3A and 3B depict in greater detail a preferred implementation of the blocks 16 through 26 of FIG. 2. In FIGS. 3A and 3B, each box labeled A/D comprises a single analog-to-digital converter; each box labeled A comprises a 16-bit adder which adds modulo a Fermat number (hereinafter: &#34;Fermat adder&#34;); each box labeled S comprises a 16-bit subtractor which subtracts modulo a Fermat number (hereinafter: &#34;Fermat subtractor&#34;); each box containing a number represents an end around shift with inverted carry (hereinafter: &#34;Fermat shift&#34;) by the number of bits indicated (recall that this is a matter only of relabeling the lines comprising a bus, not a true hardware shift); each box labeled X comprises a 16-bit Fermat multiplier while the inputs W to the multipliers X represent the terms of the beam forming function; and all buses, which are shown as single lines, comprise 17 bit data lines (16 data bits plus a zero flag bit). 
     Referring now to FIG. 3A, the Fermat adders 18a through 18h and Fermat subtractors 19a through 19h represent the 8-channel Fermat adder and subtractor 18 and 19, respectively, of FIG. 2. The remainder of FIG. 3A depicts the details of the data processors 16 and 17 of FIG. 2. As pointed out above, unit 16 receives the real components of the detected antenna signals, while unit 17 receives the imaginary components. The analog-to-digital converters are indicated in one column by reference numeral 30 of FIG. 3A. 
     The remainder of the circuitry of data processors 16 and 17 is very similar to the familiar FFT butterfly configuration for 8 data points and can be derived from that configuration, as will now be explained. A good explanation of the FFT algorithm, its derivation and its various forms is given in &#34;What is the Fast Fourier Transform?&#34; by W. T. Cochran et al, supra. In particular, FIG. 10 of that article depicts a signal flow graph for the decimation-in-frequency form of the FFT for 8 data points. Because the FFT algorithm can be used in computing the FNT, this signal flow graph is directly applicable to the present computation, as long as the differences between equations (2) and (3) above are kept in mind. (The Cochran article discusses the FFT algorithm only as it applies to computation of the DFT (discrete Fourier transform)). Therefore, multiplications are by powers of α rather than by powers of exp(-2πj/N), and arithmetic is done modulo a Fermat number. Taking this into account, the FNT circuits of FIGS. 3A and 3B can be drawn directly from FIG. 10 of the above Cochran article once a value for α is determined. 
     As explained in the Agarwal article on the FNT, supra, α is determined according to which Fermat number is used and how many data points N are being convolved. In the present embodiment, for the sake of illustration, N is chosen to be 8. The determination of F t  is based on overflow considerations. The dynamic range of an FNT is limited to ±(F t  -1)/2. Thus for F 2  =17, the range is -8 to +8, and 4-bit hardware can be used, as explained by Agarwal. A convenient Fermat number is F 4  =2 16  +1=65537, which allows a range of from -32768 to +32768 and requires 16-bit hardware. The availability of 4-bit arithmetic and logic units (ALU&#39;s) and 16-bit multipliers coupled with this maximum dynamic range makes F=2 16  +1 an attractive choice for the design. 
     α may be determined from the following equation (taken from Agarwal): 
     
         α.sup.N =1 (mod F.sub.t)                             (4) 
    
     For N=8 and F t  =2 16  +1, α is 16, since 16 8  =2 32  =1 (mod 2 16  +1). The latter may be seen to be true as follows: 
     
         2.sup.32 -1=0 (mod 2.sup.16 +1)                            (5) 
    
     
         (2.sup.16 +1)(2.sup.16 -1)=0 (mod 2.sup.16 +1)             (6) 
    
     Since 2 16  +1 divides the left side of equation (6) evenly, the equality is seen to be true. The transistion from equation (5) to equation (6) results from the equality 
     
         (X+1)(X-1)=X.sup.2 -1                                      (7) 
    
     where in this case X=2 16 . 
     Knowing that α=16=2 4 , the transition from the Cochran FFT signal flow graph to FIG. 3A can be made. Where the term exp(-2πj/N) is raised to powers of 0, 1, 2, and 3, the FNT circuit will raise 2 4  to powers of 0, 1, 2 and 3, resulting in factors of 2 0 , 2 4 , 2 8 , and 2 12 . This raising of 2 to powers of 4, 8 and 12 is seen, for example, in the boxes indicated by reference numerals 31, 32 and 33 in FIG. 3A. The pattern of add, subtract and shift follows the FFT in all respects, and is therefore not deemed needful of further explanation. The FNT circuits of units 16 and 17 in FIG. 3A are identical, and result in the formation at the outputs of the adders A and subtractors S, indicated generally by reference numeral 34, of the real and imagainary components of the antenna output signals. 
     Novel aspects of this portion of the present invention are believed to reside in the digital number system used to represent values throughout the processor, in the simplified design of the adders and subtractors, which will be explained below in conjunction with FIG. 4, and in the concept of transforming real and imagainary components separately using a real transform. 
     At this stage of the present inventive technique, the transform of a complex input sequence representing the antenna voltages has been computed and is present at the outputs of adder and subtractor column 34. The next step is to multiply the transformed sequence by the stored beam forming function. The transformed antenna voltages are of the form C i  +jD i  where i ranges from 0 to 7. The term C 0  +jD 0  could be formed from the output of adders 34a and 34b in FIG. 3A. The stored beam forming function is of the same complex form and can be represented by E i  +jF i . 
     The product which must be formed is 
     
         Product=(C.sub.i +jD.sub.i)(E.sub.i +jF.sub.i).            (8) 
    
     This product is formed in a novel way according to the present invention as follows. Rather than compute equation (8) directly, which would require four multiplications and two additions, use is made of the modulus arithmetic aspect of the FNT to reduce the number of multiplications to two. Since 2 16  +1=0 (mod F t ) in this system, 2 16  =-1 (mod F t ) and 2 8  =√-1 (mod F t ). Thus 2 8  behaves much as j, the square root of -1. (For the general case of F t  =2 b  +1, b=2 t , the number which replaces j is 2 b/2 . ) Equation (8) can then be written: 
     
         Product=(C.sub.i +2.sup.8 D.sub.i)(E.sub.i +2.sup.8 F.sub.i). (9) 
    
     Expanding the right side of equation (9) yields: 
     
         (C.sub.i +2.sup.8 D.sub.i)(E.sub.i 2.sup.8 F.sub.i)=C.sub.i E.sub.i +2.sup.16 D.sub.i F.sub.i +2.sup.8 (C.sub.i F.sub.i +D.sub.i E.sub.i). (10) 
    
     One can also form (C i  -2 8  D i ) and (E i  -2 8  F i ), whose product is: 
     
         (C.sub.i -2.sup.8 D.sub.i)(E.sub.i -2.sup.8 F.sub.i)=C.sub.i E.sub.i +2.sup.16 D.sub.i F.sub.i -2.sup.8 (C.sub.i F.sub.i +D.sub.i E.sub.i). (11) 
    
     Taking the sum and difference of equations (10) and (11): 
     
         Sum=2(C.sub.i E.sub.i +2.sup.16 D.sub.i F.sub.i)           (12) 
    
     
         Difference&gt;2·2.sup.8 (C.sub.i F.sub.i +D.sub.i E.sub.i) (13) 
    
     (Equation (12) can be rewritten: 
     
         Sum=2(C.sub.i E.sub.i -D.sub.i F.sub.i)                    (14) 
    
     since 12 16  =-1 (mod F t ). The desired result may be provided by direct expansion of equation (9): 
     
         (C.sub.i +j D.sub.i)(E.sub.i +jF.sub.i)=(C.sub.i E.sub.i -D.sub.i F.sub.i)+j(D.sub.i E.sub.i +C.sub.i F.sub.i),             (15) 
    
     which is seen to be equal to the combination of the right side of equation (14), representing the real part of the answer, and the right side of equation (13) representing the imaginary part of the answer, as long as the factors of 2 in equation (14) and 2·2 8  in equation (13) are removed. 
     The implementation of the operations of equations (10), (11), (13), and (14) in the system of FIGS. 3A and 3B will now be exemplified for a single element C 0  +jD 0  of the transformed antenna voltages, it being clear that the remaining elements are treated similarly. As above, C 0  is the output of adder 34a and D 0  is the output of adder 34b. The term C 0  +2 8  D 0  is formed in adder 18a, while the term C 0  -2 8  D 0  is formed in subtractor 19a. The multiplication of D 0  by 2 8  (which is done, as is all arithmetic in this processor, modulo 2 16  +1), is shown by box 35a which represents a hardwired Fermat shift of 8 bits, as will be explained more fully in conjunction with FIG. 6. Having formed C 0  +2 8  D 0  and C 0  -2 8  D 0 , equations (10) and (11) are implemented in Fermat multipliers 20a and 21a in FIG. 3B. From the foregoing it is clear that the weight W forming the second input term to multiplier 20a must be equal to E 0  +2 8  F 0 , and the weight W forming the second input term to multiplier 21a must be equal to E 0  -2 8  F 0 . These weights W are developed from the discrete beam forming function E i  +jF i , i=0, 1, . . ., 7. The output of multiplier 20a is equal to the right side of equation (10) and the output of multiplier 21a is equal to the right side of equation (11). The sums and differences represented by equations (13) and (14) are created by adders and subtractors indicated generally by reference numeral 36 in FIG. 3B. Thus, adder 36a forms the term 2(C 0  E 0  -D 0  F 0 ), representing the real portion of the product (C 0  +jD 0 )(E 0  +jF 0 ), while subtractor 36b forms the term 2·2 8  (C 0  F 0  +D 0  E 0 ), representing the imaginary portion of the desired product. 
     The shift units indicated generally by reference numeral 37 perform the necessary scaling to remove the factor of 2 from the real portions and the factor of 2·2 8  =2 9  from the imaginary portions. Rather than divide (i.e., shift right) this scaling is depicted as a multiplication. Of course, since the process is only a relabeling of lines, it could be thought of either way. Because 2 32  =1 (mod F t ) for this system, in which F t  =2 16  +1, a multiplication by 2 31  (mod F t ) is equivalent to division by 2. Similarly, a multiplication by 2 23  (mod F t ) is equivalent to division by 2 9  (mod F t ). Therefore, the scaling of the output of adder 36a is performed by the 31-bit shift of box 37a, and the scaling of the output of subtractor 36b is performed by the 23-bit shift of box 37b. Now the entire product has been formed, using only the two multipliers 20a and 21a, rather than the four multipliers which would be required were the product formed directly. 
     The remainder of FIG. 3B depicts the details of boxes 25 and 26 of FIG. 2. The circuits of boxes 25 and 26 are identical; one computes the inverse transform for the real portion of the product of the transformed input signals and the stored function, while the other handles the imaginary portion. In determining the pattern for these inverse transform circuits, the FFT algorithm is again directly applicable. 
     The differences between the inverse FNT and the inverse DFT are the same as the differences between the FNT and the DFT. This can be appreciated from a comparison of the definitions of these inverse transforms. The inverse DFT is ##EQU5## and the inverse FNT is ##EQU6## 
     It may be seen from a comparision of equation (17) with equation (3) that the only differences between the FNT and the inverse FNT are a factor of 1/N and a change of sign in the exponents of α. Thus, the same FFT signal flow graph can be used for the inverse FNT as was used for the FNT. However, as explained in the Cochran FFT article, supra, the form of FFT used in FIG. 3A causes the transformed sequence to emerge in shuffled order, so that if the input sequence is X 0  through X 7 , the output will be in the order of F 0 , F 4 , F 2 , F 6 , F 1 , F 5 , F 3 , F 7 . Therefore, the form of FFT used for the inverse transform must accept such a shuffled sequence and re-sort it back to the original order. The circuits 25 and 26 of FIG. 3B are thus equivalent to the circuits 16 and 17 of FIG. 3A, except for a rearrangement of nodes which is necessary to accomplish the reordering. The circuits of FIG. 3B correspond to FIG. 11 of the Cochran article, the latter being merely a rearrangement of his FIG. 10 which was discussed above in connection with the present FIG. 3A. The boxes indicated generally by reference numeral 38 perform the necessary corrections to satisfy the differences between equations (17) and (3), including dividing by N=8, which can be thought of as a 3-bit Fermat shift to the right, or a 29-bit shift to the left, as discussed above. 
     FIGS. 4, 5 and 6 show the manner in which readily available hardware may be utilized to implement the system of FIGS. 3A and 3B. Due to the availability of TTL-compatible 16-bit multiply chips, such as those manufactured by TRW, Inc., TTL is the logic family chosen for the hardware implementation illustrated. One objective in designing the Fermat hardware is to minimize the amount of logic as much as possible. This object is achieved in the present invention through the use of a special digital number system which allows standard 1&#39;s complement ALU&#39;s to perform arithmetic modulo 2 16  +1. Another object in the hardware design is to eliminate as many multiplications as possible. This is achieved through the use of the FNT to perform the desired convolution and by the special method for forming a complex linear vector product given above. The latter reduces by a factor of 2 the number of multiplications required to be performed, albeit at the cost of increased additions and subtractions. However, this is a desirable tradeoff from the standpoint of cost since presently available 16-bit LSI multiplier chips are very expensive compared to adders and subtractors. 
     An explanation of the special numbering system of the present invention shall now be set forth for the case of four bit hardware, corresponding to F 2  =2 4  +1=17, , it being understood that the following explanation is equally valid for other Fermat numbers. The object is to design an ALU which can add modulo 17. A four bit ALU can handle numbers from -7 to +7 if the most significant bit (MSB) is reserved for use as a sign bit. This is the well known one&#39;s complement representation of numbers and is illustrated in the following table: 
     
         ______________________________________Decimal            BinaryValue              Representation______________________________________+7                 0111+6                 0110+5                 0101+4                 0100+3                 0011+2                 0010+1                 0001+0                 0000-0                 1111-1                 1110-2                 1101-3                 1100-4                 1011-5                 1010-6                 1001-7                 1000______________________________________ 
    
     To perform arithmetic modulo some other number, all results must stay within the range of -7 to +7. one way to do this is to add any carry bits generated into the least significant bit (LSB) of the result as follows: 
     
         ______________________________________4             0100      -5         1010+5            0101      -6         1001         1001                 0011No Carry        0       Carry        1(-6)          1001      (4)        0100______________________________________ 
    
     These examples show that the 4-bit ALU with carry added to the LSB performs arithmetic modulo 15. Similarly, a p-bit ALU with carry added to LSB would perform arithmetic modulo 2 p  -1. 
     To enable the ALU to operate modulo 2 p  +1, the following interpretation of the binary bits of a 4-bit ALU is provided by the present invention: 
     
         ______________________________________New Inter-         Binarypretation          Representation______________________________________+8                 0111+7                 0110+6                 0101+5                 0100+4                 0011+3                 0010+2                 0001+1                 0000-1                 1111-2                 1110-3                 1101-4                 1100-5                 1011-6                 1010-7                 1001-8                 1000______________________________________ 
    
     Zero is represented by adding a fifth bit which, when equal to one, denotes that the number represented is zero. The following examples show that an ALU with carry inverted and added to the LSB operates modulo 17 with this number system: 
     
         ______________________________________ 2        0001+4        00116         0100       1       inverted carry added to LSB     0101      = 64         0011+5        01009         0111       1       inverted carry added to LSB     1000      = -8 = 9 mod 17-5        1011-6        1010-11       0101       0       inverted carry added to LSB     0101      = 6 = -11 mod 17______________________________________ 
    
     Thus a 4-bit ALU with the carry-out signal inverted and added to the LSB of the result performs arithmetic modulo 17 when binary numbers are interpreted according to the special system of the present invention. Similarly, a p-bit ALU with inverted carry added to LSB performs arithmetic modulo 2 p  +1 using the special number system. 
     Multiplication by powers of 2 is accomplished by shifting left with inverted end around carry. The following example illustrates: 
     
         ______________________________________  6 × 2.sup.3 = 6 × 8 = 48 = -3 mod 17  6     =      0101               1011 after one shift               0110 after two shifts  -3    =      1101 after three shifts.______________________________________ 
    
     Thus multiplication by 2 3  simply requires 3 left shifts with bits shifted out of the most significant bit (MSB) being inverted and entered into the vacated LSB. 
     Of course, getting into, and out of, this new number system requires some sort of conversion. This conversion is a simple operation whose details depend on the starting and final number systems to be used. More particularly, the digital representation of all numbers adheres to the formula B=A-1 is the decimal value of A is greater than or equal to 1, or B=A+1 if the decimal value of A is less than or equal to 1,wherein A denotes a one&#39;s complement binary number of p bits, and B denotes a binary number of p+1 bits wherein the (p+1)th bit equals 1 if the decimal value of A is zero. Note in particular that the representation of negative numbers in this system is identical to the representation of negative numbers in the well known two&#39;s complement binary system. In implementing the convolution processor of this invention, the initial conversion into the special number system could be performed automatically by, for example, using analog-to-digital converters designed to convert directly to the new system. 
     Referring now to FIG. 4, one preferred design for a 16-bit ALU using the new number system is shown. The design includes four 4-bit Schottky TTL ALUs, which may be, for example, of the type known as 74S181 manufactured by Signetics. Two such ALUs 40 and 41 are illustrated, although it is understood that a total of four such units are utilized in the embodiment of FIG. 4. The individual ALUs are commonly fed by a Look Ahead Carry Generator (e.g., Signetics type 74S182), designated by reference numeral 42 in FIG. 4. These devices are interconnected in a known fashion to produce a 16-bit ALU. The control lines S0, S1, S2 and S3, designated generally by numeral 43, are set to LHHL if a subtractor is desired, or to HLLH if addition is desired. The two 16-bit inputs to the ALU are denoted A 0  through A 15  and B 0  through B 15 , respectively. The 16-bit output is denoted F 0  through F 15 , and is wired to a set of D-type latches denoted by reference numeral 50. The zero flag bit of each input is A 16  and B 16 , and the zero flag bit of the result is F 16 , which is also latched. The latches 50 may comprise, for example, 74174 or 74175 type TTL circuits. The 74174 provides six D latches with only the non-inverted output available. The 74175 provides four D latches with both output polarities available. Which type is selected for use would depend, inter alia, upon whether the inverted outputs of a particular adder or subtractor in FIGS. 3A and 3B are required. For example, those units followed by a scaler block will need inverted outputs available on at least some bits, since the scaling process involves an inverted end-around shift. All latches for a given ALU are clocked together as soon as the ALU outputs are valid, and are not reclocked until the next block of data passes into the ALU from the latches of the previous ALU and the new result is determined to be valid. The clock signals may be provided by a conventional control system (not shown) for the processor. 
     The present invention modifies the standard TTL 16-bit ALU thus far described by providing the logic gates 44, 45, 46, 47, 48 and 49. The latter elements cause the ALU to add an inverted carry from the look ahead carry generator 42 to the LSB of the ALU, and additionally generate the correct zero flag bit F16 of the ALU output. More particularly, the ALU must have the carry-in input equal to the complement of the carryout in order to perform as defined above. Thus, if A plus B generates an overflow, then there should be no carry-in to the low order ALU 41 or to the look ahead carry chip 42. If A plus B generates no overflow, then a carry-in is desired to ALU 41 and look ahead carry chip 42. The carry-out signal is a function of the generate (G) and propagate (P) outputs of chip 42, where G indicates that the addition of A and B has generated a carry, and P indicates that the state of the ALU after computing A plus B is such that a carry-in would propagate through the ALU to produce a carry-out. The latter is important in determining whether a zero result has occurred, for the only way a zero sum can be obtained is by propagation of a carry through all four ALUs. I have determined that for the 74S182 chip, the true &#34;propagate&#34; condition is given by 
     
         Propagate=GP                                               (18) 
    
     while the true &#34;generate&#34; condition is given 
     
         Generate=GP.                                               (19) 
    
     The carry-in signal is a down-level active signal. Since carry-in should be generated whenever there is not a carry-out generated (except that no carry should be generated if either input is zero (i.e., if A 16  =1 or B 16  =1)), the proper Boolean expression for the carry-in is 
     
         Carry-In=A.sub.16 +B.sub.16 +(G·P)                (20) 
    
     which can be written as 
     
         Carry-In=A.sub.19 ·B.sub.16 ·GP.         (21) 
    
     NAND gate 44 forms GP, and NAND gate 45 forms the carry-in signal which is fed to the lower-order ALU 41 and C and to the carry-in input C in of the look ahead carry chip 42. 
     The output of line F 16  of the ALU represents a zero result, which can occur either as the result of a &#34;propagate,&#34; or as a result of both inputs being zero. Thus the Boolean expression for F 16  is 
     
         F.sub.16 =A.sub.16 ·B.sub.16 +GP                  (22) 
    
     
         F.sub.16 =A.sub.16 ·B.sub.16 +G·Carry-In (23) 
    
     which is implemented by NAND gates 47, 48, 49 and inverter 46. 
     FIG. 5 is a detailed schematic of one of the multipliers represented by the boxes labeled X in FIGS. 3A and 3B. The multiplier illustrated in FIG. 5 must accept two 16-bit inputs and compute their product modulo a Fermat number. This is accomplished in accordance with the present invention by using commercially available two&#39;s complement multipliers as follows: 
     (a) The negative numbers of the special number system are identical to the two&#39;s complement interpretation of these numbers. Therefore, to enter the multiplier with proper magnitudes all non-negative inputs are converted by inverting their bits. 
     This conditional inversion of bits is performed by sixteen exclusive OR gates, only two of which 52a and 52b are illustrated for the sake of simplicity. If bit 15 of the input term is zero, indicating a positive input, then inverter 53 causes the sixteen exclusive OR gates to invert the incoming bits. If the incoming term is negative, it passes the exclusive OR gates unchanged. The weight or stored beam function term 54, which may be stored in, for example, a ROM or simple hardwiring, is connected to the other input of multiplier 51. If it is positive, it is complemented before being stored (or hardwired). The sign bit of the weight 54 is exclusive-OR&#39;ed with the sign bit of the input term in gate 55 to determine the correct output polarity of the product. If the product needs to be inverted, this takes place in a bank of sixteen exclusive OR gates represented by reference numeral 56. Note that this process is inhibited, by means of AND gate 57 and inverter 58, if the zero flag bit of the input term is on. 
     (b) The product from the multiplier will be a positive two&#39;s complement number. In order to reconvert this to the number system used in the present invention, an LSB must be subtracted to achieve the proper magnitude. The circuit of FIG. 5 does this automatically. A 16-bit signed two&#39;s complement multiplier will produce a 30-bit product, with the 31st bit serving as sign bit. To reduce such products modulo 2 16  +1, they are broken up into two 16-bit numbers, both necessarily less than 2 16  +1, and the product P is represented as: 
     
         P=U×2.sup.16 +L                                      (24) 
    
     where L is the number in the 16 less significant bits and U is a 16-bit number obtained by extending the upper 15 bits to the left. Now because 2 16  =-1 (mod 2 16  +1), 
     
         U×2.sup.16 =-U (mod 2.sup.16 +1)                     (25) 
    
     
         P=L-U (mod 2.sup.16 +1).                                   (26) 
    
     Both L and U are two&#39;s complement numbers. Thus, the indicated subtraction, were a two&#39;s complement result desired, would be performed by adding the two&#39;s complement of U to L. The two&#39;s complement of U would be obtained by inverting the bits of U, then adding an LSB. Summing this complement with L would then produced a two&#39;s complement answer that, once again, would require subtraction of an LSB to make its direct interpretation as a number (adhering to the special number system of this invention) have the proper magnitude. If the second step (adding an LSB) is deleted in the formation of the two&#39;s complement of U, then the proper magnitude is automatically obtained. 
     The addition of the inverted bits of U to L is done by a Fermat subtractor 59, which operates modulo 2 16  +1 using inverted end around carry as described above. Application of U to the subtracting input is equivalent to inverting U and adding. 
     FIG. 6 is a table showing how the output lines of a given ALU may be relabeled if the output is to be scaled. This scaling is represented in FIGS. 3A and 3B by a box containing a decimal numeral. The numeral represents the power of 2 which is to be multiplied by the ALU output. As mentioned earlier, this multiplication, which in a general purpose computer would be accomplished by rotating bits in a register, is accomplished in this special-purpose processor by hardwiring. In other words, the bits are merely relabeled. Additionally, the shift must be done modulo a Fermat number. This is accomplished according to the present invention by merely inverting the carry-out bit of an end around shift before the bit is reentered into the LSB of the register. The special number system used herein allows this simple implementation. An example is given above in the discussion of the number system. 
     FIG. 6 depicts how the relabeling may be accomplished. At the left side of the table is shown the latched outputs of an ALU, with both polarities of all signals being available. The power of two which is to be the multiplier is shown across the top of the table. The proper relabeling for a given power of two is determined by the column of numerals immediately below the desired power of two. Thus, for multiplication by 2 1 , which is a one bit shift, Q15 of the input is not used. Rather, Q15 becomes the new bit 0. Q14 becomes new bit 15, Q13 becomes new bit 14, and so on. Note that the inversion of Q15 and relabeling as bit zero is due to the inverted end-around carry coupled with the one bit shift to the left. The technique for other powers of two is similar. For the sake of simplicity, the entries for powers of two between 8 and 11 and for 13 and above are not shown. These can be determined quite easily by extending the table of FIG. 6 to the right. 
     The description of the preferred embodiment of the present invention is now complete. The pipeline processor described above is capable of a throughput rate of roughly one convolution every 200 nanoseconds, the limiting factor being the time required by the 16-bit multiplier chip. If such a high throughput rate is not required in a given application, the transform hardware can be time-shared using well known techniques, thereby reducing the amount of logic required. For example, the real FNT and imaginary FNT could be performed serially by the same transform circuit. Also, because the FNT circuit and the inverse FNT circuit are actually equivalent, having only been redrawn to compensate for the reordering of terms in the sequences as discussed above, single FNT circuit could perform both the FNT and inverse FNT for both real and imaginary portions. Time sharing schemes for reducing the number of individual ALUs within a given FNT circuit can also be designed. Noting that each bank of latches feeds two ALUs, one to form a sum and the other to form a difference, it may be appreciated that if the ALUs to form the differences are eliminated, the remaining ALUs can first form the sums, to be latched into the sum registers, then form the differences to be latched into the difference registers. 
     Obviously, numerous modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.