Transformation circuit arrangement
BACKGROUND OF THE INVENTION 
     The invention relates to a transformation circuit arrangement for the transformation of a first unknown vector (A) with the aid of a matrix (B 2 ). 
     The publication, NTZ Archiv, Volume 3 (1981), No. 1, pages 9 et seq. discloses a novel function system (m functions) and its use for image transmission. With the aid of feedback connected shaft registers, bivalent periodic pulse sequences can be generated. By orthonormalizing these sequences, one obtains an orthonormal, binary function system which serves as the basic function system for a signal transformation (m transformation), similarly to a Fourier or Walsh transformation. 
     SUMMARY OF THE INVENTION 
     It is the object of the invention to provide a transformation circuit which permits the transformation of vectors with little circuitry requirements. 
     This is accomplished by the above circuit arrangement wherein the matrix B 2  has rows containing a second, known vector B which is cyclically shifted by rows, both vectors having the same length, and the first vector (A) is divided into partial vectors (P), with each partial vector (P) being fed to an addition circuit and to a longic linkage circuit connected to the outpur of the addition circuit. The output of the linkage circuits are fed to a calculating unit which furnishes the transform T as the result. Advantageous modifications of the invention are discussed further hereunder. 
     According to the invention, a known vector is transformed with an unknown vector in that a matrix is constructed in such a manner that a known vector is cyclically shifted by rows and thus a square matrix is constructed. The transformation of the unknown vector with the thus constructed matrix produces the transform in a manner realizable with simple circuitry. According to the invention, it is assumed that both vectors have the same length N and the vector constructing the matrix is present in binary form. To simplify the circuit, the unknown vector is divided into P groups each having a length R. The length of the groups is less than the total length of the vector. The last group is filled cup with zeros if necessary. 
     For parallel processing, the individual groups are each fed to an addition stage. The output signals are fed to a calculating unit, via logic linkage circuits, e.g. simple wiring, and possibly--if the linkage circuits all have the same configuration--with a decording circuit switched according to a modulo-R relationship connected to their outputs. The number to be divided in the modulo-R relationship is the number of elements of the water. The calculating unit furnishes the transform as its output signal. 
     If the individual groups are processed serially, only one addition stage and one linkage circuit are required. The individual groups of the vector are multiplexed and fed, in accordance with the modulo-R relationship, to an accumulator and to a register which furnishes the transform. 
     With these measures it is accomplished that circuitry requirements are reduced since the multiplication operations to be performed of necessity are not required for the transformation of vectors. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a better understanding of the invention, the invention will be described in greater detail below with reference to the drawing figures. It is shown in: 
     FIG. 1, a block circuit diagram for parallel processing; 
     FIG. 2, the structure of an addition circuit; 
     FIG. 3, a formation law for the addition circuit; 
     FIG. 4, a square matrix; 
     FIG. 5, a formation law for the linkage circuit; 
     FIG. 6, a logic linkage circuit; 
     FIG. 7, a calculating unit; and 
     FIG. 8, a block circuit diagram for serial processing. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     FIG. 1 is a block circuit diagram for the transformation of an unknown vector A with the aid of a matrix B 2 . All elements a(i) of vector A are read into a buffer 1 and are there divided into groups 2, 3, and 4, each having a length R. Groups 2, 3, 4 are each fed to an addition circuit 5, 6, 7. The outputs of addition circuits 5, 6, 7 are each connected with a logic linkage circuit 8, 9, 10. The outputs of linkage circuits 8-10 lead to a calculating unit 11, at whose outputs the transform is present. The characteristic properties of matrix B 2  are inherent in the wiring of the linkage circuits 8-10 (VO, VN, VP-1). 
     FIG. 2 shows the configuration of an addition circuit 5. According to the R elements of a group P, where P=INT(N/R)+1, addition circuit 5 has R inputs a(n*R+m), where m=0, 1, . . . , R-1 and n is the sequence number of the respective addition and linkage circuits, n=1, . . . , P, for the a(Rn+m) elements of vector A. Addition circuit 5 furnishes output signals U(n, j). It is configured according to the following mathematical relationship: ##EQU1## where x(j,m) is the m th  digit of the binary representation of number j, j=2exp(R)-1, a(R(n)) is the n th  group of the unknown vector and m is the ranking in the group. 
     The determination of 2exp(R) values u(n,j) in each block U(n) requires only 2exp(R)-R-1 additions since the following condition is met: 
     (a) determination of u(n,0), u(n,0)=0; 
     (b) determination of u(n,j) for j=2exp(m), u(n,j)=a(R*n+m); 
     (c) determination of n(n,j) which are the sum of two u(n,j) according to (b); the binary representation of j then has two digits of the value &#34;1&#34;; 
     (d) determination of u(n,j) expressed as the sum of two values according to (b) and (c); the binary representation of j then has three digits of the value &#34;1&#34;; 
     (e) etc. 
     FIG. 3 shows as an example the output U(n,0) shown in FIG. 2 with zero. Outputs U(n,1), U(n,2) and U(n,4) are connected directly with inputs a(n*R), a(n*R+1) and a(n*R+2). The result of the addition of a(n*R) and a(n*R+1) is present at output U(n,3); the addition result of a(n*R) and a(n*R+2) is present at output U(n,5), the addition result of a(n*R+1) and a(n*R+2) is present at output U(n,6) and the addition result of all inputs is present at output U(n,7). 
     FIG. 4 shows the square matrix B 2  of the known vector B (0 1 1 0 1) which is cyclically shifted by rows. The cyclical shift of vector B is continued beyond the range of square matrix B 2  if the N elements of vector A are not a whole number multiple of the R group elements. The number of rows is increased to the number of the whole number multiple of the R group elements according to the mathematical rule 
     
         R-(n-int(n/R)*R).                                          (3) 
    
     FIG. 5 shows the formation law for a logic linkage circuit. For the build-up of the linkage circuit, the R rows of matrix B 2  corresponding to the G group are utilized to link inputs U(n,j) with outputs d(n,k) of linkage circuit 8. The inputs and outputs are linked according to the following mathematical relationship: ##EQU2## where b are the column elements of matrix B 2 , j(k) are the inputs and k the outputs of linkage circuit 8. Inputs j(k) are connected to outputs k of linkage circuit 8 according to Equation 2. Example: column k=0 has a value, at R=3, of 0*2 0  +1*2 1  +1*2 2 . The result obtained is that input U(n,6) must be connected with output d(n,0); j(k) forms a structure which is called a V structure. This structure is independent of n. That means that all linkage circuits 8-10 have the identical configuration. 
     However, for parallel processing it is also possible to build a corresponding linkage circuit for each group G according to Equation 2, with the rows of matrix B 2  employed being each shifted by R rows. If the last group of vector A has been filled up with zeros, the rows of matrix B 2  will be expanded, according to its formation law, by the number of zeros. Then the corresponding outputs of linkage circuits 8-10 can be fed directly to an accumulator. 
     FIG. 6 shows the circuit configuration of the linkage circuit according to the formation law of Equation 2. Outputs d(n,k) are connected with inputs U(n,j) according to the formation law of FIG. 5. 
     If during parallel processing all linkage circuits 8-10 have the same configuration, the outputs of the individual linkage circuits 8-10 will be added according to n-int(n/R)*R to the preceding linkage circuit 8-10. As can be seen in FIG. 7, output d(0,0) of linkage circuit 8 is added to output d(1,2) of linkage circuit 9 and to output d(2,4) of linkage circuit 10. The transform T(0)-T(5) is present at the output of adders 18-23. However, since vector A is multiplied not only with vector B, but also with its square matrix B 2 , transform T must also be divided by the number N of elements of vector B. If, during parallel processing, a corresponding linkage circuit is constructed according to Equation 2 for each group, the corresponding outputs d(0,0) will be linked with d(1,0) and d(2,0); d(0,1) will be linked with d(1,1) and d(2,1); d(0,2) will be linked with d(1,2) and d(2,2); etc. and the values at the outputs are added to one another. 
     For serial processing according to FIG. 8, vector A is shifted into a buffer 25 having a length R. The outputs of buffer 25 are connected with addition circuit 5. This is followed by a linkage circuit 8. According to the formation law of n*(N-int(n/R)*R), outputs d(n,k) are fed, with the aid of multiplexers 36-40, to accumulators 26-30. The outputs of the accumulators are connected with registers 31-35. Buffer 25 and accumulators 26-30 are loaded after a time interval T(G), registers 31-35 after a time interval P*T(G). Multiplexers 36-40 are likewise shifted on at time intervals T(G). At the end of the calculating operation, transform T is present at the output of registers 31-35. Simultaneously with loading of registers 31-35, accumulators 26-30 are set back. 
     This circuit arrangement is used, for example, for the performance of a discrete correlation or a discrete convolution. The circuit arrangement can be used as a discrete cyclic correlator for the pulse detection during code multiplex transmission, for time delay measurements in radar, as a matched filter, to perform m-transformations in image processing, in cryptographic applications, for error corrections or for synchronization of digital data transmissions. 
     If a plurality of circuit arrangements according to FIGS. 1 and/or 8 are connected in parallel, a plurality of vectors and/or matrixes can be transformed into one spectral region. 
     The following circuits can be used as modules for a digital correlator: one or a plurality of D flip-flops in the form of integrated circuits &#39;74 can be used for reference numeral 1 (see in this connection, Texas Instruments, &#34;The TTL Data Book&#34;). One or a plurality of full adders &#39;83 can be used for reference numerals 5, 6 and 7, for reference numerals 8, 9 and 10 the conductor paths are linked or wired accordingly. For reference numeral 11, one or a plurality of full adders &#39;83 or &#39;283 can be used, for reference numerals 12-23, again one or a plurality of full adders &#39;83 or &#39;283, for reference numeral 25, one or a plurality of serial in/parallel out integrated circuits &#39;95 or &#39;164, for reference numerals 26-30, one or a plurality of D flip-flops and full adders &#39;74 and &#39;83, for reference numerals 31-35 one or a plurality of D flip-flop &#39;74 and for each of reference numerals 36-40 one or a plurality of multiplexers &#39;53. The &#34;&#39;&#34; here stands for TTL circuits belonging to families 74, 74LS, 74ALS, 74S, etc. For example, a module &#39;00 corresponds to integrated circuits 7400, 74LS00, 74AS00 or 74S00.