Orthogonal sequence generator and radar system incorporating the generator5136611Abstract An apparatus, usable in a radar system, for generating a multi-value orthogonal sequence includes a multi-element M-sequence generator and a component substituting device. The M-sequence generator comprises a shift register including delay elements, multipliers for multiplying signals output from the respective delay elements by feedback factors, and adders for adding the multiplied signals from the multipliers to provide the result to the delay element initially arranged in the shift register whereby a multi-element M-sequence is generated from the delay element finally arranged therein. The substituting device comprises a microcomputer and substitutes each component of the M-sequence with one of complex-numbers, Z.sub.o, Z.sub.i, . . . Z.sub.q-1 in such a manner that when the component is 0, it is substituted with Z.sub.o .noteq.0, and when it is .epsilon..sup.i (i=1, 2, . . . q-1), it is substituted with z.sub.i, where the set of Z.sub.o, Z.sub.i, . . . Z.sub.q-1 are the solution of the following simultaneous algebraic equations: ##EQU1## where mod.sub.q-1 (.multidot.) represents a calculation of modulo (q-1) and is expressed as q-1 when the result is 0, * represents a complex conjugate, and r=1, 2, . . ., q-2. Claims What is claimed is: Description BACKGROUND OF THE INVENTION
TABLE 1
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Calculation Modulo 3 (q = 3)
(a) Addition (b) Multiplication
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+ 0 1 2 .multidot.
0 1 2
0 0 1 2 0 0 0 0
1 1 2 0 1 0 1 2
2 2 0 1 2 0 2 1
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With the linear feedback shift register 20 arranged as mentioned above, the sequence {.alpha..sub.n } created by applying initial values other than all zero to the delay elements 22a-22c is known as a multi-element M-sequence. The multi-element M-sequence has the largest period among the sequences created from the linear feedback shift register. In the embodiment shown in FIG. 11, the delay elements in the linear feedback shift register are arranged at three stages, but they can be extended over k-stages. However, it is necessary to satisfy some limited combinations of the feedback coefficient values in order to create the multi-element M-sequence with the k-stage linear feedback shift register. Such combinations have already been obtained and an example combination is shown in FIG. 11A. Since the feedback coefficient h.sub.i is the element of GF(q), the elements other than 0 are expressed by the following expression using the primitive element .epsilon., and thus in FIG. 11A, 0-element is expressed using 0 and other elements are expressed using m.sub.i, h.sub.i =.epsilon..sup.m i(m.sub.i =1,2, . . . ,q-1) (34) The operation of the component substituting unit 21 will be described. The component substituting unit 21 is provided with means for substituting a component .theta..sub.n for the component .alpha..sub.n of the multi-element M-sequence {.alpha..sub.n } created by the linear feedback shift register 20. The substitution operation is performed in accordance with a program of the microcomputer as the substitution means. FIG. 12 is a flowchart showing the operation of the component substituting unit 21 of the embodiment. At step 25, the component .alpha..sub.n of the multi-element M-sequence {.alpha..sub.n } transferred from the linear feedback shift register 20 is successively input to the substituting unit 21, and then at steps 26a-26c, the unit 21 determines whether the component .alpha..sub.n is 0, .epsilon., . . . or .epsilon..sup.q-1. If .alpha..sub.n =0, then at step 27a, it is set at a complex number of .theta..sub.n =z.sub.0, and if .alpha..sub.n =.epsilon..sup.i (i=1, 2, . . . , q-1), then at steps 27b-27c, it is set at a complex-number .theta..sub.n =z.sub.i. In any case, at the subsequent step 28, the obtained complex-number .theta..sub.n which is one of (z.sub.0, z.sub.1, . . ., z.sub.q-1) is output to the exterior. The complex-numbers z.sub.0, z.sub.1, . . . , z.sub.q-1 are the solution of the simultaneous equations indicated by the following expressions which are the same as expressions 30 and 31 and preliminarily determined and stored in the microcomputer of the component substituting unit: ##EQU21## Following is the explanation of the sequence {.theta..sub.n } created in the above manner being an orthogonal sequence. For this purpose, the basic properties of a multi-element M-sequence {.alpha..sub.n } (simply referred to as M-sequence hereinbelow) shown in "Code Theory (Third Edition)" by Miyakawa, Iawdare, et al., Shokodo, 1976, will first be described: (1) The period N of M-sequence is N=q.sup.k -1. (2) In one period of the M-sequence, a (q.sup.k-1 -1)-number of 0-elements is included and a (q.sup.k-1)-number of other elements .epsilon..sup.i (i=1, . . . , q-1) is included. (3) If a set of the sequence having a length N including one period (.alpha..sub.0, . . . , .alpha..sub.1, . . . , .alpha..sub.N-1) of the M-sequence and its all of the cyclic permutations is added by (0, . . . , 0), the longest sequence code having a code length of N=q.sup.k -1 is obtained. (4) The longest sequence code is an equi-distance code having an equal signal distance between two arbitrary codes, and the signal distance is expressed by the following expression: d.sub.H =q.sup.k-1 (35) If two codes are indicated as A=(a.sub.0, a.sub.1, . . . , a.sub.N-1) and B=(b.sub.0, b.sub.1, . . . , b.sub.N-1), the signal distance d.sub.H is defined as follows: ##EQU22## (5) The longest sequence code forms a k-dimensional linear vector space in GF(q), where k is the number of stages of the shift register. (6) The basic vector of the k-dimensional linear vector space can obtained as follows: The primitive element .alpha. in GF(q.sup.k) is expressed by the following column vector: .alpha.=(a.sub.11,a.sub.21, . . . ,a.sub.k1).sup.T (37) where, a.sub.11, a.sub.21, . . . , a.sub.k1 .epsilon. GF(q), that is: a.sub.11 -a.sub.k1 are elements in GF(q). Its power vectors .alpha., .alpha..sup.2, . . . , .alpha..sup.N (N: period of M-sequence) are arranged to create the following matrix g. ##EQU23## The k-number of column vectors v.sub.i (i=1, 2, . . . , k) of the matrix g is the basic vectors. ##EQU24## (7) .epsilon. is the primitive element of GF(q) and t is an integer defined as t=(q.sup.k -1)/(q-1). If g' is put as indicated below, g'=[.alpha.,.alpha..sup.2, . . . ,.alpha..sup.t ] (40) then the matrix g of expression 38 is expressed as follows:ps g=[g',.epsilon.g',.epsilon..sup.2 g', . . . ,.epsilon..sup.q-1 g'](41) (8) When the longest sequence code other than 0=(0, 0, . . . , 0) corresponds to the vector expressed by the following expression: ##EQU25## B.sub.n (n=0, 1, . . . , N-1) can be expressed by a linear combination of the basic vectors v.sub.i (i=1, 2, . . . , N-1) as follows: B.sub.n =c.sub.1 (n)v.sub.1 +c.sub.2 (n)v.sub.2 + . . . +c.sub.k (n)v.sub.k(43) (n=0,1, . . . ,N-1) where, c.sub.1 (n), c.sub.2 (n), . . . , c.sub.k (n).epsilon.GF(q). The number of combinations of c.sub.1 (n), c.sub.2 (n), . . . , c.sub.k (n) is q.sup.k -1, because of excluding c.sub.1 =c.sub.2 = . . . =c.sub.k =0. The number q.sup.k-1 coincides with the period of the M-sequence. For example, if k=2, h.sub.0 =2 and h.sub.1 =1, then the content of one period of the M-sequence in GF(5) having a period length of 24=5.sup.2 -1 is as follows: (014434023313041121032242) (44) It is apparent that the M-sequence representable by expression 44 satisfies the above basic properties (1), (2) and (4). In this case, the matrix g of expression 38 is as follows: ##EQU26## It is easily confirmed that all the vectors corresponding to expression 42 are obtained by the linear combination of v.sub.1 with v.sub.2. It is further noted that one period of the M-sequence of expression 44 can be divided into four quite similar blocks each having six-length, (014434), (023313), (041121) and (032242), that is: the second block (023313) is the first block (014434) multiplied by 2, and the third and fourth blocks are the first block multiplied by 4 and 3, respectively. It is appreciated that since the primitive element of GF(5) is 2, the above basic property (7) of the M-sequence is satisfied. In order to show the sequence {.theta..sub.n } created by the component substituting unit 21 being an orthogonal sequence, the autocorrelation function .rho.(m) of the sequence {.theta..sub.n } at m.noteq.0 must first be determined and then the condition under which the sequence {.theta..sub.n } is the orthogonal sequence, i.e., .rho.(m)=0 (m=1, 2, 3, . . . , N-1) must be approved. The autocorrelation function .rho.(m) of the sequence {.theta..sub.n } at m.noteq.0 is expressed by the following expression using the basic property (2) of the M-sequence: ##EQU27## where, K.sub.m (i,j) (i, j=0, 1, . . . , q-1) is the number of the terms z.sub.i z.sub.j contained in the autocorrelation function .rho.(m) and thus is a function of m. Since the sequence {.theta..sub.n } is obtained by substituting it for the M-sequence {.alpha..sub.n }, K.sub.m (i,j) can be considered as follows: The M-sequence {.alpha..sub.n } and all the sequences obtained by cyclically permutating it are expressed corresponding to the vector B.sub.n (n=0, 1, . . . , N-1) as indicated by expression 42, and the values of the components, other than 0, of B.sub.o and B.sub.n are expressed by the power .epsilon..sup.i of the primitive element .epsilon. of GF(q). Now, as shown by 0.fwdarw.0 and .epsilon..sup.i .fwdarw.i (i =1, 2, . . . , q-1), when the values of components other than 0 are expressed by symbolizing them with the number of the power i, then K.sub.m (i,j) (i, j=0, 1, . . . , q-1) can be defined as the number of components in which the value of the component of the vector B.sub.0 is i and the value of the component of the vector B.sub.m is j. The values of K.sub.m (i,j) so defined will now be obtained using the basic property (5), i.e., the vector B.sub.n (n=0, 1, . . . , N-1) indicated by expression 42 forms the k-dimensional linear vector space. First, the vector B.sub.m (m=0, 1, . . . , N-1) is expressed by the following expressions using a suitable basic vector v.sub.i (i=1, 2, . . . , k) of the k-dimensional linear vector space based on the basic property (8): ##EQU28## where, c.sub.1 (m), c.sub.2 (m), . . . , c.sub.k (m).epsilon.GF(q). Even when the vector B.sub.0 is expressed by expression 48a, the generality is not lost. This is because the condition that v.sub.1 through v.sub.k are the basic vectors is v.sub.1 through v.sub.k being linearly independent and thus the k-number of basic vectors can be selected such that one of them is B.sub.0 with the property of the linear vector space. As is clear from expressions 48a and 48b, among an (N-1)-number of vectors B.sub.m (m=1, . . . , N-1), a (q-2)-number of vectors B.sub.m =c.sub.1 (m)v.sub.1 (in the case of c.sub.2 (m)=c.sub.3 (m)= . . . =c.sub.k (m)=0) is linearly dependent upon B.sub.0 the other N-1-(q-2) number of vectors are linearly independent of B.sub.0. First, the case of B.sub.0 and B.sub.m being linearly dependent will be considered. When B.sub.m linearly depending upon B.sub.0 is expressed by B.sub.m ' as follows: B.sub.m '=c.sub.1 (m').multidot.v.sub.1 =c.sub.1 (m').multidot.B.sub.0(49) where, c.sub.i (m') is an element of GF(q) other than 0 and 1. Thus, c.sub.1 (m') is expressed by the following expression using the primitive element .epsilon.. c.sub.1 (m')=.epsilon..sup.r (r=1,2, . . . ,q-2) (50) The relation between m' and r is expressed by the following expression based on the basic property (7) of the M-sequence: m'=(q.sup.k -1)r/(q-1) (51) (r=1,2, . . . ,q-2) Using expressions 49 and 50, K.sub.m (i,j) in the case of B.sub.0 and B.sub.m being linearly dependent is expressed by the following expression with reference to the basic property (2) of the M-sequence: ##EQU29## where, m=m'=(q.sup.k-1 -1)r/(q-1) (r=1, 2, . . . , q-2), and mod.sub.q-1 is a process operation modulo q-1 and is expressed as q-1 when the result is 0. The case of B.sub.0 and B.sub.m being linearly independent will now be considered. Since the mutually linearly independent vectors can be the basic vectors of the linear vector space, K.sub.m (i,j) regarding B.sub.0 and B.sub.m are the same as those between the basic vectors. This is because they possess an identical property, when the vectors are basic vectors. The basic vectors v.sub.i (i=1, 2, . . . , k) are the column vectors of the matrix g of expression 37, as indicated in the basic property (6) of the M-sequence. On the other hand, the matrix g has the arrangement of the column vectors of the N (=q.sup.k -1)-number of k-dimensional vectors .alpha., .alpha..sup.2, . . . , .alpha..sup.N having the elements of GF(q) as components. Since .alpha. is the primitive element of GF(q.sup.k), .alpha., .alpha..sup.2, . . . , .alpha..sup.N are all mutually different vectors. Thus, .alpha., .alpha..sup.2, . . . , .alpha..sup.N can one-by-one correspond to any one of the N (=q.sup.k -1)-number of vectors formed by arranging a k-number of elements picked up from the q-number of elements of GF(q) except the zerovector. From this, it is appreciated that in the combination of arbitrary basic vectors v.sub.x, v.sub.y (x.noteq.y), K.sub.m (i,j) is equal to the number of the permutations arranged by the (k-2)-number of elements picked up from the q-number of the elements of GF(q). From the above-mentioned consideration, in the case of B.sub.0 and B.sub.m being linearly dependent, K.sub.m (i,j) can be obtained as indicated by the following expression, independent of m: ##EQU30## where, m=1, 2, . . . , N-1, and m.noteq.m'=(q.sup.k-1 -1)r/(q-1) (r=1, 2, . . . , q-2). By substituting expressions 52 and 53 for expression 47, the following expression is obtained: ##EQU31## Since the condition under which the sequence {.theta..sub.n } is an orthogonal sequence is .rho.(m)=0 (m=1, 2, . . . , N-1), the following expressions are obtained by setting the right side of expressions 54 and 55 at 0: ##EQU32## Since expressions 56 and 57 are nothing but expressions 31 and 30, the foregoing prove that the sequence {.theta..sub.n } is an orthogonal sequence. An example of the solution of the simultaneous equations 30 and 31 of the embodiment will now be described. The simultaneous equations 30 and 31 of the proposition contain a q-number of unknown numbers z.sub.o, z.sub.1, . . . , z.sub.q-1. Since the number of equations is q-1, on the other hand, the solution of the simultaneous equations 30 and 31 is an indefinite solution. The solution can be obtained by setting one of the unknown values z.sub.o through z.sub.q-1 as a constant. Herein z.sub.o is set as a constant (=C). In this case, if z.sub.o =C=0, a trivial solution of z.sub.1 =z.sub.2 = . . . =z.sub.q-1 =0 is obtained, but this trivial solution has no physical importance and thus C is set to C.noteq.0. For z.sub.1, . . . , z.sub.q-1, further, a change of variables is performe as follows: x.sub.i =z.sub.i+1 /C(i=0,1, . . . ,q-2) (58) Then, the simultaneous equations 30 and 31 are expressed by the following expressions: ##EQU33## In expression 60, it is tacitly understood that the convolution integration .SIGMA. is cyclic, and the symbol mod.sub.q-1 (.multidot.) representing the process operation modulo q-1 is omitted. Hereinbelow, the convolution integration is all to be cyclic and thus such a symbol is omitted. First, the following expression is obtained by substituting expression 60 for expression 59: ##EQU34## Then, by setting L.sub.0 =q-1, the following discrete Fourier transformation is defined: ##EQU35## .vertline.X.sub.m .vertline..sup.2 can be expressed as indicated by the following expression using the property of the discrete Fourier transformation: ##EQU36## The following expression is obtained by substituting expressions 60 and 61 for expression 63. ##EQU37## Rewriting expression 63 using these relations 65 and 66, the following expressions are obtained. .vertline.X.sub.0 .vertline..sup.2 =-(X.sub.0 +X.sub.0 *)+(-q.sup.k-1 +q)/q.sup.k-1 (67) .vertline.X.sub.m .vertline..sup.2 =-(X.sub.0 +X.sub.0 *)+(q.sup.k -2q.sup.k-1 +1)/q.sup.k-1 (68) (m=1,2, . . . ,L.sub.0 -1) If expression 67 is deformed and arranged, it will be appreciated that X.sub.o is expressed by an arbitrary point on a circle having a center point of (-1, 0) and a radius of 1/.sqroot.q.sup.k-2, on the complex plane. On the other hand, using the arbitrary content K.sub.m (m=1, 2, . . . , q-2), X.sub.m (m=1, 2, . . . , q-2) is feed from expression 68 as follows: X.sub.m =[-(X.sub.o +X.sub.o *)+(q.sup.k -2q.sup.k-1 +1)/q.sup.k-1 ].sup.1/2 .multidot.exp (jK.sub.m) (69) In the above-mentioned principle, the solution of the simultaneous equations 30 and 31 can be determined in accordance with the procedure of a flowchart shown in FIG. 13A. The flow through the flowchart will be described. (a) At step 35, the value of one arbitrary point X.sub.o on a circumference having a center point of (-1, 0) and a radius of 1/.sqroot.q.sup.k-2 on the complex plane, as shown in FIG. 13D. (b) As step 36, the value of X.sub.m (m=1, . . . , q-2) is then set in accordance with expression 69, where K.sub.m is an artitrary real constant. (c) As step 37, inverse-discrete-Fourier transformation of X.sub.o and X.sub.m (m=1, . . . , q-2) is then performed to obtain x.sub.i (i=0, 1, . . . , q-2) as follows: ##EQU38## (d) At step 38, z.sub.i (i=0, . . . , q-1) is then determined by multiplying x.sub.i with a complex constant value c (.noteq.0). ##EQU39## In expression 72, X.sub.o is a complex constant satisfying the following expression 73: .vertline.X.sub.o +1.vertline..sup.2 =l/q.sup.k-2 (73) As described above, the orthogonal sequence created by the orthogonal sequence generator of the present invention can be obtained by substituting the solution z.sub.i (i=0, . . . , q-1) of simultaneous equations 30 and 31 for the component .alpha..sub.n of the multi-element M-sequence from the generator 20, and thus the period of the orthogonal sequence is the same as the period N (=q.sup.k-1) of the multi-element M-sequence, and each component of the orthogonal sequence has a multi-value of any one of z.sub.o, z.sub.1, . . . , z.sub.q-1. In this sense, the orthogonal sequence created by the orthogonal sequence generator of the present invention will be called a complex multi-value orthogonal sequence. As a practical example of the complex multi-value orthogonal sequence created by the orthogonal sequence generator of the present invention, the case of q=3, k=3 and N=26 is shown below: (11100202122102220010121120) The above example designates one period of the complex multi-value orthogonal sequence. Each figure (0, 1, 2) is a symbol and the corresponding complex-number z.sub.i is as follows: 0.fwdarw.z.sub.o =1.0 1.fwdarw.z.sub.1 =-1.38+j0.289=1.412exp(j168.pi./180) 2.fwdarw.z.sub.2 =0.382+j0.289=0.479exp(j37.pi./180) (74) FIG. 14A is a vector diagram of the components of the practical example of the above-described complex multi-value orthogonal sequence. Modifications of the orthogonal sequence generator as described above will next be described. As explained above, the solution (z.sub.0, z.sub.1, . . . , z.sub.q-1) of simultaneous equations 30 and 31 is represented by expressions 71 and 72 and the component substituting unit 21 converts the component .alpha..sub.n of the multi-element M-sequence {.alpha..sub.n } from the shift register 20 into one of z.sub.0, . . . , z.sub.q-1. These complex-numbers z.sub.0, z.sub.1, . . . , z.sub.q-1 contain X.sub.o and K.sub.m which are not uniquely determined, and thus there exists an infinite number of solutions depending upon the determination of the constants X.sub.o and K.sub.m. In one of the modifications, the component substituting unit 21 is so arranged that the absolute values .vertline.z.sub.0 .vertline., .vertline.z.sub.1 .vertline., . . . .vertline.z.sub.q-1 .vertline. of z.sub.0, z.sub.1, . . . , z.sub.q-1 are all equal by determining X.sub.o and K.sub.m uniquely, while in the other modification, the unit 21 is so arranged that the arguments of z.sub.0, z.sub.1, . . . , z.sub.q-1 are all equal by determining X.sub.o and K.sub.m uniquely. As to the first modification, X.sub.o and K.sub.m with which all the absolute values of z.sub.0, z.sub.1, . . . , z.sub.q-1 are equal are given by the following expressions: ##EQU40## Simultaneous equations 77 can analytically be solved as shown below, when q is relatively small: ##EQU41## If q is large, it will be difficult to analytically solve expression 77, but it will be possible to analytically solve it using a numerical analysis, such as well-known Newton-Raphson method and so on, and to obtain practical numerical values of K.sub.m (m=1, . . . , q-2). Thus, the solution of the simultaneous equations 30 and 31 in which all the absolute values are equal is determined in accordance with the procedure shown in FIG. 13B. To obtain equal absolute values of z.sub.0, z.sub.1, . . . , z.sub.q-1 with X.sub.o and K.sub.m satisfying expressions 75 or 76 and 77 will be described below. (a) First at step 39, X.sub.o is calculated from expression 75 or 76. (b) At step 40, K.sub.o, .gamma..sub.o and .gamma. are calculated from expressions 78-80. (c) At step 41, by solving simultaneous equations 77, K.sub.m is obtained. (d) At step 42, X.sub.m is calculated from the expression X.sub.m =.gamma.exp(jK.sub.m). (e) At step 43, x.sub.i is determined by the following inverse-discrete-Fourier transformation of X.sub.m : ##EQU42## (f) Finally, at step 44, z.sub.i+1 is obtained by the following transformation (expressions 71 and 72): z.sub.0 =C(.noteq.0) z.sub.i+1 =Cx.sub.i (i=0,1, . . . ,q-2) An explanation why the absolute values of z.sub.0, z.sub.1, . . . , z.sub.q-1 become equal when X.sub.o and K.sub.m satisfy expressions 75 or 76 and 77 is as follows. The fact that the absolute values of z.sub.0, z.sub.1, . . . , z.sub.q-1 are equal is expressed as below, resulting from the relation between expressions 75 or 76 and 77. From expressions 71 and 72, ##EQU43## Thus, it is equivalent to the fact that the absolute values of x.sub.i (i=0, 1, 2, . . . , q-2) are 1. On the other hand, the left side, .vertline.x.sub.i .vertline..sup.2, of expression 85 can be rewritten, using the relation of expression 70, as follow: ##EQU44## The transformation from expression 86a to expression 86b utilizes the fact that m=n+n', and the summation .SIGMA. is cyclic. Both sides of expression 86b are then subjected to discrete-Fourier-transformation to obtain the following expression: ##EQU45## The transformation from expression 87a to expression 87b utilizes the following relation: ##EQU46## From expression 85, .vertline.x.sub.i .vertline..sup.2 =1 (i=0, . . . , q-2), and then expression 87b can be expressed by the following expression: ##EQU47## Whereupon, if X.sub.0 =.gamma..sub.0 exp(jK.sub.0) (90) X.sub.m =.gamma.exp(jK.sub.m)(m=1,2, . . . ,q-2) (91) then, expression 89 can be reformed as follows: .gamma..sub.0.sup.2 +(q-2).gamma..sup.2 =(q-1).sup.2 (92) Thus, expression 77 is obtained. On the other hand, the following relations are established by comparing expressions 67 and 68 with expressions 90 and 91, respectively: ##EQU48## By simultaneously solving expressions 92, 93 and 94, expressions 78, 79 and 80 are obtained. From the foregoing, it will be appreciated that X.sub.0 and K.sub.m (m=1, 2, . . . , q-2) with which the absolute values of z.sub.0, z.sub.1, . . . , z.sub.q-1 are equal are given by expressions 75 or 76 and 77. An microprocessor, for example the microcomputer in the component substituting unit 21 previously calculates such values of X.sub.0 and K.sub.m from expressions 75 or 76 and 77, and in accordance therewith, and determines z.sub.0, z.sub.1, . . . , z.sub.q-1 so as to have the same absolute value. The obtained z.sub.0, z.sub.1, . . . , z.sub.q-1 are then preset into the component substituting unit 21 and then the unit 21 can generate such a orthogonal sequence components of which have the same absolute value, in accordance with the steps shown in FIG. 12. A practical example of the orthogonal sequence created by the first modified orthogonal sequence generator is shown below, in the case of q=3, k=3 and N=26: (11100202122102220010121120) The above example indicates one period of the orthogonal sequence. Each figure (0, 1, 2) is a symbol and the corresponding complex-number z.sub.i is as follows: 0.fwdarw.z.sub.0 =1.0 1.fwdarw.z.sub.1 =j.multidot.1.0=1.0exp(j90.pi./180) 2.fwdarw.z.sub.2 =-0.901+j.multidot.0.433=1.0exp(j205.7.pi./180)(95) FIG. 14B is a component vector diagram of an example of the orthogonal sequence generated by the modification. As shown in FIG. 14B, the absolute values of the components of the created orthogonal sequence are identical with another, and in this sense this orthogonal sequence can be called as a polyphase orthogonal sequence. Thus, with the orthogonal sequence generator of the first modification can create a polyphase orthogonal sequence as a special case. With the conventional polyphase orthogonal sequence proposed by Frank, the number of phases is given by .sqroot.N when the period is N, whereas with the orthogonal sequence of the invention it is given by .sup.k .sqroot.N+1, thereby bringing forth the effect that even when the period N is large the number of phases is small, in comparison with the conventional polyphase orthogonal sequence. The second modification will next be explained. X.sub.0 and K.sub.m with which all the arguments of z.sub.0, z.sub.1, . . . , z.sub.q-1 of an orthogonal sequence are the same are given by the following expressions: ##EQU49## Expression 98 shows that K.sub.m is an odd function represented by K.sub.m =m/(q-1) for example. z.sub.0, z.sub.1, . . . , z.sub.q-1, which are the solution of the simultaneous equations 30 and 31 and have the same argument, are obtained in accordance with a flow shown in FIG. 13C. That is: (a) At step 45, X.sub.0 is calculated from expression 96 or 97. (b) At step 46, K.sub.m satisfying expression 98 is calculated and set. (c) At step 47, using expression 69, X.sub.m is calculated as follows: ##EQU50## (d) At step 48, X.sub.m is substituted with x.sub.i by the inversediscrete-Fourier transformation. ##EQU51## (e) At step 49, z.sub.0 is set to C (.noteq.0) as expression 71 and by multiplying x.sub.i by C, z.sub.i+1 is obtained as expression 72. z.sub.0 =C z.sub.i+1 =Cx.sub.i (i=0,1, . . . ,q-2) The following is an explanation why the arguments of z.sub.0, z.sub.1, . . . , z.sub.q-1 are the same if X.sub.0 and K.sub.m satisfy expressions 96 or 97 and 98. When the arguments of z.sub.0, z.sub.1, . . . , z.sub.q-1 are the same, all of z.sub.1 /z.sub.0, z.sub.2 /z.sub.0, . . . , z.sub.q-1 /z.sub.0 are real numbers. From expressions 71 and 72, x.sub.i =z.sub.i+1 /C=z.sub.i+1 /z.sub.0 (99) (i=0,1, . . . ,q-2) Accordingly, the fact that the arguments of z.sub.0, z.sub.1, . . . , z.sub.q-1 are the same is equal to that z.sub.i+1 /C is a real number. In order to satisfy that z.sub.i+1 /C is a real number, it is necessary to satisfy the following relations in view of expression 72: ##EQU52## When X.sub.0 is a real number, expression 75 or 76 is obviously fed from expression 73. The left side of expression 101 can be rewritten as follows: ##EQU53## Accordingly, if K.sub.m is an odd function as represented by expression 98, ##EQU54## is also an odd function with regard to m and thus ##EQU55## Therefore, it is understood that when K.sub.m is an odd function, expression 101 is satisfied. From the foregoing, it will be appreciated that X.sub.0 and K.sub.m (m=1, 2, . . . , q-2) with which the argument values of z.sub.0, z.sub.1, . . . , z.sub.q-1 are equal are given by expressions 96 or 97 and 98. z.sub.0, z.sub.1, . . . , z.sub.q-1 having the same argument are determined in response to the constants K.sub.m and X.sub.0 obtained in the above manner, and are preset into the component substituting unit 21. Accordingly, the unit 21 can provide such a orthogonal sequence components of which have the same argument, in line with the flow shown in FIG. 12. A practical example of the orthogonal sequence created by the second modification is shown as follows, where q=3, k=3 and N=26: (111002021222102220010121120) This example indicates one period of the orthogonal sequence. Each figure (0, 1, 2) is a symbol and the corresponding complex-number z.sub.i is as follows: 0.fwdarw.z.sub.0 =1.0 1.fwdarw.z.sub.1 =-0.911 2.fwdarw.z.sub.2 =0.488 (103) FIG. 14C shows an example vector diagram of the orthogonal sequence generated by the second modification. Since all the components of this orthogonal sequence are real numbers, it can be called as a real multi-value orthogonal sequence which is a special case of a complex multi-value orthogonal sequence. An embodiment of a radar system of this invention will now be described. FIG. 15 is a schematic representation of the embodiment of the radar system provided with an orthogonal sequence generator. The radar system shown in FIG. 15 employs a complex multi-value orthogonal sequence generator 29 arranged as in FIG. 11, a modulator 12A which code-modulates a sinusoidal wave signal cos.omega.t using a complex multi-value orthogonal sequence {.theta..sub.n } from the generator 29 and a demodulator 19A which performs the correlation process of its input signal and the orthogonal sequence {.theta..sub.n } to obtain a demodulated signal Z(k). The other components of the radar system are substantially the same as those of the prior system shown in FIG. 6. Accordingly, the operation of the radar system can be described using the component .theta..sub.n of the complex multi-value orthogonal sequence instead of the component a.sub.n of the complex two-value orthogonal sequence with reference to expressions 16a to 24 with respect to the respective signals already described in connection with the prior art, and thus description of the common parts will be omitted. FIG. 16A shows an arrangement of the modulator 12A shown in FIG. 15. In this drawing, numeral 35 refers to phase shifters 36 to a control unit, 37 to a switching circuit and 38 to amplifiers. If the code-modulated transmission signal U(t) is referred to by A.sub.n cost(.omega.t+.phi..sub.n), the absolute value and argument of the complex number taken by the component .theta..sub.n of the complex multi-value orthogonal sequence will respectively correspond to the amplitude A.sub.n and phase .phi..sub.n of the transmission signal. A.sub.n and .phi..sub.n in the case of the complex three-value orthogonal sequence indicated by expression 74 are as follows: ##EQU56## Two phase shifters 35 are adapted to advance by .phi..sub.1 and .phi..sub.2 the quantity of phase of the sinusoidal wave signal e.sup.j.omega.t fed from the local oscillator 11 through the switching circuit 37, and two amplifiers 38 are adapted to amplify A.sub.1 and A.sub.2 times the amplitudes of the phase-shifted sinusoidal wave signal. The switching circuit 37 changes over the destination of the sinusoidal wave signal at every .tau. time interval and the operation thereof depends on a command signal C.sub.0 provided from the control unit 36. The control unit 36 creates the command signal C.sub.0 in accordance with the amplitude A.sub.n and argument, or phase .phi..sub.n of the component .theta..sub.n of the orthogonal sequence. For example, in the case of the amplitude being 1 and the phase being 0, the command signal C.sub.0 is created so as to connect terminals I and O of the switching circuit 37, the case of the amplitude A.sub.1 and the phase .phi..sub.1, the command signal C.sub.0 is created so as to connect the terminals I and A, and in the case of the amplitude A.sub.2 and the phase .phi..sub.2, the command signal C.sub.0 is created so as to connect the terminals I and B. In the above-mentioned radar system of this invention using the complex three-value orthogonal sequence, the number of change-overs of the phase and amplitude at the modulator 12A is three, and thus the number of channels comprising the phase shifters 35 and amplifiers 38 is two. Generally, with the complex multi-value orthogonal sequence, the period N thereof is expressed by q.sup.k -1, and thus the number of change-overs of the phase and amplitude at the modulator is q=(N+1).sup.1/k. Since with the conventional polyphase orthogonal sequence, the period N thereof is expressed by L.sup.2, the number of change-overs of the phase at the modulator is q=.sqroot.N. Therefore, if the period N of the sequence is large, the number of change-overs of the phase at the modulator in the case of using the complex multi-value orthogonal sequence is reduced as compared to that in the conventional case, and the reduction of the number of phase shifters provided in the modulator makes it possible to simplify the construction of the modulator. With respect to the demodulator 19A for performing the correlation process of the detection signal V input thereto from the detector 18 and the complex multi-value orthogonal sequence {.theta..sub.n }, the coefficient of the correlation process is different from that of the conventional process using the complex two-value orthogonal sequence {a.sub.n }. Thus, the demodulator 19A performs the correlation process expressed as the following expression 105, using the sampled detection signal V as indicated by expression 22 and the complex multi-value orthogonal sequence {.theta..sub.n } fed from the orthogonal sequence generator 29A, and then outputs the demodulated signal Z(k). ##EQU57## The demodulated signal Z(k) indicated by expression 105 is expressed by the following expression, similar to expression 24, when the autocorrelation function of the complex multi-value orthogonal sequence {.theta..sub.n } is represented by .rho..sub.OR (m): Z(k)=.eta..sub.a exp(-j.omega..tau.k.sub.a).rho..sub.OR (k-k.sub.a)+.eta..sub.b exp(-j.omega..tau.k.sub.b).rho..sub.OR (k-k.sub.b)(106) As described above, an autocorrelation function of a complex multi-value orthogonal sequence has a side lobe the magnitude of which is 0. Accordingly, when the complex multi-value orthogonal sequence {.theta..sub.n } is employed for the code-modulation of the radar system, such advantages as those in the prior art; even when there is a substantial difference between the radio-wave reflection intensities .eta..sub.a and .eta..sub.b on the adjacent two targets, the two-target signals Z.sub.a and Z.sub.b can be detected from the demodulated signal Z(k) without the main lobe of the smaller-magnitude signal Z.sub.b being covered by any side lobe of the larger-magnitude signal Z.sub.a, as shown in FIG. 8C. Further, in the radar system of this invention, since the sinusoidal wave signal e.sup.j.omega.t is code-modulated with the complex multi-value orthogonal sequence instead of a complex two-value orthogonal sequence, the detection, by any other electronic device, of the angular frequency .omega. of such a transmission signal will be extremely difficult. That is: The transmission signal U(t) obtained by code-modulating the sinusoidal wave signal with the complex multi-value orthogonal sequence {.theta..sub.n } can be expressed by the following expression from expression 16a. ##EQU58## where, .theta..sub.n =A.sub.n exp(j.phi..sub.n). Thus, the real component of U(t) of expression 107 can be expressed by the following expression: ##EQU59## Now, assuming that any other electronic device has received such a code-modulated transmission signal U(t) and U(t) is square-detected thereat to identify the angular frequency .omega. of U(t). If the output signal of a square detection unit referred to by Y(t), then Y(t) is expressed by the following expressions: ##EQU60## In the case of the complex multi-value orthogonal sequence, the value of 2.phi..sub.n never takes both the values of 0 and 2.pi., in contrast to the case previously described in connection with the conventional radar system using the complex two-value orthogonal sequence. For example, in the case of the example indicated by expression 104, 2.phi..sub.n takes one of the values 0, 168.pi./90 or 37.pi./90. Thus, Y(t) is not a sinusoidal wave, but a signal equivalent to that code-modulated by the code A.sub.n.sup.2 exp(j2.phi..sub.n) to cause the spectrum to be expended. Therefore, even when the frequency components of Y(t) are analyzed by a spectral analyzer, no sharp peak is detected in the waveform of the output thereof. Accordingly, it is extremely difficult for other electronic devices to detect the angular frequency .omega. of the transmission signal from the radar system using the complex multi-value orthogonal sequence in comparison with the prior art employing the complex two-value orthogonal sequence. In the polyphase orthogonal sequence as shown in the prior example, the period N is expressed by N=L.sup.2 (L: an integer equal to or more than 2), and if N is determined as a certain number, the polyphase orthogonal sequence having the period N exists only in a singular form. On the contrary, in the complex multi-value orthogonal sequence regarding the present invention, the period N can be expressed by N=q.sup.k -1 (q: a prime number or power thereof, k: an integer equal to or more than 2), and if N is determined as a certain number, the complex multi-value orthogonal sequence having the period N exits in a plural form. The number of the sequences having the period N is equal to the number of k-degree primitive polynominals in GF(q). Referring to "Code Theory (Third Edition)" mentioned above, the number of the primitive polynominals is four in the case of q=3 and k=4, for example. Thus, there exist four complex multi-value orthogonal sequences having the period of N=80. Because of the existence of a plurality of complex multi-value orthogonal sequences having the period N, if those sequences are utilized while being changed from one to another, it will be extremely difficult to detect the code-modulated transmission signal U(t) with any other electronic device, in comparison with the case of the prior polyphase orthogonal sequence. Next, arrangements of the modulator 12A of the radar system, shown in FIG. 15, respectively using the first and second modifications of the multi-value orthogonal sequence generators will be described. When the first modification is employed as the orthogonal sequence generator 29A, since the absolute values of the components of the sequence generated therefrom are the same as shown in FIG. 14B, the amplifiers 38 of the modulator 12A shown in FIG. 16 can be cancelled, and thus in this case the modulator 12A can be constituted as shown in FIG. 16B. Representing as U(t)=A.sub.n cos (.omega.t+.phi..sub.n), the amplitude A.sub.n and phase .phi..sub.n of U(t) are as follows in the case of expression 95: A.sub.0 =1 and .phi..sub.0 =0 A.sub.1 =1 and .phi..sub.1 =90.pi./180 A.sub.2 =1 and .phi..sub.2 =205.7.pi./180 (110) When the second modification is employed as the orthogonal generator 29A, since the phases of the components of the sequence generated therefrom are the same as shown in FIG. 14C, the phase shifters 35 of the modulator 12A shown in FIG. 16 can be cancelled, and thus in this case the modulator 12A can be constituted as shown in FIG. 16C. In the case of expression 103, the amplitude A.sub.n and phase .phi..sub.n of U(t)=A.sub.n cos (.omega.t+.phi..sub.n) are as follows: A.sub.0 =1 and .phi..sub.0 =0 A.sub.1 =0.911 and .phi..sub.1 =0 A.sub.2 =0.488 and .phi..sub.2 =0 (111) It is further understood by those skilled in the art that the foregoing description is a preferred embodiment of the disclosed device and that various changes and modifications may be made in the invention without departing from the spirit and scope thereof.
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