1406
`
`IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 38, NO. 4, JULY 1992
`
`occurs. Also, if q = pk and p > 5. it is necessary to have k = 1,
`as in the proof of Theorem 1.
`For primes p > 5,
`the roots of x2 — x — 1 = D are (1
`1' v/§)/2 E (1 1 \/€)((p + 1)/2) (mod p). As in the proof of
`Theorem 1,
`this restricts us to primes for which 5 is a quadratic
`residue. There is, however, the further restriction that both or and
`B : — 1/tx must be primitive in GF(p). Since 1/11 is primitive ifi‘
`or is primitive, the extra condition is that the factor of — 1 must not
`destroy primitivity. Now, multiplication by —1 preserves primitiv-
`ity iff -1 is a quadratic residue mod p, which occurs iff p E 1
`(mod 4). Combined with the requirement
`that 5 be a quadratic
`residue, namely p E :1 (mod 10),
`this restricts G4 to primes
`p E 1 (mod 20) or p 2 9 (mod 20).
`It remains only to show that all primes in these two residue
`classes modulo 20 for which the T4 construction occurs also have
`the G4 construction, and that no other such primes have the 04
`construction.
`If f(x) = x2 + x -1 then f(—x) = x2 — x — 1, so the roots
`of x2 — x — l are the negatives of the roots of X2 + x — 1. Also,
`we are considering only primes for which a factor of —l has no
`effect on primitivity. We have already seen that
`the roots of
`x2 — x — 1 are primitive or imprimitive together in the fields under
`consideration, so this also must hold for the roots of x2 + x — 1.
`By Theorem 1,
`the T4 construction occurs iff at least one of the
`roots of X2 + X — 1 is primitive in CvF( p); but for p E 1, 9 (mod
`20) this will mean that both roots are primitive, and also that both
`roots of X2 — x — 1 will be primitive. Conversely,
`if a root of
`x2 — x — 1 is not primitive, then both its roots are imprimitive, and
`the roots of x2 + x — 1 are also imprimitive.
`E]
`
`In the G4 construction for Costas arrays, since at + B = 1 and
`046 = — 1, certain symmetries can be expected relative to the main
`diagonal.
`
`Theorem 4.‘ Every Costas array given by the G4 construction
`modulo a prime p > 5 has the points (1, 1) and ((p — 3)/2, (p —
`3) /2) on the main diagonal, and the pairs of points (2, p — 2), (p
`— 2, 2) and ((p +1)/2, p — 3), (p ~ 3,(p + 1)/2) situated sym-
`metrically with respect to the main diagonal.
`
`Proof: Since or + 8 = 041+ B1=1,(1,1) is a point of the
`array. From or = ~13“ and 8 = -04", and using oz"””/2 :
`;3<rH>/2 = -1, it follows that or : (—1)(B’ ') = awe“/25 1:
`alps”/2,
`and
`similarly
`5 = (-l)(or') = u<””"/law’ =
`oz“"3’/2. Thus, 1 = or + B = fi””37/2 + cx‘”’3’/2, whence ((17 —
`3) /2, ( p —— 3) /2) is a point of the array.
`From Definition 2. 0:2 + 6 '2 1; that is, 012 + B” '2 = 1, and
`(2, p — 2) is a point of the array. From (28 = -1, 62 + of‘ =
`B2 — 6 = 1, because (as shown in the proof of Theorem 2),
`L3
`is a
`root of x2 —x— 1 = 0. Hence, (—1,2) = (p — 2.2) is also a
`point of the array. Next, a"‘3 +B“’“V2 2 of2 + B - B"”‘”/2
`= (32 — 6 :1,
`and similarly 6”’ 3 + 12"“ “/2 = [3
`2 + or
`-
`a”””/2 = :12 — or =1, from which both (1) — 3, (/7 + 1)/2) and
`((p + 1)/2, p — 3) are points of the array.
`3
`Note: Fig.
`1 shows that, except for the six points identified in
`Theorem 4, none of the other points of the G4 construction need be
`symmetrically situated relative to the main diagonal.
`
`[1]
`
`[2]
`
`REFERENCES
`S. W. Golomb and H. Taylor, “Constructions and properties of
`Costas arrays." Proc. IEEE, vol. 72, pp. 1143-1163, Sept. 1984.
`S. W. Golomb, “Algebraic constructions for Costas arrays," J.
`Combin. Theory (A), vol. 37, no. 1, pp. 13-21, July 1984.
`
`[3] 0. Moreno and J . Sotcro, “Computational approach to conjecture A
`of Golomb," Corlgressus Numerrmtium. vol. 70, pp. 7-16, 1990.
`
`Generalized Chirp-Like Polyphasc Sequences with
`Optimum Correlation Properties
`
`Branislav M. Popovic'
`
`Ab.t'lracI—A new general class of polyphase sequences with ideal
`periodic autocorrelation function is presented. The new class of se-
`quences is based on the application of Zadoll‘—Chu polyphase sequences
`of length N : smz, where s and m are any positive integers. It is shown
`that the generalized chirp-like sequences 01‘ odd length have the opti-
`mum crosscorrelation function under certain conditions. Finally,
`re-
`cently proposed generalized P4 codes are derived as a special case of the
`generalized chirp-like Sequences.
`Index Terms—Sequences, codes, spread-spectrum, radar, pulse com-
`pression.
`
`I. INTRODUCTION
`
`Sequences with ideal periodic autocorrelation function [l|—[5] are
`finding their applications in the field of spread spectrum communica-
`tions [l], construction of (super) complementary sets [6]. [7], etc.
`These sequences usually have small aperiodic autocorrelation and
`ambiguity function sidelobes, so they are very useful in the pulse
`compression radars [7] — [10] _
`On the other hand, spread spectrum multiple access systems
`demand minimum possible crosscorrelation between the sequences
`within selected set of sequences having good periodic autocorrela-
`tion function properties. Sarwate [S] has shown that the maximum
`magnitude of the periodic crosscorrelation function and the maxi-
`mum magnitudc of the periodic autocorrelation function are related
`through an inequality, which provides a lower bound on one of the
`maxima if the value of the other is specified. By using this inequal-
`ity the optimum correlation properties of the set of sequences can be
`defined. So, when the maximum magnitude of the periodic autocor—
`relation function equals zero,
`from the Sarwate’s inequality it
`follows that the lower bound for the maximum magnitude of the
`periodic crosscorrelation is equal to \/N, where N is the length of
`sequences.
`In this correspondence, we shall present a new general class of
`polyphase sequences with ideal periodic autocorrelation function,
`having at the same time the optimum crosscorrelation function. The
`new sequences can be classified as the modulatable orthogonal
`sequences, according to the terminology from [1]. The generalized
`Frank sequences, of length N = m2, where m is any positive
`integer, are presented in [1] as the only known example of the
`modulatable orthogonal scqucnccs. It is noticed that these sequences
`also have interesting aperiodic autocorrelation function properties
`[10].
`the basic definitions are given. In Section III, we
`In Section II,
`present the generalized chirp—like sequences. In Section III—A, we
`show that the generalized chirp—like sequences have the ideal peri-
`odic autoeorrelation function. Section III-B concerns the periodic
`crosscorrelation function of the generalized chirp—like sequences.
`Finally, in Section IV, we show that the recently proposed general-
`
`Manuscript received June 11, 1991', revised November 19, 1991.
`The author is with IMTEL Institute of Microwave Techniques and Elec-
`tronics, B. Lenjina 165b,, 11071 Novi Bcogred, Yugoslavia.
`IEEE Log Number 9107517.
`
`0018-9448/92$03.00
`
`© 1992 IEEE
`
`APPLE 1009
`
`APPLE 1009
`
`1
`
`

`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38. NO. 4, JULY 1992
`
`1407
`
`ized P4 codes are only a special case of the generalized chirp-like
`sequences.
`
`II. BASIC DEFINITIONS
`
`The following definitions will be usefirl.
`A primitive nth root of unity W" is defined as
`
`(GCL) sequence {sk} is defined as
`
`k=0a"'—N‘1=
`5k=‘7kb</omodms
`where (k) mod m means that index k is reduced modulo m.
`Example: For N = 8, we have that s = 2 and m = 2, so the
`generalized chirp-like sequence {sk} is given by
`
`(7)
`
`W" 2 exp(—j21rr/I1).
`
`j : V/4 1,
`
`(1)
`
`{sk} = {b0, b,W°5,b0W2,b,W“'5,b[,,
`
`where r is any integer relatively prime to n.
`It can be easily shown that for any integer u, 0 < u 5 rl — 1, the
`following relation is valid [11]
`n—1
`
`(3)
`b,W”,b,,W1,b,W°5},
`rela-
`where W= exp(~j21rr/8), j = V -— 1 ,r is any integer
`tively prime to N = 8, q = 0, while b0 and b, are arbitrary
`complex numbers with magnitude equal to 1.
`
`(2)
`W, =# 1.
`go VV_,* 1"‘ : 0.
`The sequence {sk} of length L is said to have the ideal periodic
`autocorrelation function 0( p) if it satisfies the following relation
`L—l
`
`19(17):
`
`*
`k:0
`Z 5k5:k+pinm I,
`
`A. Periodic Autocorrelation Function of the Generalized
`Chirp—Like Sequences
`
`Theorem 1: The generalized chirp-like polyphase sequences have
`the ideal periodic autocorrelation function.
`
`that Zadofi°—Chu sequences are
`Proof: It can be seen in [2]
`defined separately for N even and N odd,
`in order to satisfy the
`following condition
`
`(3)
`
`‘7/«+42/v = and-
`
`(9)
`
`—
`
`E.
`0,
`
`p = O(mod L),
`17 ¢ 0(mod L),
`
`where the asterisk denotes complex conjugation. the index (k + p)
`is computed modulo L,
`time shift p is assumed to be positive
`because 9(—p) = 0*(p), E = 0(0) is energy of the sequence {sk}.
`Besides the well—known Frank sequences, and recently proposed
`generalized Frank sequences [1],
`there is another large class of
`polyphase sequences with ideal periodic autocorrelation function.
`The so-called Zad0ff— Chu sequences [2], [3] are defined as
`
`at:
`
`k =0,1,2,---,N~ l,
`W161’/2+qk,
`w,5**’+'>i’2+4*, k=0,1,2,---,N— 1,
`q is any integer.
`
`for Neven,
`for N odd.
`
`where d is an arbitrary delay.
`It can be easily proved that generalized chirp-like polyphase
`sequence {sk}, defined by (7). also satisfy the condition (9). In that
`case.
`the periodic autocorrelation function 0(1)) of the sequence
`{sk} can be written as
`N~p~1
`ll
`
`N—1
`
`9(P)
`
`Sksilp
`
`Z Sksilf-+—p + Z
`k:0
`k:N—p
`N—1
`
`*
`k:0
`Z 5/(5/.»+p~
`
`(10)
`
`For N even, from (4), (7), and (10),
`N—1
`
`it follows
`
`(4)
`
`k:()
`6(1)) = WND2/'2/qp Z W/;pkb(k)mudmb(>l;c+p)mcxJm‘
`
`The similar construction is proposed by Ipatov [4] and is defined
`
`by
`
`ak = Wfwk. k= 0.i,2.---,N— 1;
`N is odd; q is any integer.
`
`(5)
`
`However, one of the referees pointed out that the class of Ipatov
`sequences is equal
`to the class of '/.adolT—Chu sequences of odd
`length N. Namely,
`starting from the definition of Z-adolT~Chu
`sequences of odd length N = 21‘ — 1. it follows
`
`W’5(k+1,/Hqk: (H/[:I)):lHll'ZqJk,
`
`(6)
`
`is a
`I/VA’,
`is relatively prime to N.
`where I = 2” mod N. Since I
`primitive Nth root of unity. Hence.
`the right-hand side of (6)
`corresponds to the definition of Ipatov sequences.
`Based on the ZadotT—Chu sequences,
`the new. more general,
`class of polyphase sequences with optimum periodic autocorrelation
`and crosscorrelation function properties will be presented in the next
`section.
`
`III. GENERALIZED CHIRP-LIKE P0r.Yi=1-rAsE SEQUENCES
`
`We shall introduce here the following change of variables:
`120.1,-
`k:ism+d,
`~,m—1,
`d=O,l,---,sm~1.
`
`(12)
`
`From (11) and (12) we obtain
`
`sm~l
`
`d:0
`ail’) = WNPZ/2iqp Z ['VNpdb(d)modm
`m—1
`
`.
`
`~ip
`*
`i=0
`b(d+p}mo(‘lm Z Wm '
`
`(13)
`
`If p :6 xm, x = 0,1,---, Sm — 1, the second summation in (13)
`is obviously zero, according to (2).
`For p 2 xm, x : 0, 1.---,sm — 1, from (13) we obtain
`
`6(Xm) = r"W]G(XIl1)
`
`sm—1
`2 I
`/2~qxm Z Wsjnxd’
`d=0
`
`if x t O. the summation in (14) equals zero according to (2). In
`that way, we have proved Theorem 1 when N is even. The proof
`for N odd is the same.
`C]
`
`Let {ak}, k = 0,1,-", N— 1, be a Zadoff—Chu sequence of
`length N : smz, where m and 5 are any positive integers. Let
`{b,-},
`i = 0, '
`- -, m — 1, be any sequence of m complex numbers
`having the absolute values equal to 1. The generalized chirp-like
`
`B. Periodic Crasscorrelazion Function of the Generalized
`Chirp—Like Sequences
`
`N — 1, be the two general-
`Let {xk} and {yk}, k = 0, l,-
`ized chirp—like sequences of odd length, obtained from any two
`
`2
`
`

`
`l 408
`
`IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 33. NO. 4. JULY 1992
`
`complex vectors {[71,} and {b2,.}. i = 0,1,‘-~, m — 1, and from
`the two different primitive Nth roots of unity exp( j2 -rru / N) and
`exp(j21ru/N), i.e..
`
`xk = Wl\$[k(k+ 1)/2+qk]b1
`
`(k)mudm’
`
`(15)
`
`yk : W#[k(k+])/2+qk]b2t'k)modm‘
`
`WN= exp(j2-K/N),
`
`(u,N) =1;(u,N) = l;v#=u.Nisodd.
`
`Theorem 2: The absolute value of the periodic crosscorrelation
`function between any two generalized chirp—like sequences of odd
`length N, obtained from the two different primitive Nth roots of
`unity exp (j27rv/N) and exp(j21ru/N), is constant and equal to
`\/N , if (v — u) is relatively prime to N.
`
`Proof: The squared absolute value of the periodic crosscorrela-
`tion function RXy( p) of sequences {xk} and { yk} is defined as
`Nil
`N/I
`*
`Z X] yl-+-p‘
`{=0
`
`(16)
`
`iRx,v(p) l 2 = 50 x;J’7:+p
`Substituting (15) into (l6), we obtain
`
`I R,y(1>) I 2
`
`I N—l
`
`n
`
`M?I: ll OIO
`
`L W’\(/u—u)(k—I)[(k+I+1)/2+q]—up(/(-1)
`
`
`- bl(k)mod mb1EFI)mnd m
`_
`ac
`b2(k+p) mod mb2(l+P)mod m
`
`We can introduce the following change of variables:
`
`l=k+e,e=0,l,---,N—l.
`
`From (17) and (18), we obtain
`
`lRxy(p)I2
`N—l N—l
`
`Z W’;(u-u)el(2k+e+l)/’2+q]+upe
`e=0
`A
`
`. bl(k)mod mb1)(kk+e)modm
`'
`*
`b2(k+p)m0dmb2(k-+-z+p)mod m‘
`
`(17)
`
`(13)
`
`(19)
`
`This expression can be rewritten as
`N—l
`
`e——0
`|RXy(p) | 2 : Z WN(v—u)e[(e+I)/2+q]+pueSP’
`
`(20)
`
`where Se is defined as
`N—l
`
`Se: 2 W)\—](u—u)ek
`k:O
`
`I b1(k)rnod mb1Ekls'+e)mod m
`- b2*
`(k+p)mod mb2(k+p+e)mndm'
`
`From (20) and (21), it is obvious that
`
`|RXy(p)|2 =N,
`
`ife=O.
`
`(21)
`
`(22)
`
`We shall prove now that
`
`|Rx,(p)|2=0»
`
`ife=#0.
`
`We shall introduce the following change of variables
`1':
`0,1,-~
`,,=0,,_..
`
`k=ism+d,
`
`From (21) and (24), we obtain
`sm—l
`
`*
`A( - )1’
`_
`Se - 0:0 I4/N U “E bl(d)modmbl(d+e)modm
`m~l
`.
`*
`.
`b2(d+p)mod mb2(d+p+e)mod m
`
`m
`Z W—(u—u)ei
`[=0
`
`(23)
`
`(24)
`
`.
`
`(25)
`
`As (2; — u) is relatively prime to N, it is also relatively prime to
`m, so according to (2) the inner summation in (25) is equal to zero
`ifeatxm, x=O,1,---,sm4 1.
`For e = xm, x= 0,1,--~,sm — 1, SE, also becomes equal
`zero, i.e..
`
`to
`
`sm—l
`
`S, = m X W;,f"“""" = 0,
`d=0
`
`ifx ¢ 0.
`
`(26)
`
`On that way we have proved the Theorem 2.
`
`Although the GCL sequences are defined either for odd and even
`lengths, we have seen that Theorem 2.
`is valid only for the odd
`lengths. We shall briefly discuss why it cannot be valid for even
`lengths.
`If the length of sequences N is even, and u and u are integers
`relatively prime to N, then u and u must be odd. Consequently,
`(u — u) must be an even integer, because the difference between
`any two odd integers is always an even integer. In that way, (u —— u)
`can never be relatively prime to N, and that is the reason why the
`Theorem 2 is defined only for the odd lengths.
`It is interesting to note that the same fact is true for the general~
`ized Frank sequences [1]. Namely, the generalized Frank sequences
`are defined for lengths N = m2, where m is any positive integer. It
`is shown in [1] that when (u — u) and m are relatively prime, the
`two generalized Frank sequences, corresponding to u and u, have
`optimum crosscorrelation function. However,
`from the previous
`discussion it follows that the pairs or sets of such sequences, having
`the optimum crosscorrelation function, can exist only if m is an
`odd number. This important property of the generalized Frank
`sequences is not explicitly mentioned in [1].
`If N is a second power of a prime number m, the set of m — l
`GCL polyphase sequences can be constructed using m — 1 different
`primitive Nth roots of unity. Any pair of sequences in such a set
`has the optimum crosscorrelation function, having the constant
`magnitude equal to VN.
`IV. GENERALIZED P4 CODES
`
`the authors claimed that it has been found by computer
`In [9],
`simulation that a P4 code of length N = m2 can be written in terms
`of an m X m matrix with mutually orthogonal rows, and with
`additional property that all rotations of any two columns are mutu-
`ally orthogonal. By postmultiplying such a matrix with the diagonal
`matrix B, and concatenating the rows of the resultant matrix,
`the
`s'o~called generalized P4 codes with ideal periodic autocorrelation
`function can be obtained.
`It can be easily seen that the generalized P4 codes can also be
`obtained from the definition of generalized chirp-like sequences, by
`taking the sequence {ak} in (7) to be the P4 code.
`
`3
`
`

`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, N0. 4. JULY 1992
`
`1 409
`
`We shall show below that the P4 codes are only a special case of
`the ZadolT—Chu sequences, what means that
`the generalized P4
`codes are only a special case of the generalized chirp-like sequences
`of length N = m2.
`The P4 code elements {ak} are given by [8], [9]
`
`a,,=exp[j{]%k2+ark)],
`where N is any positive integer.
`This expression can be rewritten as
`
`k=0,1,---,N— 1.
`
`,
`,
`ak : W152/2+(N/21k’
`
`WN : gxp
`
`27r
`
`_
`
`(28)
`
`For N even, the P4 code can be obtained by substituting q = N/2
`in (4). For N odd,
`the P4 code can be obtained by substituting
`q = (N — 1)/2 in (4).
`As the P4 codes are only a special case of the Zadotf—Chu
`sequences, it is not surprising that the P4 codes have ideal periodic
`autocorrelation [9].
`In [2], it was shown that all the cyclic time shifted versions of the
`Zadoff—Chu sequences have the same absolute value of the aperi-
`odic autocorrelaticn function. Consequently, the same is true for the
`P4 codes [9].
`
`V. CONCLUSION
`
`In this correspondence, we have presented the new general class
`of polyphase sequences with ideal periodic autocorrelation function,
`having at
`the same time the optimum periodic crosscorrelation
`function. If the length of sequences N is a second power of a prime
`number m,
`the set of m — 1 generalized chirp-like polyphasc
`sequences can be constructed using in — 1 different primitive Nth
`roots of unity. Any pair of sequences in such a set has the optimum
`crosscorrelation function, with the constant magnitude equal
`to
`V//V .
`Compared with the generalized Frank sequences. the generalized
`chirp-like sequences ofl‘er
`the higher degree of freedom for the
`choice of sequence length.
`ACKNOWLEDGMENT
`
`The author is grateful to the referees for their valuable comments.
`
`REFERENCES
`[1] N. Suehiro and M. Hatori. “Modulatable orthogonal sequences and
`their application to SSMA systems,” IEEE Trans. Inform. Theory,
`vol. 34, pp. 93-100, Jan. 1988.
`[2] D. C. Chu. “Polyphase codes with good periodic correlation proper-
`ties,” IEEE Trans. Inform. Theory, vol. IT~18, pp. 531-532, July
`1972.
`[3] R. L. Frank, “Comments on Polyphase codes with good correlation
`properties," IEEE Trans. Inform. Theory, vol. 1T—l9, p. 244, Mar.
`1973.
`[4] V. P. Ipatov, “Multiphase sequences spectrums,” Izvestiya VUZ.
`Radioelektronika (Radioelectronics and Communications Sys-
`tems), vol. 22, no. 9, pp. 80-82, 1979.
`[5] D. V. Sarwate, “Bounds on crosscorrelation and autocorrclation of
`sequences,” IEEE Trans. Inform. Theory, vol. IT—25. pp. 720~724,
`Nov. 1979.
`[6] B. M. Popovié, “Complementary sets based on sequences with ideal
`periodic autocorrelation,” Electron. Lett., vol. 26, no. 18, pp.
`1428-1430, Aug. 30, 1990.
`[7] —,
`“Complementary sets of chirp—like polyphase sequences,"
`Electron. Lett., vol. 27, no. 3, pp. 254-255, Jan. 31, 1991.
`
`[9]
`
`[8] B. L. Lewis and F, F. Kretschmer, “Linear frequency modulation
`derived polyphase pulse
`compression codes,”
`IEEE Trans.
`Aerospace Electron. Syst., vol. AES—18. no. 5, pp. 637-641, Sept.
`1982.
`F. F. Kretschmer, Jr. and K. Gerlach, “Low sidelobe radar wave-
`forms derived from orthogonal matrices," IEEE Trans. on AES,
`vol. 27, no. 1, pp. 92-101, Jan. 1991.
`factor of Frank and Chu
`B. M. Popovié, “Comment on Merit
`sequences," Electron. Lett., vol. 27, no. 9, pp. 776-777, April 25,
`1991.
`R. C. Hcimiller, “Author's comment,” IRE Trans, vol. 1'1‘-8, p.
`382, Oct. 1962.
`
`Nonbinary Kasami Sequences over GF( p)
`
`Shyh—Chang Liu and John J. Komo, Senior Member, IEEE
`
`Abstracr—The correlation values and the distribution of these correla-
`tion values are presented for the small set of nonbinary Kasami se-
`quences over GF( p) (p prime). The correlation results are an extension
`of the binary results and have p + 2 correlation levels. This nonbinary
`Kasami set
`is asymptotically optimum with respect to its correlation
`properties. These sequences are obtained, as in the binary case, from a
`large primitive polynomial of degree n = 2m and ll small primitive
`polynomial of degree m that yields a sequence length of p" — 1 and
`maximum nontrivial correlation value of 1 + p”'. Nonbinary Kasami
`sequences are directly implemented using shift registers and are applica-
`ble for code division multiple access systems.
`
`Index Terms—Kasami sequence, nonbinary, correlation levels and
`distribution, code division multiple access.
`
`I. INTRODUCTION
`
`The set of binary Kasami sequences over GF(2) can be expressed
`as [1]. [2]
`
`S={s,(t)|0stsN—1,1sis2"'},
`
`where
`
`s,-(t) = trf" {tr} (of) + 7,-aT’} ,
`
`(1)
`
`(2)
`
`n=2m, N=2"— 1, T=(2"— 1)/(2"'— 1) =2'"+ 1, ozisa
`primitive element of GF(2 ”), and 7, takes on each value of GF(2“)
`for 1 S i 5 2'". Since a7 is a primitive element of GF(2“), one
`si(t) is an m-sequence with period N and each other S,-(t) is the
`sum of an m-sequence with period N and a different phase of an
`m-sequence with shorter period 2"’ — 1. The shorter m-sequence is
`a decimation by T of the longer m-sequence and the period of the
`shorter m-sequence divides the period of the longer m-sequence.
`The correlation function, R,j(r), 1 S i, j 5 2"‘, of the ith and
`jth sequences of S is given by
`N-l
`
`t=O
`Rm = 2 (-1)"“*”**"”,
`
`057'SN—l,
`
`(3)
`
`is modulo N addition. If i and j are not cyclically
`where 1+ 1'
`distinct, R,-j(r) reduces to the autocorrelation function. The correla-
`
`Manuscript received January 15, 1991; revised November 21, 1991. This
`work was presented in part at the IEEE International Symposium on lnfor»
`mation Theory, Budapest, Hungary. June 24-28, 1991.
`The authors are with the Department of Electrical and Computer Engi-
`neering, Clemson University, Clemson. SC 29634-0915.
`IEEE Log Number 9108026.
`
`001899448/92$03.00
`
`© 1992 IEEE
`
`4