1406
`
`IEEF. TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992
`
`occurs. Also, if q = 17" 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
`i v€)/2 E (1 i v€)((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, die further restriction that both at and
`B : — l/oz must be primitive in GF(p). Since 1/11 is primitive ifir
`or is primitive, the extra condition is that the factor of — 1 must not
`destroy primitivity. Now, multiplication by 71 preserves primitiv~
`ity iff —1 is a quadratic residue mod p, which occurs itf p E 1
`(mod 4). Combined with the requirement
`that 5 be a quadratic
`residue, namely p E :1 (mod 10),
`this restricts 04 to primes
`p E 1 (mod 20) or p a 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.
`then f(—x) = x2 — x — 1, so the roots
`If f(x) = x2 + x — 1
`of x2 — x — l are the negatives of the roots of x2 + x — I. Also,
`we are considering only primes for which a factor of 71 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 — l is primitive in GF( p); but for p E l, 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 — l is not primitive, then both its roots are imprimitive, and
`the roots of x2 + x — l are also imprimitive.
`E]
`
`In the 64 construction for Costas arrays, since a + £3 = 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 (I, l) and ((p — 3)/2, (p —
`3) /2) on the main diagonal, and the pairs of points (2, p 7 2), (p
`— 2, 2) and ((p +1)/2, p — 3), (p ~ 3,(p + 1)/2) situated sym-
`metrically with respect to the main diagonal.
`
`is a point of the
`Proof: Since at + 8 = 041+ B1 = 1,(1, 1)
`array. From a = ~6“ and B = —oc‘1, and using a(”’”/2 :
`W’le : —1, it follows that or :(—1)(/3*1)= open/25 1:
`arm,
`and
`similarly
`5 = (~l)(a“) = u(”")/2a"’ =
`o‘p‘wz. Thus, 1 = 01 + B = fiw’wz + a(”’3’/2, whence ((17 —
`3) /2. ( p —— 3) /2) is a point of the array.
`From Definition 2. a2 + 6 '2 1; that is, 012 + B” '2 :1, and
`(2, p — 2) is a point of the array. From all? = — l, (3’2 + or" =
`62 — 6 = 1, because (as shown in the proof of Theorem 2),
`L3
`is a
`root of x2 —x7 1= 0. Hence, (—1,2) = (p — 2.2) is also a
`point of the array. Next, 019—3 +B‘p+wz 2 a4 + B - 3‘7")”
`: [32 — 6 z 1,
`and similarly t3” 3 + 01”“ W2 = [3
`2 + at
`'
`(NW/2 = (12 — (x = I, from which both (1) — 3. (p + 1)/2) and
`3
`((p + l)/2, p — 3) are points of the array.
`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]
`
`[21
`
`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] O. Moreno and J . Sotcro, “Computational approach to conjecture A
`of Golomb," Congressus Numerantium. vol. 70, pp. 7—16, 1990.
`
`Generalized Chirp-Like Polyphasc Sequences with
`Optimum Correlation Properties
`
`Branislav M. Popovic'
`
`Abstract—A new general class of polyphase sequences with ideal
`periodic autocorrelation function is preSenled. The new class of se-
`quences is based on the application of Zadofl—Chu polyphase sequences
`of length N : smz, where s and m are any positive integers. It is shown
`that the generalized chirp-like sequences of 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 comr
`pression.
`
`I. INTRODUCTION
`
`Sequences with ideal periodic autocorrelation function [ll—[5] are
`finding their applications in the field of spread spectrum communica-
`tions [1], 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] 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 v/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 = 1722, where m is any positive
`integer, are presented in [1] as the only known example of the
`modulatable orthogonal sequences. 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 autocorrelation 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 lMTEL Institute of Microwave Techniques and Elec-
`tronics, B. Lenjina 165b,, 11071 Novi Bcogred, Yugoslavia.
`IEEE Log Number 9107517.
`
`cameos/9303.00
`
`© 1992 IEEE
`
`PETITIONERS 1069-0001
`
`|PR2016—OO758
`
`PETITIONERS 1069-0001
`IPR2016-00758
`
`

`

`
`
`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.
`
`11. BASIC DEFINITIONS
`
`The following definitions will be useful.
`A primitive nth root of unity W" is defined as
`
`(GCL) sequence {5k} is defined as
`
`k=0,“',N—l,
`Sk=akb(k)modma
`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 {5k} is given by
`
`(7)
`
`anexp(—j27rr/n),
`
`j: i— 1,
`
`(l)
`
`{5k} = {boa bIWOSaboW2>brWvaoi
`
`where r is any integer relatively prime to n.
`It can be easily shown that for any integer u, 0 < u s n 7 l, the
`following relation is valid [ll]
`"—1
`
`(3)
`le”,bOW2,b1W°5},
`rela-
`where W: exp(~j21rr/8), j = v - l ,r is any integer
`tively prime to N = 8, q = 0, while b0 and b1 are arbitrary
`complex numbers with magnitude equal to l.
`
`(2)
`W, =# l.
`go W,,*“" = 0.
`The sequence {5k} of length L is said to have the ideal periodic
`autocorrelation function 0( p) if it satisfies the following relation
`L»l
`
`0(1)):
`
`*
`k:0
`Z Sk5ik+ptmod l,
`
`A. Periodic Autocorrelation Function of the Generalized
`Chirp-Like Sequences
`
`Theorem I: The generalized chirp-like polyphase sequences have
`the ideal periodic autocorrelation function.
`
`that ZadolT—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)
`
`ak+d~N = amu-
`
`(9)
`
`-
`
`E.
`0,
`
`p : 0(mod L),
`p ¢ 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(7p) = 6*(p), E = 0(0) is energy of the sequence {5k}.
`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 Zadoff— Chu sequences [2], [3] are defined as
`
`Wit/2+“,
`“k: Wlfl‘ik+1)/2+qk,
`
`k = 0,1,2,~-, Nv r,
`k = 0,1,2,‘-‘, N71,
`q is any integer.
`
`for Neven,
`for N odd,
`
`(4)
`
`The similar construction is proposed by Ipatov [4] and is defined
`
`by
`
`a, = WW, k= 0.1,2,---,N71;
`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 ZadoflLChu sequences of odd
`length N. Namely,
`starting from the definition of Zadofi~Chu
`sequences of odd length N = 21‘ — 1, it follows
`
`"(gun/2+qu (pr/[Uk‘mtzmka
`
`(6)
`
`is a
`is relatively prime to N, WA’,
`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 ZadofiLChu 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 POLYPHASE SEQUENCES
`
`where d is an arbitrary delay.
`It can be easily proved that generalized chirp-like polyphase
`sequence {5k}, defined by (7). also satisfy the condition (9), In that
`case.
`the periodic autocorrelation function 0(1)] 0f the sequence
`{3,} can be written as
`N>p~1
`ll
`
`N71
`
`6(p)
`
`Sksz+p
`
`Z Skst-t-p + Z
`k:0
`k:N7p
`N71
`
`*
`k:0
`Z 5k5k+p~
`
`(10)
`
`For N even, from (4), (7), and (10),
`N71
`
`it follows
`
`9(1’) = WWI/2"” [(20 WNpkbtk)mudmb:;6+p)modm‘
`We shall introduce here the following change of variables:
`120,1,-
`k:ism+d,
`“,m-l,
`d=0,l,---,sm~l.
`
`(11)
`
`(12)
`
`From (11) and (12) we obtain
`
`smil
`
`d:0
`6(p) = WNpZ/‘qu Z u/dibtd)modm
`m—l
`
`,
`
`"ip
`*
`1'20
`btd+pimodm Z Wm '
`
`(13)
`
`If p :6 xm, x = 0, l,- r -, sm — l, the second summation in (13)
`is obviously zero, according to (2).
`For p 2 xm, x : 0, l,- - -,sm — 1, from (13) we obtain
`sm — l
`
`6(Xm) = mWIGtxrri)2/2~qu Z WS—mxd’
`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 = O,1,-‘-,N— 1, be a Zadofi1Chu 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 l. The generalized chirp—like
`
`B. Periodic Crosscorrelazion Function of the Generalized
`Chirp-Like Sequences
`
`N 7 1, be the two general-
`Let {xk} and {yk}, k = 0, l,-
`ized chirp—like sequences of odd length, obtained from any two
`
`PETITIONERS 1069-0002
`|PR2016—OO758
`
`PETITIONERS 1069-0002
`IPR2016-00758
`
`

`

`l 408
`
`JEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 33. NO. 4. JULY 1992
`
`compch vectors {bli} and {b2i}, i = 0,1,~-, m — 1, and from
`the two different primitive Nth roots of unity exp( j2 1ru / N) and
`exp(j21ru/N), i.e..
`
`We shall prove now that
`
`Inytp)|2=0,
`
`ifeaéO.
`
`(k)mudm‘
`
`xk = Wb;[k(k+ 1)/2+qk]b1
`,_
`k k+1 2+ k
`yk " WNHI
`(
`V
`q lbzt‘k)modm‘
`11
`
`§
`
`exp(j27r/N),
`
`(Is)
`
`We shall introduce the following change of variables
`k=ism+d
`i=0 1
`m—l
`a
`a
`g
`7
`d=0,l,"',sm l
`
`,
`
`From (21) and (24), we obtain
`sm~l
`
`(23)
`
`24
`
`(
`
`)
`
`(u,N) = l;(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 dilferent primitive Nth roots of
`unity exp (j27rv/N) and exp(j21ru/N), is constant and equal to
`x/N , if (v — u) is relatively prime to N.
`
`Proof: The squared absolute value of the periodic crosscorrela-
`tion function ny( p) of sequences {xk} and { yk} is defincd as
`Nil
`Nil
`
`d=0
`Se = Z VVNW—WEdblUJhnod mblEkd-i-e)modm
`
`m~|
`I:
`
`I
`(25)
`Wr;(uiu)m'
`.bzfd+p)modmb2(d+p+e)modm‘
`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
`ife¢xm,x=0,1,---,smgl.
`For 9 = xm, x: 0, l,--~,sm — 1,85, also becomes equal
`zero, i.e..
`
`to
`
`ley(p)|2 = Z xkyz-l-p Z xTyl-l-p'
`k=0
`[=0
`
`smil
`
`(26)
`
`
`
`
`S, = m X raw/“W = 0,
`d=0
`
`ifx a: 0.
`
`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 v 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 — 1
`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 v/N.
`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
`so‘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,
`
`Substituting (15) into (16), we obtain
`
`ley(p)|2
`N—l N—l
`
`Wéu—u)(k—I)[(k+l+1)/2+q]—up(k—l)
`A
`
`= Z
`k=0 (=0
`_
`
`=I=
`bl(k)mod mb1(l)mnd m
`=1:
`b2(k+p) mod mb2(l+p)mod m
`
`_
`
`We can introduce the following change of variables:
`
`l=k+e,e=0,1,---,Nil.
`
`From (17) and (18), we obtain
`
`|ny(iv)|2
`N—l N—l
`
`Z Wgw—u)el(2k+e+l)/’2+q]+upe
`e=0
`A
`*
`' bl(k)mod rnb1(k+e)modm
`'
`*
`b2(k+p)modmb2(k+z+p)mod m‘
`
`(17)
`
`(18)
`
`(19>
`
`This expression can be rewritten as
`N—l
`
`(=0
`ley(p) | 2 = Z W~("’"”’““*”/2“““MS”
`
`(20)
`
`where Se is defined as
`N—l
`
`Se = Z Wfi(u—u)ek
`[6:0
`*
`' b1(k)rnod mb](k+e)mod m
`~ b2*
`(k+p)mod mb2(k+p+e)mndm'
`
`From (20) and (21), it is obvious that
`
`|ny(p)|2 =N,
`
`ife=0.
`
`(21)
`
`(22)
`
`PETITIONERS 1069-0003
`|PR2016—OO758
`
`PETITIONERS 1069-0003
`IPR2016-00758
`
`

`

`IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 38, N0. 4. JULY 1992
`
`l 409
`
`We shall show below that the P4 codes are only a special case of
`the Zadoff—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]
`1r
`ak=exp[j{fik2+7rk)],
`where N is any positive integer.
`This expression can be rewritten as
`
`k=0,1,---,N71.
`
`[9]
`
`[8] B. L. Lewis and F. F. Kretschmer, “Linear frequency modulation
`derived polyphase pulse
`compression codes,”
`IEEE Trans.
`Aerospace Electron. Syst., vol. ABS-18. no. 5, pp. 637—641, Sept.
`1982.
`F. F. Kretsehmer, 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. IT-8, p.
`382, Oct. 1962.
`
`(28)
`
`Nonbinary Kasami Sequences over GF( p)
`
`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 Zadoff—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 autocorrelation 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 polyphase
`sequences can be constructed using m w 1 different primitive Nth
`roots of unity. Any pair of sequences in such a set has the optimum
`crosseorrelation function, with the constant magnitude equal
`to
`[A7 .
`Compared with the generalized Frank sequences. the generalized
`chirp-like sequences offer the higher degree of freedom for the
`choice of sequence length.
`ACKNOWLEDGMENT
`
`The author is grateful to the referees for their valuable comments.
`
`REFERENCES
`[l] 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-19, 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. Sarware, “Bounds on crosscorrelation and autocorrclation of
`sequences." IEEE Trans. Inform. Theory, vol. IT—25. pp. 720A724,
`Nov. 1979.
`[6] B. M. Popovic', “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 chirpelike polyphase sequences,"
`Electron. Lett., vol. 27. no. 3, pp. 254-255, Jan. 31, 1991.
`
`Shyh—Chang Liu and John J. Komo, Senior Member, IEEE
`
`AbstractiThe correlation values and the distribution of these correla-
`tion values are presented for the small set of nonbinary Kasami se-
`quences over GF( 1)) (p prime). The correlation results are an extension
`of the binary results and have p + 2 correlation levels. This nonhinary
`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 a small primitive
`polynomial of degree m that yields a sequence length of p" — 1 and
`maximum nontrivial correlation value of l + p’". Nonbinury Kasami
`sequences are directly implemented using shift registers and are applica—
`ble for code division multiple access systems.
`
`Index Terms—Kasami sequence, nonhinary, 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)|05tsN—1,1siszm},
`
`where
`
`so) = tr.“ {n.7, (or) + 7.01"}.
`
`(l)
`
`(2)
`
`n=2m, N=2"— l, T=(2"— l)/(2’”— 1)=2’"+1, aisa
`primitive element of GF(2 "), and 7, takes on each value of GF(Z’")
`for 1 s i S 2’". Since air is a primitive element of GF(Z’"), one
`slit) is an m-sequence with period N and each other si(t) is the
`sum of an m-sequenee with period N and a different phase of an
`m-sequenee with shorter period 2’" — 1. The shorter m-scquenee 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, RUM),
`l s i, j s 2’", of the ith and
`jth sequences of S is given by
`N—l
`
`R..(7>=
`
`r:
`
`(mm—1.
`
`(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»
`motion 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.
`
`0018A9448/92$03.00
`
`© 1992 IEEE
`
`PETITIONERS 1069-0004
`|PR2016—OO758
`
`PETITIONERS 1069-0004
`IPR2016-00758
`
`