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`IEEE Xplore Abstract - Generalized chirp-like polyphase sequences with optimum correlation properties
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`Generalized chirp-like polyphase
` sequences with optimum
` correlation properties
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`1
`
` Author(s)
`
`B. M. Popovic ; IMTEL Inst. of Microwave Tech. & Electron., Novi Beogred,
` Yugoslavia
`
`Abstract
`
`Authors
`
`References
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`Cited By
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`A new general class of polyphase sequences with ideal periodic autocorrelation function is
` presented. The new class of sequences is based on the application of Zadoff-Chu polyphase
`2
`sequences of length N=sm , where s and m are any positive integers. It is shown that the
` generalized chirp-like sequences of odd length have the optimum crosscorrelation function under
` certain conditions. Finally, recently proposed generalized P4 codes are derived as a special case of
` the generalized chirp-like sequence
`
`Published in:
` IEEE Transactions on Information Theory (Volume:38 ,
`
`Issue: 4 )
`
`Page(s):
`1406 - 1409
`ISSN :
`0018-9448
`INSPEC Accession Number:
`4251126
`DOI:
`10.1109/18.144727
`
`Date of Publication :
`Jul 1992
`Date of Current Version :
`Tue Aug 06 00:00:00 EDT 2002
`Issue Date :
`Jul 1992
`Sponsored by :
`IEEE Information Theory Society
`Publisher:
`
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`1406
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`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY I992
`
`occurs. Also, if q = pk and p > 5. it is necessary to have k = 1,
`as in the proof of Theorem 1.
`x2 — x — 1 = 0 are (1
`For primes p > 5,
`the roots of
`: v’5‘)/2 E(1i fi)((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/ 01 must be primitive in GF(p). Since 1/or is primitive iff
`or is primitive, the extra condition is that the factor of ~ I must not
`destroy primitivity. Now, multiplication by —l preserves primitiv-
`ity iff —l
`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 ii (mod 10),
`this restricts G4 to primes
`p E 1 (mod 20) or p 5 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 olher such primes have the G4
`construction.
`If f(x) = xx + x -1 then f(—x) = X2 — x — 1, so the roots
`of x2 — x — 1 are the negatives of the roots of xi + X — 1. Also.
`we are considering only primes for which a factor of -1 has no
`effect on primitivity. We have already seen that
`the roots of
`x2 — )6 ~ 1 are primitive or imprimitive together in the fields under
`consideration, so this also must hold for the roots of x2 + X — I.
`By Theorem 1,
`the T4 construction occurs iff at least one of the
`roots of X2 + x — 1 is primitive in GF( 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 — l is not primitive, then both its roots are imprimitive, and
`the roots of xi + X — 1 are also imprimitive.
`E.‘
`In the G4 construction for Costas arrays, since at + ,8 = 1 and
`016 = — 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.
`
`is a point of the
`-4- B’ = l,(l. I)
`Proof: Since or + (3 = oz‘
`array. From oz : -5“ and 6 = —ot", and using rx“"”’/Z =
`6”")/2 == -1, it follows that or = (— l)(B“ 1) = [i""‘”"26 ‘=
`6‘”’3’/2,
`and
`similarly
`[3 2 (-1)(or") = a"””’/2a“' =
`tx("‘“/2. Thus, 1 = or
`-+- 6’ = fi“"3’/2 + a‘”’3’/Z, whence “.0 —‘
`3)/2, (p — 3)/2) is a point of the array.
`From Definition 2, 012 + 6*‘ = ; that is, of + 5/” = Land
`(2, p — 2) is a point of the array. From 043 = -l, B2 + of‘ =
`B2 — 5 : 1, because (as shown in the proof of Theorem 2),
`[3 is a
`root of X2 —X—-
`l = 0. Hence, (—l,2) = (p — 2.2) is also a
`point of the array. Next, or”'3 +B“’*"/2 2 of} + B ‘ /3”"””3
`= (32 — 5 =1,
`and similarly 6""3 + u”’+'”Z = I3‘: + oc-
`a<”-W2 = of — oz : 1, from which both (p — 3, (p + 13/2) and
`((p + 1)/2, p — 3) are points ofthe 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. Golomh, “Algebraic constructions for Costas arrays." J.
`Combin. Theory (A), vol. 37, no. 1, pp. 13-21. July 1984.
`
`[3] 0. Moreno and J. Sotero. “Computational approach to conjecture A
`of Golotnb." Congressus Numeranlium. vol. 70. pp. 7-16, 1990.
`
`Generalized Chirp-Like Polyphase Sequences with
`Optimum Correlation Properties
`
`Branislav M. Popovic
`
`Ab.ttracz—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 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 Term5—Sequences, codes, spread-spectrum, radar, pulse com-
`pression.
`
`I. INTRODUCTION
`
`Sequences with ideal periodic autocorrelation function [1]-[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]—[l0].
`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 [5] has shown that the maximum
`magnitude of the periodic crosscorrelation function and the maxi-
`mum magnitude 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 \/TV‘, 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 modu/arable 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 sequences. It is noticed that these sequences
`also have interesting aperiodic autocorrelation function properties
`[10].
`In Section II, the basic definitions are given. In Section III, we
`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
`crnsscorrelation function of the generalized chirp-like sequences.
`Finally, in Section IV, we show that the recently proposed general-
`
`Manuscript received June II, 1991; revised November 19, 1991.
`The author is with IMTEL Institute of Microwave Techniques and Elec-
`tronics, B. Lenjina l65b.. 1l07l Novi Beogred, Yugoslavia.
`IEEE Log Number 9107517.
`
`0018-9448 /92$03.00
`
`152
`
`1992 IEEE
`
`ZTE/SAMSUNG/HTC 1010-0006
`
`

`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4. JULY 1992
`
`ized P4 codes are only a special case of the generalized chirp—like
`sequences.
`
`ll. BASIC DEFINITIONS
`
`The following definitions will be useful.
`A primitive nth root of unity W" is defined as
`
`j= v’— 1,
`W'"=exp(—j27rr/I1)‘
`where r is any integer relatively prime to n.
`It can be easily shown that for any integer u, 0 < u 5 n — 1, the
`following relation is valid [1 1]
`n—l
`
`(1)
`
`(2)
`W" ;= l.
`A20 W,f""' 2 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
`>l<
`k=O
`Z 5k5<k lp)mod 1_
`
`0(1))
`
`E,
`0.
`
`p =O(mod L),
`p¢0(modL),
`
`(3)
`
`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) = 9*(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 Zadolf—Chu sequences [2], [3] are delined as
`k 1 /2 +4”:
`,
`WM
`kk-I-l)’§'.+ k
`I4/NI
`,/
`4'1
`
`k=0,l,2,---.Nr— l.
`k = 0,1,2."-,N— 1,
`q is any integer.
`
`H
`
`for N even,
`for N odd,
`
`at:
`
`(4)
`The similar construction is proposed by Ipatov [41 and is defined
`
`by
`
`aA.=
`
`Wk1+qk
`N
`
`, k=0,l,2,~--.N—l;
`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 Zadotf—Chu sequences of odd
`length N. Namely,
`starting from the definition of Zadol'f—Chu
`sequences of odd length N = 21 — 1. it follows
`
`(6)
`W/A]’¢v(k+1)/2+qk : (W[£)k1t(l I721/bk.
`is a
`where I 2 2 1 mod N. Since I
`is relatively prime to N.
`primitive Nth root of unity. Hence,
`the right—hand side of (6)
`corresponds to the definition of Ipatov sequences.
`Based on the Zadofl"—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.
`
`111. GENERALIZED CHIR1>»LII<E POLYPI-IASE SEQUENCES
`
`(GCL) sequence {sk} is defined as
`
`k‘0"“vN‘ 1-
`5k=‘7I.-[7(A~;nmd»n—
`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 {S5,} is given by
`
`(7)
`
`{Sic} : iboabtwojaboW2»biW4‘5vbo-
`m
`(8)
`b,W‘-5,b0W2,b1W"5},
`rela-
`where W = exp(—j21rr/8),] = \/A l . r
`is any integer
`tively prime to N= 8, q = 0, while b0 and b, are arbitrary
`complex numbers with magnitude equal to 1.
`
`A. Periodic Autocorrelation Function of the Generalized
`Chirp—Like Sequences
`Theorem I : The gencrali7ed chirp—like polyphase sequences have
`the ideal periodic autocorrelation function.
`
`Proof: It can be seen in [2] that Zadofi‘~Chu sequences are
`defined separately for N even and N odd,
`in order to satisfy the
`following condition
`
`ak+d—N : a/(+45
`
`(9)
`
`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 6(p) of the sequence
`{sk} can be written as
`N— p—I
`Zk'0
`N—I
`
`N~l
`
`*
`sks,(+p+ I‘ g__P
`*
`k—o
`Z Sks/\'+P'
`
`3|!
`5k5k+p
`
`9(9)
`
`For N even, from (4), (7). and (10).
`N—l
`
`it follows
`
`as
`— k
`— Z'2—
`_
`BU’) ’ Wzvp / W [(20 W/vp b(k)mLIdmb(k+p)modrn'
`We shall introduce here the following change of variables:
`k=I'sm+d,
`‘=0.l.~'-,m——l,
`=0,l,-A-,sm~ 1.
`
`From (ll) and (12) we obtain
`
`5m»!
`
`I/:0
`0(17) = W,G”Z’°“”’ Z W7pdb(d)modm
`nI—l
`
`,_
`I b(*vi+{7)Inod m Zn W,;'p-
`If p zit xm, x = 0, l,- -
`sm — 1, the second summation in (13)
`is obviously zero, according to (2).
`For p 2 xm. X = 0,1,---,5m — 1. from (13) we obtain
`sm — l
`
`6(Xm) = mW/\~l(.wn)3 /1 arm 2 W/S—mxd’
`d=0
`
`(14)
`
`If x at 0, 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.
`Cl
`
`Let {ak}, k : 0,1,-~,N— 1. be a Zadofi‘—Chu sequence of
`length N 2 smz, where m and s 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 geiieralized chirp-like
`
`B. Periodic Crosscorrelation Function of the Generalized
`Chi'rp—Like Sequences
`
`Let {xk} and {yk}, k 2 0, 1,--i, N — 1, be the two general-
`ized chirp—like sequences of odd length, obtained from any two
`
`ZTE/SAMSUNG/HTC 1010-0007
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`

`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38. NO. 4. JULY 1992
`
`complex vectors {b1,-} and {b2,}, 1' = 0,1,‘--,m — 1, and from
`the two difierent primitive Nth roots of unity exp ( j27ru/ N) and
`exp (j27ru/N), i.e.,
`_
`— W;
`
`[k(It'+l)/2+ k]
`" b1(k,......,..
`
`(15)
`
`W[:i[k(k+ I)/2+qk]b2(k)mud m ’
`
`exp(j21r/N),
`
`(u,N) - l;(u,N) = l;u=#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(j27ru/N) and exp (j27ru/N), is constant and equal to
`\/N , if (u — 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
`N—l
`N—l
`*
`[=0
`Z xI«vI+p'
`
`(16)
`
`iRxy(P) '2 = ‘go xkyi-€+p
`Substituting (15) into (16), we obtain
`
`I Rxy(p) I 2
`N—l N—l
`
`= Z Z H/A(]v—u)(k—I)[(k+l+1)/2+q]—up(k—l)
`k=0 [=0
`_
`*
`b1(k)modmbl(l)modm
`
`‘b2ikk+p>mod mb2(!+17)mod m
`
`We can introduce the following change of variables:
`
`I:/t+e,e=0,1,--~,N— 1.
`
`From (17) and (18), we obtain
`
`lRx,v(p) I 2
`
`N—l
`
`Z W;(u—u)e[(2k+e+t)/2+q]+upe
`9:0
`
`2 ‘:1=0
`‘ b1(k)rnod mb1ikk+e)mod m
`,
`*
`b2(k+p) mod mb2(k+e+p)mod m‘
`
`This expression can be rewritten as
`N—l
`
`.7:
`|RXJ/(P) I 2 = Z0 W1;(tr—u)t'I(e+I)/2+q]-fpuesey
`where S,, is defined as
`N—l
`
`Se : Z Wfi(u—u,)ek
`k=0
`.
`=i=
`b1(k)modmb1(k+e)modm
`*
`b2(Ic+p)mod mb2(k+p+e)mut1 m‘
`From (20) and (21), it is obvious that
`
`,
`
`|RU(p)|2=N,
`
`ife=O.
`
`We shall prove now that
`
`ife$O.
`
`|Rxy(p)|2 =0:
`We shall introduce the following change of variables
`j =
`k=ism+d,
`0,1,"-,m—l,
`d =
`,1,---,sm~l'
`
`From (21) and (24), we obtain
`smwl
`
`*
`I
`—( — )
`_
`=0
`Se — (1: Wm "
`" Hbl(d)mudmb1(d+e)modm
`m —l
`
`(23)
`
`(24)
`
`1:
`
`“';("7")“>
`' b2Td+p)modmb2(d+p+e)mnd m '
`As (v - 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=0.1.~--,sm— 1.
`For e = xm, x = 0,1,---,sm — 1,5, also becomes equal
`zero, ie.
`
`to
`
`5771-]
`
`E
`S =m Z W‘:n(u—u)xd=0’
`11:0
`
`ifx¢O.
`
`(25)
`
`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 = mg, 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 crosscorrclation 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 »/N.
`IV. GENERALIZED P4 Comes
`
`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,
`
`ZTE/SAMSUNG/HTC 1010-0008
`
`

`
`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, N0. 4, JULY 1992
`
`I409
`
`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]
`
`a,(=exp[j(%k2+1rk)],
`where N is any positive integer.
`This expression can be rewritten as
`
`k=0,l,"'
`
`(28)
`
`21
`WAV = exp _
`ak = W[\,;2/«2+(N/2)k’
`For N even, the P4 code can be obtained by substituting q 2 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 Zadofl'—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 In V 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
`(IV .
`Compared with the generalized Frank sequences. the generalized
`chirp-like sequences olfer 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—l8, pp. 531-532, July
`1972.
`[3] R. L. Frank, “Comments on Polyphase codes with good correlation
`properties,” IEEE Trans. Inform. Theory, vol. IT-19, p. 244, Mar.
`1973.
`[4] V. P. Ipatov, “Multiphase sequences spectrums,” Izvestiya VUZ.
`Radioeleklronika (Radioelecrronics and Communications Sys-
`tems), vol. 22, no. 9, pp. 80-82, 1979.
`[5] D. V. Sarwate, “Bounds on crosscorrelation and autocorrelation 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. Lent, vol. 26, no. 18, pp.
`1428-1430, Aug. 30, 1990.
`—, “Complementary sets of chirp-like polyphase sequences,"
`Electron. Lett., vol. 27, no. 3, pp. 254-255, Jan. 31. 1991.
`
`[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. Popovic', “Comment on Merit
`sequences," Electron. Letl., vol. 27, no. 9, pp. 776-777, April 25,
`1991.
`R. C. Heimiller, “Author's comment,” IRE Trans., vol. IT-8, p.
`382, Oct. 1962.
`
`Nonbinary Kasami Sequences over GF( p)
`
`Shyh—Chang Liu and John J. Komo, Senior Member, IEEE
`Abstract—Thc correlation values and the distribution of these correla-
`tion values are presented for the small set of nonbinary Kasami se-
`quences over GF( 11) (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 a small primitive
`polynomial of degree In that yields :1 sequence length of p” — l 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—KasImi 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)|OstsN—1,1sis2”‘},
`
`5i(’) = "1," {trrii (‘Y’) ‘l’ 71'“?-I}!
`
`(1)
`
`(2)
`
`n=2m, N=2"— 1, T=(2”— 1)/(2'"— 1)=2'"+ 1, or isa
`primitive element of GF(2"), and 71- takes on each value of GF(2’”)
`for l S i S 2”’. Since ozr is a primitive element of GF(2’”), one
`s,(t) is an m—sequence with period N and each other S,-(I) 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-(7),
`l S i, j 5 2"‘, of the ith and
`jth sequences of S is given by
`N —l
`
`1:0
`R,.,.(1) = Z (—1)""*”*’1"’,
`
`0STSN— 1,
`
`(3)
`
`is modulo N addition. If 1' and j are not cyclically
`where t + -r
`distinct, R ,-J-(7) 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 Infor-
`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.
`
`0018-9448/92$03.00 © 1992 IEEE
`
`ZTE/SAMSUNG/HTC 1010-0009