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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012
Gaussian Integer Sequences With Ideal Periodic Autocorrelation Functions Wei-Wen Hu, Sen-Hung Wang, and Chih-Peng Li
Abstract—A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if the out-of-phase value of the periodic autocorrelation function is equal to zero. This paper presents two novel classes of perfect sequences constructed using two groups of base sequences. The nonzero elements of these base sequences belong to the set ±1 ± . A perfect sequence can be obtained by linearly combining these base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal magnitudes. In general, the elements of the constructed sequences are not Gaussian integers. However, if the complex coefficients are Gaussian integers, then the resulting perfect sequences will be Gaussian integer perfect sequences (GIPSs). In addition, a periodic cross-correlation function is derived, which has the same mathematical expression as the investigated sequences. Finally, the maximal energy efficiency of the proposed GIPSs is investigated. Index Terms—Gaussian integer, perfect sequence, periodic auto-correlation function (PACF), periodic cross-correlation function (PCCF).
I. INTRODUCTION Time-discrete sequences with ideal periodic autocorrelation function (PACF) [1]–[3] are widely adopted in a variety of applications, such as channel estimation [4], synchronization [5], peak-to-average power ratio (PAPR) reduction [6], and cell search [7]. Sequences with ideal PACF are usually called perfect or ideal sequences. In general, perfect binary and perfect quadriphase sequences are preferable because of the simplicity of their implementation and their high-energy efficiency. 4 and perfect Unfortunately, perfect binary sequences of length 16 are unknown [8]. Hence, perquadriphase sequences of length fect ternary sequences with elements 1, 0, and 01 can be utilized [9]. In addition, two well-known perfect polyphase sequences with complex-valued elements of unit magnitude are constructed. These are the Frank-Zadoff-Chu (FZC) sequence [1], [2] and the generalized chirplike sequence, obtained by modifying the FZC sequence [10]. Selected mapping (SLM) schemes are widely employed for PAPR reduction in orthogonal frequency division multiplexing systems. However, the traditional SLM scheme requires a bank of inverse fast Fourier transforms (IFFTs) to generate various candidate signals, resulting in a dramatic increase in computational complexity. This drawback can be overcome by adopting time-domain conversion vectors instead of conventional IFFTs [11], where the conversion vectors are obtained using the IFFT of the frequency-domain phase rotation vectors. To ensure that the elements of the frequency-domain phase rotation vectors have equal magnitudes so that the signals of different sub-carriers have the same gain, [11] demonstrated that the
N>
N>
conversion vectors should have the form of a perfect sequence, i.e., the PACF must be a delta function. The computational complexity in generating various candidate signals can also be substantially reduced if the elements of the conversion vectors are Gaussian integers. Hence, Gaussian integer sequences with ideal PACF should be investigated. In contrast to traditional perfect sequences, equal magnitude is not a desired feature for these investigated sequences. Only a few studies have focused on sequence design over Gaussian integers. Fan and Darnell [12] studied the maximal length sequences over Gaussian integers with a “quasi-perfect” PACF. The sequence of length is defined to be quasi-perfect in [12] if and only if its PACF has three nonzero sidelobes at shifts N4 , N2 , and 34N . A few attempts that addressed the applications of Gaussian integers to the construction of different types of codes were derived in [13]–[15]. In addition, multilevel sequences with good correlation properties have been developed in recent literature [16]. To the best of the authors’ knowledge, the present study is the first work that investigates Gaussian integer sequences with ideal PACFs. The current paper investigates two novel classes of sequences with ideal PACFs. The sequences are generated from two groups of base sequences with two and four base sequences in each group, comprising a total of six base sequences. The nonzero elements of these base sequences belong to the set f61 6 g. The proposed sequence can be constructed by linearly combining these six base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal magnitudes. In particular, if the complex coefficients are Gaussian integers, the resulting sequences are Gaussian integer sequences with ideal PACFs, called as Gaussian integer perfect sequences (GIPSs) in the present paper. Moreover, the periodic cross-correlation functions (PCCFs) of the proposed sequences are derived in close-form expressions. This study shows that the PCCFs have the same mathematical representation as the proposed sequences. Finally, although equal magnitude is not of primary interest, this research studies the maximal energy efficiency of the investigated GIPSs, defined as the inverse of PAPR.
N
; j
II. BASIC NOTATIONS AND DEFINITIONS
x y
In this paper, the bold lowercase a denotes a vector. a[ : ] is the th to the th elements of a. [ ] denotes the th element of a. (1)N and j 1 j denote the modulo operation and the absolute values of the arguments, respectively. b1c gives the largest integer less than or equal to the arguments. The superscript * stands for complex conjugate operation. The following basic definitions will be used in the subsequent discourse. The PACF of a time-discrete sequence g of length is defined as
x
an N
y
n
N
Rgg [ ] = N1 1
N01 n=0
g[n] 1 g3 [(n 0 )N ]; = 0; 1; . . . ; N 0 1:
(1)
The PCCF of two arbitrary discrete-time sequences f and g of length
Manuscript received February 20, 2012; revised May 27, 2012; accepted July 07, 2012. Date of publication July 27, 2012; date of current version October 09, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Sofiene Affes. The authors would like to thank the National Science Council of Taiwan for financially supporting this research under Contract No. NSC 99–2628-E-110-001, NSC 100–2628-E-110004, and NSC 100–2219-E-007-010. C.-P. Li is with both the Institute of Communications Engineering and the Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan (e-mail:
[email protected]). W.-W. Hu and S.-H. Wang are with the Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TSP.2012.2210550
N is given by
Rfg [ ] = N1 1
N01 n=0
f [n] 1 g3 [(n 0 )N ]; = 0; 1; . . . ; N 0 1:
To evaluate the time-discrete sequence g with length ficiency g is used and defined by [17]
g where
Eg = N1 1
1053-587X/$31.00 © 2012 IEEE
Eg
max
0nN01
jg[n]j2 ;
(2)
N , energy ef(3)
N01 jg[n]j2 is the average energy of a sequence. n=0
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012
III. CONSTRUCTING THE GAUSSIAN INTEGER SEQUENCES WITH IDEAL PACFS This paper first presents two groups of base sequences, each having two and four base sequences. A perfect sequence can be obtained by linearly combining these base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal are magnitude. These two groups of base sequences of length denoted by u [ ] and v [ ], respectively; = 1 2, = 1 2 3 4, 0 1. The length of each base sequence is and = 0 1 2 . . . a multiple of two. = 1 2 can be written as The first group of base sequences xu vectors two 1 2 x1 = [1 0 . . . 0 1 0 . . . 0] (4)
x n y n ; ; ; ;N
n
N
N ; ; ;
u ; v N ;u ; ;
;
; ; ;
01
(5)
;v
; ; ; ; can be written
= 1 2 3 4
y1 = [1; 1; 1; 1; 1; 1; 1; 1; . . . ; 1; 1; 1; 1] ; y2 = [1; j; 01; 0j; 1; j; 01; 0j; . . . ; 1; j; 01; 0j ] ; y3 = [1; 01; 1; 01; 1; 01; 1; 01; . . . ; 1; 01; 1; 01] ; y4 = [1; 0j; 01; j; 1; 0j; 01; j; . . . ; 1; 0j; 01; j ] :
;v
(6) (7) (8) (9)
; ; ;
= 1 2 3 4, can be obIn particular, the first four elements of yv tained from the generalized complex Hadamard matrix with a dimension of 4 2 4 [18]. The th element of each of the base sequences in the first and the second groups can be respectively represented as
n
xu n ej 1 n 0 Nn 1 N ; u ; ; (10) yv n ej (v01)n ; v ; ; ; ; (11) where n ; ; ; N 0 and is a delta function, which is equal to one when . [
] =
2
2
= 1 2
= 1 2 3 4
= 0 1 ...
1
[
]
= 0
Only two nonzero elements are present in each xu , and all the elements of yv are nonzero. The nonzero elements of both groups of base sequences belong to the set f1 01 0 g. In addition, x1 , x2 , y1 , and y3 have real values, whereas y2 and y4 have complex values. Theorem 1: Let zo be a sequence of length , where = 2 and is any positive odd integer. The th element of zo is given by
; ; j; j
p
N
n
2
zo n [
] =
u=1
N
p
N 1 co;u 1 xu n 0 so;u N (
2
n
)
Xu m [
c
; ; ;N u ; s ;s
)
(12)
; ; N
1 m 0 m 1 0 u 0 2
2
(
1)
;m
; ; ;N 0 :
= 0 1 ...
1
(13)
Similarly, the DFT results of the second group of base sequences in (11) can be represented as
Yu m [
] =
N 1 m0 N 1 u0 4
(
1)
;m
co;u 1 xu n 0 so;u N (
)
)
bo;u 0 co;u 1 y2u01 n 0 so;u N
+ DFT((
(
)
N 1 co;u 1 e0j
2 =
)
1 Xu m [
2
u=1
)
bo;u 0 co;u 1 e0j
]
1 Y2u01 m : [
)
]
(15)
Equation (15) can be rewritten as
Zo m Po m Qo m ; (16) N 1 X1 m (17) Po m 1 co;1 1 e0j bo;1 0 co;1 1 e0j 1 Y1 m ; N 0 j 1 X2 m Qo m 1 co;2 1 e 1 Y3 m : (18) bo;2 0 co;2 1 e0j According to (17), Po m is a combination of X1 m and Y1 m , where the nonzero elements of X1 m are located at even indexes according to (13), and the only nonzero element of Y1 m N m is located at m according to (14). Therefore, the nonzero elements of Po m are located at even indexes. On the other hand, Qo m is a combination of X2 m and Y3 m according to (18), where the nonzero elements of X2 m are located at odd indexes according to (13), and the nonzero elements of Y3 m NN m 0 N2 are loN according to (14). As m p is an odd cated at m 2 2 integer, conclusion was made that the nonzero elements of Qo m are located at odd indexes. Thus, conclusion was arrived at that the nonzero elements of Po m and Qo m do not overlap, leading to when p is odd. As a rePo m 1 Q3o m Po3 m 1 Qo m sult, the absolute value of Zo m is given by jZo m j jZo m j2 2 2 p jP2o m2j jQo m j E 1 N E 1 N; (19) where E is the magnitude of bo;u and co;u , u ; . Equation (19) demonstrates that the elements of Zo m have equal magnitude. The PACF of zo n can be obtained by performing inverse discrete Fourier [
] =
[
] =
[
] +
[
[
2
+ (
[
]
] =
]
)
[
2
+ (
]
[
]
]
)
[
[
]
[
[
]
[
]
]
[
] =
[
]
[
]
[
]
= 0
[
]
[
]
[
[
]
]
[
]
=
=
[
[
]
=
=
] =
[
]
[
]
[
] = 0
=
[
]
=
[
]
[
[
]
]
[
]
+
[
]
=
= 1 2
(
] = 2
DFT(
2
u=1
[
where = 0 1 . . . 0 1. The sequence zo is a perfect sequence if o;u and o;u , = 1 2, are arbitrary nonzero complex numbers of equal magnitude and o;1 o;2 2 f0 1 . . . 0 1g. Proof: Performing discrete Fourier transform (DFT) on the first group of base sequences in (10) results in the following expression:
b
] =
=
)
bo;u 0 co;u 1 y2u01 n 0 so;u N ;
+(
[
N1
2
Zo m
+ (
01
The second group of base sequences yv as four 1 2 vectors
] =
m
01
x2 = [1; 0 . . . ; 0; 01; 0 . . . ; 0]:
[
Consequently, performing DFT on the sequences in (12) and using the fact that the cyclic shifts of time-domain signals are equivalent to the phase rotations of the frequency-domain signals, the th element of the DFT results is given by
; ;
01
N
6075
; ; ;N 0 :
= 0 1 ...
1
(14)
[
]
]
transform (IDFT) to both sides of (19), i.e.,
Rz z IDFT jZo m j E 1 : (20) As the PACF of zo n has a delta function form, zo n can be proven to ; , are arbitrary nonzero be a perfect sequence if bo;u and co;u , u complex numbers of equal magnitude (i.e., jbo;u j jco;u j E ). [
[
] =
(
[
] ) =
]
[
[
]
]
= 1 2
=
=
Although both groups of base sequences are Gaussian integer sequences, the elements of zo given in Theorem 1 are not Gaussian integers in general because o;u and o;u are arbitrary nonzero complex numbers. = 1 2, are nonzero Gaussian Corollary 1a: If o;u and o;u , integers of equal magnitude, the perfect sequence zo shown in Theorem 1 is a Gaussian integer sequence. This sequence is termed as the GIPS in this paper.
b
b
c
c
u
;
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012
TABLE I ILLUSTRATIVE EXAMPLES OF THE PROPOSED SEQUENCES (
Corollary 1b: The perfect sequence can be obtained by using only the first group base sequences, i.e., 2
zo [n]
=
u=1
N 2
1
co;u 1 xu
n 0 so;u )N :
(
(21)
If co;1 and co;2 are nonzero Gaussian integers of equal magnitude, the perfect sequence zo is a GIPS. Proof: The proof can be easily obtained by letting bo;1 = co;1 and bo;2 = co;2 in (12). Corollary 1c: All elements of the perfect sequences zo constructed in Theorem 1 have real values if bo;u and co;u , u = 1; 2. The proof is quite straightforward and is omitted for brevity. Corollary 1d: All elements of the perfect sequences zo constructed in Theorem 1 are purely imaginary if bo;u and co;u , u = 1; 2, are all purely imaginary. The proof is quite straightforward and is omitted for brevity. Table I demonstrate some illustrative examples of the GIPS presented in Corollaries 1a, 1b, 1c, and 1d for the case of odd p. Theorem 2: Let ze be a sequence of length N , where N = 2p and p is any positive even integer. The nth element of ze is given by 2 N ze [n] = 1 ce;u 1 xu (n 0 se;u ) N u=1
2
4 +
v=1
be;v
0
ce;d(v)
1
yv
n 0 se;d(v) N ;
(22)
where n = 0; 1; . . . ; N 0 1 and d(v ) = (v 0 1) 1 N4 2 + 1. The sequence ze is a perfect sequence if be;v , v = 1; 2; 3; 4, and ce;u , u = 1; 2, are arbitrary nonzero complex numbers of equal magnitude, and se;1 ; se;2 2 f0; 1; . . . N 0 1g. Proof: Performing DFT on the sequences in (22) and using the fact that cyclic shifts of time-domain signals are equivalent to the phase rotations of frequency-domain signals, the mth element of the DFT results is given by 2 N 0j Ze [m] = 1 Xu [m] 1 ce;u 1 e u=1
2
4 +
v=1
be;v
0
ce;d(v)
1
e0j
1
Yv [m] :
(23)
N = 2p, p IS ODD)
Equation (23) can be rewritten as Ze [m] = Pe [m] + Qe [m] ; N 0j Pe [m] = 1 ce;2 1 e 2
Qe [m]
=
N
1
2
(24)
ce;1 1 e0j
1
X2 [m] ;
1
X1 [m]
1
e0j
4
be;v
+
v=1
ce;d(v)
0
(25)
1
Yv [m] : (26)
From (25) and (26), the nonzero elements of Pe [m] and Qe [m] can be easily seen as not overlapping, leading to Pe [m] 1 Q3e [m] = Pe3 [m] 1 Qe [m] = 0. Thus, the absolute value of Ze [m] is given by Ze [m]j
j
Ze [m]j2
=
j p
=
E2 1 N 2
Pe [m] j2 + jQe [m] j2
=
=
j
E 1 N;
(27)
where E is the magnitude of be;v and ce;u . Equation (27) demonstrates that the elements of Ze [m] have equal magnitude. The PACF of ze [n] can be obtained by performing IDFT to both sides of (27), i.e., Rz z [ ]
=
IDF T
Ze [m]j)
(j
=
E 1 [ ] :
(28)
As the PACF of ze [n] has a delta function form, ze [n] can be proven to be a perfect sequence if be;v and ce;u , are arbitrary nonzero complex numbers of equal magnitude (i.e., jbe;v j = jce;u j = E ). The elements of ze given in Theorem 2 are not Gaussian integers, in general. Corollary 2a: If be;v , v = 1; 2; 3; 4, ce;u , u = 1; 2, are nonzero Gaussian integers of equal magnitude, the sequence demonstrated in Theorem 2 is a GIPS. Corollary 2b: The perfect sequence ze presented in Theorem 1 can be obtained by using only the first group base sequences, i.e., 2
ze [n]
=
u=1
N 2
1
ce;u 1 xu
n 0 se;u )N :
(
(29)
If ce;u , u = 1; 2, are Gaussian integers of equal magnitude, the perfect sequence ze is a GIPS.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012
6077
TABLE II ILLUSTRATIVE EXAMPLES OF THE PROPOSED SEQUENCES (
N = 2p, p IS EVEN)
Corollary 2c: All nonzero elements of the perfect sequences ze constructed in Theorem 2 have real values if i. be;1 ; be;3 and ce;u ; u = 1; 2; have real values. ii. be;2 = 0be;4 = wj are purely imaginary; w 2 f61g. Corollary 2d: All nonzero elements of the perfect sequences ze constructed in Theorem 2 are purely imaginary if i. be;1 ; be;3 and ce;u ; u = 1; 2; are purely imaginary. ii. be;2 = 0be;4 = w are real-valued; w 2 f61g. Table II demonstrates some illustrative examples of the GIPS presented in Corollaries 2a, 2b, 2c, and 2d for the case of even p. IV. PCCFS This section describes the PCCFs of the investigated perfect sequences for both even and odd p. The PCCF of the two arbitrary sequences f and g is given by (2). Performing DFT to both sides of (2) yields the following result:
f [ ]g = F [m] 1 G [m] ; (30) where F [m] and G[m] are the DFTs of f [n] and g [n], respectively. DF T Rfg
Consequently, PCCF can be obtained by performing the IDFT on [ ] 1 G3 [m]. Theorem 3: Considering two different sequences zo;1 and zo;2 constructed in Theorem 1, their PCCFs are given by z
[ ] =
2
N2
1 co;u;1 1 c3o;u;2xu [( 0 so;u;1 + so;u;2)N ] 2 u=1 3 3 + N 1 bo;u;1 1 bo;u; 2 0 co;u;1 1 co;u;2 1 y2u01 ( 0 so;u;1 + so;u;2 )N
;
(31)
where bo;u;i and co;u;i ; u = 1; 2; i = 1; 2, are arbitrary nonzero complex numbers of equal magnitude. so;u;i 2 f0; 1; . . . ; N 0 1g; u = 1; 2; i = 1; 2. Proof: Given the two different sequences constructed in Theorem 1 zo;1 and zo;2 . Performing DFT on the sequences in (12), the mth element of the DFT results for zo;i is given by
[ ]=
Zo;i m
2
0j 2 1 co;u;i 1 e
N
u=1
+ (bo;u;i 0 co;u;i ) 1 e0j
[ ] = Po;i [m] + Qo;i [m] ; N 0j 1 X1 [m] Po;i [m] = 2 1 co;1;i 1 e 1 Y1 [m] ; + (bo;1;i 0 co;1;i ) 1 e0j N 0j Qo;i [m] = 1 X2 [m] 2 1 co;2;i 1 e 1 Y3 [m] : + (bo;2;i 0 co;2;i ) 1 e0j Zo;i m
[ ]
Xu m
[ ]
Y2u01 m
;
(32)
(33) (34)
(35)
Obviously, the nonzero elements of Po;i [m] and Qo;i [m] do not 3 3 [m] = Po;i [m] 1 Qo;i [m] = 0. overlap, leading to Po;i [m] 1 Qo;i Therefore,
[ ]1
[ ] = Po;1 [m]1 Po;3 2 [m]+ Qo;1 [m]1 Qo;3 2 [m] :
3 Zo;1 m Zo;2 m
3
F m
Rz
where Xu [m] and Y2u01 [m] are defined by (13) and (14), respectively. Equation (32) can be rewritten as
(36)
Finally, performing IDFT on both sides of (36), the PCCF of zo;1 and can be represented by (31). 3 3 ~ Let c~o;u N 1 co;u;1 1 co;u; ~o;u 2 , bo;u N 1 bo;u;1 1 bo;u;2 , and s 3 N 1 so;u;1 1 so;u;2 . The PCCFs of zo;1 and zo;2 can be rewritten as
zo;2
Rz
z
[ ]=
2 u=1
2 1~co;u 1 xu [( 0~so;u )N ] + ~bo;u 0 c~o;u 1 y2u01 ( 0 s~o;u )N N
:
(37)
By comparing (37) with (12), interestingly, the PCCFs of zo;1 and zo;2 have the same mathematical expression as that of zo . Corollary 3a: Considering two different sequences zo;1 and zo;2 3 3 with PCCF given by (31), if bo;u;1 1 bo;u; 2 = co;u;1 1 co;u;2 = , u = 1; 2, and so;1;1 0 so;1;2 = so;2;1 0 so;2;2 , where is any arbitrary nonzero number, the number of nonzero PCCF elements is one, and the magnitude of the nonzero element is j N j. Proof: The proof is straightforward and is omitted for brevity. Corollary 3b: Considering two different sequences zo;1 and zo;2 3 3 with PCCF given by (31), if bo;u;1 1 bo;u; 2 = co;u;1 1 co;u;2 = ; u = 1; 2, where is an arbitrary nonzero number, and so;1;1 0 so;1;2 6= so;2;1 0 so;2;2 , the number of nonzero PCCF elements between two sequences is four, and the magnitude of the corresponding nonzero elements is j N 2 j. Proof: The proof is straightforward and is omitted for brevity.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012
TABLE III ILLUSTRATIVE EXAMPLES OF THE PCCFS FOR DIFFERENT COEFFICIENT COMBINATIONS (
TABLE IV ILLUSTRATIVE EXAMPLES OF THE PCCFS FOR DIFFERENT COEFFICIENT COMBINATIONS (
Table III demonstrates some illustrative examples of the PCCF presented in Corollaries 3a and 3b for the case of odd p. Theorem 4: Considering two different sequences ze;1 and ze;2 constructed in Theorem 2, their PCCFs are given by
Rz
z
[
] =
2
N
u=1
2
2
4
+
v =1 1
yv
1
ce;u;1 1 c3e;u;2 1 xu [( 0 se;u;1 + se;u;2 )N ]
N
1
3 3 be;v;1 1 be;v; 2 0 ce;d(v );1 1 ce;d(v );2
3 n 0 se;d(v);1 + se;d (v );2
N
;
(38)
where be;v;i , ce;d(v);i and ce;u;i , v = 1; 2; 3; 4, i = 1; 2, and u = 1; 2, are arbitrary nonzero complex numbers of equal magnitude. se;u;i ; se;d(v);i 2 f0; 1; . . . ; N 0 1g; u = 1; 2; i = 1; 2. The proof of Theorem 4 is similar to that of Theorem 3 and is omitted in this paper. The PCCF given in (38) has the same mathematical representation as that of ze . Corollary 4a: Considering two different sequences ze;1 and ze;2 3 3 with PCCF given by (38), if be;v;1 1 be;v; 2 = ce;d(v );1 1 ce;d(v );2 = ; v = 1; 2; 3; 4, where is an arbitrary nonzero number, and se;1;1 0 se;1;2 = se;2;1 0 se;2;2 , the number of nonzero PCCF elements between two sequences is one, and the magnitude of the corresponding nonzero elements is j N j. Proof: The proof is straightforward and is omitted here. Corollary 4b: Considering two different sequences ze;1 and ze;2 3 3 with PCCF given by (38), if be;v;1 1 be;v; 2 = ce;d(v );1 1 ce;d(v );2 = ; v = 1; 2; 3; 4, where is an arbitrary nonzero number, and se;1;1 0 se;1;2 6= se;2;1 0 se;2;2 , the number of nonzero PCCF elements between two sequences is four, and the magnitude of the corresponding nonzero elements is j N2 j. Proof: The proof is straightforward and is omitted here.
N = 2p, p IS ODD)
N = 2p, p IS EVEN)
Table IV provides some illustrative examples of the PCCF presented in Corollaries 4a and 4b for the case of even p. V. ENERGY EFFICIENCY OF THE PROPOSED SEQUENCES In this section, the energy efficiency of the proposed sequences is evaluated. Energy efficiency, as defined in (3) is basically the inverse of PAPR. As with (12) and (22), all the coefficients bo = [bo;1 ; bo;2 ], co = [co;1 ; co;2 ], be = [be;1 ; be;2 ; be;3 ; be;4 ], and ce = [ce;1 ; ce;2 ] must be nonzero values of equal magnitude to construct the proposed perfect sequences. However, larger magnitudes of these coefficients result in lower energy efficiency. In addition, the selections of so = [so;1 ; so;2 ] and se = [se;1 ; se;2 ] also affect energy efficiency. Without loss of generality, bo , co , be , and ce are selected from the set f61; 6j g in the following discussions, and exhaustive computer searches are conducted to determine the coefficients that maximize energy efficiency. The results are summarized as follows: 1) Case 1: The GIPS zo constructed in Corollary 1a has length N = 2p, where p is any arbitrary positive odd integer. i. For p = 3, zo achieves the maximal energy efficiency z = N8 if bo;2 = 6jbo;1 , co = 0bo , and so are arbitrary values. ii. For p 5, zo achieves the maximal energy efficiency z = 4N (N 08N +32) if bo;2 = 6jbo;1 , co = 0bo , and jso;1 0 so;2 j 6= 0 N or 2 . The sequences ze constructed in Corollary 2a of length N = 2p, where p is even, will be discussed for two different cases: N = 8p1 + 4 and N = 8p2 , where p1 and p2 are any positive integers. 2) Case 2a: The GIPS ze constructed in Corollary 2a has length N = 8p1 + 4, where p1 is any positive integer. ze achieves the maximal energy efficiency z if be;2 2 [6be;1 ; 6jbe;1 ], [be;3 ; be;4 ] = [be;1 ; be;2 ], ce = 0be [1 : 2], and jse;1 0 se;2 j has an odd value. N . i. For p1 = 1, z = 16
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012
TABLE V COEFFICIENTS THAT ACHIEVE MAXIMAL ENERGY EFFICIENCY FOR THE INVESTIGATED GIPSS OF LENGTH
N = 8p
TABLE VI MAXIMAL ENERGY EFFICIENCY OF VARIOUS GIPSS
4N . ii. For p1 2, z = (N0 8) 3) Case 2b: The GIPS ze constructed in Corollary 2a has length N = 8p2 , where p2 is any positive integer. ze achieves the maximal energy efficiency z if be and se fulfill one of the eight conditions listed in Table V, and ce is arbitrary. N. i. For p2 = 1, z = 16 ii. For p2 2, z = N4 . Table VI summarizes the maximal energy efficiency of the investigated GIPSs. Obviously, the maximum energy efficiency of the proposed sequences with lengths N = 6, N = 8, and N = 12 are 0.75, 0.5, and 0.75, respectively. For a sequence with length N 10 (N = 2p, p is odd), its maximum energy efficiency is equal to (N 048NN +32) . For a sequence with length N 20 (N = 8p1 + 4; p1 2), its 4N . For a sequence with maximum energy efficiency is equal to (N0 8) length N 16 (N = 8p2 ; p2 2), its maximum energy efficiency is equal to N4 . In general, energy efficiency decreases with sequence length N . GIPSs are adopted as the conversion vectors in a low-complexity SLM PAPR reduction scheme. Therefore, as opposed to traditional perfect sequences, equal magnitude is not a desired feature for the investigated GIPSs. Further applications of the GIPSs are to be discovered.
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VI. CONCLUSIONS The current study has constructed two novel classes of perfect sequences generated by linearly combining two groups of base sequences or their cyclic shift equivalents with arbitrary nonzero complex coefficients of equal magnitudes. If the complex coefficients are Gaussian integers, the resulting sequences are Gaussian integer sequences with ideal PACFs. In addition, the PCCFs of the proposed perfect sequences were derived in close-form expressions and have been found to have the same mathematical expression as that of the investigated sequences. Finally, the maximal energy efficiency of the proposed GIPSs was investigated.
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