1
Shared Autocorrelation Property of Sequences Corneliu Bodea, Calina Copos, Matt Der, David O’Neal, James A. Davis
Abstract—Due to the low Peak-to-Mean Envelope Power Ratio (PMEPR) of Golay sequences together with their connection to generalized Reed-Muller codes, many authors have proposed using Golay sequences as codewords in orthogonal frequency-division multiplexing (OFDM). Golay sequences that unexpectedly share the same aperiodic autocorrelation function can be used to construct longer Golay sequences. We study the shared autocorrelation property of general sequences with the ultimate goal of using those results to find other low power sequences. In this paper, we provide several new constructions of families of pairs of sequences with the shared autocorrelation property that enable us to explain nearly all of the binary shared autocorrelation property for sequences of lengths up to 24. Index Terms—aperiodic autocorrelation function, Golay sequence, quaternary, shared autocorrelation property.
I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) has many advantageous features due to the increased capabilities of digital signal processors, but it suffers from the difficulty of controlling the power of uncoded signals. Many publications ([1], [2], [3]) have proposed restricting the allowable transmitted sequences to Golay sequences and related objects as these sequences have predictably low Peak-to-Mean Envelope Power Ratios (PMEPR). Understanding Golay sequences and their C. Bodea was an undergraduate at the University of Richmond, 28 Westhampton Way, University of Richmond, VA 23173. Supported in part by NSF grant EMSW21-MCTP 0636528. Email:
[email protected] C. Copos was an undergraduate at the University of Richmond, 28 Westhampton Way, University of Richmond, VA 23173. Supported in part by NSF grant EMSW21-MCTP 0636528. Email:
[email protected] M. Der was an undergraduate at the University of Richmond, 28 Westhampton Way, University of Richmond, VA 23173. Supported in part by NSF grant EMSW21-MCTP 0636528. Email:
[email protected] D. O’Neal was an undergraduate at the University of Richmond, 28 Westhampton Way, University of Richmond, VA 23173. Supported in part by NSF grant EMSW21-MCTP 0636528. Email:
[email protected] J. Davis is a Professor of Mathematics at the University of Richmond, 28 Westhampton Way, University of Richmond, VA 23173. Supported in part by NSF grant EMSW21-MCTP 0636528. Email:
[email protected]
generalizations requires the introduction of the aperiodic autocorrelation function of a sequence. Whitehead [4] investigated the properties of the aperiodic autocorrelation function of binary sequences, and he observed that binary sequences of composite lengths occasionally shared aperiodic autocorrelation functions even though those sequences were not reversals, negations, or negative reversals of each other. He demonstrated that certain tensor products of shorter sequences could produce longer sequences with shared autocorrelation functions, and he was able to explain many of the shorter binary sequence shared autocorrelation pairs. More recently, Li and Chu [5] discovered 1024 additional non-standard quaternary Golay sequences of length 16 that were not explained by Davis and Jedwab’s [1] standard construction. Fiedler and Jedwab [6] later showed that all 1024 are either concatenations or interleavings of standard Golay sequences of length 8 whose aperiodic autocorrelation functions are unexpectedly equal. Motivated by these prior studies, Davis and Pohl [7] extended Whitehead’s work to quaternary and produced several infinite families of quaternary sequences that had the shared autocorrelation property. These new families were not tensor products of smaller sequences, thus establishing that new techniques will be needed to understand when two sequences will have the same aperiodic autocorrelation. This paper provides further results that explain the shared autocorrelation property of sequences. The paper is organized as follows. In Section 2 we establish the notation that will be used in the rest of the paper and we provide details of the background material. Section 3 contains our first general result about the shared autocorrelation property, and Section 4 contains our other main constructions. The results in Sections 3 and 4 enable us to explain almost all of the binary shared autocorrelation property up to length 24 and much of the quaternary up to length 14. Section 5 establishes some general results about the aperiodic cross correlation of sequences, results that are useful in building sequences with the shared autocorrelation property.
Table 1 Length Binary SAP Quaternary SAP
1 0 0
Length Binary SAP
2 0 0
3 0 0 15 14
4 0 0 16 12
5 0 0 17 1
6 0 4
7 0 0 18 42
8 0 12 19 0
9 1 120 20 44
10 0 92 21 67
11 0 0 22 0
12 6 2034 23 1
13 0 0
14 0 792
24 422
our main results.
II. BACKGROUND AND N OTATION
Given an M -ary sequence a of length n and a scalar c ∈ ΞM , we define the scalar multiple of a by c as the M -ary sequence c · a = {c · a0 , c · a1 , . . . , c · an−1 }. Similarly, we define the reversal conjugate of a as the M -ary sequence R(a) = {an−1 , an−2 , . . . , a0 }. The following lemma shows that all sequences of the form c·a and c·R(a) have the same aperiodic autocorrelation.
An M -ary sequence a of length n is the vector a = {a0√ , a1 , . . . , an−1 } where aj ∈ {ξ k |k ∈ Z, ξ = 2π −1/M } (for the remainder of the paper √we will e use the notation ΞM = {ξ k |k ∈ Z, ξ = e2π −1/M } for the set of scalars being used in the sequences). In this context, binary sequences have terms consisting of ±1 and √ quaternary sequences have terms consisting of ±1, ± −1. The following two definitions provide tools for studying the relationship between two sequences (the cross correlation) and the self-similarity of a sequence (the autocorrelation).
Lemma 2.3: Let a and b be M -ary sequences of length n. If b ∈ {c · a|c ∈ ΞM } ∪ {d · R(a)|d ∈ ΞM }, then A(a) = A(b). The proof is trivial and is left to the reader. The fact that the converse of Lemma 2.3 is not true is what requires a formal definition of what we call the shared autocorrelation property.
Definition 2.1: Suppose m ≥ n. The aperiodic cross correlation function of M -ary sequences a of length n and b of length m is C(a, b) = {C(a, b)(−(m − 1)), C(a, b)(−(m − 2)), . . . , C(a, b)(n − 1)}, where Pm+u−1 C(a, b)(u) = k=0 if −m < u < −(m − a b Pn−1 k k−u n); C(a, b)(u) = k=0 ak bk−u if −(m − n) ≤ u ≤ 0; Pn−u−1 and C(a, b)(u) = k=0 ak+u bk if 0 < u < n (the definition for m < n is similar).
Definition 2.4: Let a and b be M -ary sequences of length n. The sequences a and b possess the shared autocorrelation property (SAP) if A(a) = A(b) but b 6∈ {c · a|c ∈ ΞM } ∪ {d · R(a)|d ∈ ΞM }. This definition is due to Fiedler and Jedwab [6], but the idea goes back to Whitehead [4]. Whitehead ran computer searches for binary sequences up to length 15 and found examples of pairs of sequences (up to the equivalence described in Lemma 2.3) with the SAP at the composite lengths 9, 12, and 15. Table 1 extends the computer searches for the number of equivalence classes of pairs of binary sequences with the SAP up to length 24, and Table 1 also includes the results from computer searches for the number of equivalence classes of pairs of quaternary sequences with the SAP up to length 14. We observe that we found binary pairs of sequences with the SAP for lengths 17 and 23, both prime lengths, and these pairs cannot be explained by any known results (including results in this paper).
We have allowed for different length sequences in the definition of the aperiodic cross correlation function since our constructions will allow this possibility. Definition 2.2: The aperiodic autocorrelation function of the M -ary length n sequence a = {a0 , a1 , . . . , an−1 } is A(a) = {A(a)(1), A(a)(2), . . . , A(a)(n − 1)}, where Pn−u−1 A(a)(u) = C(a, a)(u) = k=0 ak+u ak . We note that the vector A(a) only includes C(a, a)(u) for 0 < u < n. We do this for simplicity since C(a, a)(0) = n and C(a, a)(−u) = C(a, a)(u). The associated generating polynomial of a is a(x) = a0 + a1 x + a2 x2 + · · · + an−1 xn−1 . If a is a length n sequence and b is a length m sequence, then Pn−1 a(x)b(x−1 ) = k=−(m−1) C(a, b)(k)xk . Similarly, for a given length n sequence a we have a(x)a(x−1 ) = Pn−1 n + k=1 (A(a)(k)xk + A(a)(k)x−k ). The polynomial viewpoint will be used in Sections III and IV to prove
This paper will explain many of the observed occurrences of SAP in Table 1. Whitehead began the explanation with the concept of a
2
tensor product construction, which we now define. The tensor product of two sequences a = {a0 , a1 , . . . , an−1 } and b = {b0 , b1 , . . . , bm−1 } is a ⊗ b = {a0 b0 , a1 b0 , . . . , an−1 b0 , a0 b1 , . . . , an−1 b1 , . . . , a0 bm−1 , a1 bm−1 , . . . , an−1 bm−1 }. The generating polynomial of a ⊗ b is a(x)b(xn ). The following theorem from Whitehead shows why tensor products are useful in explaining SAP. Theorem 2.5: Let a and b be sequences of length m such that A(a) = A(b), and let c and d be sequences of length n such that A(c) = A(d). Then A(a ⊗ c) = A(b ⊗ d).
a(x)c(x
)a(x−1 )c(x−n )
=
1) A(a1 ) = {3, 0, −1, −2, −1} = A(b1 ) (by Lemma 2.3 since b1 = R(a1 )) 2) A(a2 ) = {−3, 0, 1, −2, 1} = A(b2 ) (by Lemma 2.3 since b2 = −R(a2 )) 3) C(a2 , a1 ) = {−1, 0, 1, 0, 3, 0, −3, 0, −1, 0, 1} = C(b2 , b1 ) The following theorem generalizes Example 3.1.
Proof: n
a2 = {1, −1, 1, −1, −1, 1}, and b is formed by concatenating the two sequences b1 = {−1, −1, 1, 1, 1, 1} and b2 = {−1, 1, 1, −1, 1, −1}. We see that a1 , a2 , b1 , and b2 have the following properties:
Theorem 3.2: Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be M -ary sequences. If
a(x)a(x−1 )c(xn )c(x−n )
= b(x)b(x−1 )d(xn )d(x−n )
1) A(ak ) = A(bk ), 1 ≤ k ≤ n; 2) C(ak , aj ) = C(bk , bj ), j 6= k;
= b(x)d(xn )b(x−1 )d(x−n )
Then A({a1 |a2 | . . . |an }) = A({b1 |b2 | . . . |bn }). Proof:
Example 2.6: We demonstrate that the single occurrence of the shared autocorrelation property in binary sequences of length 9 can be explained by tensor products. The sequences e = {1, 1, −1, 1, 1, −1, −1, −1, 1} and f = {1, 1, −1, −1, −1, 1, −1, −1, 1} exhibit the SAP since A(e) = A(f ) but f 6∈ {c·e|c ∈ Ξ2 } ∪ {d·R(e)|d ∈ Ξ2 }. Theorem 2.5 explains this instance of SAP by letting a = b = c = {1, 1, −1} and d = {1, −1, −1}, leading to the construction of e = a ⊗ c and f = b ⊗ d. Since A(a) = A(c) and A(b) = A(d), we have A(e) = A(f ). In addition to the lone instance of binary length 9 SAP, Theorem 2.5 explain 36 of the 120 instances of the SAP for quaternary length 9 sequences in Table 1.
Let `1 , `2 , . . . `n be the lengths of the sequences a1 , a2 , . . . , an respectively (and hence also the lengths of b1 , b2 , . . . , bn respectively). The two conditions imply ak (x)aj (x−1 ) = bk (x)bj (x−1 ), 1 ≤ j, k ≤ n. The polynomial for the sequence a = {a1 |a2 | · · · |an } is a(x) = a1 (x) + x`1 a2 (x) + x`1 +`2 a3 (x) + · · · + x`1 +···+`n−1 an (x), and similarly we get the polynomial b(x) = b1 (x) + x`1 b2 (x) + x`1 +`2 b3 (x) + · · · + x`1 +···+`n−1 bn (x) for the sequence b = {b1 |b2 | · · · |bn }. A simple calculation yields the result.
Theorem 3.2 provides a recipe for constructing sequences with the SAP if we can find shorter sequences with the aperiodic and cross-correlation properties. Section V will address the issue of determining how to create a collection of sequences that has all the cross correlation properties required.
III. C ONCATENATION OF SEQUENCES If a = {a0 , a1 , . . . , a`−1 } and b = {b0 , b1 , . . . , bm−1 }, then the concatenation of a and b is defined as {a|b} = {a0 , a1 , . . . , a`−1 , b0 , b1 , . . . , bm−1 }. The following binary length 12 example uses concatenation to construct sequences with the SAP.
IV. R EVERSAL CONCATENATION Example 3.1: Let a = {1, 1, 1, 1, −1, −1, 1, −1, 1, −1, − 1, 1} and b = {−1, −1, 1, 1, 1, 1, −1, 1, 1, −1, 1, −1}. We notice that a and b have the SAP although a and b do not follow the construction described in Theorem 2.5. The key observation is that a is formed by concatenating the two sequences a1 = {1, 1, 1, 1, −1, −1} and
As in the previous section, we list an example to illustrate our next (and most general) construction technique. Example 4.1: Let a = {1, 1, 1, 1, −1, −1, 1, 1} and b = {1, −1, 1, −1, −1, 1, 1, −1}. We observe that C(a, b) = 3
s(x)s(x−1 ) = (a(x)t(x) + b(x)u(x))(a(x−1 )t(x−1 ) + b(x−1 )u(x−1 )) = a(x)t(x)a(x−1 )t(x−1 ) + a(x)t(x)b(x−1 )u(x−1 )+ b(x)u(x)a(x−1 )t(x−1 ) + b(x)u(x)b(x−1 )u(x−1 ) = a(x)t(x)a(x−1 )t(x−1 ) + c · b(x)t(x)a(x−1 )u(x−1 )x`−m + c · a(x)u(x)b(x−1 )t(x−1 )xm−` + b(x)u(x)b(x−1 )u(x−1 ) = (a(x)t(x−1 )xn−` + c · b(x)u(x−1 )xn−m ) (a(x−1 )t(x)x`−n + c · b(x−1 )u(x)xm−n ) = s0 (x)s0 (x−1 )
−C(b, a). Using a and b as subsequences, we build the sequences S = {a|b| − b} and S0 = {b| − b|a}. We can easily verify that S and S0 have the SAP. Theorem 8 of Whitehead [4] includes Example 4.1. The theorem below generalizes Whitehead’s result by allowing (i) different length sequences a and b; (ii) larger alphabets than binary; and (iii) different scalar multiples than −1 for the cross-correlation property. The sequences S and S0 from Example 4.1 cannot be constructed via Theorem 2.5 or Theorem 3.2, so Theorem 4.2 is a new tool for constructing sequences with the SAP. Given M -ary sequences a of length ` and b of length m satisfying C(a, b)(u) = c · C(b, a)(m − ` + u), −(m − 1) ≤ u ≤ (` − 1), c ∈ ΞM , define two sequences S and S0 as follows: S = {S1 |S2 | · · · |Sn } where each Si is the sequence di ·a or di ·b for di ∈ ΞM . The partner sequence is S0 = {S0n |S0n−1 | · · · |S01 }, where S0i = di · a if Si = di · a and S0i = c · di · b if Si = di · b.
Thus the sequences S and S0 have the same aperiodic autocorrelation. A more general version of Theorem 4.2 is the following, which we only state. For j = 1, 2, . . . , n, suppose we have M -ary sequences aj (not necessarily distinct) of length `j satisfying C(ai , aj )(u) = ci,j · C(aj , ai )(`j − `i + u), −(`j − 1) ≤ u ≤ (`i − 1), ci,j ∈ ΞM . Define two sequences S and S0 similar to the previous result: S = {S1 |S2 | · · · |Sn } where each Si is the sequence di · ai for di ∈ ΞM , 1 ≤ i ≤ n. The partner sequence is S0 = {S0n |S0n−1 | · · · |S01 }, where S0i = ci,j · di · aj for some j.
Theorem 4.2: Let a be a length ` M -ary sequence and b be a length m M -ary sequence, and suppose that C(a, b)(u) = c · C(b, a)(m − ` + u), −(m − 1) ≤ u ≤ (` − 1), c ∈ ΞM . Then the sequences S and S0 defined above satisfy A(S) = A(S0 ).
Theorem 4.3: Suppose the M -ary sequences aj satisfy C(ai , aj )(u) = ci,j · C(aj , ai )(`j − `i + u), 1 ≤ j ≤ n, cj ∈ ΞM , −(`j − 1) ≤ u ≤ (`i − 1). Then the sequences S and S0 defined above satisfy A(S) = A(S0 ).
Proof: Let a(x) be the generating polynomial for the sequence a and b(x) for the sequence b, and suppose C(a, b)(u) = c·C(b, a)(m−`+u), c ∈ ΞM , −(m−1) ≤ u ≤ (` − 1). The generating polynomial of S can be written as
There still remain instances of SAP that Theorems 4.2 and 4.3 do not explain, but we can use these theorems in a new construction technique that ultimately relies upon Theorem 3.2. Recall that Theorem 3.2 concatenates subsequences with two key properties to form a pair of sequences {a1 |a2 | . . . |an } and {b1 |b2 | . . . |bn } that have identical autocorrelations. Our next technique builds these sequences using Theorems 4.2 and 4.3 to satisfy A(ak ) = A(bk ) and in such a way that guarantees C(ak , aj ) = C(bk , bj ), j 6= k. As in the previous sections, we start with an example to illustrate the construction technique.
s(x) = a(x)t(x) + b(x)u(x).
Similarly, if the total length of the sequence S is n then the partner sequence S0 has polynomial
Example 4.4: Let b1 = {1, 1, 1} and b2 = {−1, 1, 1}, and let a1 = {1, 1} and a2 = {1, −1}. Observe that C(a1 , a2 ) = −C(a2 , a1 ). Now, build the sequences
s0 (x) = a(x)t(x−1 )xn−` + c · b(x)u(x−1 )xn−m .
T1 = {a1 ⊗ b1 |a2 ⊗ b1 } T01 = {−a2 ⊗ b1 |a1 ⊗ b1 } We know that C(a, b)(u) = c·C(b, a)(m−`+u) implies a(x)b(x−1 ) = c · b(x)a(x−1 )x`−m . The polynomial computations for the aperiodic autocorrelation functions of S and S0 go as follows.
T2 = {a1 ⊗ b2 |a2 ⊗ b2 } T02 = {−a2 ⊗ b2 |a1 ⊗ b2 }
We can verify that A({T1 |T2 }) = A({T01 |T02 }). This result is straightforward since
4
Table 2 Pair no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sequence 1 + + − − + + − + + + + + − − + − + − + − − + +− + + + + + + − + + − − + − + − + + + − − − + ++ + − + + + − + + + − + + + + + − − + − + + − +− + − + + + − + + − − + − − + + + − + − + + − +− + + + + + + + + − + − + + − − + + + + + + − −− + + − − + + + − − − − − + + + − + − + − + − −+ + − + + − + + + − + − + − + + − − − − + − + −+ + + + + − − + − + + + + − − − + − + − + − + +− + + + + + + − + − − − − + + − − + + − − − + ++ + + + + − − − + − − − − + + − + − + − + + − −+ + + − − + + + + + + + − − + − + − − − + + − −+ + + + + − − + − + + + + + + − + − + + − − + +− + + + + − − − + − − − − − − − + − + + − + − −+ + + − − + + − + + + + + + + + − + − − + − + +−
Sequence 2 + + − − + + − − − − − + − + − + − − + − − + +− + + + + − − − − + + − + − + + + − + + + − + ++ + − + + − + + − + + + + + + − − + + + − + − +− + − + + − + + − − + + − − + − + + + + − + − +− + + + + + + − + − − + + + + + + + − + + + − −− + + − − − − + + + + − + − + − + − − − + + − −+ + − + + − + − + − − + + − − − − − + − + − + −+ + + − − − − + + + + − + − + − + + + − + + − +− + + + + − − − − − + − − + + + − − + + + − + ++ + + − − + + − − − − − + − + − + + + + − − + −+ + + − − + + + − − − − − − − + − + − − + + − −+ + + − − + + + + + + + − − + − + + + − + + − +− + + − − − − − − − − + − − + − + + + + − − + −+ + + − − − − − − − − + − − + − + − − + − − + +−
Table 3 Pair no. 1 2 3 4 5 6 7 8 9 10
Sequence 1 +1, +i, −1, +1, −i, −i, +1, −1, +1 +1, +1, +1, −1, −1, +i, −1, +i, +1 +1, +1, +1, +i, +1, +1, −1, −1, +1 +1, +1, −1, −1, +1, +i, +1, −1, +1 +1, +1, −1, −1, +1, +i, +i, −i, +i +1, +1, +1, +i, −i, −1, −i, +i, −i +1, +1, +1, −1, +i, +i, −i, −i, +i +1, +1, +1, +i, +1, +1, +i, −i, +i +1, +1, +1, −1, −i, −i, +1, −1, +1 +1, +1, +1, +i, −i, +1, +1, −1, +1
1) A(Ti ) = A(T0i ), 1 ≤ i ≤ 2 by Theorem 4.2, and 2) C(T2 , T1 ) = C(T02 , T01 ).
Sequence 2 +1, +i, −1, +i, −1, +1, +1, −1, +1 +1, +1, +1, −i, +i, −1, −1, +i, +1 +1, +i, −1, +i, +i, −1, −1, +i, +1 +1, −i, −1, −i, +i, +1, −1, −i, +1 +1, +i, −1, +1, +1, +i, −i, −1, +i +1, +i, −1, +i, −i, +1, −i, +1, −i +1, −i, −1, −1, +1, −i, −i, , +1, +i +1, +i, −1, +i, +i, −1, +i, −1, +i +1, +i, −1, −1, +1, +i, +1, +i, +1 +1, +i, −1, +1, +1, +i, +1, +i, +1
To illustrate the power of the theorems presented in this paper, we refer the reader back to Table 1. According to the table, there are 422 instances of SAP for binary sequences of length 24. Theorems 2.5, 3.2, 4.2, 4.3, and Corollary 4.5 enable us to explain 408 of these instances. Table 2 contains representatives for the 14 binary pairs of length 24 that are SAP pairs but can’t be explained by results in this paper (note that we use the notation + + −+ to represent the sequence {1, 1, −1, 1}). Similarly, 64 more of the instances of the SAP for quaternary sequences of length 9 are explained by Theorems 4.2 and 4.3. When combined with the 36 instances of the SAP for quaternary sequences of length 9 already explained in Section 2, this leaves 20 instances of the length 9 quaternary SAP unexplained by results in this paper. Using the observation that A(a) = A(b) whenever A(a) = A(b), Table 3 contains representatives for 10 of the SAP pairs that remain unexplained (the other 10 are simply conjugates of the pairs in the table).
Hence, Theorem 3.2 guarantees the autocorrelation equality. To get the generalization of this example, suppose we have the sequences b1 , b2 , . . . , bn of equal length k and the sequences a1 , a2 , . . . , am of equal length ` with C(a1 , aj )(u) = cj · C(aj , a1 )(u), 2 ≤ j ≤ m, cj ∈ ΞM . For 1 ≤ i ≤ n, define Ti = {d1 · x1 ⊗ bi |d2 · x2 ⊗ bi | · · · |dp · xp ⊗ bi }, where xj ∈ {a1 , a2 , . . . , am }, 1 ≤ j ≤ p and dj ∈ ΞM , 1 ≤ j ≤ p. Similarly, define T0i = {dp · yp ⊗ bi |dp−1 · yp−1 ⊗ bi | · · · |d1 · y1 ⊗ bi } for yj = cj · xj . The following corollary shows that T = {T1 |T2 | · · · |Tn } and T0 = {T01 |T02 | · · · |T0n } have the same aperiodic autocorrelation function. Corollary 4.5: For the M -ary sequences ai , 1 ≤ i ≤ m of length ` and bj , 1 ≤ j ≤ n of length k, if C(a1 , aj )(u) = cj · C(aj , a1 )(u) for 1 ≤ j ≤ m, cj ∈ ΞM , −(` − 1) ≤ u ≤ (` − 1), then A(T) = A(T0 ) for T and T0 defined above.
V. S EQUENCES WITH THE C ROSS -C ORRELATION P ROPERTY
The proof uses Theorems 3.2 and 4.2, and is a polynomial argument similar to the proofs of those theorems and hence is left to the reader.
Theorems 3.2 and 4.2 and Corollary 4.5 require that we find sequences a1 and a2 satisfying C(a1 , a2 )(u) = 5
c · C(a2 , a1 )(u) for some c ∈ ΞM . The following example illustrates sequences that have this cross correlation property, and we use it to set up our final theoretical result.
2n, where c1 , c2 ∈ ΞM . The proof is similar to Theorem 5.2 and is left to the reader. These two theorems can be used to identify pairs of sequences satisfying a cross-correlation property that enables them to be used in Theorems 3.2 and 4.2 as well as Corollary 4.5.
Example 5.1: Let b1 = {1, i, −1, 1, i, 1} and b2 = {i, i, 1, 1, −1, −1} be quaternary sequences of length 6. We use those to construct a1 = {b1 |i · R(b1 )} = {1, i, −1, 1, i, 1, i, 1, i, −i, 1, i} and a2 = {b2 | − R(b2 )} = {i, i, 1, 1, −1, −1, 1, 1, −1, −1, i, i}. It is easy to verify that C(a1 , a2 )(u) = −i · C(a2 , a1 )(u), −11 ≤ u ≤ 11. Similarly, if we define a3 = {b1 |i|(i)2 · R(b1 )} = {1, i, −1, 1, i, 1, i, −1, i, −1, 1, i, −1} and a4 = {b2 | − 1|(−1)2 · R(b2 )} = {i, i, 1, 1, −1, −1, −1, −1, −1, 1, 1, −i, −i}, we can verify that C(a3 , a4 )(u) = (i)2 (−1)2 C(a4 , a3 )(u), −12 ≤ u ≤ 12.
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The first construction in Example 5.1 can be generalized to generate even-length pairs of sequences with the cross correlation property as follows. Theorem 5.2: Suppose b1 and b2 are length n M -ary sequences. Then the sequences a1 = (b1 |c1 · R(b1 )) and a2 = (b2 |c2 · R(b2 )) satisfy C(a1 , a2 )(u) = c1 · c2 · C(a2 , a1 )(u), −(2n − 1) ≤ u ≤ (2n − 1), where c1 , c2 ∈ ΞM . Proof: The polynomial for a1 is a1 (x) = b1 (x) + xn c1 · b1 (x−1 ), and the polynomial for a2 is a2 (x) = b2 (x) + xn c2 ·b2 (x−1 ). Using these to check the cross correlation yields the following polynomial equation. a1 (x)a2 (x−1 ) =
[b1 (x) + xn c1 · b1 (x−1 )][b2 (x−1 ) + x−n c2 · b2 (x)]
= b1 (x)b2 (x−1 ) + c2 · x−n b1 (x)b2 (x−1 ) + c1 · xn b1 (x−1 )b2 (x−1 ) + c1 · c2 · b2 (x)b1 (x−1 ) = c1 · c2 · a2 (x)a1 (x−1 )
The following theorem, based on the second construction in Example 5.1, provides a construction for a pair of odd-length sequences satisfying C(a1 , a2 )(u) = (c1 )2 (c2 )2 C(a2 , a1 )(u), −2n ≤ u ≤ 2n. Theorem 5.3: Suppose b1 and b2 are length n M -ary sequences. Then the sequences a1 = (b1 |c1 |(c1 )2 · R(b1 )) and a2 = (b2 |c2 |(c2 )2 · R(b2 )) satisfy C(a1 , a2 )(u) = (c1 )2 (c2 )2 C(a2 , a1 )(u), −2n ≤ u ≤ 6
