Fast Correlation Algorithm for an Orthogonal Set of Real-ValuedPerfect Sequences from the Huffman Sequences

TAKAHIRO MATSUMOTO 1, HIDEYUKI TORII 2, SHINYA MATSUFUJI 1

1 Graduate School of Science and Engineering, Yamaguchi University2-16-1 Tokiwadai, Ube-shi, Yamaguchi, 755-8611, Japan

matugen@yamaguchi-u.ac.jp, s-matsu@yamaguchi-u.ac.jp2 Department of Information Network and Communication, Kanagawa Institute of Technology

1030 Shimo-ogino, Atsugi-shi, Kanagawa, 243-0292, Japantorii@nw.kanagawa-it.ac.jp

Abstract: A perfect sequence has the optimal periodic autocorrelation function where all out-of-phase values arezero, and an orthogonal set has orthogonality that the periodic correlation function for any pair of distinct sequencesin the set takes zero at the zero shift. The real-valued perfect sequences of period N = 2n are derived from a real-valued Huffman sequence of length 2ν + 1 with ν ≥ n whose out-of-phase aperiodic autocorrelation function takeszero except at the left and right shift-ends. This paper proposes fast periodic auto- and cross-correlation algorithmsfor an orthogonal set of real-valued perfect sequences of period 2n. As a result, the number of multiplications andadditions can be suppressed on the order N log2 N.

Key–Words: periodic sequence, finite-length sequence, perfect sequence, Huffman sequence, orthogonal set, fastcorrelation algorithm

1 Introduction

The perfect sequence (also called the periodic orthog-onal sequence) [1] has the optimal periodic autocorre-lation function where all out-of-phase values are zero,and an orthogonal set has orthogonality that the pe-riodic correlation function for any pair of distinct se-quences in the set takes zero at the zero shift. The or-thogonal set of perfect sequences is useful in varioussystems, such as synchronous code division multipleaccess (CDMA) systems, pulse compression radarsand digital watermarkings[1].

A binary perfect sequence consisting of elements1 and −1 is only (1, 1, 1,−1) of period 4. On the otherhand, any real-valued perfect sequence consisting ofelements of real numbers is generated by its generalsolution [2]. In addition, the real-valued perfect se-quence of period 2n is derived from a real-valuedHuffman sequence (also called shift-orthogonal finite-length sequence) [3, 4] of length 2ν + 1 with positiveintegers n and ν, and ν ≥ n whose out-of-phase aperi-odic auto correlation function takes zero except at theleft and right shift-ends [5, 6]. Previously, a fast peri-odic auto correlation algorithm for a perfect sequenceof period 2n derived from a real-valued Huffman se-quence of length 2n + 1 has been proposed [6].

In this paper, we propose fast periodic auto- andcross-correlation algorithms for an orthogonal set of

real-valued perfect sequences of period 2n derivedfrom the Huffman sequence of length 2ν + 1, ν ≥ n.

2 The Huffman Sequence

2.1 Definition of the Huffman sequence

Let aℓM be a real-valued finite-length sequence oflength M consisting of real-elements , written as

aℓM = (aℓM,0, · · · , aℓM,i, · · · , aℓM,M−1), aℓM,i ∈ R,

where aℓM,i = 0, i < 0, i > M − 1, i denotes the ordervariable, ℓ the sequence number and R the set of realnumbers. Let A be a set of real-valued finite-lengthsequences aℓM, written as

A = {a0M , · · · , aℓM , · · · , aL−1

M }= {{a0

M,i}, · · · , {aℓM,i}, · · · , {aL−1M,i }},

where L denotes the number of sequences in a se-quence set , i.e. family size.

The aperiodic correlation function between se-quences aℓM and aℓ

′M at shift i′ is defined by

ρℓ,ℓ′

M,i′ =1M

M−1∑i=0

aℓM,iaℓ′M,i−i′ =

1M

(aℓM,i′ ⊗ aℓ

′M,−i′),

(1)

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where ⊗ denotes the convolution and ρℓ,ℓ′

M,i′ = 0 for|i′| > M − 1. If the aperiodic autocorrelation functionsatisfies

ρℓ,ℓM,i′ =

1 ; i′ = 0,0 ; i′ , 0,±(M − 1),εM−1 ; i′ = ±(M − 1),

(2)

where εM−1 is called a shift-end value and |εM−1| ≤1/2, the sequence aℓM is called a real-valued Huffmansequence [3] or a real-valued shift-orthogonal finite-length sequence [4].

2.2 The Huffman sequence of length 2

Let aℓM be a real-valued Huffman sequence with a neg-ative shift-end value εM−1 < 0 and length M. LetbℓM be a real-valued Huffman sequence with a posi-tive shift-end value ε′M−1 > 0. It is easy to give theHuffman sequence of short length from definition ofthe sequence. From Eqs. (1) and (2), the Huffman se-quence a0

2 of length 2 and a shift-end value ε1 < 0 issolved as

a02,0 = (

√1 + 2ε1 +

√1 − 2ε1)/

√2

a02,1 = (

√1 + 2ε1 −

√1 − 2ε1)/

√2

. (3)

Similarly, from Eqs. (1) and (2), the Huffman se-quence b0

2 of length 2 and a shift-end value ε′1 > 0is solved as

b02,0 = (

√1 + 2ε′1 +

√1 − 2ε′1)/

√2

b02,1 = (

√1 + 2ε′1 −

√1 − 2ε′1)/

√2

. (4)

The sequence b0m of length m ≥ 3 and a shift-end value

ε′m−1 > 0 is obtained by insertion of zero values be-tween neighboring values of the sequence b0

2 of length2 and multiplication by the constant for normalizationas follows.

b0m =

√m2

(b0

2,0, 0, 0, · · · , 0︸ ︷︷ ︸m−2

, b02,1

). (5)

Note that a shift-end value ε′m−1 of the sequence b0m of

length m ≥ 3 is equal to a shift-end value ε′1 of thesequence b0

2 of length 2.Sequences a1

2 and b1m, m ≥ 2 with a sequence

number 1 can be replaced by reversed sequences

a12 = (a0

2,1, a02,0),

b1m = (b0

m,m−1, · · · , b0m,i, · · · , b0

m,0).

2.3 The Huffman sequence of length 2ν + 1derived from the sequence of length 2

The real-valued Huffman sequence aℓM of length M =2ν + 1,ν = 1, 2, · · · and a shift-end value εM−1 < 0 isgiven by the ν-multiple convolution of a real-valuedHuffman sequence aℓ22 of length 2 and a shift-end

value ε1 < 0 and the sequences bℓ′22 , b

ℓ′33 , · · · , b

ℓ′M+12

M+12

of

length 2, 3, · · · , M+12 (= 2ν−1 + 1) and shift-end values

ε′1, ε′2, · · · , ε′M−1

2> 0 as follows [4].

aℓM,i = K(aℓ22,i ⊗ b

ℓ′22,i ⊗ b

ℓ′33,i ⊗ · · ·

⊗bℓ′m

m,i ⊗ · · · ⊗ bℓ′M+1

2M+1

2 ,i

), (6)

K = K3K5 · · ·K2m−1 · · ·KM ,

K2m−1 =

√2m − 1

m√

1 − 2ε2m−1

, (7)

where m = 2µ + 1, µ = 0, 1, · · · , ν − 1, ⊗ denotes theconvolution as

aℓm,i ⊗ bℓ′

n,i =

m−1∑k=0

aℓm,kbℓ′

n,i−k

and ℓ2, ℓ′2, ℓ′3, · · · , ℓ′M+12

denote the sequence num-

bers. The sequences aℓ22 and bℓ′m

m , m = 2µ + 1, µ =0, 1, · · · , M+1

2 are called elementary sequences. Froma shift-end value εM−1 of the Huffman sequence aℓMof length M, shift-end values ε′M−1

2, · · · , ε′2, ε′1, ε1 of

elementary sequences bℓ′M+1

2M+1

2, · · · , b

ℓ′33 , b

ℓ′22 and aℓ22 are

solved asεm−1

2= −

√−εm−1

1−2εm−1< 0,

ε′m−12= −εm−1

2> 0,

m = M, M+12 , · · · , 5, 3.

(8)

These elementary sequences are derived from theHuffman sequence of length 2 shown in Eqs. (3), (4)and (5).

A set of the Huffman sequences aℓM is constructedby combinations of original elementary sequenceswith a sequence number 0 and reversed elementary se-quences with a sequence number 1. A sequence num-ber ℓ of the Huffman sequence aℓM is expressed in abinary notation as

ℓ = (ℓ′M+12, · · · , ℓ′3, ℓ′2, ℓ2)2 = ℓ2 + 2

ν−1∑k=0

2kℓ′2k+1.

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3 A Perfect Sequence

3.1 Definition of a perfect sequence

Let a j,sN be a real-valued periodic sequence of period

N consisting of real elements, written as

a j,sN = {a j,s

N,i} = (a j,sN,0, · · · , a

j,sN,i, · · · , a

j,sN,N−1), a j,s

N,i ∈ R,

where a j,sN,i+kN = a j,s

N,i with an integer k, i denotes theorder variable, j the sequence number, s the set num-ber and R the set of real numbers. Let As be a set ofreal-valued periodic sequences a j,s

N , written as

As = {a0,sN , · · · , a

j,sN , · · · , a

L−1,sN },

where L is the number of sequences in a sequence set,i.e. family size.

A periodic correlation function between se-quences a j,s

N and a j′,sN at a shift i′ is defined by

ρj, j′,sN,i′ =

1N

N−1∑i=0

a j,sN,ia

j′,sN,(i−i′) mod N (9)

where ρ j, j′,sN,i′+kN = ρ

j, j′,sN,i′ for any integer k. Equation (9)

means the autocorrelation function for j = j′ and thecross one for j , j′. If the periodic autocorrelationfunction satisfies

ρj, j,sN,i′ =

{1 ; i′ = 0,0 ; i′ , 0,

the sequence a j,sN is called a real-valued perfect se-

quence [1] or a real-valued periodic orthogonal se-quence [2]. In addition, if the periodic correlationfunction satisfies

ρj, j′,sN,i′ =

1 ; i′ = 0, j = j′,0 ; i′ , 0, j = j′,0 ; i′ = 0, j , j,

(10)

a set As of sequences a j,sN is called an orthogonal set

of real-valued perfect sequences.

3.2 Construction of an Orthogonal set ofperfect sequences

An orthogonal set of real-valued perfect sequencesa j,s

N of period N = 2n is derived from the Huffmansequences aℓM of length M = 2ν + 1 with ν ≥ n. Aperfect sequence a j,s

N of period N in an orthogonal set

is expressed as

a j,sN,i =

√N

M(1 + 2εM−1)

{aℓM,i ⊗ ∆N,i− j

}= K

{∆N,i− j ⊗ aℓ22,i ⊗ b

ℓ′22,i ⊗ b

ℓ′33,i ⊗ · · ·

⊗bℓ′m

m,i ⊗ · · · ⊗ bℓ′M+1

2M+1

2 ,i

}= K

{∆N,i− j ⊗ αℓ22,i ⊗ β

ℓ′22,i ⊗ β

ℓ′33,i ⊗ · · ·

⊗βℓ′m

m,i ⊗ · · · ⊗ βℓ′M+1

2M+1

2 ,i

}, (11)

∆N,i =

∞∑k=−∞

δi,kN =

{1 ; i = kN0 ; i , kN

,

K =

√N

2(1 + 2εM−1)∏ν−1µ=0(2µ + 1)(1 − 2ε2

2µ),

where ∆N,i denotes the impulse sequence of period Nand δi,kN the Kronecker delta, and

{αℓ22,i} = {aℓ22,i ⊗ ∆N,i} =(aℓ22,0, a

ℓ22,1, 0, · · · , 0︸ ︷︷ ︸

N−2

),

{βℓ′m

m,i} = {bℓ′m

m,i ⊗ ∆N,i}

=

(bℓ′m

m,0, 0, · · · , 0︸ ︷︷ ︸m−2

, bℓ′m

m,1, 0, · · · , 0︸ ︷︷ ︸N−m

); m ≤ N,(

bℓ′m

m,0 + bℓ′m

m,1, 0, · · · , 0︸ ︷︷ ︸N−1

), ; m > N,

and αℓ22,i+kN = αℓ22,i, β

ℓ′mm,i+kN = β

ℓ′mm,i for any integer k.

A shift-end value εM−1 of the Huffman sequence aℓMshould be chosen a value to reduce the absolute valueof perfect sequences a j,s

N in the set.A sequence number j and a family number s are

expressed by

j = (ℓ′M+12, · · · , ℓ′m, · · · , ℓ′3)2 =

ν−2∑k=0

2kℓ′2k+1+1,(12)

s = ℓ′2, (13)

respectively, where ℓ′2 , ℓ2. Therefore, we can con-struct two orthogonal sets A0 and A1 of perfect se-quences of period N = M − 1 or (M − 1)/2, familysize L = (M − 1)/2 and set number s = ℓ′2 = 0 and 1with ℓ′2 , ℓ2 . This orthogonal set with N = (M−1)/2can reach the upper bound L = N on family size [1].

As an example of orthogonal sets, let us constructtwo orthogonal sets A0 and A1 of 8 perfect sequencesa j,s

8 of period N = 8 derived from the Huffman se-quences aℓ17 of length M = 17 and a shift-end valueε16 = −0.00073. From Eq. (11), a perfect sequence

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Table 1: Two orthogonal sets A0 and A1 of 8 real-valued perfect sequences a j,s8 of period N = 8 derived from the

real-valued Huffman sequences aℓ17 of length M = 17 and the shift-end value ε16 = −0.00073.i 0 1 2 3 4 5 6 7

a0,08 -1.72761 1.72840 1.10053 0.70074 0.44619 0.28410 0.18090 0.11518

a1,08 0.28410 -0.44619 0.70074 -1.10053 1.72840 1.72761 0.11518 -0.18090

a2,08 1.10053 0.70074 0.44619 0.28410 0.18090 0.11518 -1.72761 1.72840

A0 a3,08 0.70074 -1.10053 1.72840 1.72761 0.11518 -0.18090 0.28410 -0.44619

a4,08 0.44619 0.28410 0.18090 0.11518 -1.72761 1.72840 1.10053 0.70074

a5,08 1.72840 1.72761 0.11518 -0.18090 0.28410 -0.44619 0.70074 -1.10053

a6,08 0.18090 0.11518 -1.72761 1.72840 1.10053 0.70074 0.44619 0.28410

a7,08 0.11518 -0.18090 0.28410 -0.44619 0.70074 -1.10053 1.72840 1.72761

a0,18 1.72761 1.72840 -1.10053 0.70074 -0.44619 0.28410 -0.18090 0.11518

a1,18 0.28410 0.44619 0.70074 1.10053 1.72840 -1.72761 0.11518 0.18090

a2,18 -1.10053 0.70074 -0.44619 0.28410 -0.18090 0.11518 1.72761 1.72840

A1 a3,18 0.70074 1.10053 1.72840 -1.72761 0.11518 0.18090 0.28410 0.44619

a4,18 -0.44619 0.28410 -0.18090 0.11518 1.72761 1.72840 -1.10053 0.70074

a5,18 1.72840 -1.72761 0.11518 0.18090 0.28410 0.44619 0.70074 1.10053

a6,18 -0.18090 0.11518 1.72761 1.72840 -1.10053 0.70074 -0.44619 0.28410

a7,18 0.11518 0.18090 0.28410 0.44619 0.70074 1.10053 1.72840 -1.72761

a j,s8 is generated by convolution of the Huffman se-

quence aℓ17 of length 17 and the impulse sequence∆8,i− j of period 8. Table 1 shows two orthogonal setsA0 and A1 of 8 perfect sequences a j,s

8 of period N = 8.

4 Fast Correlation Algorithm foran Orthogonal Set of Perfect Se-quences

4.1 Fast autocorrelation algorithm

Let d be an input sequence of length N consisting ofreal-elements, written as

d = {di} = (d0, · · · , di, · · · , dN−1), di ∈ R,

where di = 0, i < 0, i > N − 1. From Eq. (9), the pe-riodic autocorrelation function between a perfect se-quence a j,s

N and an input sequence d is given by

xi′ =1N

N−1∑i=0

diaj,sN,(i−i′) mod N , (14)

where x = {xi′} = (x0, · · · , xi′ , · · · , xN−1), xi′ ∈ R.In addition, Eq. (14) is replaced by a determinant ofmatrix as

X =1N

A j,sN · D, (15)

where X = [x0, · · · , xi′ , · · · , xN−1]T , D =

[d0, · · · , di, · · · , dN−1]T , XT denotes a transposedmatrix of X, and

A j,sN =

[{a j,s

N,i}, {aj,sN,i ⊗ ∆N,i−1}, · · · , {a j,s

N,i ⊗ ∆N,i−(N−1)}]T

=

a j,s

N,0, a j,sN,1, · · · a j,s

N,N−1a j,s

N,N−1, a j,sN,0, · · · a j,s

N,N−2...

.... . .

...

a j,sN,1, a j,s

N,2, · · · a j,sN,0

.

Therefore, the number of multiplications Nmul and ad-ditions Nadd are given by N2 + N and N(N − 1), re-spectively.

From Eq. (11), Eq. (15) is factorized as

X =KN

R j · Bℓ′ M+1

2M+1

2· · · Bℓ

′m

m · · · Bℓ′33 · B

ℓ′22 · A

ℓ22 · D,

(16)

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where

Aℓ22 =[{αℓ22,i}, {α

ℓ22,i ⊗ ∆N,i−1}, · · · , {αℓ22,i ⊗ ∆N,i−(N−1)}

]T

=

aℓ22,0, aℓ22,1, · · · 00, aℓ22,0, · · · 0...

.... . .

...

aℓ22,1, 0, · · · , aℓ22,0

,Bℓ′m

m =[{βℓ

′m

m,i}, {βℓ′mm,i ⊗ ∆N,i−1}, · · · , {βℓ

′m

m,i ⊗ ∆N,i−(N−1)}]T,

R =[{∆N,i−1}, {∆N,i−2} · · · , {∆N,i−(N−1)}, {∆N,i},

]T,

=

0, 1, 0, · · · 00, 0, 1, · · · 0.......... . .

...

0, 0, 0, · · · , 11, 0, 0, · · · , 0

,

R0 = 1,Rk = R · · ·R︸ ︷︷ ︸

k

.

Therefore, the number of multiplications Nmul and ad-ditions Nadd are given by 2N{log2(N) + 1} + N andN{log2(N) + 1}, respectively.

As an example, Fig. 1 shows a signal flow offast periodic autocorrelation algorithm for a perfectsequence a2,1

8 of period N = 8 , sequence numberj = 2 and set number s = 1 derived from the real-valued Huffman sequences a10

17 of length M = 17 andsequence number ℓ = (ℓ′9, ℓ

′5, ℓ′3, ℓ′2, ℓ2)2 = (01010)2 =

10. Therefore, the numbers of multiplications Nmuland additions Nadd are 72(= 2 × 8(3 + 1) + 8) and32(= 8(3 + 1)), respectively.

4.2 Fast crosscorrelation algorithm

From Eq. (9), the periodic crosscorrelation functionbetween a perfect sequence a j,s

N and an input sequenced is given by

y j =1N

N−1∑i=0

diaj,sN,i, (17)

where y = {y j} = (y0, · · · , y j, · · · , yL−1), y j ∈ R. Inaddition, Eq. (17) is replaced by a determinant of ma-trix as

Y =1N

CsN · D, (18)

i i′

a0

2,0

di xi′

a0

2,1

a0

2,1

a0

2,0b1

2,0

b12,1

b1

2,1

b12,0

b03,0

b03,2

b03,2

b0

3,0b1

5,0

b15,4

b15,4

b15,0

KN

(b09,0

+ b09,8

)

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

KN

(b09,0

+ b0

9,8)

Figure 1: A signal flow of fast autocorrelation algo-rithm for a perfect sequence a2,1

8 of period N = 8,sequence number j = 2 and set number s = 1 derivedfrom the Huffman sequence a10

17 of length M = 17 andsequence number ℓ = 10.

where Y = [y0, · · · , y j, · · · , yL−1]T , CsN is a matrix of

size L × N and

CsN =

[a0,s

N , · · · , aj,sN , · · · , a

L−1,sN

]T

=

a0,s

N,0, a0,sN,1, · · · a0,s

N,N−1a1,s

N,0, a1,sN,1, · · · a1,s

N,N−1...

.... . .

...

aL−1,sN,0 , aL−1,s

N,1 , · · · aL−1,sN,N−1

.Therefore, the number of multiplications Nmul and ad-ditions Nadd are given by LN+L and L(N−1), respec-tively.

From Eq. (11), Eq. (18) is factorized as

Y =KN

F M+12· C2ν−2+1 · · · C2k+1 · · · C3 · B

ℓ′22 · A

ℓ22 · D,

(19)

where set number s = ℓ′2 and ℓ2 , ℓ′2, and

C2k+1

=[{β0

2k+1,i ⊗ ∆N,i}, · · · , {β02k+1,i ⊗ ∆N,i−(2k−1−1)},

·{β12k+1,i ⊗ ∆N,i−2k−1}, · · · {β1

2k+1,i ⊗ ∆N,i−(2k−1)},·{β0

2k+1,i ⊗ ∆N,i−2k }, · · · {β02k+1,i ⊗ ∆N,i−(2k+1−1)},

· · ··{β1

2k+1,i ⊗ ∆N,i−(N−2k−1)}, · · · {β12k+1,i ⊗ ∆N,i−(N−1)}

]T,

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i j

aℓ22,0

di yj

aℓ22,1

aℓ22,1

aℓ22,0

bℓ′2

2,0

bℓ′2

2,1

bℓ′2

2,1

bℓ′2

2,0 bℓ′3

3,0

bℓ′3

3,2

bℓ′3

3,2

bℓ′3

3,0b05,0

b05,4

b05,4

b0

5,0

KN

(b09,0 + b0

9,8)

0

1

2

3

4

5

6

7

0

2

4

6

b1

5,0

b15,4

b15,4

b15,0

KN

(b19,0 + b1

9,8)

1

3

7

5

b13,0

b13,2

b1

3,2

b1

3,0

Figure 2: A signal flow of fast crosscorrelation algo-rithm for an orthogonal set of perfect sequences of pe-riod N = 8, family size L = 8 and set number s = ℓ′2derived from the Huffman sequence of length M = 17.

F M+12=[{β0

M+12 ,i⊗ ∆N,i}, ·{β0

M+12 ,i⊗ ∆N,i−1}, · · · ,

{β0M+1

2 ,i⊗ ∆N,i−( L

2−1)}, ·{β1M+1

2 ,i⊗ ∆N,i− L

2},

· · · {β1M+1

2 ,i⊗ ∆N,i−(L−1)},

]T,

and Fm is a matrix of size L× N and Cm is a matrix ofsize N × N. Therefore, the number of multiplicationsNmul and additions Nadd are given by 2N{log2(N) +1} + L and N{log2(N) + 1}, respectively.

As an example, Fig. 2 shows a signal flow offast crosscorrelation algorithm for a ZCZ set of real-valued perfect sequences a j,s

8 of period N = 8, fam-ily size L = 8 and set number s = ℓ′2 derived fromthe real-valued Huffman sequences aℓ17 of M = 17.Note that b0

9,0 + b09,8 is equal to b1

9,0 + b19,8 because

b09 is a reversed sequence of b1

9. Therefore, the num-bers of multiplications Nmul and additions Nadd are72(= 2 × 8(3 + 1) + 8) and 32(= 8(3 + 1)), respec-tively.

5 ConclusionIn this paper, we have proposed fast periodic auto- andcross-correlation algorithms for an orthogonal set ofreal-valued perfect sequences of period N = 2n whichare derived from the real-valued Huffman sequencesof length M = 2ν + 1 with n ≤ ν. The sequence issynthesized by multiple convolutions of ν+ 1 elemen-tary sequences which are generated by combinationsof original and reversed elementary sequences.

As a result, the number of multiplications and ad-ditions can be suppressed from the order of N2 to theorder of N log2 N by proposed algorithms.

Acknowledgements: This research was supported bythe Ministry of Education,Culture,Sports,Science andTechnology(MEXT), Grant-in-Aid for Young Scien-tists (B)(21760294).

References:

[1] P. Z. Fan, M. Darnell, “Sequence design forcommunications applications,” Research StudiesPress, 1996.

[2] Y. Tanada, “General solution for orthogonal pe-riodic real-number sequences,” IEICE Trans. onFundamentals, vol. 68-A, no. 5, pp. 451–457,May 1985 (in Japanese).

[3] D. A. Huffman, “The generation of impulseequivalent pulse trains,” IRE Trans. InformationTheory, vol. IT-8, pp. s10–s16, Sept. 1962.

[4] Y. Tanada, “Synthesis of a set of real-valuedshift-orthogonal finite-length PN sequences,”Proc. of the IEEE 4th International Symposiumon Spread Spectrum Techniques and Applica-tions, pp. 58–62, Sept. 1996.

[5] T. Matsumoto, Y. Tanada, “A real-valued peri-odic orthogonal sequence derived from a real-valued shift-orthogonal finite-length sequenceand its fast periodic correlation algorithm,” IE-ICE Trans. on Fundamentals, vol. 89-A, no. 5,pp. 323–332, May 2006 (in Japanese).

[6] T. Matsumoto, S. Matsufuji, T. Kojima, U.Parampalli, “Generation method of an orthog-onal set of real-valued periodic orthogonal se-quences from Huffman sequences,” Proc. of the2011 Australian Communication Theory Work-shop, pp. 66–70, Jan. 2011.

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