A New Construction of QAM Golay Complementary Sequence Pair 2020 - - PowerPoint PPT Presentation

SLIDE 1

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

A New Construction of QAM Golay Complementary Sequence Pair

2020 IEEE International Symposium on Information Theory

Zilong Wang1, Erzhong Xue1, Guang Gong2

1 State Key Laboratory of Integrated Service Networks, Xidian University 2 Department of Electrical and Computer Engineering, University of Waterloo

21-26 June 2020 Los Angeles, California, USA

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 2

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Contents

1

• 1. Introduction

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

2

• 2. Main Results

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

3

• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 3

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

GCP & GCS

Let F(y) = (F(0), F(1), · · · , F(L − 1)) be a complex valued sequence of length L. Suppose that F(y) = 0 if y < 0 and y ≥ L. The aperiodic auto-correlation of sequence F(y) at shift τ (1 − L ≤ τ ≤ L − 1) is defined by CF (τ) =

• y

F(y + τ) · F(y). A pair of sequences {F(y), G(y)} of length L is said to be a Golay complementary pair (GCP) if CF (τ) + CG(τ) = 0 (∀τ = 0). And either sequence in a GCP is called a Golay complementary sequence (GCS).

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 4

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

Example F1(y) = (1, 1, 1, −1, 1, 1, −1, 1) and F2(y) = (1, ξ, ξ, −1, 1, −ξ, −ξ, −1). (CF1(τ))7

0 = (8, −1, 0, 3, 0, 1, 0, 1)

(CF2(τ))7

0 = (8, 1, 0, −3, 0, −1, 0, −1)

CF1(τ) + CF2(τ) = 0 (∀τ = 0) ⇒ F1(y) and F2(y) are GCP.

1

1ξ = √−1 is a fourth primitive root of unity.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 5

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

GBF

Generalized Boolean Function

For 0 ≤ y < L = 2m, y can be uniquely written in a binary expansion as y = m

k=1 xk ·2k−1

where xk ∈ F2. Let x = (x1, x2, · · · xm). Then a sequence F(y) of length 2m over QPSK can be associated with a generalized Boolean function (GBF) f(x) over Z4 by F(y) = ξf(x).

Example f(x1, x2, x3) = 2(x2x3 + x1x3) + x1 + x2 Z4 sequence:(0, 1, 1, 2, 0, 3, 3, 2). QPSK sequence:(1, ξ, ξ, −1, 1, −ξ, −ξ, −1).

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 6

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

Standard GCPs

[Davis and Jedwab, 1999]

. . . xπ(1) xπ(2) xπ(3) xπ(m−1) xπ(m) Figure: The Graph of Coset Representatives For GBF f(x) = 2 ·

m−1

• k=1

xπ(k)xπ(k+1) +

m

• k=1

ck · xk + c0, where ck ∈ Z4(0 ≤ k ≤ m), and c′ ∈ Z4, the sequence pair associated with the GBFs over Z4

• f(x),

f(x) + 2xπ(1) + c′,

• r
• f(x),

f(x) + 2xπ(m) + c′, form a GCP over QPSK.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 7

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

QAM Sequence

Figure: eg: v = 4α + 2β + γ A sequence over 4q-QAM can be viewed as the weighted sums of q sequences over QPSK, with weights in the ratio of 2q−1 : 2q−2 : · · · : 1.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 8

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

V-GBF and QAM Sequence

A vectorial generalized Boolean function (V-GBF) is a function from Fm

2 to Zq 4,

denoted by

• f(x) = (f (0)(x), f (1)(x), · · · , f (q−1)(x)),

where each component function f (p)(x)(0 ≤ p < q) is a GBF over Z4. A sequence over 4q-QAM of length 2m can be associated with a V-GBF

• f(x) = (f (0)(x), f (1)(x), · · · , f (q−1)(x)) over Z4 by

F(y) =

q−1

• p=0

2p · ξf (p)(x), where y = m

k=1 xk · 2k−1. Obviously, a sequence over QPSK can be viewed as a

sequence over 4q-QAM when q = 1.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 9

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

QAM GCP Structure

Denote the q-dimension vector (1, 1, · · · , 1) by

• 1. For the generalized constructions of cases

I-III given by Li [Li, 2010] and the generalized constructions for cases IV-V given by Liu et

• al. [Liu et al., 2013], all the GCPs {F(y), G(y)} of length 2m over 4q-QAM are associated

with V-GBFs f(x) = 1 · f(x) + s(x),

• g(x) =

f(x) + µ(x), where f(x) is a standard GCS.

• s(x) =
• s(0)(x) = 0, s(1)(x), · · · , s(q−1)(x)
• is called a offset V-GBF,
• µ(x) =
• µ(0)(x), µ(1)(x), · · · , µ(q−1)(x)
• is called a pairing difference V-GBF.
• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 10

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

all the GCPs {F(y), G(y)} of length 2m over 4q-QAM are associated with V-GBFs f(x)

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 11

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

all the GCPs {F(y), G(y)} of length 2m over 4q-QAM are associated with V-GBFs f(x)· 1 = (f (0), f (0), f (0), · · · , f (0)),

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 12

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

all the GCPs {F(y), G(y)} of length 2m over 4q-QAM are associated with V-GBFs f(x)· 1 = (f (0), f (0), f (0), · · · , f (0)), + (s(0), s(1), s(2), · · · , s(q−1)),

• s(x)

= (f (0), f (1), f (2), · · · , f (q−1)),

• f(x)
• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 13

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

all the GCPs {F(y), G(y)} of length 2m over 4q-QAM are associated with V-GBFs f(x)· 1 = (f (0), f (0), f (0), · · · , f (0)), + (s(0), s(1), s(2), · · · , s(q−1)),

• s(x)

= (f (0), f (1), f (2), · · · , f (q−1)),

• f(x)

+ (µ(0), µ(1), µ(2), · · · , µ(q−1)),

• µ(x)

= (g(0), g(1), g(2), · · · , g(q−1)),

• g(x)
• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 14

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

all the GCPs {F(y), G(y)} of length 2m over 4q-QAM are associated with V-GBFs f(x)· 1 = (f (0), f (0), f (0), · · · , f (0)), + (s(0), s(1), s(2), · · · , s(q−1)),

• s(x)

= (f (0), f (1), f (2), · · · , f (q−1)),

• f(x)

+ (µ(0), µ(1), µ(2), · · · , µ(q−1)),

• µ(x)

= (g(0), g(1), g(2), · · · , g(q−1)),

• g(x)

where f(x) is a standard GCS.

• s(x) =
• s(0)(x) = 0, s(1)(x), · · · , s(q−1)(x)
• is called a offset V-GBF,
• µ(x) =
• µ(0)(x), µ(1)(x), · · · , µ(q−1)(x)
• is called a pairing difference V-GBF.
• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 15

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs

The generalized cases I-III [Li, 2010]

For d(p)

i

∈ Z4 (1 ≤ p ≤ q − 1, 0 ≤ i ≤ 2), and 2d(p) + d(p)

1

+ d(p)

2

= 0. (1) The generalized case I: s(p)(x) = d(p) + d(p)

1 xπ(1), 1 ≤ p ≤ q − 1,

• µ(x) = 2xπ(m) ·

1; (2) The generalized case II: s(p)(x) = d(p) + d(p)

1 xπ(m), 1 ≤ p ≤ q − 1,

• µ(x) = 2xπ(1) ·

1; (3) The generalized case III: s(p)(x) = d(p) + d(p)

1 xπ(ω) + d(p) 2 xπ(ω+1), 1 ≤ p ≤ q − 1,

with µ(x) = 2xπ(1) · 1 or 2xπ(m) · 1, 1 ≤ ω ≤ m − 1.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 16

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Section 2. Main Results

2.1 Cases ++

New framework Cases I-III in short Case for q = q0 × q1

2.2 Main Construction

Case for q = q0 · q1 · · · qm

• s(x),

µA(x) and µB(x)

2.3 Enumeration

Enumeration for q = 6 Conclusion

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 17

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

New Structure

• f(x)
• f0,0(x)
• µA(x)
• µB(x)
• f0,1(x)
• µB(x)
• f1,0(x)
• µA(x)
• f1,1(x)
• g(x)

Figure: A Diagram of the Relationship       

• f(x) =

1 · f(x) + s(x),

• g(x) =

f(x) + µA(x),

• r

f(x) + µB(x),

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 18

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Cases I-III in short

For 0 ≤ p ≤ q − 1, let d(p)

i

∈ Z4 ( 0 ≤ i ≤ 2) satisfying 2d(p) + d(p)

1

+ d(p)

2

= 0. Let xπ(0) = xπ(m+1) ≡ 0 be “fake” variables for convenience. s(p)(x) =        d(p)

0 +d(p) 1 xπ(ω)+d(p) 2 xπ(ω+1),

(0 < ω < m), d(p) + d(p)

2 xπ(1),

(ω = 0), d(p) + d(p)

1 xπ(m),

(ω = m),        = d(p)

0 +d(p) 1 xπ(ω)+d(p) 2 xπ(ω+1)

• µ(x) is chosen from either

µA(x) or µB(x), where µ(p)

A (x) =

• 2xπ(1) + d(p)

1 ,

(ω = 0), 2xπ(1), (ω = 0), and µ(p)

B (x) =

• 2xπ(m) + d(p)

2 ,

(ω = m), 2xπ(m), (ω = m),

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 19

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Case for q = q0 × q1

For factorization q = q0 · q1, let p0 ∈ Zq0 and p1 ∈ q0 · Zq1 Any p ∈ Zq can be uniquely decomposed as p = p0 + p1, where p0 and p1 can be determined by:

• p0 ≡ p (mod q0),

p1 ≡ p − p0. Let q = 6. For factorization q = q0 · q1 = 2 × 3 = 6, the decomposition of p = p0 + p1 is given as: p Possible value 1 2 3 4 5 p0 {0, 1} 1 1 1 p1 {0, 2, 4} 2 2 4 4

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 20

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

• s(x)

Suppose q = 6 = q0 · q1 = 2 × 3, Let

• s(p0)

ω

(x) = d(p0) + d(p0)

1

xπ(ω) + d(p0)

2

xπ(ω+1), s(p1)

κ

(x) = d(p1) + d(p1)

1

xπ(κ) + d(p1)

2

xπ(κ+1), For 1 ≤ p ≤ 5, the offset s(p)(x) is simplified to s(p)(x) = s(p0)

ω

(x) + s(p1)

κ

(x)

• r more clearly:

p p0 p1

• ffset : s(p)(x)

1 1 s(1)

ω (x)

2 2 s(2)

κ (x)

3 1 2 s(1)

ω (x) + s(2) κ (x)

4 4 s(4)

κ (x)

5 1 4 s(1)

ω (x) + s(4) κ (x)

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 21

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

• µA(x) and

µB(x)

µ(p)

A (x) =

       2xπ(1) + d(p0)

1

, (ω = 0), 2xπ(1) + d(p1)

1

, (κ = 0), 2xπ(1), (0 = κ or ω). µ(p)

B (x) =

       2xπ(m) + d(p0)

2

, (ω = m), 2xπ(m) + d(p1)

2

, (κ = m), 2xπ(m), (m = κ or ω).

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 22

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Case for q = q0 · q1 · · · qm

For factorization q = q0 · q1 · · · qm, let p0 ∈ Zq0 and pk ∈ k−1

• i=0

qi

• · Zqk.

Any p ∈ Zq can be uniquely decomposed as p = p0 + p1 + · · · + pm, where pk (0 ≤ k ≤ m) can be uniquely determined by the following recursive formula:        p0 ≡ p (mod q0), pk ≡ p −

k−1

• i=0

pi (mod

k

• i=0

qk), 1 ≤ k ≤ m.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 23

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

• s(x),

µA(x) and µB(x)

For p = p0 + p1 + · · · + pm, let s(pk)

ωk (x) = d(pk)

+ d(pk)

1

xπ(ωk) + d(pk)

2

xπ(ωk+1). For 0 ≤ p ≤ q − 1, s(p)(x) = m

k=0 s(pk) ωk (x)

µ(p)

A (x) =

• 2xπ(1) + d(pk)

1

, (ωk = 0) 2xπ(1), (0 = ∀ωk)

• = 2xπ(1) + d(pk)

1

, (ωk = 0) µ(p)

B (x) =

• 2xπ(m) + d(p0)

2

, (ωk = m) 2xπ(m), (m = ∀ωk)

• = 2xπ(m) + d(pk)

2

, (ωk = m)

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 24

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Enumeration

The number of the GCSs over 4q-QAM of length 2m equals the product #{ s(x)} × #{f(x)}. It is well known that the number of the standard GCSs over QPSK is given by #{f(x)} = (m!/2)4(m+1) It was shown in [Li, 2010] that the number of the compatible offsets in the generalized cases I-III is N 123

m,q = (m + 1)42(q−1) − (m + 1)4(q−1) + 2(q−1), m ≥ 2.

The enumeration of the compatible offsets in the generalized cases IV-V N 45

m,q was given in

[Liu et al., 2013] for q ≥ 3, m ≥ 3, especially N 45

m,4 = 14 · (m − 2)(m + 1) and N 45 m,6 = 584 · (m − 2)(m + 1).

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 25

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Enumeration for q = 6

For q = 2 · 3 and Z6 = Z2 + 3Z2, or q = 3 · 2 and Z6 = Z3 + 2Z3, the offset s(x) in our construction can be expressed in the form of s(p)(x) =

• d(p0)

1

xω + d(p0)

2

xω+1 + d(p0)

• +
• d(p1)

1

xκ + d(p1)

2

xκ+1 + d(p1)

• , (0 ≤ p < q),

where p = p0 + p1 (0 ≤ ω = κ ≤ m) are given below respectively. p 1 2 3 4 5 p0 1 1 1 p1 2 2 4 4 p 1 2 3 4 5 p0 3 3 3 p1 1 2 1 2 Case (1) Z6 = Z2 + 3Z2 Case (2) Z6 = Z3 + 2Z3

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• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 26

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Table: Coefficients of offsets for q = 6

s(p)(x) in Case (1) s(p)(x) in Case (2) p xω xω+1 xκ xκ+1 xω′ xω′+1 xκ′ xκ′+1 1 d(1)

1

d(1)

2

d(1)

1

d(1)

2

2 d(2)

1

d(2)

2

d(2)

1

d(2)

2

3 d(1)

1

d(1)

2

d(2)

1

d(2)

2

d(3)

1

d(3)

2

4 d(4)

1

d(4)

2

d(3)

1

d(3)

2

d(1)

1

d(1)

2

5 d(1)

1

d(1)

2

d(4)

1

d(4)

2

d(3)

1

d(3)

2

d(2)

1

d(2)

2

Define the values of − → d (p) = (d(p)

0 , d(p) 1 , d(p) 2 ). The 16 possible values of −

→ d (p) can be classified into: C1 = {(1, 1, 1), (3, 1, 1), (0, 1, 3), (2, 1, 3), (0, 2, 2), (2, 2, 2), (0, 3, 1), (2, 3, 1), (1, 3, 3), (3, 3, 3)}, C2 = {(1, 0, 2), (3, 0, 2)}, C3 = {(1, 2, 0), (3, 2, 0)}, C4 = {(0, 0, 0), (2, 0, 0)}.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 27

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Take Case (1) as an example. Conditions: − → d (p) (p = 1, 2, 4) and (ω, κ) satisfy:

1

− → d (1) ∈ C1;

2

− → d (2), − → d (4) / ∈ C4, (− → d (2), − → d (4)) / ∈ (C2, C2) (C3, C3);

3

(ω, κ) = (0, m) or (m, 0) and |ω − κ| ≥ 2. Number:

1

10;

2

(142 − 2 × 22) = 188;

3

(m + 1)(m − 2). Total: #{Case (1) & Case (2)} = 3760(m2 − m − 2).

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 28

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration

Conclusion Table: Comparisons of the number of the compatible offsets q = 4 q = 6 Cases I-III [Li, 2010] 4032m + 4040 1047552m + 1047584 Case IV-V [Liu et al., 2013] 14(m2 − m − 2) 584(m2 − m − 2) New in this paper ≥ 100(m2 − m − 2) ≥ 3760(m2 − m − 2)

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 29

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

• 3. A Sketch of Proof

3.1 Viewpoint of Array 3.2 Generating Function 3.3 Para-unitary Matrix

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 30

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

Mapping from Array to Sequence

A complex-valued array of size 2 × 2 × · · · × 2 can be expressed by a function F(x1, x2, · · · , xm) (or F(x) for short) from Zm

2 to C. For example, an array for m = 3 can

be viewed as the following cube: F(x1, x2, x3) =

F (001) F (011) F (000) F (010) F (101) F (111) F (100) F (110)

A sequence F(y) of length 2m can be connected with an array F(x) by setting y = m

k=1 xk · 2k−1.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 31

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

Aperiodic auto correlation and GAP

The aperiodic auto-correlation of F(x) at shift τ = (τ1, τ2, · · · τm) (τk = −1, 0 or 1) is CF (τ) =

• x

F(x + τ) · F(x).

Example: 2 × 2 array F(x0, x1) F(x0, x1) =

F (0,1) F (1,1) F (0,0) F (1,0)

CF (0, 1) = F(0, 1)F(0, 0) + F(1, 0)F(1, 1) CF (1, 1) = F(1, 1)F(0, 0) Aperiodic auto-correlation of F(x0, x1): CF (τ0, τ1) =

CF (−1,1) CF (0,1) CF (1,1) CF (−1,0) CF (0,0) CF (1,0) CF (−1,−1) CF (0,−1) CF (1,−1)

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 32

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

Generating Function

For a complex-valued array F(x) of size 2 × 2 × · · · × 2, we can define its generating function by F(z1, z2, · · · , zm) =

• x1,x2,··· ,xm

F(x1, x2, · · · , xm)zx1

1 zx2 2 · · · zxm m ,

• r denoted by F(z) =

x F(x) · zx for short.

{F(x), G(x)} form a GAP if their generating functions {F(z), G(z)} satisfy F(z) · F(z−1) + G(z) · G(z−1) = c, where c is a real constant.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 33

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

Para-unitary Matrix

Suppose M(z) = F0,0(z) F0,1(z) F1,0(z) F1,1(z)

• .

be the generating-function matrix of array matrix M(x) = F0,0(x) F0,1(x) F1,0(x) F1,1(x)

• .

If M(z) is a para-unitary (PU) matrix, i.e., M(z)M†(z−1) = c · I, where c is a real number, (·)† denotes the Hermitian transpose and I is an identity matrix

• f order 2, then the arrays over QAM described by every row (or column) of

M(x) form a GAP.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 34

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

PU over QPSK

Example ([Z. Wang and Ma, 2016]) M(z) = H · D(z1) · H · D(z2) · · · H · D(zm) · H.

where H = 1 1 1 −1

• is Butson-type Hadamard matrices, and D(z) =

1 z

• is delay

matrix.

• M(x) = 2x1 ·

1 1

• + 2xm ·

1 1

• + f(x) ·

1 1 1 1

• .

where f(x) = 2 · m−1

k=1 xkxk+1.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 35

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

PU over 4q-QAM

M(z) is a PU matrix over 4q-QAM constellation. Suppose that M(z) =

q−1

• p=0

2p · M (p)(z) where M (p)(z) is multivariate polynomial matrix over QPSK. For 0 ≤ p ≤ q − 1, the corresponding GBF matrices of M (p)(z) is given by

• M (p)(x) = µ(p)

A (x) ·

1 1

• + µ(p)

B (x) ·

1 1

• +
• f(x) + s(p)(x)
• ·

1 1 1 1

• .

(1)

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 36

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

PU over QAM

Example ([Budiˇ sin and Spasojevi´ c, 2018]) M(z) = H · D(z1) · · · D(zω) · H · D(zω+1) · · · D(zm) · H. M (p)(z) = H · D(z1) · · · D(zω) · Hp · D(zω+1) · · · D(zm) · H.

where H =

p∈Zq 2p · H(d(p) 0 , d(p) 1 , d(p) 2 ) where

H(d0, d1, d2) = ξd0 · 1 ξd1 1 1 1 −1 1 ξd2

• =

ξd0 ξd0+d2 ξd0+d1 −ξd0+d1+d2

• , (2d0+d1+d2 = 0)

µ(p)

A (x) = 2x1 and µ(p) B (x) = 2xm.

f(x) = 2 · m−1

k=1 xkxk+1.

s(p)(x) = d(p)

0 +d(p) 1 xω+d(p) 2 xω+1.

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 37

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

Example ([Wang et al., ]) M(z) = H · · · D(zω) · Hω · D(zω+1) · · · D(zκ) · Hκ · D(zκ+1) · · · H. M (p)(z) = H · · · D(zω) · Hp1 · D(zω+1) · · · D(zκ) · Hp2 · D(zκ+1) · · · H.

where

• Hω = 20 · H0 + 21 · H1,

Hκ = 20 · H0 + 22 · H2 + 24 · H4. where s(p1)(x) = (d(p)

0 +d(p1) 1

xω+d(p1)

2

xω+1) + (d(p2) +d(p2)

1

xκ+d(p2)

2

xκ+1) p Possible value 1 2 3 4 5 p0 {0, 1} 1 1 1 p1 {0, 2, 4} 2 2 4 4

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 38

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

References

Budiˇ sin, S. Z. and Spasojevi´ c, P. (Aug. 2018). Paraunitary-based Boolean generator for QAM complementary sequences of length 2K. IEEE Trans. Inf. Theory, 64(8):5938–5956. Davis, J. A. and Jedwab, J. (1999). Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes. IEEE Trans. Inf. Theory, 45(7):2397–2417. Li, Y. (2010). A construction of general QAM Golay complementary sequences. IEEE Trans. Inf. Theory, 56(11):5765–5771. Liu, Z., Li, Y., and Guan, Y. L. (2013). New constructions of general QAM Golay complementary sequences. IEEE Trans. Inf. Theory, 59(11):7684–7692. Wang, Z., Xue, E., and Gong, G. New Constructions of Complementary Sequence Pairs over 4q-QAM. [Online]. Available: https://arxiv.org/abs/2003.03459.

• Z. Wang, G. W. and Ma, D. (2016).

A new method to construct golay complementary.

• Proc. Sequences and Their Appl., pages 252–263.
• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)

SLIDE 39

• 1. Introduction
• 2. Main Results
• 3. A Sketch of Proof

Viewpoint of Array Generating Function Para-unitary Matrix

Author information

Zilong Wang, Ph.D, Professor State Key Laboratory of Integrated Service Networks Xidian University Xi’an 710071, China Tel: +86-29-88202528 Email: zlwang@xidian.edu.cn, Erzhong Xue, Ph.D Candidate Cyberspace Security, School of Cyber Engineering Xidian University Xi’an 710071, China Tel: +86-18700196786 Email: 2524384374@qq.com, Guang Gong, Ph.D, Professor, University Research Chair, IEEE Fellow Department of Electrical and Computer Engineering (ECE) University of Waterloo 200 University Ave West Waterloo, Ontario N2L 3G1, Canada Phone: +1 519 888 - 4567 x45650. Fax: +1 (519) 746-3077 Email: ggong@uwaterloo.ca, https://uwaterloo.ca/scholar/ggong/home

• Z. Wang
• E. Xue G. Gong

QAM GCPs (ISIT2020)