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 Wang 1 , Erzhong Xue 1 , Guang Gong 2 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)
1. Introduction 2. Main Results 3. A Sketch of Proof Contents 1. Introduction 1 1.1 GCS & GCP 1.2 QPSK GCPs 1.3 QAM GCPs 2. Main Results 2 2.1 Cases ++ 2.2 Main Construction 2.3 Enumeration 3. A Sketch of Proof 3 Viewpoint of Array Generating Function Para-unitary Matrix Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 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 � C F ( τ ) = F ( y + τ ) · F ( y ) . y A pair of sequences { F ( y ) , G ( y ) } of length L is said to be a Golay complementary pair (GCP) if C F ( τ ) + C G ( 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)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs Example F 1 ( y ) = (1 , 1 , 1 , − 1 , 1 , 1 , − 1 , 1) and F 2 ( y ) = (1 , ξ, ξ, − 1 , 1 , − ξ, − ξ, − 1) . ( C F 1 ( τ )) 7 0 = (8 , − 1 , 0 , 3 , 0 , 1 , 0 , 1) ( C F 2 ( τ )) 7 0 = (8 , 1 , 0 , − 3 , 0 , − 1 , 0 , − 1) C F 1 ( τ ) + C F 2 ( τ ) = 0 ( ∀ τ � = 0) ⇒ F 1 ( y ) and F 2 ( y ) are GCP. 1 1 ξ = √− 1 is a fourth primitive root of unity. Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs GBF Generalized Boolean Function For 0 ≤ y < L = 2 m , y can be uniquely written in a binary expansion as y = � m k =1 x k · 2 k − 1 where x k ∈ F 2 . Let x = ( x 1 , x 2 , · · · x m ) . Then a sequence F ( y ) of length 2 m over QPSK can be associated with a generalized Boolean function (GBF) f ( x ) over Z 4 by F ( y ) = ξ f ( x ) . Example f ( x 1 , x 2 , x 3 ) = 2( x 2 x 3 + x 1 x 3 ) + x 1 + x 2 Z 4 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)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs Standard GCPs [Davis and Jedwab, 1999] x π (1) x π (2) x π (3) x π ( m − 1) x π ( m ) . . . Figure: The Graph of Coset Representatives For GBF m − 1 � � m f ( x ) = 2 · x π ( k ) x π ( k +1) + c k · x k + c 0 , k =1 k =1 where c k ∈ Z 4 (0 ≤ k ≤ m ) , and c ′ ∈ Z 4 , the sequence pair associated with the GBFs over Z 4 � � f ( x ) , f ( x ) , or f ( x ) + 2 x π (1) + c ′ , f ( x ) + 2 x π ( m ) + c ′ , form a GCP over QPSK. Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs QAM Sequence A sequence over 4 q -QAM can be viewed as the weighted sums of q sequences over QPSK, with weights in the ratio of 2 q − 1 : 2 q − 2 : · · · : 1 . Figure: eg: � v = 4 α + 2 β + γ Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs V-GBF and QAM Sequence 2 to Z q A vectorial generalized Boolean function (V-GBF) is a function from F m 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 Z 4 . A sequence over 4 q -QAM of length 2 m can be associated with a V-GBF � f ( x ) = ( f (0) ( x ) , f (1) ( x ) , · · · , f ( q − 1) ( x )) over Z 4 by q − 1 � 2 p · ξ f ( p ) ( x ) , F ( y ) = p =0 where y = � m k =1 x k · 2 k − 1 . Obviously, a sequence over QPSK can be viewed as a sequence over 4 q -QAM when q = 1 . Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 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 2 m over 4 q -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 (0) ( x ) = 0 , s (1) ( x ) , · · · , s ( q − 1) ( x ) � s ( x ) = is called a offset V-GBF , � � µ (0) ( x ) , µ (1) ( x ) , · · · , µ ( q − 1) ( x ) µ ( x ) = � is called a pairing difference V-GBF . Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs all the GCPs { F ( y ) , G ( y ) } of length 2 m over 4 q -QAM are associated with V-GBFs f ( x ) Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs all the GCPs { F ( y ) , G ( y ) } of length 2 m over 4 q -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)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs all the GCPs { F ( y ) , G ( y ) } of length 2 m over 4 q -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)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs all the GCPs { F ( y ) , G ( y ) } of length 2 m over 4 q -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)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs all the GCPs { F ( y ) , G ( y ) } of length 2 m over 4 q -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 (0) ( x ) = 0 , s (1) ( x ) , · · · , s ( q − 1) ( x ) � s ( x ) = is called a offset V-GBF, � � µ (0) ( x ) , µ (1) ( x ) , · · · , µ ( q − 1) ( x ) � µ ( x ) = is called a pairing difference V-GBF. Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 1.1 GCS & GCP 2. Main Results 1.2 QPSK GCPs 3. A Sketch of Proof 1.3 QAM GCPs The generalized cases I-III [Li, 2010] For d ( p ) ∈ Z 4 ( 1 ≤ p ≤ q − 1 , 0 ≤ i ≤ 2 ), and 2 d ( p ) + d ( p ) + d ( p ) = 0 . i 0 1 2 (1) The generalized case I: s ( p ) ( x ) = d ( p ) + d ( p ) µ ( x ) = 2 x π ( m ) · � 1 x π (1) , 1 ≤ p ≤ q − 1 , � 1; 0 (2) The generalized case II: s ( p ) ( x ) = d ( p ) + d ( p ) µ ( x ) = 2 x π (1) · � 1 x π ( m ) , 1 ≤ p ≤ q − 1 , � 1; 0 (3) The generalized case III: s ( p ) ( x ) = d ( p ) + d ( p ) 1 x π ( ω ) + d ( p ) 2 x π ( ω +1) , 1 ≤ p ≤ q − 1 , 0 µ ( x ) = 2 x π (1) · � 1 or 2 x π ( m ) · � with � 1 , 1 ≤ ω ≤ m − 1 . Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
1. Introduction 2.1 Cases ++ 2. Main Results 2.2 Main Construction 3. A Sketch of Proof 2.3 Enumeration Section 2. Main Results 2.1 Cases ++ New framework Cases I-III in short Case for q = q 0 × q 1 2.2 Main Construction Case for q = q 0 · q 1 · · · q m � s ( x ) , � µ A ( x ) and � µ B ( x ) 2.3 Enumeration Enumeration for q = 6 Conclusion Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
� � � � � � 1. Introduction 2.1 Cases ++ 2. Main Results 2.2 Main Construction 3. A Sketch of Proof 2.3 Enumeration New Structure � f ( x ) µ A ( x ) � � � � f 0 , 0 ( x ) f 0 , 1 ( x ) f ( x ) = � � 1 · f ( x ) + � s ( x ) , � µ B ( x ) µ B ( x ) � g ( x ) = � � f ( x ) + � µ A ( x ) , or � f ( x ) + � µ B ( x ) , � � f 1 , 0 ( x ) f 1 , 1 ( x ) � µ A ( x ) � g ( x ) Figure: A Diagram of the Relationship Z. Wang E. Xue G. Gong QAM GCPs (ISIT2020)
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