Cross Z-Complementary Pairs (CZCPs) for Optimal Training in Broadband Spatial Modulation Systems Zilong Liu (University of Essex, UK) Ping Yang (University of Electronic Science and Technology of China) Yong Liang Guan (Nanyang Technological University, Singapore) Pei Xiao (University of Surrey, UK) June 2020 (ISIT) Zilong LIU (U. Essex, CSEE) June 2020 (ISIT) 1 / 26
Full Paper of This Work “Cross Z-Complementary Pairs for Optimal Training in Spatial Modulation Over Frequency Selective Channels," Accepted by IEEE TRANSACTIONS ON SIGNAL PROCESSING , 2020. IEEE Link: https://ieeexplore.ieee.org/document/8993844 Arxiv Link: https://arxiv.org/abs/1909.10206 Zilong LIU (U. Essex, CSEE) 2 / 26
Outline Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training Zilong LIU (U. Essex, CSEE) 3 / 26
Background Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training
Background Aperiodic correlation function Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training
Background Aperiodic correlation function Aperiodic correlation function Suppose two length- N sequences a (or { a t } ) and b (or { b t } ). The aperiodic function of a and b for time-shift τ is defined as N − 1 − τ � 0 ≤ τ ≤ ( N − 1 ); a t + τ b ∗ t , t = 0 N − 1 + τ ρ a , b ( τ ) = � a t b ∗ t − τ , −( N − 1 ) ≤ τ ≤ − 1 ; t = 0 0 , | τ | ≥ N . Zilong LIU (U. Essex, CSEE) 4 / 26
Background Golay complementary pairs (GCPs) Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training
Background Golay complementary pairs (GCPs) Golay complementary pairs (GCPs) Binary GCPs were introduced by Marchel J. E. Golay in his study of infrared multislit spectrometry in the late 1940s. M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., vol. 39, pp. 437-444, 1949. M. J. E. Golay, Static multislit spectrometry and its application to the panoramic display of infrared spectra, J. Opt. Soc. Amer., vol. 41, pp. 468-472, 1951. Zilong LIU (U. Essex, CSEE) 5 / 26
Background Golay complementary pairs (GCPs) Golay complementary pairs (GCPs) ( a , b ) , a pair of length- N sequences, is called a Golay complementary pair (GCP) if they have zero out-of-phase aperiodic autocorrelation sums, i.e., ρ a ( τ ) + ρ b ( τ ) = 0 , for all τ � = 0 . Requirement for the Transmission of a GCP Two orthogonal channels (one for each constituent sequence) with no interference from each other. Zilong LIU (U. Essex, CSEE) 6 / 26
Background Golay complementary pairs (GCPs) Applications of Complementary Sequences Multislit spectrometry; Ultrasound measurements, acoustic measurements; Channel estimation and Synchronization; Radar pulse compression (Doppler Resilient Radar Waveform); Digital watermarking (detecting the embedded info from the watermarked images); Complementary Code Keying (CCK) for high rates in 802.11b (generalized Hadamard transform encoding); Orthogonal design (e.g., OSTBC, ZCZ sequences, etc); PAPR reduction in multicarrier systems. Zilong LIU (U. Essex, CSEE) 7 / 26
Background Golay complementary pairs (GCPs) Applications of Complementary Sequences Multislit spectrometry; Ultrasound measurements, acoustic measurements; Channel estimation and Synchronization; Radar pulse compression (Doppler Resilient Radar Waveform); Digital watermarking (detecting the embedded info from the watermarked images); Complementary Code Keying (CCK) for high rates in 802.11b (generalized Hadamard transform encoding); Orthogonal design (e.g., OSTBC, ZCZ sequences, etc); PAPR reduction in multicarrier systems. Motivation of this work : GCP (and complementary sequences) may not work well when the two constituent sequences are transmitted over two non-orthogonal channels. Zilong LIU (U. Essex, CSEE) 7 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Cross Z-Complementary Pair (CZCP) Definition Let ( a , b ) be a pair of sequences of identical length N . For a proper integer Z < N , define T 1 � { 1 , 2 , · · · , Z } and T 2 � { N − Z , N − Z + 1 , · · · , N − 1 } . ( a , b ) is called an ( N , Z ) -CZCP if the following two conditions are satisfied: C1 : ρ ( a ) ( τ ) + ρ ( b ) ( τ ) = 0 , for all | τ | ∈ T 1 ∪ T 2 ; (1) C2 : ρ ( a , b ) ( τ ) + ρ ( b , a ) ( τ ) = 0 , for all | τ | ∈ T 2 . Zilong LIU (U. Essex, CSEE) 8 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Correlation properties of ( N , Z ) -CZCP ACC sum a b , b a , a b AAC sum ... Tail-end Front-end Tail-end ... ZCCZ ZACZ ZACZ N Z 1 N 0 N N 0 N Z 2 N 1 1 2 ... Z Z 1 N Z 1 N Z 2 1 1 2 ... ... ... N 1 2 Z N Z 1 Z Z Z Zilong LIU (U. Essex, CSEE) 9 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Example Consider the length-9 quaternary pair ( a , b ) below. a = ω [ 0 , 1 , 1 , 2 , 0 , 2 , 1 , 1 , 3 ] , b = ω [ 0 , 1 , 1 , 0 , 1 , 0 , 3 , 3 , 1 ] . (2) 4 4 ( a , b ) is a ( 9 , 3 ) -CZCP because √ � 8 �� � � ρ ( a ) ( τ ) + ρ ( b ) ( τ ) τ = 0 =( 18 , 0 1 × 3 , 2 2 , 2 , 0 1 × 3 ) , � � � (3) √ √ √ � 8 �� � � � � ρ ( a , b ) ( τ ) + ρ ( b , a ) ( τ ) τ = 0 = 4 , 4 2 , 2 2 , 2 2 , 4 , 2 , 0 1 × 3 . � � � Zilong LIU (U. Essex, CSEE) 10 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Binary ( N , Z ) -CZCPs (with maximum Z ) of lengths up to 26 Zilong LIU (U. Essex, CSEE) 11 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Relationship Between CZCP and GCP Perfect CZCPs (i.e., Strengthened GCPs) CZCPs GCPs Z N 2 Z N 2 Zilong LIU (U. Essex, CSEE) 12 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Property (selected) Every q -ary ( N , Z ) -CZCP ( a , b ) is equivalent to ( N , Z ) -CZCP ( c , d ) by dividing a by a 0 and b by b 0 , respectively, i.e., c = a / a 0 , d = b / b 0 , where the latter CZCP satisfies c i = d i , c N − 1 − i = − d N − 1 − i , for all i ∈ { 0 , 1 , · · · , Z − 1 } . (4) By utilizing (4), one can readily show that Z ≤ N / 2. Zilong LIU (U. Essex, CSEE) 13 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties CZCP → Strengthened GCP Definition Every q -ary CZCP ( a , b ) is called perfect if Z = N / 2 ( N even). In this case, a perfect ( N , N / 2 ) -CZCP reduces to a sequence pair, called strengthened GCP , whose equivalent CZCP ( c , d ) is given below: � � � � c c 0 c 1 c 2 · · · c N / 2 − 1 c N / 2 c N / 2 + 1 · · · c N − 1 = . d c 0 c 1 c 2 · · · c N / 2 − 1 − c N / 2 − c N / 2 + 1 · · · − c N − 1 (5) Zilong LIU (U. Essex, CSEE) 14 / 26
Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Systematic Construction Background 1 Aperiodic correlation function Golay complementary pairs (GCPs) Cross Z-Complementary Pairs (CZCPs): Properties and 2 Constructions Definition & Properties Systematic Construction CZCP for Optimal SM Training 3 Principle of Spatial Modulation (SM) Optimal SM Training
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