Constructions of complementary sequences from 2-level autocorrelation sequences and permutation polynomials Guang Gong Department of Electrical and Computer Engineering University of Waterloo CANADA <https://uwaterloo.ca/scholar/ggong> SEquences and Their Applications 2020 (SETA 2020) Virtual Conference, Sep 22 - 25, 2020, Russia, Saint-Petersburg Joint work with Zilong Wang G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 1 / 57
Outline Basic definitions of sequences and correlation Complementary sequences sets (CSS) and complete mutually orthogonal CSS (or shorted as complete complementary code (CCC)) Paraunitary (PU) matrices and CSS/CCC Explicit algebraic normal forms of CSS/CCC from DFT Hadamard matrices Sequences with idea 2-level autocorrelation and cyclic Hadamard matrices Explicit algebraic normal forms of constructed CSS/CCC from 2-level autocorrelation sequences G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 2 / 57
Historic development of Golay sequences There are a large volume of the publications for those work, which are not shown in the above figure. Please refer to those in our earlier paper (Wang-Ma-Gong-Xue 2019). G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 3 / 57
Sequences, periodic and aperiodic correlation q -ary sequence of length L (or period L ) is given by a = ( a (0) , a (1) , · · · , a ( L − 1)) , a ( t ) ∈ Z q Let ω be a primitive q th root of unity, a periodic autocorrelation function at shift τ is defined as L − 1 ω a ( i + τ ) − a ( i ) , τ = 0 , ± 1 , ± 2 , · · · � R ( τ ) = i =0 where i + τ is reduced by modulo L . So, R ( τ ) is also periodic with period L . G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 4 / 57
Example This can be generated by an LFSR with 3 An m -sequence of period 7 generated by stages the characteristic polynomial w ( x ) = x 3 + x + 1 is a = 1110100 0 0 1 Continuous-time signal of the m-sequence: s c ( t ) +1 t − 1 T c NT c The autocorrelation has all out-of-phase autocorrelation values to equal − 1 . G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 5 / 57
Remarks Perfect sequences: all out-of-phase periodic autocorrelation is equal to zero! For binary case, only known sequence is a = 0111 . For an m -sequence with period L = 2 n − 1 , we have the normalized autocorrelation: � 1 R ( τ ) τ ≡ 0 mod L = − 1 L → 0 τ �≡ 0 mod L ( n → large ) L This property of m -sequences resembles white Gaussian noise, which makes it so popular in digital communications. After Solomon Golomb successfully used for detecting returning signal from the first satellite launched by US in 1958, it finds many applications such as detection of transmitted signals, GPS, radar distance range, spread spectrum, hardware testing, etc. G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 6 / 57
Aperiodic autocorrelation An aperiodic autocorrelation function at shift τ is defined as L − 1 − τ � ω a ( i + τ ) − a ( i ) , − L < τ < L. i =0 This is the functionality of matched filters in signal process with applications in wireless communication. G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 7 / 57
Example We take q = 4 , N = 4 , √ − 1 , { ω k | k = 0 , 1 , , 2 , 3 } = { 1 , i, − 1 , − i } a = (0 , 1 , 0 , 3) = ⇒ ω = i, i = Aperiodic autocorrelation computation at the fashion of matched filter detection: 0 − 1 0 − 3 τ C ( τ ) 0 1 0 3 0 1 0 3 − 3 − i 0 1 0 3 − 2 0 0 1 0 3 − 1 − i 0 1 0 3 0 4 ( matched ) 0 1 0 3 1 i 0 1 0 3 2 0 0 1 0 3 3 i G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 8 / 57
Sequences and generating functions The given sequence can be evaluated from a given generalized boolean function (GBF) in two variables: f ( x 0 , x 1 ) = 2 x 0 x 1 + x 0 , x i ∈ { 0 , 1 } The truth table of f yields a : x 1 x 0 f ( x 0 , x 1 ) 0 0 0 0 1 1 1 0 0 1 1 3 So a = ( f (0) , f (1) , f (2) , f (3)) where t = x 0 + x 1 2 , x i ∈ F 2 is the binary representation of the integer t . Now we can associate the sequence a with its Z transform or generating function: � 3 i f ( t ) Z t = 1 + iZ + Z 2 − iZ 3 . F ( Z ) = t =0 G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 9 / 57
Aperiodic correlation and generating function Suppose that Z is a complex variable. The squared magnitude of F ( Z ) is L − 1 � L − 1 � | F ( Z ) | 2 = F ( Z ) F ( Z ) = L + C a ( τ ) Z τ + C a ( τ ) Z − τ , τ =1 τ =1 where C a ( τ ) is aperiodic autocorrelation of sequence a at shift τ . G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 10 / 57
Golay Complementary Pairs A pair of sequences ( a , b ) of length L is called a Golay complementary pair if C a ( τ ) + C b ( τ ) = 0 , 0 < τ < L. Each sequence of such a pair is called a Golay sequence . If ( a , b ) form a Golay pair of length L , and let G ( Z ) be b ’s generating function, then | F ( Z ) | 2 + | G ( Z ) | 2 = 2 L In other words, if a is a Golay sequence, then | F ( Z ) | 2 � 2 L for all | Z | = 1 , which leads to PMEPR ( a ) � 2 . This property of Golay sequences can be used in OFDM transmission systems for reducing the peak to average power ratio. G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 11 / 57
Sequences and arrays For the previous example, f = ( f (0) , f (1) .f (2) , f (3)) = (0 , 1 , 0 , 3) , it is evaluated from a GBF f ( y 0 , y 1 ) = 2 y 0 y 1 + y 0 , f ( t ) = f ( t 0 , t 1 ) where t = t 0 + t 1 2 , t i ∈ Z 2 , a binary representation of an integer t . f ( y 0 , y 1 ) is referred to as an array. We can associate it with a (multivariable) generating function: � i f ( y 0 ,y 2 ) z y 0 0 z y 1 f ( z 0 , z 1 ) = = 1 + iz 0 + z 1 − iz 0 z 1 1 ( y 0 ,y 1 ) ∈ F 2 2 G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 12 / 57
Relationship of sequences and arrays A q -ary sequence f of length L Corresponding function : f ( t ) : Z L → Z q Generating function : L − 1 � ω f ( t ) Z t F ( Z ) = t =0 An m -dimensional q -ary array of size p × p × · · · × p Corresponding function : f ( y ) = f ( y 0 , y 1 , · · · , y m − 1 ) : Z m p → Z q Generating function : p − 1 p − 1 p − 1 � � � ω f ( y ) z y 0 0 z y 1 y m − 1 F ( z ) = · · · 1 · · · z m − 1 y 0 =0 y 1 =0 y m − 1 =0 G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 13 / 57
Aperiodic correlation of arrays We can also define the aperiodic cross-correlation of two arrays f 1 ( y ) and f 2 ( y ) from Z m p to Z q at shift τ = ( τ 0 , τ 1 , · · · , τ m − 1 ) by � ω f 1 ( y + τ ) − f 2 ( y ) C f 1 ,f 2 ( τ ) = y ∈ Z m p where “ y + τ ”is the element-wise addition of vectors over Z , and ω f 1 ( y + τ ) − f 2 ( y ) = 0 if f 1 ( y + τ ) or f 2 ( y ) is not defined. The aperiodic autocorrelation of array f at shift τ is C f ( τ ) = C f,f ( τ ) . Array correlation is a stronger condition than a sequence correlation. G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 14 / 57
Example on array correlation For previous example f ( y 0 , y 1 ) = 2 y 0 y 1 + y 0 , the array autocorrelation of f ( z 0 , z 1 ) � i f (( y 0 ,y 1 )+( τ 0 ,τ 1 )) − f ( y 0 ,y 1 ) C f ( τ 0 , τ 1 ) = y ∈ Z 2 2 y 1 y 1 f ( y 0 ,y 1 ) f ( y 0 ,y 1 ) y 0 y 0 f ( y 0 + τ 0 ,y 1 + τ 1 ) ( τ 0 ,τ 1 )=( − 1 , 0) f ( y 0 + τ 0 ,y 1 + τ 1 ) ( τ 0 ,τ 1 )=( − 1 , − 1) C f ( − 1 . − 1) = i f (0 , 0) − f (1 , 1) = − i C f ( − 1 . 0) = i f (0 , 0) − f (1 , 0) + i f (1 , 0) − f (1 , 1) = i f (0) − f (2) + i f (1) − f (3) = 0 G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 15 / 57
(Cont.) The autocorrelation of array f ( y 0 , y 1 ) is the sum of the overlapping points of two squares. Thus the autocorrelation of the sequence (0, 1, 0, 3) can be computed by the corresponding array correlation. C a ( − 3) = C f ( − 1 , − 1) , C a ( − 2) = C f (0 , − 1) , C a ( − 1) = C f ( − 1 , 0) + C f (1 , − 1) , C a (0) = C f (0 , 0) , C a (1) = C f (1 , 0) + C f ( − 1 , 1) , C a (2) = C f (0 , 1) , C a (3) = C f (1 , 1) . G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 16 / 57
Complementary sequence sets (CSS) A set of sequences S = { f 0 , f 1 , · · · , f N − 1 } is called a complementary sequence set ( CSS ) of size N if the sum of their autocorrelation values at τ is equal to zero at all τ � = 0 , i.e., N − 1 � C f j ( τ ) = 0 for τ � = 0 . j =0 Equivalently using their generating functions: N − 1 � F j ( Z ) F j ( Z − 1 ) = NL. j =0 G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 17 / 57
Complete complementary codes Mutually orthogonal CSSs : Two CSSs S 1 = { f 1 , 0 , f 1 , 1 , · · · , f 1 ,N − 1 } S 2 = { f 2 , 0 , f 2 , 1 , · · · , f 2 ,N − 1 } are said to be mutually orthogonal if N − 1 � C f 1 ,j , f 2 ,j ( τ ) = 0 , ∀ τ. j =0 Equivalently, N − 1 � F 1 ,j ( Z ) F 2 ,j ( Z − 1 ) = 0 . j =0 CCC : Let S i = { f i, 0 , f i, 1 , · · · , f i,N − 1 } be CSSs of size N for 0 ≤ i < N . If any of two sets are mutually orthogonal, then the collection of S i is called complete mutually orthogonal complementary sets (CMOCS) or complete complementary codes ( CCC ). G. Gong (UW) CSS/CCC - 2-level auto - PP SETA2020 18 / 57
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