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Feb 03, 2023 669 likes •1.06k views

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

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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

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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

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Outline

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

Zilong LIU (U. Essex, CSEE) 3 / 26

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Background

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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Background Aperiodic correlation function

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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Background Aperiodic correlation function

Aperiodic correlation function Suppose two length-N sequences a (or {at}) and b (or {bt}). The aperiodic function of a and b for time-shift τ is defined as ρa,b(τ) =             

N−1−τ

  • t=0

at+τb∗

t ,

0≤τ≤(N − 1);

N−1+τ

  • t=0

atb∗

t−τ,

−(N − 1)≤τ≤ − 1; 0, |τ| ≥ N.

Zilong LIU (U. Essex, CSEE) 4 / 26

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Background Golay complementary pairs (GCPs)

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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Background Golay complementary pairs (GCPs)

Golay complementary pairs (GCPs) Binary GCPs were introduced by Marchel J. E. Golay in his study

  • f 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

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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., ρa(τ) + ρb(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

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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

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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

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Cross Z-Complementary Pairs (CZCPs): Properties and Constructions

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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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 T1 {1, 2, · · · , Z} and T2 {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| ∈ T1 ∪ T2; C2 : ρ (a, b) (τ) + ρ (b, a) (τ) = 0, for all |τ| ∈ T2. (1)

Zilong LIU (U. Essex, CSEE) 8 / 26

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Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties

Correlation properties of (N, Z)-CZCP

     

     a b

Z

1 2 ...

...

1 N  2 N  N Z 

...

1 2 N Z  

     

, ,      a b b a

... ... ...

1 Z  1 N Z  

N Z  2 N  1 N  1 N Z  

1 2

 

AAC sum ACC sum Front-end ZACZ Tail-end ZACZ Tail-end ZCCZ

Z Z 1 N Z   Z

Zilong LIU (U. Essex, CSEE) 9 / 26

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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]

4

, b = ω[0,1,1,0,1,0,3,3,1]

4

. (2) (a, b) is a (9, 3)-CZCP because

  • ρ (a) (τ) + ρ (b) (τ)
  • 8

τ=0 =(18, 01×3, 2

√ 2, 2, 01×3),

  • ρ (a, b) (τ) + ρ (b, a) (τ)
  • 8

τ=0 =

  • 4, 4

√ 2, 2 √ 2, 2 √ 2, 4, 2, 01×3

  • .

(3)

Zilong LIU (U. Essex, CSEE) 10 / 26

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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

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Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties

Relationship Between CZCP and GCP

CZCPs GCPs

Perfect CZCPs (i.e., Strengthened GCPs)

2 N Z  2 N Z  Zilong LIU (U. Essex, CSEE) 12 / 26

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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 a0 and b by b0, respectively, i.e., c = a/a0, d = b/b0, where the latter CZCP satisfies ci = di, cN−1−i = −dN−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

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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

d

  • =
  • c0

c1 c2 · · · cN/2−1 cN/2 cN/2+1 · · · cN−1 c0 c1 c2 · · · cN/2−1 −cN/2 −cN/2+1 · · · −cN−1

  • .

(5)

Zilong LIU (U. Essex, CSEE) 14 / 26

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Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Systematic Construction

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Systematic Construction

Systematic Construction Let (e, f) be q-ary GCP of length N/2. Then, every sequence pair (arranged in matrix form with two rows) in (6) is a perfect CZCP (i.e., strengthened GCP).

  • ωυ1

q · e, ωυ1+υ q

· f ωυ2

q · e, −ωυ2+υ q

· f

  • ,
  • ωυ1

q · e, −ωυ1+υ q

· f ωυ2

q · e, ωυ2+υ q

· f

  • ,
  • ωυ1

q · f, ωυ1+υ q

· e ωυ2

q · f, −ωυ2+υ q

· e

  • ,
  • ωυ1

q · f, −ωυ1+υ q

· e ωυ2

q · f, ωυ2+υ q

· e

  • (6)

where υ1, υ2, υ ∈ Zq(q even) and υ1 − υ2 ∈ {0, q/2}(mod q).

Zilong LIU (U. Essex, CSEE) 15 / 26

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CZCP for Optimal SM Training

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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CZCP for Optimal SM Training Principle of Spatial Modulation (SM)

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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CZCP for Optimal SM Training Principle of Spatial Modulation (SM)

Structure of SM Tx-Rx

MIMO channel

Input bits Constellation Symbol Gen. Spatial Symbol Gen.

Spatial modulation Transmitter

Add CP/ZP RF chain

b(k) b1(k)

n(k)

s

… …

n(k)

e

Bits-to-Symbol Mapping Input bits Active antenna BPSK symbol

00 01 10 11 1 2 1 2

  • 1

+1

  • 1

+1

Example: SM with Nt=2，BPSK

… …

RF chain RF chain RF chain RF chain

Remove CP/ZP Spatial Demodulation Time/Frequency domain equalization P/S

Output bits

Receiver

n(k)

S/ P

SM symbols d1 dk dK

... ...

b2(k) 2 Nt Nr 1 1 2

TA Selection switch

Zilong LIU (U. Essex, CSEE) 16 / 26

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CZCP for Optimal SM Training Principle of Spatial Modulation (SM)

An Example of SM Transmission Example

1

Consider an SC-SM system with Nt = 4 TAs using 8-PSK modulation (MSM = 8).

2

For illustration purpose, we consider natural mapping for TA selection and 8-PSK modulation, i.e., each index (for a TA or an 8-PSK phase) can be obtained from its natural binary representation (with the left-most bit being the most significant bit).

3

Suppose each SC-SM block constitutes K = 4 SM symbols.

4

Suppose further these symbols correspond to K log2(8Nt) = 20 message bits [10010110110111000100], with b(1) = [10010], b(2) = [11011], b(3) = [01110], and b(4) = [00100].

Zilong LIU (U. Essex, CSEE) 17 / 26

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CZCP for Optimal SM Training Principle of Spatial Modulation (SM)

An Example of SM Transmission (Cont’d) Example

1

Taking the first SM symbol ([10010]) for example, we have b(1) = [b1(1), b2(1)], where b1(1) = [10] and b2(1) = [010]. This means that during the first time-slot, only the third TA will be activated for the sending of 8-PSK symbol ω2

  • 8. Therefore, the first

SM symbol can be written as d1 = [0, 0, ω2

8, 0]T.

2

The entire SC-SM block can be expressed by the sparse matrix as follows. [d1, d2, d3, d4] =

     

ω4

8

ω6

8

ω2

8

ω3

8

     

. (7)

Zilong LIU (U. Essex, CSEE) 18 / 26

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CZCP for Optimal SM Training Optimal SM Training

1

Background Aperiodic correlation function Golay complementary pairs (GCPs)

2

Cross Z-Complementary Pairs (CZCPs): Properties and Constructions Definition & Properties Systematic Construction

3

CZCP for Optimal SM Training Principle of Spatial Modulation (SM) Optimal SM Training

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CZCP for Optimal SM Training Optimal SM Training

SM Training Structure

1

x

2

x x

Training Seq.

L 

1 L 

t

1

D

2

D D

CP

…

2 Nt 1

… … …

Nt Nt

Define the following training matrix Ω: Ω =   x1 x2 . . . xNt   =   x1,0 x1,1 · · · x1,L−1 x2,0 x2,1 · · · x2,L−1 . . . . . . ... . . . xNt,0 xNt,1 · · · xNt,L−1  

Nt×L

. (8)

Zilong LIU (U. Essex, CSEE) 19 / 26

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CZCP for Optimal SM Training Optimal SM Training

Received Signal The k-th received signal at a RA can be written as yk =

Nt

  • n=1

λ

  • l=0

hn,lxn,k−l + wk, (9) where wk a discrete uncorrelated white complex Gaussian noise sample with zero-mean and variance σ2

w/2 per dimension.

Zilong LIU (U. Essex, CSEE) 20 / 26

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CZCP for Optimal SM Training Optimal SM Training

System Equation Xn =

      

xn,0 xn,L−1 · · · xn,L−λ xn,1 xn,0 · · · xn,L−λ+1 . . . . . . ... . . . xn,L−1 xn,L−2 · · · xn,L−λ−1

      

L×(λ+1)

, h =

      

h1 h2 . . . hNt

      

Nt(λ+1)×1

, (10) and y = [y0, y1, · · · , yL−1]T, X = [X1, X2, · · · , XNt]L×(Ntλ+Nt), w = [w0, w1, · · · , wL−1]. (11)

Zilong LIU (U. Essex, CSEE) 21 / 26

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CZCP for Optimal SM Training Optimal SM Training

System Equation (Cont’d) The system equation can be expressed in matrix form as follows: y = Xh + w. (12) Applying the least-squares (LS) channel estimator (unbiased), the estimated CIR vector is given by ^ h =

  • XHX

−1 XHy,

(13)

Zilong LIU (U. Essex, CSEE) 22 / 26

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CZCP for Optimal SM Training Optimal SM Training

Optimal Training Matrix MSE = 1 (Ntλ + Nt)Tr

  • E
  • h − ^

h

  • h − ^

h

H

= σ2

w

(Ntλ + Nt)Tr

  • XHX

−1

, XHX =

      

XH

1 X1

XH

1 X2

· · · XH

1 XNt

XH

2 X1

XH

2 X2

· · · XH

2 XNt

. . . . . . ... . . . XH

NtX1

XH

NtX2

· · · XH

NtXNt

      

(Ntλ+Nt)×(Ntλ+Nt)

. (14)

Zilong LIU (U. Essex, CSEE) 23 / 26

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CZCP for Optimal SM Training Optimal SM Training

Optimal Training Matrix MSE = 1 (Ntλ + Nt)Tr

  • E
  • h − ^

h

  • h − ^

h

H

= σ2

w

(Ntλ + Nt)Tr

  • XHX

−1

, XHX =

      

XH

1 X1

XH

1 X2

· · · XH

1 XNt

XH

2 X1

XH

2 X2

· · · XH

2 XNt

. . . . . . ... . . . XH

NtX1

XH

NtX2

· · · XH

NtXNt

      

(Ntλ+Nt)×(Ntλ+Nt)

. (14) Minimum MSE is achieved if and only if XHX is a diagonal matrix whose elements on the diagonal are identical.

Zilong LIU (U. Essex, CSEE) 23 / 26

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CZCP for Optimal SM Training Optimal SM Training

Channel Estimation Performance Let us consider a perfect (N = 8, Z = 4)-CZCP and the following two training matrices (4, 2, 8)-Ω1. Ω1 =

     

a b a b a b a b

     

. (15) where 0 in (15) stands for 01×8.

Remark With regard to a training matrix with structure of Ω1, the front-end ZACZ and the tail-end ZACZ of CZCP are helpful in mitigating the ISI of each training sequence and the inter-antenna interference (IAI) between the i-th and the (i + 1)-th TAs (i = 1, 2, · · · , Nt − 1), respectively. Furthermore, the tail-end ZCCZ of CZCP will help eliminate the IAI between the 1-th and the Nt-th TAs.

Zilong LIU (U. Essex, CSEE) 24 / 26

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CZCP for Optimal SM Training Optimal SM Training

MSE comparison with other training sequences at EbNo=16dB:

3 4 5 6 7 8 9 10 11 12 13

  • 34.5
  • 34
  • 33.5
  • 33
  • 32.5
  • 32

Zilong LIU (U. Essex, CSEE) 25 / 26

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CZCP for Optimal SM Training Optimal SM Training

Thank You! (zilong.liu@essex.ac.uk)

Zilong LIU (U. Essex, CSEE) 26 / 26