Phase noise sensitivity analysis of lattice constellation

The Journal of China Universities of Posts and Telecommunications April 2010, 17(2): 36–40 www.sciencedirect.com/science/journal/10058885

www.buptjournal.cn/xben

Phase noise sensitivity analysis of lattice constellation YU Guang-wei ( ), NIU Kai, HE Zhi-qiang, WANG Xu-zhen, LIN Jia-ru Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract

Based on the assumption of large number of constellation points and high signal-to-noise ratio (SNR), phase noise sensitivity of lattice constellation is analyzed. The upper bound of symbol error rate (SER) in additive white Gaussian noise (AWGN) channel is derived from pairwise error probability. For small phase noise, phase noise channel is transformed to AWGN channel. With the aid of Wiener model, the obtained upper bound can be extended to phase noise channel. The proposed upper bound can be used as performance criterion to analyze the sensitivity of phase noise in multi-dimensional lattice constellation. Simulation results show that with the same normalized spectral efficiency, higher dimensional lattice constellations are more sensitive than lower ones in phase noise channel. It is also shown that with the same dimension of constellation, larger normalized spectral efficiency means more performance loss in phase noise channel. Keywords lattice constellation, symbol error rate (SER), phase noise

1

Introduction

With the development of communication technologies, high data rate is required urgently, which implies high spectral efficiency due to limited frequency resources. Besides multilevel modulation, lattice constellations are commonly accepted as good methods for transmission with high spectral efficiency [1]. Lattice constellation is modified from infinite lattice, and a lattice is a discrete set of points in real number field. Furthermore, the linearity and highly symmetrical geometry structure of lattice usually simplify the decoding task, such as sphere decoding [2–3] that utilizes the symmetrical structure of lattice and approaches the performance of maximum likelihood with polynomial decoding complexity. Consequently, lattice becomes a research focus in communication field. Sphere lower bound has been used in Ref. [4] as a benchmark for comparing multi-dimensional constellations in the block-fading channel. The application and approximation of sphere lower bound has been enumerated in Ref. [4]. A new signal model has been introduced in Ref. [5]. Based on the signal model, the exact SER using polar coordinates has been Received date: 21-05-2009 Corresponding author: YU Guang-wei, E-mail: [email protected] DOI: 10.1016/S1005-8885(09)60443-X

computed in 2-demensional (2-D) constellation. However, to the best of the authors’ knowledge, performance analysis of multi-dimensional lattice constellation in phase noise channel has never been addressed. Phase noise is one of the primary factors that limit the performance of many communication systems. Phase noise is mainly caused by oscillator instability [6–7]. Sensitivity of phase noise is also the measure standard for different communication systems. Consequently, sensitivity analysis of phase noise for multi-dimensional lattice constellation is meaningful. Generally speaking, phase noise is usually modeled as Gaussian model or Wiener model [8]. In this contribution, Wiener model is employed to model the phase noise, and transmitted vector sets are modified from infinitely multi-dimensional lattice. Wiener model signifies the correlation between the phase noise per nearby 2-D component of a lattice constellation. In this contribution, SER upper bound of lattice constellation in Gaussian channel is derived based on the same assumption of total number of constellation points and average energy per constellation point as in Ref. [1], and offset vector is considered to minimize the average energy per constellation point. Furthermore, the derivation of the upper bound can be extended to phase noise channel in consideration of SNR degradation, and the obtained upper

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YU Guang-wei, et al. / Phase noise sensitivity analysis of lattice constellation

bound is tight for larger number of lattice constellation points and higher SNR. Therefore, this upper bound can be used as a criterion to analyze the sensitivity of phase noise of different dimension lattice constellation. The remainder of this article is organized as follows. The signal model and basic definitions of lattice are introduced in Sect. 2, where phase noise channel with small phase noise is transformed to Gaussian channel by approximation of one-order maclaurin polynomial. The SER upper bound of lattice constellation in Gaussian channel is derived in Sect. 3. Sect. 4 shows the simulation results and conclusions are presented in Sect. 5.

2 2.1

Definitions of lattice and signal model Definition of lattice

The basic definition of lattice will be reviewed [9–11]. Let {v1, v2 ,..., vm } be a linearly independent set of vectors in n m ⎧ ⎫ (so that m≤n ). The set of points Λ = ⎨ x = ∑ λi vi , λi ∈ Z ⎬ i =1 ⎩ ⎭ is called a lattice of dimension m , and {v1 , v2 ,..., vm } is

called a basis of the lattice. v1n ⎞ ⎛ v11 v12 ⎜ ⎟ v v v2 n ⎟ The matrix A = ⎜ 21 22 is called a generator ⎜ ⎟ ⎜⎜ ⎟⎟ vmn ⎠ ⎝ vm1 vm 2 matrix for the lattice. More concisely, the lattice can be defined by its generator matrix as Λ = { x = λA, λ ∈ Z m } .

The Gram matrix for a lattice is defined as the matrix G = AAT , where (i)T denotes transposition. In following sections, full-rank lattice is considered, that is, m = n , and A is a square matrix. The determinant of a lattice is defined to be the determinant of the Gram matrix G. For full-rank lattices, the square root of the determinant is the fundamental volume of the lattice. Kissing number of a lattice is the number of spheres that touch one sphere in the sphere packing problem [11], i.e., the number of points nearest to one lattice point, where the ‘nearest’ means the minimum Euclidean distance of a lattice. For a lattice, the kissing number is the same for every lattice point. 2.2

Signal model

The transmitted vector is polluted by both phase noise and AWGN, and the signal model is denoted by

37

y T = Hx T + nT

(1)

where H = diag(h) ∈

N×N

jθ1

jθ 2

jθ n

, with h = (e ,e ,......,e ) ∈

is the phase noise diagonal matrix, and y ∈

N

N

,

is the

N-dimensional complex received signal vector, with is the y = ( Re y1 ,Im y1 ,...,Re y N ,Im y N ) ∈ 2 N . x ∈ N N-dimensional complex transmitted signal vector, with x = ( Re x1 ,Im x1 ,...,Re xN ,Im xN ) ∈ 2 N , where Re i and Im i denote the real part and imaginary part, respectively. n ∈ N is the N-dimensional complex Gaussian noise vector. The real transmitted signal vector x is assumed to belong to a 2N-dimensional signal constellation S ∈ 2N , where S is carved from 2N-dimension infinite lattice Λ = {uA,

u∈

2N

}

with full rank generator matrix A ∈

2 N ×2 N

[10].

With the consideration of normalization purpose, fundamental volume is fixed to 1. The simplest labeling operation can be used for lattice constellation, that is, S = {uA + v , u ∈ 2mN } , where m =

{0,1,..., m − 1}

[9], lb m is the number of bits per dimension

and v is the offset vector used to minimize the average transmitted energy of lattice constellation S . Therefore, the rate of such lattice constellation is R = lb m bit per dimension, which is usually referred to as full-rate uncoded transmission and R is the so-called normalized spectral efficiency per dimension. For the transmitted symbol xi , the received symbol yi after the phase noise channel is yi = xi e jθi + ni where xi ,

yi

(2)

and ni are complex symbols, and ni

represents the complex AWGN whose real and imaginary parts both have zero mean and a variance N 0 . Furthermore,

θi is defined as Wiener model [8], and discrete time Wiener phase noise can be expressed as θi +1 = θi + wθ (i )

(3)

where wθ (i ) is stationary Gaussian process with zero mean and a variance σ θ2 , and independent of ni . Meanwhile, θi and ni are independent of xi . Without loss of generality, θ 0 is assumed to be zero and i

θi +1 = ∑ wθ (k )

(4)

k =0

2 Ei = E ⎡ xi ⎤ is the average energy per symbol and SNR ⎣ ⎦ is ΠSNRi = Ei (2 N 0 ) . If θi is far less than 1, e jθi can be

approximated by one-order Maclaurin polynomial

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The Journal of China Universities of Posts and Telecommunications

e jθi ≈ 1 + jθi ; θi

1

Therefore Eq. (2) is transformed to yi = xi + jxiθi + ni = xi + nˆi

(5)

where V (Λ ) is fundamental volume of a lattice, and V (Λ ) = det( A) for full rank lattice [11]. The minimum

(6)

average energy per constellation point can be approximated by continuous integral with consideration of offset vector v , and identical component v is used because of cubic shape

where nˆi = jxiθi + ni is the interference plus noise and the autocorrelation of ni is given by

Rnˆi ( nˆi , nˆk ) = E ( nˆi nˆk * ) = ( iEiσ θ2 + 2 N 0 ) δik

(7)

where δik is the well-known Dirac-delta function. Hence, the variance of the random process nˆi is σ n2ˆi = iEiσ θ2 + 2 N 0 . Correspondingly, the phase noise channel is transformed to AWGN channel with the equivalent SNR ΠSNRi = Ei ( iEiσ θ2 + 2 N 0 ) , and 1

ΠSNRi

=

1

ΠSNRi

+ iσ θ2

2010

(8)

constellation, thus average energy is 2

x 2 E = E⎡ x ⎤ ≈ ∫ dx = [ v , m −1+ v ]2 N m 2 N ⎣ ⎦ 2 2 m −1+ v m −1+ v ( x1 + ... + x2 N ) 2 N ( m − 1) 2 N Ψ (m, v) ... dx1...dx2 N = 2N ∫v ∫v m 3m 2 N (12) where 2 Ψ ( m, v ) = [3v 2 + 3v ( m − 1) + ( m − 1) ] (13) approaches minimum value Ψ (m, v) min =

and Ψ ( m, v)

( m − 1) 4 when v = − ( m − 1) 2. Hence the minimum average 2

(12m ) , and E ≈ 2 Nm

3 Upper bound of symbol error rate

energy E ≈ 2 N ( m − 1)

Because of geometrically uniform performance of lattice, it may simply be written as Pe ( Λ ) = Pe ( Λ | x ) for any

sufficiently large m, Fig. 1 illustrates the details. From Eq. (11), m 2 can be denoted by m 2 = M 1/ NV ( Λ )1/ N = 22 R V (Λ )1/ N (14)

transmitted lattice point x ∈ Λ . For convenience, x is usually taken to be the all zero vector. In Ref. [1], the union bound is applied to provide an upper bound of error probability of lattice constellation Pe ( S )≤Pe (Λ )≤∑ Pr( x → y ) (9) y≠ x

where Pr( x → y ) is the pairwise error probability, the probability that the received vector is closer to y than to x , when x is transmitted. The first inequality takes into account the edge effects of the finite constellation S compared to the infinite lattice Λ [1]. For the AWGN channel, Pe (Λ ) can be obtained by [1,11]

⎛ d E min ⎞ ⎜ ⎟ (10) Pe (Λ )≤ erfc ⎜ 2 ⎟ ⎜ 2N ⎟ 2 0 ⎠ ⎝ Eq. (10) is the case of high SNR and higher SNR means more precise, when SNR → ∞ , the right-hand side of Eq. (10) approaches Pe (Λ ) [9]. In Eq. (10), τ is the kissing number

τ

and d E min is the minimum Euclidean distance of a lattice. The inequality can be modified to a function of Eb N 0 , where the derivation uses the same assumption of total number of constellation points and average energy per constellation point as Ref. [1]. In this article, however, offset vector is considered to minimize the average energy. The total number of points in constellation S can be approximated by m2 N M= (11) V (Λ)

2N +2

2N

due to M = 22 NR and the average energy per bit is 0.5E m 2 22 R V ( Λ )1/ N = = Eb = 2 NR 24 R 24 R

thus

( d Emin 2 )

2

12 for

(15)

2 N 0 in Eq. (10) is transformed to

d Emin 2 2 2 = d Emin = 3R Eb d Emin 8N0 22 R N 0 V (Λ )1/ N 2 N0

(16)

and Eq. (10) is transformed to ⎛ 3R E d 2 ⎞ τ Emin ⎟ Pe (Λ )≤ erfc ⎜ 2 R b (17) 1/ N ⎟ ⎜ 2 ⎝ 2 N 0 V (Λ) ⎠ where Eb is the narrowband average energy per bit and N 0 2 is the narrowband noise power spectral density. For the phase noise channel with approximation of e jθi ≈ 1 + jθi ,

θi

1 , the kissing number, minimum Euclidean distance and

fundamental volume of a lattice keep invariant. Hence, the upper bound of Pe (Λ ) just varies with the variance of interference plus noise, and the evaluation of equivalent SNR becomes necessary. Similar to Eq. (12), the minimum average energy Ei can be approximated by 2

xi 2 dx = Ei = E ⎡ xi ⎤ ≈ ∫ ⎣ ⎦ [ − ( m −1) 2,( m −1) 2]2 m 2

+ x22i ) (m − 1) 4 (18) ∫ −( m −1) 2 ∫ −( m −1) 2 m2 dx2i −1dx2i = 6m2 Ei ≈ m 2 6 for sufficiently large m, namely E = NEi . The ( m −1) 2

( m −1) 2

(x

2 2 i −1

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YU Guang-wei, et al. / Phase noise sensitivity analysis of lattice constellation

39

equivalently average SNR of Eq. (1) can be obtained by Eb Eb Eb = = = N 1 N0 2 N +1 N0 + Eiσ θ2 ∑ k N 0 + REbσ θ 2 2N k =1 1 (19) N0 R + ( N + 1)σ θ2 Eb 2 According to Eqs. (17) and (19), the upper bound Pe (Λ ) of phase noise channel is transformed to

τ

Pe (Λ )≤ erfc 2

2 3R Eb d Emin 2R 2 N 0 V (Λ )1/ N

(20)

2 V (Λ )1/ N is the asymptotic gain of a lattice Λ where d Emin

over integer grid lattice Z n (fundamental gain of Z n is 1 [9]). From the simulation results in Sect. 4, it is noted that the upper bound in Eq. (20) is tight for larger m and higher SNR. Therefore, the upper bound in Eq. (20) can be used as performance criterion to analyze the sensitivity of phase noise of lattice constellation of different dimensions.

4

Performance analysis

In Sect. 3, sufficiently large m is assumed to evaluate average energy per constellation point, hence smaller value of m means larger gap of the upper bound. Meanwhile, higher SNR means tighter upper bound. Two aspects above are illustrated in Fig. 1, where a 2-D integral grid Z2 is used to compare the performance of the upper bound. The lattice constellation carved from Z2 lattice is square M-ary quadrature amplitude modulation (QAM) with adjustment of average energy in 2-D Euclidian space. In Fig. 1 the solid line denotes SER of AWGN channel by simulation, and the dashed

Fig. 1 Upper bound comparison of different m of Z2 lattice

Figs. 2 and 3 illustrate the SER evaluated by the upper bound of lattice constellation of different dimensions, with the same normalized spectral efficiency R = 1 and R = 2 bit per dimension, respectively. In these figures, the solid line denotes SER of AWGN channel (approximated by the upper bound in Eq. (17)), and the dashed line denotes AWGN and phase noise channel (approximated by the upper bound in Eq. (20)), with 2° variance of phase noise in Wiener model. As shown in Figs. 2 and 3, it can be noted that the higher dimension constellations are more sensitive than lower ones in the phase noise channel. In Fig. 3, an almost 1 dB difference of SNR loss can be observed between Λ24 and A2 lattice constellation at SER of 10−4 . Furthermore, larger normalized spectral efficiency means worse robust than smaller one. From Figs. 2 and 3, the SNR difference at 0.75 dB over different R for Λ24 lattice can be observed at SER of 10−4 .

line denotes upper bound in Eq. (10), which is effected by the value of m and SNR , thus sufficiently large m is assumed and higher SNR means tighter upper bound. From Fig. 1, it can be noted that the hypothesis of sufficiently large m is satisfied when m is larger than 4. Even for m = 2 , an SNR gap of upper bound is less than 2dB at a SER of 10−4 . When SNR → ∞ , the right-hand side of Eq. (10) approaches Pe (Λ ) accurately for larger m. Simulation results of Fig. 1 show that the upper bound in Eq. (17) can be used for SER analysis with larger m and SNR. When θi is far smaller than 1, the phase noise channel is equivalent to AWGN channel. Likewise, the upper bound of phase noise channel derived in Eq. (20) can be used for SER analysis with larger m and SNR.

Fig. 2 SER of lattice constellation of different dimension when R=1

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The Journal of China Universities of Posts and Telecommunications

Fig. 3 SER of lattice constellation of different dimensions when R=2

Fig. 4 depicts SER comparison of E8 lattice constellation corrupted by both AWGN and phase noise at different variance of phase noise σ θ2 and normalized spectral

2010

authors analyze the impact of phase noise on lattice constellations, and the phase noise is modeled as Wiener model. The SNR performance degradation introduced by phase noise is evaluated. The symbol error rate upper bound is used as criterion to analyze the sensitivity of phase noise of lattice constellation of different dimensions, with larger number of constellation points and higher SNR. Simulation results show that lattice constellations of higher dimension are more sensitive than those of lower dimensions in the phase noise channel with the same normalized spectral efficiency. It is also shown that with the same dimension of constellation, larger normalized spectral efficiency means more performance loss in phase noise channel. Future work will be focused on finding more new methods of combating the phase noise and relevant receiver construction. Acknowledgements

efficiency R. It is worth pointing out that the constellation points are only perturbed by AWGN when σ θ2 = 0 . In Fig. 4,

of China (2007CB310604, 2009CB320401), the National Natural

the solid line denotes SER of AWGN channel with σ θ2 = 0 ,

Science Foundation of China (60772108, 60702048).

and the dashed line denotes AWGN and phase noise channel, with different phase noise variance of 1° and 2° , respectively. From Fig. 4, similar results can be obtained, that is, larger R means more performance loss.

Fig. 4 SER comparison of E8 lattice constellation of different σ θ2 and R

5

Conclusions

Lattice constellations are accepted as meaningful means for improving spectral efficiency because of the fundamental gain over integral lattices in AWGN channel. In this article, the

This work was supported by the National Basic Research Program

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(Editor: WANG Xu-ying)