李巨虎　聶娟

(1.北京林業(yè)大學(xué) 信息學(xué)院, 北京 100083; 2.北京農(nóng)學(xué)院 計(jì)算機(jī)與信息工程學(xué)院, 北京 102206)

載波頻偏對(duì)OFDM AF中繼系統(tǒng)性能的影響*

李巨虎1聶娟2

(1.北京林業(yè)大學(xué) 信息學(xué)院, 北京 100083; 2.北京農(nóng)學(xué)院 計(jì)算機(jī)與信息工程學(xué)院, 北京 102206)

摘要：研究了存在載波頻偏的正交頻分復(fù)用(OFDM)放大轉(zhuǎn)發(fā)(AF)中繼系統(tǒng)的中斷概率和誤碼率.首先給出了存在載波頻偏時(shí)系統(tǒng)信噪比及其概率分布,然后推導(dǎo)出瑞利信道中存在載波頻偏時(shí)系統(tǒng)中斷概率和誤碼率的閉合表達(dá)式.為了更加直觀,文中還給出了大信噪比時(shí)系統(tǒng)中斷概率和誤碼率的錯(cuò)誤下界,并通過數(shù)值仿真來驗(yàn)證理論分析的正確性.理論分析和仿真結(jié)果表明:存在載波頻偏時(shí),即使在大信噪比情形下系統(tǒng)的中斷概率和誤碼率并不趨于0,而是趨于錯(cuò)誤下界;OFDM AF系統(tǒng)對(duì)載波頻偏很敏感,隨著信噪比的增大,系統(tǒng)性能隨著載波頻偏的增大迅速變差.

關(guān)鍵詞：放大轉(zhuǎn)發(fā);正交頻分復(fù)用;載波頻偏;中斷概率;誤碼率

中繼系統(tǒng)能夠擴(kuò)展傳輸距離及對(duì)抗信道衰落,成為無線通信的研究熱點(diǎn),在近幾年得到了飛快發(fā)展.中繼系統(tǒng)的傳輸機(jī)制主要包括放大轉(zhuǎn)發(fā)(AF)模式和譯碼轉(zhuǎn)發(fā)(DF)模式.采用AF模式時(shí)中繼節(jié)點(diǎn)不需要對(duì)接收數(shù)據(jù)進(jìn)行檢測(cè)和譯碼,而是簡單的放大轉(zhuǎn)發(fā),這樣能夠減小數(shù)據(jù)的傳輸時(shí)延而且實(shí)現(xiàn)方式簡單,因而得到了廣泛的應(yīng)用.正交頻分復(fù)用(OFDM)因能夠?qū)剐诺赖念l率選擇性衰落并消除符號(hào)間的干擾而得到了廣泛的應(yīng)用.OFDM AF中繼系統(tǒng)綜合了AF和OFDM兩種技術(shù)的優(yōu)點(diǎn),成為新一代無線通信的標(biāo)準(zhǔn).很多文獻(xiàn)對(duì)OFDM AF中繼系統(tǒng)進(jìn)行了研究.文獻(xiàn)[1]研究了瑞利信道中OFDM AF中繼系統(tǒng)的性能,給出了中繼節(jié)點(diǎn)采用固定放大增益時(shí)系統(tǒng)的誤碼率.文獻(xiàn)[2- 4]對(duì)基于子載波配對(duì)的OFDM AF中繼系統(tǒng)做了深入研究,首先研究了瑞利信道中基于子載波配對(duì)的OFDM AF中繼系統(tǒng)的中斷概率、中斷容量和系統(tǒng)誤碼率,然后研究了Nakagami-m信道中基于子載波配對(duì)的OFDM AF系統(tǒng)的誤碼率和信道容量.

與OFDM相同,OFDM AF中繼系統(tǒng)對(duì)載波頻率偏移(CFO)非常敏感,載波頻率偏移會(huì)造成子載波間干擾(ICI),導(dǎo)致系統(tǒng)性能下降.文獻(xiàn)[5- 6]針對(duì)載波頻偏對(duì)OFDM系統(tǒng)性能的影響進(jìn)行了研究,但載波頻偏對(duì)OFDM AF中繼系統(tǒng)性能影響的相關(guān)研究不多.文獻(xiàn)[7]研究了載波頻偏對(duì)OFDM AF中繼系統(tǒng)信噪比的影響,給出了存在載波頻偏時(shí)系統(tǒng)信噪比的表達(dá)式及系統(tǒng)誤碼率的仿真結(jié)果,但對(duì)系統(tǒng)誤碼率沒有做理論分析.文獻(xiàn)[8]研究了存在共信道干擾(CCI)和信道估計(jì)誤差時(shí)AF中繼系統(tǒng)的中斷概率和誤碼率,給出了其閉合表達(dá)式,但共信道干擾和載波頻偏引起的干擾有很大的區(qū)別.文獻(xiàn)[9]研究了由載波頻偏引起的子載波干擾,給出了消除子載波干擾和符號(hào)間干擾的方法.文獻(xiàn)[10]首先利用前導(dǎo)碼估計(jì)OFDM AF系統(tǒng)的載波頻偏,然后根據(jù)估計(jì)值消除載波頻偏.文獻(xiàn)[11]通過差分調(diào)制(DM)方法來消除載波頻偏和信道估計(jì)誤差對(duì)系統(tǒng)性能的影響.

OFDM AF系統(tǒng)存在源節(jié)點(diǎn)和中繼節(jié)點(diǎn)間、中繼節(jié)點(diǎn)和目的節(jié)點(diǎn)間的載波頻偏,其對(duì)OFDM AF系統(tǒng)性能的影響比OFDM系統(tǒng)復(fù)雜得多,不可能借用OFDM系統(tǒng)的結(jié)論.另外,現(xiàn)有研究主要集中在消除載波頻偏,沒有定量分析載波頻偏對(duì)系統(tǒng)誤碼率和中斷概率的影響.為此,文中研究了瑞利信道中載波頻偏對(duì)OFDM AF中繼系統(tǒng)性能的影響.首先給出了存在載波頻偏時(shí)的系統(tǒng)模型,然后證明載波頻偏引起的子載波干擾的高斯性,推導(dǎo)出系統(tǒng)信噪比表達(dá)式及其概率分布,進(jìn)而根據(jù)信噪比的概率分布推導(dǎo)出OFDM AF中繼系統(tǒng)存在載波頻偏時(shí)的中斷概率表達(dá)式及其下界,并推導(dǎo)了M元正交幅度調(diào)制(MQAM)時(shí)的系統(tǒng)誤碼率及其下界,最后對(duì)理論分析值和仿真結(jié)果進(jìn)行了比較.

1存在載波頻偏時(shí)的系統(tǒng)模型

OFDM AF系統(tǒng)采用時(shí)分雙工模式,其傳輸模型如圖1所示.源節(jié)點(diǎn)將數(shù)據(jù)調(diào)制成OFDM信號(hào)發(fā)送到中繼節(jié)點(diǎn),中繼節(jié)點(diǎn)放大并轉(zhuǎn)發(fā)信號(hào),目的節(jié)點(diǎn)接收中繼節(jié)點(diǎn)的信號(hào)并解調(diào)恢復(fù)數(shù)據(jù).兩跳傳輸發(fā)生在不同時(shí)刻、不同節(jié)點(diǎn)之間,并假定每跳信號(hào)都經(jīng)歷瑞利衰落和加性高斯白噪聲(AWGN),這樣源節(jié)點(diǎn)-中繼節(jié)點(diǎn)(S-R)信道和中繼節(jié)點(diǎn)-目的節(jié)點(diǎn)(R- D)信道是相互獨(dú)立的,互不相干的.

圖1　OFDM AF中繼系統(tǒng)結(jié)構(gòu)圖Fig.1　Structure diagram of OFDM AF relay system

在時(shí)間域,存在載波頻率偏移時(shí)第1跳的基帶信號(hào)可以表示為[12]

yR[n]=cSR[n](hSR[n]?x[n])+wR[n]

(1)

YR[i]=CSR[i]?(HSR[i]X[i])+WR[i]=

CSR[0]HSR[i]X[i]+IR[i]+WR[i]

(2)

式中:CSR[i]為cSR[n]的DFT變換,其表達(dá)式見式(7);HSR[i]為信道系數(shù)hSR[n]的DFT變換;WR[i]為wR[n]的DFT變換;IR[i]為由載波頻偏引起的子載波間干擾(ICI),可表示為

(3)

在時(shí)間域,存在載波頻率偏移時(shí)第2跳的基帶信號(hào)可以表示為

yD[n]=GcRD[n](hRD[n]?yR[n])+wD[n]

(4)

(5)

式中,ES為源節(jié)點(diǎn)發(fā)送功率,ER為中繼節(jié)點(diǎn)發(fā)送功率.中繼節(jié)點(diǎn)接收到數(shù)據(jù)時(shí),把載波數(shù)據(jù)下變頻到基帶進(jìn)行子載波映射.常見的子載波映射有動(dòng)態(tài)子載波映射和隨機(jī)子載波映射,都是將第1跳的某子載波映射到第2跳的另一子載波.假定將第1跳的第i個(gè)子載波映射到第2跳的第k個(gè)子載波.在式(4)中,對(duì)第k個(gè)子載波做DFT,可得到第2跳的頻譜表達(dá)式,然后把式(2)代入第2跳的頻譜表達(dá)式,可得

YD[k]=GCRD[k]?(HRD[k]YR[i])+WD[k]=

GCSR[0]CRD[0]HRD[k]HSR[i]X[i]+

GCRD[0]HRD[k]WR[i]+GCSR[0]·

GCSR[0]CRD[0]HRD[k]HSR[i]X[i]+

GCRD[0]HRD[k]WR[k]+I1[k]+I2[k]+

I3[k]+WD[k]

(6)

式中:Hij[k]、Wi[k]分別是hi,j[n]和wi[n]的DFT變換;i,j∈{S,R,D};Cij[k]是cij[n]的DFT變換,可表示為

(7)

從式(6)的最后一個(gè)等式可知,其第1項(xiàng)為目標(biāo)信號(hào),但增加了乘性干擾GCSR[0]CRD[0]HRD[k]·HSR[i],最后3項(xiàng)I1[k]、I2[k]和I3[k]是由載波頻偏引起的子載波間干擾,可表示為

(8)

2子載波干擾的高斯性

在OFDMAF系統(tǒng)中,通常采用M-QAM調(diào)制,因此X[i]是零均值的.因?yàn)镠SR[i]和X[i]是相互獨(dú)立的,故式(3)中IR[i]的均值為0,方差計(jì)算式為

(9)

(10)

很多文獻(xiàn)把子載波干擾看作高斯噪聲處理,但從未證明.文中根據(jù)Lyapunov’s中心極限定理[15]證明子載波干擾的高斯性.從式(8)可知,I3[k]是高斯白噪聲的加權(quán)和,因此I3[k]是高斯噪聲,只需證明I1[k]和I2[k]的高斯性即可.要證明I2[k]的高斯性,只需證明IR[i]是高斯噪聲即可.為了證明IR[i]的高斯性,首先寫出Lyapunov’s中心極限定理.

(11)

以下根據(jù)引理1來證明IR[i]的高斯性.

證明根據(jù)式(3),IR[i]可重新寫為

(12)

(13)

根據(jù)式(11)和(13),可得

(14)

證畢.

根據(jù)以上證明,可以得到IR[i]是高斯隨機(jī)變量,從而得到I2[k]為高斯隨機(jī)變量.與證明IR[i]類似,也可以證明I1[k]為高斯隨機(jī)變量.

3端到端信噪比的概率分布

(15)

(16)

當(dāng)系統(tǒng)采用瞬時(shí)功率縮放時(shí),根據(jù)式(10)和式(5),式(16)可化簡為

(17)

引理2[17]X1和X2是相互獨(dú)立的隨機(jī)變量,分別服從參數(shù)為β1和β2的指數(shù)分布,則隨機(jī)變量X=X1X2/(X1+X2+b)的概率分布函數(shù)為

FX(x)=1-2exp(-(β1+β2)x)·

(18)

式中,b為常數(shù),K1(·)為一階修正的貝塞爾函數(shù),其定義見文獻(xiàn)[18]中式(9.6.22).根據(jù)引理2,則信噪比γend[k]的概率分布函數(shù)可表示為

(19)

式(19)對(duì)γ求導(dǎo),利用貝塞爾函數(shù)求導(dǎo)等式[19],則信噪比γend[k]的概率密度函數(shù)可表示為

(20)

4性能分析

4.1　中斷概率

當(dāng)系統(tǒng)鏈路容量不能滿足所要求的用戶速率時(shí),就會(huì)產(chǎn)生中斷.對(duì)于信噪比來說,當(dāng)信噪比低于門限值γth后就會(huì)發(fā)生中斷.根據(jù)式(19)中信噪比的概率分布和中斷概率定義,系統(tǒng)的中斷概率表示為

Pout=P[R≤Rth]=P[γend[k]≤γth]=

(21)

(22)

(23)

從式(23)可以看出,系統(tǒng)存在載波頻偏且大信噪比時(shí),系統(tǒng)的中斷概率并不趨于0,而是趨于錯(cuò)誤下界.另外可以看出,源節(jié)點(diǎn)和中繼節(jié)點(diǎn)之間的載波頻偏、中繼節(jié)點(diǎn)與目的節(jié)點(diǎn)之間的載波頻偏對(duì)中斷概率錯(cuò)誤下界的貢獻(xiàn)是不等同的.

4.2　平均差錯(cuò)率

(24)

常規(guī)的方法是根據(jù)γend[k]的概率分布求得系統(tǒng)誤碼率.這種方法物理概念清楚,計(jì)算復(fù)雜.另一種方法是計(jì)算信噪比γend[k]的矩生成函數(shù)(MGF)Mγend[k](s)=E{exp(-sγend[k])},根據(jù)誤碼率和矩生成函數(shù)的關(guān)系來計(jì)算系統(tǒng)差錯(cuò)率.文中采用矩生成函數(shù)方法.利用式(24),則系統(tǒng)的矩生成函數(shù)可以表示為

(25)

式中,ζ是歐拉常數(shù).當(dāng)系統(tǒng)采用M-QAM調(diào)制時(shí),根據(jù)差錯(cuò)率與矩生成函數(shù)的關(guān)系[20]:

(26)

(27)

式(27)對(duì)γ求導(dǎo),可以得到系統(tǒng)信噪比的概率密度函數(shù)

(28)

(29)

把式(29)代入式(26)可以得到大信噪比時(shí)系統(tǒng)差錯(cuò)率的下界表達(dá)式:

(30)

5仿真及結(jié)果分析

圖2　不同載波頻偏時(shí)OFDM AF中繼系統(tǒng)的中斷概率隨信噪比的變化Fig.2　Changes of outage probability of OFDM AF relay system with different CFOs

圖3　不同鏈路速率時(shí)中斷概率的錯(cuò)誤下界Fig.3　Error floor of outage probability with different link rates

載波頻率偏移不同時(shí)OFDMAF系統(tǒng)的中斷概率隨信噪比的變化如圖2所示,圖中分別給出了理論分析、數(shù)值仿真和中斷概率的錯(cuò)誤下界.當(dāng)OFDMAF中繼系統(tǒng)的鏈路速率不同時(shí),中斷概率的錯(cuò)誤下界隨載波頻率偏移的變化如圖3所示.

OFDMAF系統(tǒng)采用4QAM、16QAM調(diào)制且不同的載波頻偏時(shí),系統(tǒng)誤碼率隨信噪比的變化如圖4所示.OFDMAF中繼系統(tǒng)調(diào)制方式不同時(shí)系統(tǒng)誤碼率的錯(cuò)誤下界隨載波頻率偏移的變化如圖5所示.

圖4　不同載波頻偏和調(diào)制時(shí)OFDM AF中繼系統(tǒng)的誤碼率隨信噪比的變化Fig.4　Changes of BER of OFDM AF system with different CFOs and different modulations

圖5　不同載波頻偏時(shí)系統(tǒng)誤碼率的錯(cuò)誤下界Fig.5　Error floor of system BER with different CFOs

6結(jié)論

文中研究了載波頻偏對(duì)OFDMAF中繼系統(tǒng)性能的影響.根據(jù)存在載波頻偏時(shí)的系統(tǒng)模型,推導(dǎo)出系統(tǒng)端到端信噪比的表達(dá)式及其概率分布,又推導(dǎo)出系統(tǒng)的中斷概率和誤碼率的表達(dá)式,并進(jìn)行了數(shù)值仿真.文中還對(duì)存在載波頻偏時(shí)系統(tǒng)中斷概率及其錯(cuò)誤下界、誤碼率及其錯(cuò)誤下界做了深入的研究.結(jié)果表明:在大信噪比時(shí),中斷概率和誤碼率不趨于0,而是趨于其錯(cuò)誤下界;OFDMAF中繼系統(tǒng)對(duì)載波頻偏很敏感,隨著信噪比的增大,系統(tǒng)性能隨著載波頻偏的增大而迅速變差.

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Effect of Carrier Frequency Offsets on Performance of OFDM AF Relay System

LiJu-hu1NieJuan2

(1. School of Information, Beijing Forestry University, Beijing 100083, China; 2. College of Computer and Information

Engineering, Beijing University of Agriculture, Beijing 102206, China)

Abstract:This paper deals with the bit error rate (BER) and outage probability of a dual hop OFDM AF relay system in the presence of carrier frequency offsets (CFO). First, the signal-to-noise ratio (SNR) of the system and its probability distribution are put forward in the presence of CFO, and then the close-form expressions of the BER and outage probability of the system are derived with CFO in Rayleigh fading channels. In order to provide more insights, the error floors of the BER and outage probability of the system are also obtained. Moreover, a theoretical analysis is proved to be correct by simulation. The theoretical analysis and simulation results indicate that, in the presence of CFO, the BER and outage probability of the system tend to be the error floor but not zero even at a high SNR, and that the performance of OFDM AF relay system is very sensitive to the CFO, and the system performance gets worse rapidly with the increases of the CFO and the SNR.

Key words:amplify-and-forward; orthogonal frequency division multiplexing; carrier frequency offsets; outage probability; bit error rate

中圖分類號(hào)：TP79

doi：10.3969/j.issn.1000-565X.2015.09.007

作者簡介：李巨虎(1978-),男,博士,講師,主要從事物聯(lián)網(wǎng)、無線通信等研究.E-mail: lijuhu@bjfu.edu.cn

*基金項(xiàng)目：北京林業(yè)大學(xué)中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金資助項(xiàng)目(TD2014- 01);北京市教委科技面上資助項(xiàng)目(KM20151002)

收稿日期：2015-03-16

文章編號(hào)：1000-565X(2015)09-0039-08

Foundation item： Supported by the General Program of Beijing Committee of Education Science and Technology(KM20151002)