無線通信外文翻譯

1、<p>　　附件1：外文資料翻譯譯文</p><p>　　7．mimo：空間多路復(fù)用與信道建模</p><p>　　本書我們已經(jīng)看到多天線在無線通信中的幾種不同應(yīng)用。在第3章中，多天線用于提供分集增益，增益無線鏈路的可靠性，并同時(shí)研究了接受分解和發(fā)射分解，而且，接受天線還能提供功率增益。在第5章中，我們看到了如果發(fā)射機(jī)已知信道，那么多采用多幅發(fā)射天線通過發(fā)射波束成形還可以提供功率

2、增益。在第6章中，多副發(fā)射天線用于生產(chǎn)信道波動(dòng)，滿足機(jī)會(huì)通信技術(shù)的需要，改方案可以解釋為機(jī)會(huì)波束成形，同時(shí)也能夠提供功率增益。</p><p>　　這章以及接下來的幾章將研究一種利用多天線的新方法。我們將會(huì)看到在合適的信道衰落條件下，同時(shí)采用多幅發(fā)射天線和多幅接收天線可以提供用于通信的額外的空間維數(shù)并產(chǎn)生自由度增益，利用這些額外的自由度可以將若干數(shù)據(jù)流在空間上多路復(fù)用至MIMO信道中，從而帶來容量的增加：采用n副

3、發(fā)射天線和接受天線的這類MIMO信道的容量正比于n。</p><p>　　過去一度認(rèn)為在基站采用多幅天線的多址接入系統(tǒng)允許若干個(gè)用戶同時(shí)與基站通信，多幅天線可以實(shí)現(xiàn)不同用戶信號(hào)的空間隔離。20世紀(jì)90年代中期，研究人員發(fā)現(xiàn)采用多幅發(fā)射天線和接收天線的點(diǎn)對(duì)點(diǎn)信道也會(huì)出現(xiàn)類似的效應(yīng)，即使當(dāng)發(fā)射天線相距不遠(yuǎn)時(shí)也是如此。只要散射環(huán)境足夠豐富，使得接受天線能夠?qū)碜圆煌l(fā)射天線的信號(hào)分離開，該結(jié)論就成立。我們已經(jīng)了解到了機(jī)

4、會(huì)通信技術(shù)如何利用信道衰落，本章還會(huì)看到信道衰落對(duì)通信有益的另一例子。</p><p>　　將機(jī)會(huì)通信與MIMO技術(shù)提供的性能增益的本質(zhì)進(jìn)行比較和對(duì)比是非常的有遠(yuǎn)見的。機(jī)會(huì)通信技術(shù)主要提供功率增益，改功率增益在功率受限系統(tǒng)的低信噪比情況下相當(dāng)明顯，但在寬帶受限系統(tǒng)的高信噪比情況下則很不明顯。正如我們將看到的，MIMO技術(shù)不僅能夠提供功率增益，還可以提供自由度增益，因此，MIMO技術(shù)成為在高信噪比情況下大幅度增加容

5、量的主要工具。</p><p>　　MIMO通信是一個(gè)內(nèi)容非常豐富的主題，對(duì)它的研究將覆蓋本書其余章節(jié)。本章集中研究能夠?qū)崿F(xiàn)空間多路復(fù)用的物理環(huán)境的屬性，并闡明如何在MIMO統(tǒng)計(jì)信道模型中簡(jiǎn)明扼要地俘獲這些屬性。具體分析過程如下：首先通過容量分析，明確確定確定性MIMO信道多路復(fù)用容量的關(guān)鍵參數(shù)，之后介紹一系列MIMO物理信道，評(píng)估其空間多路復(fù)用性能；根據(jù)這些實(shí)例的結(jié)果，我們認(rèn)為在角域?qū)IMO信道進(jìn)行建模是非常

6、自然地，同時(shí)討論了基于該方法的統(tǒng)計(jì)模型。本章采用的方法與第2章的方法是平行的，第2章就是從多徑無線信道的幾個(gè)理想實(shí)例著手進(jìn)行分析，從中了解了基本物理現(xiàn)象，進(jìn)而研究更適用于通信方案設(shè)計(jì)與性能分析的統(tǒng)計(jì)衰落模型。實(shí)際上，在特定的信道建模技術(shù)中，我們將會(huì)看到大量的類似方法。</p><p>　　我們貫穿始終的研究焦點(diǎn)是平坦衰落MIMO信道，但也可以直接擴(kuò)展到頻率選擇性MIMO信道，這方面的內(nèi)容會(huì)在習(xí)題中加以介紹。<

7、;/p><p>　　7.1確定性mimo信道的多路復(fù)用容量</p><p>　　包括nt副發(fā)射天線和nt接受天線的窄帶時(shí)不變無線信道可以用一個(gè)nt*nt階確定性矩陣H描述，H具有哪些決定信道空間多路復(fù)用容量的重要屬性呢？我們通過對(duì)信道容量的分析來回答這個(gè)問題。</p><p>　　7.1.1通過奇異值分解分析容量</p><p>　　時(shí)不變信道可

8、以表示為：y = Hx+w_</p><p>　　其中x、y與w分別表示一個(gè)碼元時(shí)刻的發(fā)射信號(hào)、接受信號(hào)與高斯白噪聲（為簡(jiǎn)單起見省略了時(shí)標(biāo)），信道矩陣H為確定性的，并假定在所有時(shí)刻都保持不變，而且對(duì)于發(fā)射機(jī)和接收機(jī)是已知的。這里的hij為發(fā)射天線j到接受天線i的信道增益，對(duì)發(fā)射天線的信號(hào)的總功率約束為P。</p><p>　　這就是矢量高斯信道，將矢量信道分解為一組并行的、相互獨(dú)立的標(biāo)量高

9、斯子信道就可以計(jì)算出該信道的容量。油線性代數(shù)的基本原理可知，每個(gè)線性變換都能夠表示為三種運(yùn)算的組合：旋轉(zhuǎn)運(yùn)算、比例運(yùn)算和另一次旋轉(zhuǎn)運(yùn)算。用矩陣符號(hào)表示，矩陣H具有如下奇異值分解（SVD）：</p><p>　　其中，與為（旋轉(zhuǎn)）酉矩陣1，是對(duì)角元素為非負(fù)實(shí)數(shù)、非對(duì)角線元素為零的矩形矩陣2。對(duì)角線元素為矩陣H的有序奇異值，其中nmin：=min（nt，nr）。因?yàn)?lt;/p><p>　　所以平

10、方奇異值為矩陣HH*的特征值，同時(shí)也是矩陣H*H的特征值。注意，奇異值共有nmin個(gè)，可以將SVD重新寫成為：</p><p>　　SVD分解可以解釋為2個(gè)坐標(biāo)變換：即如果輸入用V的各種定義的坐標(biāo)系統(tǒng)表示，并且輸出用U的各列定義的坐標(biāo)系統(tǒng)表示，那么輸入/輸出關(guān)系是非常簡(jiǎn)單的。</p><p>　　我們已經(jīng)在第5章討論時(shí)不變頻率選擇性信道以及具有完整CSI的時(shí)變衰落信道時(shí)看到了高斯并并行信道

11、的例子。時(shí)不變MIMO信道也是另外一個(gè)例子，這里空間維所起的作用與其他問題中時(shí)間維和頻率維的作用是相同的。大家熟知的容量表達(dá)式為：</p><p>　　其中，P1*,…,Pnmin*為注水功率分配：</p><p>　　通過選擇滿足總功率約束，各對(duì)應(yīng)于信道的一個(gè)特征模式（也稱特征信道）。各非零特征信道能夠支持一路數(shù)據(jù)流，因此，MIMO信道能夠支持多路數(shù)據(jù)流的空間多路復(fù)用?；赟VD的可靠通

12、信結(jié)構(gòu)與第三章介紹的OFDM系統(tǒng)之間存在明顯的相似之處，在這2種情況下，都是利用變換將矩陣信道轉(zhuǎn)換為一組并行的獨(dú)立子信道。在OFDM系統(tǒng)中，矩陣信道由上式中的輪換矩陣C給出，該矩陣由ISI信道和加在輸入碼元上的循環(huán)前綴定義，ISI信道與MIMO信道的重要區(qū)別在于，前者的U、V矩陣不依賴與ISI信道的特定實(shí)現(xiàn)，而后者的U、V矩陣則依賴與MIMO信道的特定實(shí)現(xiàn)。</p><p>　　7.2 MIMO信道的物理建模&l

13、t;/p><p>　　通過本節(jié)的內(nèi)容我們將了解到MIMO信道的空間多路復(fù)用性能對(duì)于物理環(huán)境的依賴程度，為此，我們將研究一系列理想化實(shí)例并分析騎信道矩陣的秩和條件數(shù)，這些確定性實(shí)例同時(shí)表明了下一節(jié)中討論的MIMO信道統(tǒng)計(jì)建模的常規(guī)方法。具體地講，本節(jié)的討論局限于均勻線性天線陣列，即天線一均勻的間隔分布于一條直線上，分析的細(xì)節(jié)取決于特定的天線結(jié)構(gòu)，但是我們要表達(dá)的概念于此無關(guān)。</p><p>　

14、　7.2.1 視距SIMO信道</p><p>　　最簡(jiǎn)單的SIMO信道只有一條視距信道（如下所示），圖中為不存在任何反射體和散射體的自由空間，并且各天線對(duì)之間僅存在直接信號(hào)路徑，天線間隔為，其中為載波波長(zhǎng)，為歸一化接受天線間隔，即歸一化為載波波長(zhǎng)的單位，天線陣列的尺寸比發(fā)射機(jī)與接收機(jī)之間的距離小得多。</p><p>　　發(fā)射天線與第i副接受天線之間信道的連續(xù)時(shí)間沖激響應(yīng)為：</p

15、><p>　　其中，di為發(fā)射天線與第i副接受天線之間的距離，c為光速，a為路徑衰減，假定路徑衰減對(duì)所有天線對(duì)都相同。設(shè)di/c《1/W,其中W為傳輸帶寬，則可得基帶信道增益為：</p><p>　　其中，fc為載波頻率。SIMO信道可以寫成：y=hx+w。其中，x為發(fā)射碼元，w為噪聲，y為接受矢量。有時(shí)將信道增益矢量h=[h1,…h(huán)nt]t稱為信號(hào)方向或由發(fā)射信號(hào)在接收天線陣列上感應(yīng)出的空間

16、特征圖。</p><p>　　由于發(fā)射機(jī)與接收機(jī)之間的距離遠(yuǎn)大于接收天線陣列的尺寸，所以從發(fā)射天線到各接收天線的路徑為1階并行的，并且</p><p>　　其中，d為從發(fā)射天線到第一副接收天線之間的距離，為視距路徑到接收天線陣列的入射角，為在視距方向上接收天線i相對(duì)于接受天線1的位移。并且</p><p>　　通常被稱為相對(duì)于接收天線陣列的方向余弦。因此，空間特征圖

17、h=[h1,…h(huán)nt]t為</p><p>　　即有相對(duì)時(shí)延引起的相位差為的連續(xù)天線處的接收信號(hào)。為了符號(hào)表示方便，定義</p><p>　　為方向余弦上的單位空間特征圖。</p><p>　　最佳接收機(jī)只是將有噪聲接收信號(hào)投影到該信號(hào)方向上，也就是最大比合并或接收波束成形，對(duì)不同的時(shí)延進(jìn)行調(diào)整，從而使天線的接收信號(hào)能夠進(jìn)行相長(zhǎng)合并，得到nt倍的功率增益，所獲取的容

18、量為：</p><p>　　于是，SIMO信道提供了功率增益，但沒有提供自由度增益。</p><p>　　在介紹視距信道時(shí)，有時(shí)將接收天線陣列稱為相位陣列天線。</p><p>　　8. MIMO:容量與多路復(fù)用結(jié)構(gòu)</p><p>　　本章研究MIMO衰落信道的容量，討論能夠從信道中提取所期望的多路復(fù)用增益的收發(fā)信機(jī)結(jié)構(gòu)，特別是集中研究發(fā)射

19、機(jī)未知信道的情況。在快衰落MIMO信道中，可以證明：</p><p>　　1 在高信噪比時(shí)，獨(dú)立同分布瑞利快衰落信道的容量有nminlogSNRb/s/Hz確定，其中nmin為發(fā)射天線數(shù)nt與接收天線數(shù)nr的最小值，這是自由度增益。</p><p>　　2 在低信噪比時(shí)，容量近似為nrSNRlog2eb/s/Hz，這是接收波束成形功率增益。</p><p>　　3

20、在所有信噪比時(shí)，容量與nmin呈線性比例關(guān)系，這是由于功率增益與自由度增益合并造成的。</p><p>　　此外，如果發(fā)射機(jī)也能夠跟蹤信道，那么還存在發(fā)射波束成形增益以及機(jī)會(huì)通信增益。</p><p>　　利用確定性時(shí)不變MIMO信道的容量獲取收發(fā)信機(jī)，其結(jié)構(gòu)比較簡(jiǎn)單：在適當(dāng)?shù)淖鴺?biāo)系統(tǒng)中對(duì)獨(dú)立數(shù)據(jù)流進(jìn)行多路復(fù)用，接收機(jī)將接收矢量變換到另一個(gè)適當(dāng)?shù)淖鴺?biāo)系統(tǒng)中，分別對(duì)不同的數(shù)據(jù)流進(jìn)行譯碼。如果

21、發(fā)射機(jī)未知信道，那么必須事先固定獨(dú)立數(shù)據(jù)流被多路復(fù)用所選取的坐標(biāo)系統(tǒng)。連同聯(lián)合譯碼，這種發(fā)射機(jī)結(jié)構(gòu)實(shí)現(xiàn)了快衰落信道的容量，在文獻(xiàn)中也將改結(jié)構(gòu)稱為V-BLAST結(jié)構(gòu)1。</p><p>　　8．3節(jié)討論比獨(dú)立數(shù)據(jù)流的聯(lián)合最大似然譯碼更簡(jiǎn)單的接收機(jī)結(jié)構(gòu)，雖然可以支持信道全部自由度的接收機(jī)結(jié)構(gòu)有若干種，其中的一種特殊結(jié)構(gòu)是合并使用最小均方誤差估計(jì)與串行干擾消除，即MMSE-SIC接收機(jī)可以獲取容量。</p>

22、<p>　　慢衰落MIMO信道的性能可以通過中斷概率和相應(yīng)的中斷容量來表征。在低信噪比時(shí)，一個(gè)時(shí)刻利用一副發(fā)射天線就可以獲取中斷容量，實(shí)現(xiàn)滿分集增益ntnr和功率增益nr。</p><p>　　另一方面，高信噪比時(shí)的中斷容量還受益于自由度增益，要簡(jiǎn)潔地刻畫其特征更加困難，此問題留到第9章再分析。</p><p>　　雖然采用V-BLAST結(jié)構(gòu)可以實(shí)現(xiàn)快衰落信道的容量，但該結(jié)構(gòu)

23、對(duì)于慢衰落信道則是嚴(yán)格次最優(yōu)的，實(shí)際上，它甚至還沒有實(shí)現(xiàn)MIMO信道期望的滿分集增益。為了說明這一問題，考慮通過發(fā)射天線直接發(fā)送獨(dú)立數(shù)據(jù)流，在這種情況下，各數(shù)據(jù)流的分集僅限于接收分集，為了從信道中獲取滿分集，須對(duì)發(fā)射天線進(jìn)行編碼。將發(fā)射天線編碼與MMSE-SIC結(jié)合起來的一種修正結(jié)構(gòu)D-BLAST2不僅能夠從信道中獲取滿分集，而且其性能還接近于中斷容量。</p><p>　　8.1 V-BLAST結(jié)構(gòu)</p

24、><p>　　首先考慮時(shí)不變信道y[m]=Hx[m]+w[m] m=1,2,…</p><p>　　當(dāng)發(fā)射機(jī)已知信道矩陣H時(shí)，有7.1.1節(jié)可知，最優(yōu)策略是在H*H的特征矢量的方向上發(fā)射獨(dú)立數(shù)據(jù)流，即在由矩陣V定義的坐標(biāo)系統(tǒng)中發(fā)射，該坐標(biāo)系統(tǒng)與信道有關(guān)?？紤]到要處理發(fā)射機(jī)未知信道矩陣時(shí)的衰落信道，歸納出入如下圖所示的結(jié)構(gòu)，圖中nt個(gè)獨(dú)立的數(shù)據(jù)流在由酉矩陣Q確定的任意坐標(biāo)系統(tǒng)中進(jìn)行多路復(fù)用，該

25、酉矩陣未必與信道矩陣H有關(guān)，這就是V-BLAST結(jié)構(gòu)。對(duì)數(shù)據(jù)流進(jìn)行聯(lián)合譯碼，為第k個(gè)數(shù)據(jù)流分配的功率為Pk（使得功率之和P1+…+Pnt等于P，即發(fā)射總功率約束），并利用速率為Rk的容量獲取高斯碼進(jìn)行編碼，總的速率為</p><p><b>　　幾種特殊情況如下：</b></p><p>　　1 如果Q=V并且通過注水分配的方式確定功率，則得到如圖7-2所示的容量獲取

26、結(jié)構(gòu)。</p><p>　　2 如果Q=Int，則獨(dú)立數(shù)據(jù)流被發(fā)送到不同的發(fā)射天線。</p><p>　　下面利用與第5章關(guān)于球體填充的類似論述，討論最高可靠通信速率的上界：</p><p>　　其中，Kx為發(fā)射信號(hào)x的協(xié)方差矩陣，是多路復(fù)用坐標(biāo)系和功率分配的函數(shù)：</p><p>　　考慮在長(zhǎng)度為N的碼元時(shí)間塊內(nèi)的通信，長(zhǎng)度為nrN的接收矢

27、量一高概率位于體積與下式成比例的橢圓體內(nèi)：</p><p>　　該公式是與并行信道相對(duì)應(yīng)的體積公式的直接推廣，并在習(xí)題8-2中加以證明。由于必須考慮到各碼字周圍為非混疊噪聲球空間才能卻?？煽客ㄐ?，所以能夠填充的碼字的最大數(shù)量為比值：</p><p>　　現(xiàn)在就可以得出結(jié)論，可靠通信速率的上界為上式。</p><p>　　采用V-BLAST結(jié)構(gòu)能夠達(dá)到該上界嗎？注意到

28、獨(dú)立數(shù)據(jù)流在V-BLAST結(jié)構(gòu)中多路復(fù)用，是否可能需要對(duì)數(shù)據(jù)流進(jìn)行編碼才能達(dá)到上界式？為了解決這個(gè)問題，考慮MISO信道的特殊情況（nt=1），并在該結(jié)構(gòu)中設(shè)Q=Int，即獨(dú)立數(shù)據(jù)流由各發(fā)射天線發(fā)送。這恰好就是6.1節(jié)介紹的上行鏈路信道，發(fā)射天線類似于用戶，由這一節(jié)的內(nèi)容可知，該上行鏈路信道的總?cè)萘繛椋?lt;/p><p>　　這恰恰是特殊情況下的上界式。因此，數(shù)據(jù)流獨(dú)立的V-BLAST結(jié)構(gòu)完全能夠達(dá)到上界式。在一般

29、情況下，可以將V-BLAST結(jié)構(gòu)與包括nt副接收天線、信道矩陣為HQ的上行鏈路信道進(jìn)行類比，與一副發(fā)射天線的情況相同，上界式就是該上行鏈路信道的總?cè)萘?，因此采用V-BLAST結(jié)構(gòu)可以達(dá)到。這種上行鏈路信道的詳細(xì)研究見第10章。</p><p>　　8．2 快衰落MIMO信道</p><p>　　快衰落MIMO信道為y[m]=H[m]x[m]+w[m] m=1,2,…</p>

30、<p>　　其中，{H[m]}為隨機(jī)衰落過程。為了恰當(dāng)?shù)囟x容量（由隨時(shí)間變化的信道衰落取平均獲得的）的概念，現(xiàn)做出如下（與前幾章相同的）假定，即假定{H[m]}為平穩(wěn)遍歷過程，作為歸一化處理，設(shè)E[|hij|2=1，與前面的研究方法一樣，考慮相干通信：接收機(jī)準(zhǔn)確地跟蹤信道衰落過程。首先研究發(fā)射機(jī)僅具有衰落信道統(tǒng)計(jì)特征的情況，最后研究發(fā)射機(jī)也能夠準(zhǔn)確跟蹤衰落信道的情況（完整CSI），這種情況非常類似于時(shí)不變MIMO信道的情況

31、。</p><p><b>　　附件2：外文原文</b></p><p>　　7. MIMO I: spatial multiplexing</p><p>　　and channel modeling</p><p>　　In this book, we have seen several different uses

32、 of multiple antennas in wireless communication. In Chapter 3, multiple antennas were used to provide diversity gain and increase the reliability of wireless links. Both receive and transmit diversity were considered. Mo

33、reover, receive antennas can also provide a power gain. In Chapter 5, we saw that with channel knowledge at the transmitter, multiple transmit antennas can also provide a power gain via transmit beamforming. In Chapter 6

34、, multiple </p><p>　　In this and the next few chapters, we will study a new way to use multiple antennas. We will see that under suitable channel fading conditions, having both multiple transmit and multiple

35、 receive antennas (i.e., a MIMO channel) provides an additional spatial dimension for communication and yields a degree-of- freedom gain. These additional degrees of freedom can be exploited by spatially multiplexing sev

36、eral data streams onto the MIMO channel, and lead to an increase in the capacity: the capacity</p><p>　　Historically, it has been known for a while that a multiple access system with multiple antennas at the

37、 base-station allows several users to simultaneously</p><p>　　communicate with the base-station. The multiple antennas allow spatial separation of the signals from the different users. It was observed in the

38、 mid 1990s that a similar effect can occur for a point-to-point channel with multiple transmit and receive antennas, i.e., even when the transmit antennas are not geographically far apart. This holds provided that the sc

39、attering environment is rich enough to allow the receive antennas to separate out the signals from the different transmit antennas. We </p><p>　　It is insightful to compare and contrast the nature of the per

40、formance gains offered by opportunistic communication and by MIMO techniques，Opportunistic communication techniques primarily provide a power gain.This power gain is very significant in the low SNR regime where systems a

41、re power-limited but less so in the high SNR regime where they are bandwidthlimited. As we will see, MIMO techniques can provide both a power gain and a degree-of-freedom gain. Thus, MIMO techniques become the primary &l

42、t;/p><p>　　MIMO communication is a rich subject, and its study will span the remaining chapters of the book. The focus of the present chapter is to investigate the properties of the physical environment which e

43、nable spatial multiplexing and show how these properties can be succinctly captured in a statistical MIMO channel model. We proceed as follows. Through a capacity analysis, we first identify key parameters that determine

44、 the multiplexing capability of a deterministic MIMO channel. We then go through </p><p>　　Our focus throughout is on flat fading MIMO channels. The extensions to frequency-selective MIMO channels are straig

45、htforward and are developed in the exercises.</p><p>　　7．1 Multiplexing capability of deterministic MIMO channels</p><p>　　A narrowband time-invariant wireless channel with nt transmit and nr re

46、ceive antennas is described by an nr by nt deterministic matrix H. What are the key properties of H that determine how much spatial multiplexing it can support? We answer this question by looking at the capacity of the c

47、hannel.</p><p>　　7.1.1 Capacity via singular value decomposition</p><p>　　The time-invariant channel is described by</p><p>　　y = Hx+w_ (7.1)</p><p>　　where x,yand wden

48、ote the transmitted signal,</p><p>　　received signal and white Gaussian noise respectively at a symbol time (the time index is dropped for simplicity). The channel matrix H is deterministic and assumed to be

49、 constant at all times and known to both the transmitter and the receiver. Here, hij is the channel gain from transmit antenna j to receive antenna i. There is a total power constraint, P, on the signals from the transmi

50、t antennas.</p><p>　　This is a vector Gaussian channel. The capacity can be computed by decomposing the vector channel into a set of parallel, independent scalar Gaussian sub-channels. From basic linear alge

51、bra, every linear transformation can be represented as a composition of three operations: a rotation operation, a scaling operation, and another rotation operation. In the notation of matrices, the matrix H has a singula

52、r value decomposition (SVD):</p><p>　　Where and are (rotation) unitary matrices1 and is a rectangular matrix whose diagonal elements are non-negative real numbers and whose off-diagonal elements are zero.2 T

53、he diagonal elements are the ordered singular values of the matrix H, where nmin：=min（nt，nr）. Since</p><p>　　the squared singular values _2i are the eigenvalues of the matrix HH* and also of H*H. Note that

54、there are nmin singular values. We can rewrite the SVD as</p><p>　　The SVD decomposition can be interpreted as two coordinate transformations: it says that if the input is expressed in terms of a coordinate

55、system defined by the columns of V and the output is expressed in terms of a coordinate system defined by the columns of U, then the input/output relationship is very simple. Equation (7.8) is a representation of the ori

56、ginal channel (7.1) with the input and output expressed in terms of these new coordinates.</p><p>　　We have already seen examples of Gaussian parallel channels in Chapter 5, when we talked about capacities o

57、f time-invariant frequency-selective channels and about time-varying fading channels with full CSI. The time-invariant MIMO channel is yet another example. Here, the spatial dimension plays the same role as the time and

58、frequency dimensions in those other problems. The capacity is by now familiar:</p><p>　　where P1*,…,Pnmin*are the waterfilling power allocations:</p><p>　　with chosen to satisfy the total power

59、 constraint corresponds to an eigenmode of the channel (also called an eigenchannel). Each eigenchannel can support a data stream; thus, the MIMO channel can support the spatial multiplexing of multiple streams. Figure

60、7.2 pictorially depicts the SVD-based architecture for reliable communication.</p><p>　　There is a clear analogy between this architecture and the OFDM system introduced in Chapter 3. In both cases, a transf

61、ormation is applied to convert a matrix channel into a set of parallel independent sub-channels. In the OFDM setting, the matrix channel is given by the circulant matrix C in (3.139), defined by the ISI channel together

62、with the cyclic prefix added onto the input symbols. The important difference between the ISI channel and the MIMO channel is that, for the former, the U and V m</p><p>　　7.2 Physical modeling of MIMO channe

63、ls</p><p>　　In this section, we would like to gain some insight on how the spatial multiplexing capability of MIMO channels depends on the physical environment. We do so by looking at a sequence of idealized

64、 examples and analyzing the rank and conditioning of their channel matrices. These deterministic examples will also suggest a natural approach to statistical modeling of MIMO channels, which we discuss in Section 7.3. To

65、 be concrete, we restrict ourselves to uniform linear antenna arrays, where the anten</p><p>　　7.2.1Line-of-sight SIMO channel</p><p>　　The simplest SIMO channel has a single line-of-sight (Figu

66、re 7.3(a)). Here, there is only free space without any reflectors or scatterers, and only a direct signal path between each antenna pair. The antenna separation is where is the carrier wavelength and is the normalized re

67、ceive antenna separation, normalized to the unit of the carrier wavelength. The dimension of the antenna array is much smaller than the distance between the transmitter and the receiver.</p><p>　　The continu

68、ous-time impulse response between the transmit antenna and the ith receive antenna is given by</p><p>　　where di is the distance between the transmit antenna and ith receive antenna, c is the speed of light

69、and a is the attenuation of the path, which we assume to be the same for all antenna pairs. Assuming di/c 1/W, where W is the transmission bandwidth, the baseband channel gain is given by (2.34) and (2.27):</p>&

70、lt;p>　　where fc is the carrier frequency. The SIMO channel can be written as y = hx+w where x is the transmitted symbol, w is the noise and y is the received vector. The vector of channel gains h=[h1,…h(huán)nt]t is sometim

71、es called the signal direction or the spatial signature induced on the receive antenna array by the transmitted signal.</p><p>　　Since the distance between the transmitter and the receiver is much larger tha

72、n the size of the receive antenna array, the paths from the transmit antenna to each of the receive antennas are, to a first-order, parallel and</p><p>　　where d is the distance from the transmit antenna to

73、the first receive antenna and _ is the angle of incidence of the line-of-sight onto the receive antenna array. (You are asked to verify this in Exercise 7.1.) The quantity is the displacement of receive antenna i from r

74、eceive antenna1 in the direction of the line-of-sight. The quantity</p><p>　　is often called the directional cosine with respect to the receive antenna array. The spatial signature h=[h1,…h(huán)nt]t is therefore

75、given by </p><p>　　i.e., the signals received at consecutive antennas differ in phase bydue to the relative delay. For notational convenience, we define</p><p>　　as the unit spatial signature in

76、 the directional cosine .</p><p>　　The optimal receiver simply projects the noisy received signal onto the signal direction, i.e., maximal ratio combining or receive beamforming (cf. Section 5.3.1). It adjus

77、ts for the different delays so that the received signals at the antennas can be combined constructively, yielding an nr-fold power gain. The resulting capacity is</p><p>　　The SIMO channel thus provides a po

78、wer gain but no degree-of-freedom gain.</p><p>　　In the context of a line-of-sight channel, the receive antenna array is sometimes called a phased-array antenna.</p><p>　　8. MIMO II: capacity an

79、d multiplexing architectures</p><p>　　In this chapter, we will look at the capacity of MIMO fading channels and discuss transceiver architectures that extract the promised multiplexing gains from the channel

80、. We particularly focus on the scenario when the transmitter does not know the channel realization. In the fast fading MIMO channel, we show the following:</p><p>　　? At high SNR, the capacity of the i.i.d.

81、Rayleigh fast fading channel scales like nminlogSNRb/s/Hz. where nmin is the minimum of the number of transmit antennas nt and the number of receive antennas nr . This is a degree-of-freedom gain.</p><p>　　?

82、 At low SNR, the capacity is approximately nrSNR log2 e bits/s/Hz. This is a receive beamforming power gain.</p><p>　　? At all SNR, the capacity scales linearly with nmin. This is due to a combination of a p

83、ower gain and a degree-of-freedom gain.</p><p>　　Furthermore, there is a transmit beamforming gain together with an opportunistic communication gain if the transmitter can track the channel as well.</p>

84、;<p>　　Over a deterministic time-invariant MIMO channel, the capacity-achieving transceiver architecture is simple (cf. Section 7.1.1): independent data streams are multiplexed in an appropriate coordinate system

85、(cf. Figure 7.2). The receiver transforms the received vector into another appropriate coordinate system to separately decode the different data streams. Without knowledge of the channel at the transmitter the choice of

86、the coordinate system in which the independent data streams are multiplexe</p><p>　　In Section 8.3, we discuss receiver architectures that are simpler than joint ML decoding of the independent streams. While

87、 there are several receiver architectures that can support the full degrees of freedom of the channel, a particular architecture, the MMSE-SIC, which uses a combination of minimum mean square estimation (MMSE) and succes

88、sive interference cancellation (SIC), achieves capacity.</p><p>　　The performance of the slow fading MIMO channel is characterized through the outage probability and the corresponding outage capacity. At low

89、 SNR, the outage capacity can be achieved, to a first order, by using one transmit antenna at a time, achieving a full diversity gain of nt nr and a power gain of nr . The outage capacity at high SNR, on the other hand,

90、benefits from a degree-of-freedom gain as well; this is more difficult to characterize succinctly and its analysis is relegated until Chapt</p><p>　　Although it achieves the capacity of the fast fading chann

91、el, the V-BLAST architecture is strictly suboptimal for the slow fading channel. In fact, it does not even achieve the full diversity gain promised by the MIMO channel. To see this, consider transmitting independent data

92、 streams directly over the transmit antennas. In this case, the diversity of each data stream is limited to just the receive diversity. To extract the full diversity from the channel, one needs to code across the transmi

93、t </p><p>　　8.1 The V-BLAST architecture </p><p>　　We start with the time-invariant channel (cf. (7.1))</p><p>　　y[m]=Hx[m]+w[m] m=1,2,…</p><p>　　When the channel matr

94、ix H is known to the transmitter, we have seen in</p><p>　　Section 7.1.1 that the optimal strategy is to transmit independent streams in the directions of the eigenvectors of H*H, i.e., in the coordinate sys

95、tem defined by the matrix V, where H is the singular value decomposition of H. This coordinate system is channel-dependent. With an eye towards dealing with the case of fading channels where the channel matrix is unknown

96、 to the transmitter, we generalize this to the architecture in Figure 8.1, where the independent data streams, nt of them, are multi</p><p>　　As special cases:</p><p>　　? If Q = V and the powers

97、 are given by the waterfilling allocations, then we have the capacity-achieving architecture in Figure 7.2.</p><p>　　? If Q = Inr , then independent data streams are sent on the different transmit antennas.&

98、lt;/p><p>　　Using a sphere-packing argument analogous to the ones used in Chapter 5, we will argue an upper bound on the highest reliable rate of communication:</p><p>　　Here Kx is the covariance m

99、atrix of the transmitted signal x and is a function of the multiplexing coordinate system and the power allocations:</p><p>　　Considering communication over a block of time symbols of length N, the received

100、vector, of length nrN, lies with high probability in an ellipsoid of volume proportional to</p><p>　　This formula is a direct generalization of the corresponding volume formula (5.50) for the parallel channe

101、l, and is justified in Exercise 8.2. Since we have to allow for non-overlapping noise spheres around each codeword to ensure reliable communication, the maximum number of codewords that can be packed is the ratio</p&g