外文翻譯無線通信基礎(Fundamentals_of_wireless_communications_by_David_Tse)頂淘寶(小鄧充值)講解

1、附件1：外文資料翻譯譯文7mimo：空間多路復用與信道建模 本書我們已經看到多天線在無線通信中的幾種不同應用。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增益。在第6章中，多副發(fā)射天線用于生產信道波動，滿足機會通信技術的需要，改方案可以解釋為機會波束成形，同時也能夠提供功率增益。這章以及接下來的幾章將研究一種利用多天線的新方法。我們將會看到在合適的信道衰落條件下，同時采用多幅發(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

2、維數(shù)并產生自由度增益，利用這些額外的自由度可以將若干數(shù)據(jù)流在空間上多路復用至MIMO信道中，從而帶來容量的增加：采用n副發(fā)射天線和接受天線的這類MIMO信道的容量正比于n。過去一度認為在基站采用多幅天線的多址接入系統(tǒng)允許若干個用戶同時與基站通信，多幅天線可以實現(xiàn)不同用戶信號的空間隔離。20世紀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對點信道也會出現(xiàn)類似的效應，即使當發(fā)射天線相距不遠時也是如此。只要散射環(huán)境足夠豐富，使得接受天線能夠將來自不同發(fā)射天線的信號分離開，該結論就成立。我們已經了解到了機會通信技術如何利用信道衰落，本章還會看到信道衰落對通信有益的另一例子。將機會通信與MI

3、MO技術提供的性能增益的本質進行比較和對比是非常的有遠見的。機會通信技術主要提供功率增益，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技術成為在高信噪比情況下大幅度增加容量的主要工具。MIMO通信是一個內容非常豐富的主題，對它的研究將覆蓋本書其余章節(jié)。本章集中研究能夠實現(xiàn)空間多路復用的物理環(huán)境的屬性，并闡明如何在MIMO統(tǒng)計信道模型中簡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信道多路復用容量的關鍵參數(shù)，之后介紹一系列M

4、IMO物理信道，評估其空間多路復用性能；根據(jù)這些實例的結果，我們認為在角域對MIMO信道進行建模是非常自然地，同時討論了基于該方法的統(tǒng)計模型。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理想實例著手進行分析，從中了解了基本物理現(xiàn)象，進而研究更適用于通信方案設計與性能分析的統(tǒng)計衰落模型。實際上，在特定的信道建模技術中，我們將會看到大量的類似方法。我們貫穿始終的研究焦點是平坦衰落MIMO信道，但也可以直接擴展到頻率選擇性MIMO信道，這方面的內容會在習題中加以介紹。7.1確定性mimo信道的多路復用容量包括nt副發(fā)射天線和nt接受天線的窄帶時不變無線信道可以用一個nt*nt

5、階確定性矩陣H描述，H具有哪些決定信道空間多路復用容量的重要屬性呢？我們通過對信道容量的分析來回答這個問題。7.1.1通過奇異值分解分析容量時不變信道可以表示為：y = Hx+w_其中x、y與w分別表示一個碼元時刻的發(fā)射信號、接受信號與高斯白噪聲（為簡單起見省略了時標），信道矩陣H為確定性的，并假定在所有時刻都保持不變，而且對于發(fā)射機和接收機是已知的。這里的hij為發(fā)射天線j到接受天線i的信道增益，對發(fā)射天線的信號的總功率約束為P。這就是矢量高斯信道，將矢量信道分解為一組并行的、相互獨立的標量高斯子信道就可以計算出該信道的容量。油線性代數(shù)的基本原理可知，每個線性變換都能夠表示為三種運算的組合：

6、旋轉運算、比例運算和另一次旋轉運算。用矩陣符號表示，矩陣H具有如下奇異值分解（SVD）： 其中，與為（旋轉）酉矩陣1，是對角元素為非負實數(shù)、非對角線元素為零的矩形矩陣2。對角線元素為矩陣H的有序奇異值，其中nmin：=min（nt，nr）。因為 所以平方奇異值為矩陣HH*的特征值，同時也是矩陣H*H的特征值。注意，奇異值共有nmin個，可以將SVD重新寫成為： SVD分解可以解釋為2個坐標變換：即如果輸入用V的各種定義的坐標系統(tǒng)表示，并且輸出用U的各列定義的坐標系統(tǒng)表示，那么輸入/輸出關系是非常簡單的。我們已經在第5章討論時不變頻率選擇性信道以及具有完整CSI的時變衰落信道時看到了高斯并并行信

7、道的例子。時不變MIMO信道也是另外一個例子，這里空間維所起的作用與其他問題中時間維和頻率維的作用是相同的。大家熟知的容量表達式為： 其中，P1*,Pnmin*為注水功率分配： 通過選擇滿足總功率約束，各對應于信道的一個特征模式（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)流的空間多路復用。基于SVD的可靠通信結構與第三章介紹的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換為一組并行的獨立子信道。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

8、重要區(qū)別在于，前者的U、V矩陣不依賴與ISI信道的特定實現(xiàn)，而后者的U、V矩陣則依賴與MIMO信道的特定實現(xiàn)。7.2 MIMO信道的物理建模 通過本節(jié)的內容我們將了解到MIMO信道的空間多路復用性能對于物理環(huán)境的依賴程度，為此，我們將研究一系列理想化實例并分析騎信道矩陣的秩和條件數(shù)，這些確定性實例同時表明了下一節(jié)中討論的MIMO信道統(tǒng)計建模的常規(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，分析的細節(jié)取決于特定的天線結構，但是我們要表達的概念于此無關。7.2.1 視距SIMO信道 最簡單的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

9、散射體的自由空間，并且各天線對之間僅存在直接信號路徑，天線間隔為，其中為載波波長，為歸一化接受天線間隔，即歸一化為載波波長的單位，天線陣列的尺寸比發(fā)射機與接收機之間的距離小得多。 發(fā)射天線與第i副接受天線之間信道的連續(xù)時間沖激響應為：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，假定路徑衰減對所有天線對都相同。設di/c1/W,其中W為傳輸帶寬，則可得基帶信道增益為： 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。有時將信道增益矢量h=h1,hntt稱為信號方向或由發(fā)射信號在接收天線陣列上感應出的空間特征圖。由于

10、發(fā)射機與接收機之間的距離遠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 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相對于接受天線1的位移。并且通常被稱為相對于接收天線陣列的方向余弦。因此，空間特征圖h=h1,hntt為 即有相對時延引起的相位差為的連續(xù)天線處的接收信號。為了符號表示方便，定義 為方向余弦上的單位空間特征圖。最佳接收機只是將有噪聲接收信號投影到該信號方向上，也就是最大比合并或接收波束成形，對不同的時延進行調整，從而使天線的接收信號能夠進行相長合并，得到nt倍的功率增益，所獲取的容量為：

11、于是，SIMO信道提供了功率增益，但沒有提供自由度增益。在介紹視距信道時，有時將接收天線陣列稱為相位陣列天線。8. MIMO:容量與多路復用結構本章研究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用增益的收發(fā)信機結構，特別是集中研究發(fā)射機未知信道的情況。在快衰落MIMO信道中，可以證明：1 在高信噪比時，獨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。2 在低信噪比時，容量近似為nrSNRlog2eb/s/Hz，這是接收波束成形功率增益。3 在所有信噪比時，容量與nmin呈線性比例關系，這

12、是由于功率增益與自由度增益合并造成的。此外，如果發(fā)射機也能夠跟蹤信道，那么還存在發(fā)射波束成形增益以及機會通信增益。利用確定性時不變MIMO信道的容量獲取收發(fā)信機，其結構比較簡單：在適當?shù)淖鴺讼到y(tǒng)中對獨立數(shù)據(jù)流進行多路復用，接收機將接收矢量變換到另一個適當?shù)淖鴺讼到y(tǒng)中，分別對不同的數(shù)據(jù)流進行譯碼。如果發(fā)射機未知信道，那么必須事先固定獨立數(shù)據(jù)流被多路復用所選取的坐標系統(tǒng)。連同聯(lián)合譯碼，這種發(fā)射機結構實現(xiàn)了快衰落信道的容量，在文獻中也將改結構稱為V-BLAST結構1。83節(jié)討論比獨立數(shù)據(jù)流的聯(lián)合最大似然譯碼更簡單的接收機結構，雖然可以支持信道全部自由度的接收機結構有若干種，其中的一種特殊結構是合并使

13、用最小均方誤差估計與串行干擾消除，即MMSE-SIC接收機可以獲取容量。慢衰落MIMO信道的性能可以通過中斷概率和相應的中斷容量來表征。在低信噪比時，一個時刻利用一副發(fā)射天線就可以獲取中斷容量，實現(xiàn)滿分集增益ntnr和功率增益nr。另一方面，高信噪比時的中斷容量還受益于自由度增益，要簡潔地刻畫其特征更加困難，此問題留到第9章再分析。雖然采用V-BLAST結構可以實現(xiàn)快衰落信道的容量，但該結構對于慢衰落信道則是嚴格次最優(yōu)的，實際上，它甚至還沒有實現(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

14、，須對發(fā)射天線進行編碼。將發(fā)射天線編碼與MMSE-SIC結合起來的一種修正結構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。8.1 V-BLAST結構 首先考慮時不變信道ym=Hxm+wm m=1,2,當發(fā)射機已知信道矩陣H時，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)射獨立數(shù)據(jù)流，即在由矩陣V定義的坐標系統(tǒng)中發(fā)射，該坐標系統(tǒng)與信道有關?？紤]到要處理發(fā)射機未知信道矩陣時的衰落信道，歸納出入如下圖所示的結構，圖中nt個獨立的數(shù)據(jù)流在由酉矩陣Q確定的任意坐標系統(tǒng)中進行多路復用，該酉矩陣未必與信道矩陣H有關，這就是V-BLAST結構。對數(shù)據(jù)流進行聯(lián)合譯碼，為

15、第k個數(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行編碼，總的速率為幾種特殊情況如下：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所示的容量獲取結構。2 如果Q=Int，則獨立數(shù)據(jù)流被發(fā)送到不同的發(fā)射天線。下面利用與第5章關于球體填充的類似論述，討論最高可靠通信速率的上界： 其中，Kx為發(fā)射信號x的協(xié)方差矩陣，是多路復用坐標系和功率分配的函數(shù)： 考慮在長度為N的碼元時間塊內的通信，長度為nrN的接收矢量一高概率位于體積與下式成比例的橢圓體內： 該公式是與并行信道相對應的體積公式的直接推廣，并在習題8-2中加以證

16、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牛阅軌蛱畛涞拇a字的最大數(shù)量為比值： 現(xiàn)在就可以得出結論，可靠通信速率的上界為上式。采用V-BLAST結構能夠達到該上界嗎？注意到獨立數(shù)據(jù)流在V-BLAST結構中多路復用，是否可能需要對數(shù)據(jù)流進行編碼才能達到上界式？為了解決這個問題，考慮MISO信道的特殊情況（nt=1），并在該結構中設Q=Int，即獨立數(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)的內容可知，該上行鏈路信道的總容量為： 這恰恰是特殊情況下的上界式。因此，數(shù)據(jù)流獨立的V-BLAST結構完全能夠達到上界式。在一般情況下，可

17、以將V-BLAST結構與包括nt副接收天線、信道矩陣為HQ的上行鏈路信道進行類比，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容量，因此采用V-BLAST結構可以達到。這種上行鏈路信道的詳細研究見第10章。82 快衰落MIMO信道快衰落MIMO信道為ym=Hmxm+wm m=1,2,其中，Hm為隨機衰落過程。為了恰當?shù)囟x容量（由隨時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Hm為平穩(wěn)遍歷過程，作為歸一化處理，設E|hij|2=1，與前面的研究方法一樣，考慮相干通信：接收機準確地跟蹤信道衰落過程。首先研究發(fā)射機僅具有衰落信道統(tǒng)計特征的情況，最后研究

18、發(fā)射機也能夠準確跟蹤衰落信道的情況（完整CSI），這種情況非常類似于時不變MIMO信道的情況。附件2：外文原文7. MIMO I: spatial multiplexingand channel modeling In this book, we have seen several different uses of multiple antennas in wireless communication. In Chapter 3, multiple antennas were used to provide diversity gain and increase the reliabilit

19、y of wireless links. Both receive and transmit diversity were considered. Moreover, 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, mu

20、ltiple transmit antennas were used to induce channel variations, which can then be exploited by opportunistic communication techniques. The scheme can be interpreted as opportunistic beamforming and provides a power gain as well. In this and the next few chapters, we will study a new way to use mult

21、iple antennas. We will see that under suitable channel fading conditions, having both multiple transmit and multiple 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 exp

22、loited by spatially multiplexing several data streams onto the MIMO channel, and lead to an increase in the capacity: the capacity of such a MIMO channel with n transmit and receive antennas is proportional to n.Historically, it has been known for a while that a multiple access system with multiple

23、antennas at the base-station allows several users to simultaneouslycommunicate with the base-station. The multiple antennas allow spatial separation of the signals from the different users. It was observed in the mid 1990s that a similar effect can occur for a point-to-point channel with multiple tr

24、ansmit and receive antennas, i.e., even when the transmit antennas are not geographically far apart. This holds provided that the scattering environment is rich enough to allow the receive antennas to separate out the signals from the different transmit antennas. We have already seen how channel fad

25、ing can be exploited by opportunistic communication techniques. Here, we see yet another example where channel fading is beneficial to communication. It is insightful to compare and contrast the nature of the performance gains offered by opportunistic communication and by MIMO techniques，Opportunist

26、ic communication techniques primarily provide a power gain.This power gain is very significant in the low SNR regime where systems are 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-freed

27、om gain. Thus, MIMO techniques become the primary tool to increase capacity significantly in the high SNR regime. 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 enviro

28、nment which enable 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 the multiplexing capability of a deterministic MIMO channel. We th

29、en go through a sequence of physical MIMO channels to assess their spatial multiplexing capabilities. Building on the insights from these examples, we argue that it is most natural to model the MIMO channel in the angular domain and discuss a statistical model based on that approach. Our approach he

30、re parallels that in Chapter 2, where we started with a few idealized examples of multipath wireless channels to gain insights into the underlying physical phenomena, and proceeded to statistical fading models, which are more appropriate for the design and performance analysis of communication schem

31、es. We will in fact see a lot of parallelism in the specific channel modeling technique as well. Our focus throughout is on flat fading MIMO channels. The extensions to frequency-selective MIMO channels are straightforward and are developed in the exercises.71 Multiplexing capability of deterministi

32、c MIMO channels A narrowband time-invariant wireless channel with nt transmit and nr receive 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 o

33、f the channel.7.1.1 Capacity via singular value decompositionThe time-invariant channel is described by y = Hx+w_ (7.1)where x,yand wdenote the transmitted signal,received signal and white Gaussian noise respectively at a symbol time (the time index is dropped for simplicity). The channel matrix H i

34、s deterministic and assumed to be 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 transmit antennas. This is a vector Gaussian channel.

35、The capacity can be computed by decomposing the vector channel into a set of parallel, independent scalar Gaussian sub-channels. From basic linear algebra, every linear transformation can be represented as a composition of three operations: a rotation operation, a scaling operation, and another rota

36、tion operation. In the notation of matrices, the matrix H has a singular value decomposition (SVD): 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 The diagonal elements are the or

37、dered singular values of the matrix H, where nmin：=min（nt，nr）. Since the squared singular values _2i are the eigenvalues of the matrix HH* and also of H*H. Note that there are nmin singular values. We can rewrite the SVD as The SVD decomposition can be interpreted as two coordinate transformations:

38、it says that if the input is expressed in terms of a coordinate 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 original channel (7

39、.1) with the input and output expressed in terms of these new coordinates.We have already seen examples of Gaussian parallel channels in Chapter 5, when we talked about capacities of time-invariant frequency-selective channels and about time-varying fading channels with full CSI. The time-invariant

40、MIMO channel is yet another example. Here, the spatial dimension plays the same role as the time and frequency dimensions in those other problems. The capacity is by now familiar: where P1*,Pnmin*are the waterfilling power allocations: with chosen to satisfy the total power constraint corresponds to

41、 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 7.2 pictorially depicts the SVD-based architecture for reliable communication.There is a clear analogy betw

42、een this architecture and the OFDM system introduced in Chapter 3. In both cases, a transformation 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

43、 together 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 matrices (DFTs) do not depend on the specific realization of the ISI channel, while for the latter, they do depend on the specific

44、 realization of the MIMO channel.7.2 Physical modeling of MIMO channelsIn 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 examples and analyzing the rank a

45、nd 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 be concrete, we restrict ourselves to uniform linear antenna arrays, where the antennas are evenly spaced on a st

46、raight line. The details of the analysis depend on the specific array structure but the concepts we want to convey do not.7.2.1Line-of-sight SIMO channelThe simplest SIMO channel has a single line-of-sight (Figure 7.3(a). Here, there is only free space without any reflectors or scatterers, and only

47、a direct signal path between each antenna pair. The antenna separation is where is the carrier wavelength and is the normalized receive 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 a

48、nd the receiver.The continuous-time impulse response between the transmit antenna and the ith receive antenna is given by where di is the distance between the transmit antenna and ith receive antenna, c is the speed of light and a is the attenuation of the path, which we assume to be the same for al

49、l 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):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

50、 of channel gains h=h1,hntt is sometimes called the signal direction or the spatial signature induced on the receive antenna array by the transmitted signal.Since the distance between the transmitter and the receiver is much larger than the size of the receive antenna array, the paths from the trans

51、mit antenna to each of the receive antennas are, to a first-order, parallel andwhere d is the distance from the transmit antenna to 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 quant

52、ity is the displacement of receive antenna i from receive antenna1 in the direction of the line-of-sight. The quantity is often called the directional cosine with respect to the receive antenna array. The spatial signature h=h1,hntt is therefore given by i.e., the signals received at consecutive ant

53、ennas differ in phase bydue to the relative delay. For notational convenience, we define as the unit spatial signature in the directional cosine . The optimal receiver simply projects the noisy received signal onto the signal direction, i.e., maximal ratio combining or receive beamforming (cf. Secti

54、on 5.3.1). It adjusts 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 The SIMO channel thus provides a power gain but no degree-of-freedom gain. In the context of a line-of-sight channel,

55、the receive antenna array is sometimes called a phased-array antenna.8. MIMO II: capacity and multiplexing architecturesIn 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. We partic

56、ularly focus on the scenario when the transmitter does not know the channel realization. In the fast fading MIMO channel, we show the following: At high SNR, the capacity of the i.i.d. Rayleigh fast fading channel scales like nminlogSNRb/s/Hz. where nmin is the minimum of the number of transmit ante

57、nnas nt and the number of receive antennas nr . This is a degree-of-freedom gain. At low SNR, the capacity is approximately nrSNR log2 e bits/s/Hz. This is a receive beamforming power gain. At all SNR, the capacity scales linearly with nmin. This is due to a combination of a power gain and a degree-

58、of-freedom gain.Furthermore, there is a transmit beamforming gain together with an opportunistic communication gain if the transmitter can track the channel as well.Over a deterministic time-invariant MIMO channel, the capacity-achieving transceiver architecture is simple (cf. Section 7.1.1): indepe