外文翻译无线通信基础(Fundamentals_of_wireless_communications_by_David_Tse)顶淘宝(小邓充值)

附件1：外文资料翻译译文

7．mimo：空间多路复用与信道建模

本书我们已经看到多天线在无线通信中的几种不同应用。在第3章中，多天线用于提供分集增益，增益无线链路的可靠性，并同时研究了接受分解和发射分解，而且，接受天线还能提供功率增益。在第5章中，我们看到了如果发射机已知信道，那么多采用多幅发射天线通过发射波束成形还可以提供功率增益。在第6章中，多副发射天线用于生产信道波动，满足机会通信技术的需要，改方案可以解释为机会波束成形，同时也能够提供功率增益。

这章以及接下来的几章将研究一种利用多天线的新方法。我们将会看到在合适的信道衰落条件下，同时采用多幅发射天线和多幅接收天线可以提供用于通信的额外的空间维数并产生自由度增益，利用这些额外的自由度可以将若干数据流在空间上多路复用至MIMO 信道中，从而带来容量的增加：采用n副发射天线和接受天线的这类MIMO信道的容量正比于n。

过去一度认为在基站采用多幅天线的多址接入系统允许若干个用户同时与基站通信，多幅天线可以实现不同用户信号的空间隔离。20世纪90年代中期，研究人员发现采用多幅发射天线和接收天线的点对点信道也会出现类似的效应，即使当发射天线相距不远时也是如此。只要散射环境足够丰富，使得接受天线能够将来自不同发射天线的信号分离开，该结论就成立。我们已经了解到了机会通信技术如何利用信道衰落，本章还会看到信道衰落对通信有益的另一例子。

将机会通信与MIMO技术提供的性能增益的本质进行比较和对比是非常的有远见的。机会通信技术主要提供功率增益，改功率增益在功率受限系统的低信噪比情况下相当明显，但在宽带受限系统的高信噪比情况下则很不明显。正如我们将看到的，MIMO技术不仅能够提供功率增益，还可以提供自由度增益，因此，MIMO技术成为在高信噪比情况下大幅度增加容量的主要工具。

MIMO通信是一个内容非常丰富的主题，对它的研究将覆盖本书其余章节。本章集中研究能够实现空间多路复用的物理环境的属性，并阐明如何在MIMO统计信道模型中简明扼要地俘获这些属性。具体分析过程如下：首先通过容量分析，明确确定确定性MIMO信道多路复用容量的关键参数，之后介绍一系列MIMO物理信道，评估其空间多路复用性能；根据这些实例的结果，我们认为在角域对MIMO信道进行建模是非常自然地，同时讨论了基于该方法的统计模型。本章采用的方法与第2章的方法是平行的，第2章就是从多径无线信道的几个理想实例着手进行分析，从中了解了基本物理现象，进而研究更适用于通信方案设计与性能分析的统计衰落模型。实际上，在特定的信道建模技术中，我们将会看到大量的类似方法。

我们贯穿始终的研究焦点是平坦衰落MIMO信道，但也可以直接扩展到频率选择性MIMO 信道，这方面的内容会在习题中加以介绍。

7.1确定性mimo信道的多路复用容量

包括n

t 副发射天线和n

t

接受天线的窄带时不变无线信道可以用一个n

t

*n

t

阶确定性矩阵

H描述，H具有哪些决定信道空间多路复用容量的重要属性呢？我们通过对信道容量的分析来回答这个问题。

7.1.1通过奇异值分解分析容量

时不变信道可以表示为：y = Hx+w_

其中x、y与w分别表示一个码元时刻的发射信号、接受信号与高斯白噪声（为简单起见省略了时标），信道矩阵H为确定性的，并假定在所有时刻都保持不变，而且对于发射机和接收机是已知的。这里的h

ij

为发射天线j到接受天线i的信道增益，对发射天线的信号的总功率约束为P。

这就是矢量高斯信道，将矢量信道分解为一组并行的、相互独立的标量高斯子信道就可以计算出该信道的容量。油线性代数的基本原理可知，每个线性变换都能够表示为三种运算的组合：旋转运算、比例运算和另一次旋转运算。用矩阵符号表示，矩阵H具有如下奇异值分解（SVD）：

其中，与为（旋转）酉矩阵1，是对角元素为非负实数、

非对角线元素为零的矩形矩阵2。对角线元素为矩阵H的有序奇异值，

其中n

min ：=min（n

t

，n

r

）。因为

所以平方奇异值为矩阵HH*的特征值，同时也是矩阵H*H的特征值。注意，奇异值共有n

min 个，可以将SVD重新写成为：

SVD分解可以解释为2个坐标变换：即如果输入用V的各种定义的坐标系统表示，并且输出用U的各列定义的坐标系统表示，那么输入/输出关系是非常简单的。

我们已经在第5章讨论时不变频率选择性信道以及具有完整CSI的时变衰落信道时看到了高斯并并行信道的例子。时不变MIMO信道也是另外一个例子，这里空间维所起的作用与其他问题中时间维和频率维的作用是相同的。大家熟知的容量表达式为：

其中，P

1*,…,P

nmin

*为注水功率分配：

通过选择满足总功率约束，各对应于信道的一个特征模式（也称特征信

道）。各非零特征信道能够支持一路数据流，因此，MIMO信道能够支持多路数据流的空间多路复用。基于SVD的可靠通信结构与第三章介绍的OFDM系统之间存在明显的相似之处，在这2种情况下，都是利用变换将矩阵信道转换为一组并行的独立子信道。在OFDM系统中，矩阵信道由上式中的轮换矩阵C给出，该矩阵由ISI信道和加在输入码元上的循环前缀定义，ISI信道与MIMO信道的重要区别在于，前者的U、V矩阵不依赖与ISI信道的特定实现，而后者的U、V矩阵则依赖与MIMO信道的特定实现。

7.2 MIMO信道的物理建模

通过本节的内容我们将了解到MIMO信道的空间多路复用性能对于物理环境的依赖程度，为此，我们将研究一系列理想化实例并分析骑信道矩阵的秩和条件数，这些确定性实例同时表明了下一节中讨论的MIMO信道统计建模的常规方法。具体地讲，本节的讨论局限于均匀线性天线阵列，即天线一均匀的间隔分布于一条直线上，分析的细节取决于特定的天线结构，但是我们要表达的概念于此无关。

7.2.1 视距SIMO信道

最简单的SIMO信道只有一条视距信道（如下所示），图中为不存在任何反射体和散射体的自由空间，并且各天线对之间仅存在直接信号路径，天线间隔为，其中为

载波波长，为归一化接受天线间隔，即归一化为载波波长的单位，天线阵列的尺寸比发射机与接收机之间的距离小得多。

发射天线与第i副接受天线之间信道的连续时间冲激响应为：

为发射天线与第i副接受天线之间的距离，c为光速，a为路径衰减，假定路径衰其中，d

i

/c《1/W,其中W为传输带宽，则可得基带信道增益为：

减对所有天线对都相同。设d

i

其中，f

c

为载波频率。SIMO信道可以写成：y=hx+w。其中，x为发射码元，w为噪声，y

为接受矢量。有时将信道增益矢量h=[h

1,…h

nt

]t称为信号方向或由发射信号在接收天线阵

列上感应出的空间特征图。

由于发射机与接收机之间的距离远大于接收天线阵列的尺寸，所以从发射天线到各接收天线的路径为1阶并行的，并且

其中，d为从发射天线到第一副接收天线之间的距离，为视距路径到接收天线阵列的入

射角，为在视距方向上接收天线i相对于接受天线1的位移。并且

通常被称为相对于接收天线阵列的方向余弦。因此，空间特征图h=[h

1,…h

nt

]t为

即有相对时延引起的相位差为的连续天线处的接收信号。为了符号表示方便，定义

为方向余弦上的单位空间特征图。

最佳接收机只是将有噪声接收信号投影到该信号方向上，也就是最大比合并或接收波束成形，对不同的时延进行调整，从而使天线的接收信号能够进行相长合并，得到n

t

倍的功率增益，所获取的容量为：

于是，SIMO信道提供了功率增益，但没有提供自由度增益。

在介绍视距信道时，有时将接收天线阵列称为相位阵列天线。

8. MIMO:容量与多路复用结构

本章研究MIMO衰落信道的容量，讨论能够从信道中提取所期望的多路复用增益的收发信机结构，特别是集中研究发射机未知信道的情况。在快衰落MIMO信道中，可以证明：

1 在高信噪比时，独立同分布瑞利快衰落信道的容量有n

min

logSNRb/s/Hz确定，其中

n min 为发射天线数n

t

与接收天线数n

r

的最小值，这是自由度增益。

2 在低信噪比时，容量近似为n

r

SNRlog

2

eb/s/Hz，这是接收波束成形功率增益。

3 在所有信噪比时，容量与n

min

呈线性比例关系，这是由于功率增益与自由度增益合

并造成的。

此外，如果发射机也能够跟踪信道，那么还存在发射波束成形增益以及机会通信增益。

利用确定性时不变MIMO信道的容量获取收发信机，其结构比较简单：在适当的坐标系统中对独立数据流进行多路复用，接收机将接收矢量变换到另一个适当的坐标系统中，分别对不同的数据流进行译码。如果发射机未知信道，那么必须事先固定独立数据流被多路复用所选取的坐标系统。连同联合译码，这种发射机结构实现了快衰落信道的容量，在文献中也将改结构称为V-BLAST结构1。

8．3节讨论比独立数据流的联合最大似然译码更简单的接收机结构，虽然可以支持信道全部自由度的接收机结构有若干种，其中的一种特殊结构是合并使用最小均方误差估计与串行干扰消除，即MMSE-SIC接收机可以获取容量。

慢衰落MIMO信道的性能可以通过中断概率和相应的中断容量来表征。在低信噪比时，

一个时刻利用一副发射天线就可以获取中断容量，实现满分集增益n

t n

r

和功率增益n

r

。

另一方面，高信噪比时的中断容量还受益于自由度增益，要简洁地刻画其特征更加困难，此问题留到第9章再分析。

虽然采用V-BLAST结构可以实现快衰落信道的容量，但该结构对于慢衰落信道则是严格次最优的，实际上，它甚至还没有实现MIMO信道期望的满分集增益。为了说明这一问题，考虑通过发射天线直接发送独立数据流，在这种情况下，各数据流的分集仅限于接收分集，为了从信道中获取满分集，须对发射天线进行编码。将发射天线编码与MMSE-SIC 结合起来的一种修正结构D-BLAST2不仅能够从信道中获取满分集，而且其性能还接近于中断容量。

8.1 V-BLAST结构

首先考虑时不变信道y[m]=Hx[m]+w[m] m=1,2,…

当发射机已知信道矩阵H时，有7.1.1节可知，最优策略是在H*H的特征矢量的方向上发射独立数据流，即在由矩阵V定义的坐标系统中发射，该坐标系统与信道有关。考虑到要处理发射机未知信道矩阵时的衰落信道，归纳出入如下图所示的结构，图中n

t

个独立的数据流在由酉矩阵Q确定的任意坐标系统中进行多路复用，该酉矩阵未必与信道矩阵H

有关，这就是V-BLAST结构。对数据流进行联合译码，为第k个数据流分配的功率为P

k

（使

得功率之和P

1+…+P

nt

等于P，即发射总功率约束），并利用速率为R

k

的容量获取高斯码进

行编码，总的速率为

几种特殊情况如下：

1 如果Q=V并且通过注水分配的方式确定功率，则得到如图7-2所示的容量获取结构。

2 如果Q=I

nt

，则独立数据流被发送到不同的发射天线。

下面利用与第5章关于球体填充的类似论述，讨论最高可靠通信速率的上界：

其中，K

x

为发射信号x的协方差矩阵，是多路复用坐标系和功率分配的函数：

考虑在长度为N的码元时间块内的通信，长度为n

r

N的接收矢量一高概率位于体积与下式成比例的椭圆体内：

该公式是与并行信道相对应的体积公式的直接推广，并在习题8-2中加以证明。由于必须考虑到各码字周围为非混叠噪声球空间才能却保可靠通信，所以能够填充的码字的最大数量为比值：

现在就可以得出结论，可靠通信速率的上界为上式。

采用V-BLAST结构能够达到该上界吗？注意到独立数据流在V-BLAST结构中多路复用，是否可能需要对数据流进行编码才能达到上界式？为了解决这个问题，考虑MISO信

道的特殊情况（n

t =1），并在该结构中设Q=I

nt

，即独立数据流由各发射天线发送。这恰好

就是6.1节介绍的上行链路信道，发射天线类似于用户，由这一节的内容可知，该上行链路信道的总容量为：

这恰恰是特殊情况下的上界式。因此，数据流独立的V-BLAST结构完全能够达到上界式。在一般情况下，可以将V-BLAST结构与包括n

t

副接收天线、信道矩阵为HQ的上行链路信道进行类比，与一副发射天线的情况相同，上界式就是该上行链路信道的总容量，因此采用V-BLAST结构可以达到。这种上行链路信道的详细研究见第10章。

8．2 快衰落MIMO信道

快衰落MIMO信道为y[m]=H[m]x[m]+w[m] m=1,2,…

其中，{H[m]}为随机衰落过程。为了恰当地定义容量（由随时间变化的信道衰落取平均获得的）的概念，现做出如下（与前几章相同的）假定，即假定{H[m]}为平稳遍历过程，作

|2=1，与前面的研究方法一样，考虑相干通信：接收机准确地跟为归一化处理，设E[|h

ij

踪信道衰落过程。首先研究发射机仅具有衰落信道统计特征的情况，最后研究发射机也能够准确跟踪衰落信道的情况（完整CSI），这种情况非常类似于时不变MIMO信道的情况。

附件2：外文原文

7. MIMO I: spatial multiplexing

and 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 reliability 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, multiple 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 multiple 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 exploited 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 antennas at the base-station allows several users to simultaneously communicate 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 transmit 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 fading 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，Opportunistic 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-freedom 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 environment 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 then 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 here 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 schemes. 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.

7．1 Multiplexing capability of deterministic MIMO channels

transmit and nr receive A narrowband time-invariant wireless channel with n

t

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 channel.

7.1.1 Capacity via singular value decomposition

The time-invariant channel is described by

y = Hx+w_ (7.1)

where x,y and w denote 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 is 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. 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 rotation 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 ordered 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 n

min

singular values. We can rewrite the SVD as

The SVD decomposition can be interpreted as two coordinate transformations: 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 (7.1) 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 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 P

1*,…,P

nmin

*are the waterfilling power allocations:

with chosen to satisfy the total power 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 7.2 pictorially depicts the SVD-based architecture for reliable communication.

There is a clear analogy between 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 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 realization of the MIMO channel.

7.2 Physical modeling of MIMO channels

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 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 be concrete, we restrict ourselves to uniform linear antenna arrays, where the antennas are evenly spaced on a straight 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 channel

The 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 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 and 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 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):

where fc is the carrier frequency. The SIMO channel can be written as y = h x+w where x is the transmitted symbol, w is the noise and y is the received vector.

The vector of channel gains h=[h

1,…h

nt

]t 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 transmit antenna to each of the receive antennas are, to a first-order, parallel and

where 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 quantity 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=[h

1,…h

nt

]t is therefore given by

i.e., the signals received at consecutive antennas differ in phase by

due 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. Section 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, the receive antenna array is sometimes called a phased-array antenna.

8. MIMO II: capacity and multiplexing architectures

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. 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: ? At high SNR, the capacity of the i.i.d. Rayleigh fast fading channel scales like

n min logSNRb/s/Hz. where n

min

is the minimum of the number of transmit antennas n

t

and

the number of receive antennas nr . This is a degree-of-freedom gain.

? At low SNR, the capacity is approximately n

r

SNR log2 e bits/s/Hz. This is a receive beamforming power gain.

? At all SNR, the capacity scales linearly with n

min

. This is due to a combination of a power gain and a degree-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): independent data streams are multiplexed in an appropriate coordinate system (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 the coordinate system in which the independent data streams are multiplexed has to be fixed a priori. In conjunction with joint decoding, we will see that this transmitter architecture achieves the capacity of the fast fading channel. This architecture is also called V-BLAST1 in the literature.

In Section 8.3, we discuss receiver architectures that are simpler than joint ML decoding of the independent streams. While 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 successive interference cancellation (SIC), achieves capacity.

The performance of the slow fading MIMO channel is characterized through the outage probability and the corresponding outage capacity. At low 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, benefits from a degree-of-freedom gain as well; this is more difficult to characterize succinctly and its analysis is relegated until Chapter 9.

Although it achieves the capacity of the fast fading channel, 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 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 transmit antennas. A modified architecture, D-BLAST2, which combines transmit antenna coding with MMSE-SIC, not only extracts the full diversity from the channel but its performance also comes close to the outage capacity.

8.1 The V-BLAST architecture

We start with the time-invariant channel (cf. (7.1))

y[m]=Hx[m]+w[m] m=1,2,…

When the channel matrix H is known to the transmitter, we have seen in 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 system 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 to the transmitter, we generalize this to the architecture in Figure 8.1, where the independent data streams, n

t

of them, are multiplexed in some arbitrary coordinate system given by a unitary matrix Q, not necessarily dependent on the channel matrix H. This is the V-BLAST architecture. The data streams are decoded jointly. The k th data stream is

allocated a power P

t (such that the sum of the powers, P

1

+···+P

nt

, is equal

to P, the total transmit power constraint) and is encoded using a

capacity-achieving Gaussian code with rate R

k

. The total rate is

As special cases:

? If Q = V and the powers are given by the waterfilling allocations, then we have the capacity-achieving architecture in Figure 7.2.

? If Q = I

nr

, then independent data streams are sent on the different transmit antennas.

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:

is the covariance matrix of the transmitted signal x and is a function of Here K

x

the multiplexing coordinate system and the power allocations:

Considering communication over a block of time symbols of length N, the received N, lies with high probability in an ellipsoid of volume vector, of length n

r

proportional to

This formula is a direct generalization of the corresponding volume formula (5.50) for the parallel channel, 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

We can now conclude the upper bound on the rate of reliable communication in (8.2).

Is this upper bound actually achievable by the V-BLAST architecture? Observe that independent data streams are multiplexed in V-BLAST; perhaps coding across the streams is required to achieve the upper bound (8.2)? To get some insight on this question, consider the special case of a MISO channel (n

= 1) and set Q =

r

in the architecture, i.e., independent streams on each of the transmit antennas. I

rt

This is precisely an uplink channel, as considered in Section 6.1, drawing an analogy between the transmit antennas and the users. We know from the development there that the sum capacity of this uplink channel is

This is precisely the upper bound (8.2) in this special case. Thus, the V-BLAST architecture, with independent data streams, is sufficient to achieve the upper bound (8.2). In the general case, an analogy can be drawn between the V-BLAST architecture and an uplink channel with nr receive antennas and channel matrix HQ; just as in the single receive antenna case, the upper bound (8.2) is the sum capacity of this uplink channel and therefore achievable using the V-BLAST architecture. This uplink channel is considered in greater detail in Chapter 10 and its information theoretic analysis is in Appendix B.9.

8.2 Fast fading MIMO channel

The fast fading MIMO channel is

y[m]=H[m]x[m]+w[m] m=1,2,…

Where {H[m]} is a random fading process. To properly define a notion of capacity (achieved by averaging of the channel fading over time), we make the technical assumption (as in the earlier chapters) that {H[m]} is a stationary and

|2=1. As in our ergodic process. As a normalization, let us suppose that E[|h

ij

earlier study, we consider coherent communication: the receiver tracks the channel fading process exactly. We first start with the situation when the transmitter has only a statistical characterization of the fading channel. Finally, we look at the case when the transmitter also perfectly tracks the fading channel (full CSI); this situation is very similar to that of the time-invariant MIMO channel.