1. Multiple Antennas

2. Outline Introduction to MIMO Systems MIMO Channel Decomposition
MIMO Channel Capacity
Beamforming
Diversity vs. Multiplexing Tradeoffs
MIMO Receiver Design
Maximum-Likelihood
Decision Feedback
Sphere Decoder
Space-Time Processing
Other MIMO Design Issues
Space/time Coding
Adaptive techniques
Limited feedback

3. MIMO Systems Use of multiple transmit and receive antennas
Improve data rate transmission (spatial multiplexing) or reliability (diversity)
Additional RF hardware costs and more complex signal processing
Alamouti’s space-time code is one instance of MIMO system
本章主要研究多天线的各种不同用法，并找到各自的性能优势。

4. 1. Narrowband MIMO Model
Complex dada symbols {X1, X2, …, XMt} transmitted through Mt antennas during one symbol duration
Received signal at Mr antennas: {Y1, Y2, …, YMr}
Flat fading channel gains: {hij, i=1,2,…, Mr; j=1,2,…, Mt}
窄带MIMO模型.

5. Narrowband MIMO Model (cont)
The received signal at ith antenna is
In a matrix form, we get Or
假设信道带宽为B，复高斯噪声的均值为0，协方差矩阵为σ2
信道参数在ZMSW模型中，假设为独立同分布，零均值，复高斯随机变量，方差为1
H的分布和信道信息的假设不同，信道容量和空时编码方法也不同

6. 2. Parallel Decomposition of H
MIMO systems have multiple (r) transmit and receiver antennas
With perfect channel estimates at TX and RX, decomposes into r independent channels
RH-fold capacity increase over SISO system
Demodulation complexity reduction
Can also use antennas for diversity (beamforming)
Leads to capacity versus diversity tradeoff in MIMO
MIMO信道的并行分解
MIMO系统的复用增益来源于MIMO信道可以分解成R各并行的独立子信道。在这些独立信道上传输多路数据，数据率可以提高R倍，这就是复用增益。
考虑一个MIMO信道，发送端和接收端都已知信道增益矩阵H。令RH表示H的秩。

7. 2. Parallel Decomposition of H
对任意的H，可进行如下的奇异值分解， SVD
H的奇异值构成的对角阵。其中有RH个不为零。奇异值是HHH的第i大特征值的开平方。
因为矩阵的秩不可能超过它的行数或者列数，因此RH<min(Mr,Mt)
满秩的情形称为富散射环境
若H的元素高度相关，其秩可能会降为1.
用发送预编码和接收成形，对信道的输入输出分别进行变换，就可以实现信道的并行分解。
发送预编码和接收成形将MIMO信道变换成了RH个独立的并行信道。

8. MIMO: Parallel Channels
通过发送预编码和接受成形对信道的输入和输出进行变换，可以实现信道的并行分解

9. MIMO Decomposition
Decompose channel through transmit precoding (x=Vx) and receiver shaping (y=UHy)
Leads to RHmin(Mt,Mr) independent channels with gain si (ith singular value of H) and AWGN
Independent channels lead to simple capacity analysis and modulation/demodulation design ~ ~ ~ ~ ~
y=Hx+n
y=S x+n
H=USVH ~ ~ ~
yi=six+ni

10. Example

11. 3. MIMO Channel Capacity
静态信道的信道容量由 输入和输出的互信息确定。可用香农容量表示。

12. MIMO Capacity with Informed Transmitter
发送端已知信道：用注水法进行功率分配
信道容量等于总发送功率在各个信道之间最优分配后，各个独立并行信道的容量之和。
最优功率分配就是通过优化协方差矩阵使信道容量最大化。
进行预编码可将MIMO信道分解成RH个并行信道。

13. Example

14. MIMO Capacity with Unformed Transmitter
发送端未知信道：用平均功率分配
发送端未知，如果H分布符合ZMSW信道增益模型，其均值和方差对各个天线来说是对称的。因此把功率平均分配到每个发送天线上，输入协方差矩阵是酉矩阵乘上了一个系数，在上述假设下，输入协方差矩阵确实能使信道互信息最大化。
在ZMSW模型下，当Mt趋于无穷时，互信息随着M=min(Mt,Mr)的增大而线性增大。
在天线数较少时，也是随着M线性增加。
当信噪比很大时，对于任意Mt和Mr，容量也随M线性增长。无CSIT时，高信噪比或者天线数很多时，信道容量随信道自由度线性增长。
信噪比非常低时，增加发送天线并无益处，容量只与接收天线数有关。因为信噪比非常低时，MIMO系统知识在集中能量，不能利用所有可用的自由度。无论是把能量扩展到所有发送天线上，还是集中在一两根天线上都能达到信道容量。
随着信噪比的增加，限制因素不再是功率，而是信道的自由度。
天线数很多时，无CSIT不影响容量的增长率，但会增加解调的复杂度。

15. MIMO Ergodic Capacity 如果信道增益矩阵经历了平坦衰落，即增益随时间变化，用遍历容量来描述信道的容量。
发送端已知信道时，可以用注水法。
短期功率约束是沿天线的空间注水，而长期功率约束是沿时间和空间的二维注水。

16. MIMO Ergodic Capacity (cont)
如果发送端未知信道，假设H服从ZMSW分布。零均值空间白模型，zero mean spatially white ， ZMSW
遍历容量是按照所有的信道实现平均后的最大可传输速率，取决于H的分布。
可在功率约束的条件下优化信道输入的协方差矩阵。
对于ZMSW模型，H是独立同分布的零均值循环对称的，方差为1， 能治ZMSW信道的遍历容量最大化的输入协方差矩阵为Rx=（P/Mt）Imr，就是将发送功率平分到各个天线上，每个天线发送独立的符号。
容量随M=min(Mt,Mr)的增大而增大。
遍历容量是静态信道容量的简单平均。

17. MIMO Ergodic Capacity: High SNR Regime

18. MIMO Ergodic Capacity: High SNR Regime (cont)

19. MIMO Ergodic Capacity: With and Without Transmitter Knowledge
Very slight difference between the ergodic capacities of informed and uniformed transmitter. At high SNR, virtually no difference.
Eogodic capacity is min(Mt, Mr) that of SISO channel.
当发送端未知信道矩阵，而矩阵元素为复高斯但非独立同分布时，发送端可以利用信道的均值或方差来提高容量。基本思想是根据均值或协方差来分配功率。
假设接收端理想已知CSI，相关衰落的影响取决于发送端已知信道的情况：
如果发送端知道信道的实现，如果发送端知道信道的相关性，则容量可以变大。
或者发送端既不知道信道的实现，也不知道信道的相关特性，则天线之间的相关性会使信道容量下降。

20. Slow Fading MIMO Channels: Outage Capacity
对于静态信道，如果发送端不知道信道状态或者信道的平均互信息，那就无法确定该以什么样的速率发送方能保证数据的正确接收。因此，用中断容量来定义，发送端以固定速率R发送，中断率表示接收端不能正确接收的概率。
假设信道是慢变的，信道矩阵在较长时间内保持不变。如静态信道一样，发送端并不知道信道的状态和相应的容量，但它必须以一个固定的速率发送数据。无论发送端选择的速率是多少，都会有中断发生。
给一个或多个天线不分配发送功率有时可以提高中断容量，尤其在中断率较高的时候。因为中断容量取决于概率分布的拖尾，天线数较少时，所做的平均就越少，拖尾的扩展相应变大。

21. MIMO Outage Capacity
Can see substantially higher capacity for MIMO compared to SISO case for slow flat fading channel.
随着信噪比的增加，不同中断率对应的中断容量的差距在增大。

22. Capacity of MIMO Systems
Depends on what is known at TX and RX and if channel is static or fading
For static channel with perfect CSI at TX and RX, power water-filling over space is optimal:
In fading waterfilling over space (based on short-term power constraint) or space-time (long-term constraint)
Without transmitter channel knowledge, capacity metric is based on an outage probability
Pout is the probability that the channel capacity given the channel realization is below the transmission rate.
No CSI at either the transmitter and receiver, the linear growth in capacity as a function of the number of transmit and receive antennas disappears, as in some cases adding additional antennas provides negligible capacity gain.
Low SNR, capacity is limited by noise and linearly with the number of channel degrees of freedom
Moderate to high SNR, capacity is limited by estimate error, and its growth is also linear in the number of channel degrees of freedom
Very high SNR, the capacity grows double logarithmically with SNR, the degrees of freedom cease to matter.
收发都为之CSI的情况，低信噪比时容量受限于噪声，且随信道自由度线性增长；中信噪比时，容量受限于估计误差，且随信道自由度线性增长；高信噪比时，自由度不再起作用，容量与信噪比成二重对数增长关系。
慢衰落及收发都为之CSI的情形，信噪比高时增加天线并不能带来复用增益。

23. Beamforming Scalar codes with transmit precoding y=uHHvx+uHn
Transforms system into a SISO system with diversity.
Array and diversity gain
Greatly simplifies encoding and decoding.
Channel indicates the best direction to beamform
Need “sufficient” knowledge for optimality of beamforming
Precoding transmits more than 1 and less than RH streams
Transmits along some number of dominant singular values
MIMO分集增益
除了容量增益外，可以用发送端和接收端的多天线获得阵列和分集增益。
同一符号经过不同的复数因子加权之后送到每个发送天线，输入协方差矩阵的秩为1.也称为波束成形。
实质上是将矩阵形式的预编码和接受成形变成了列向量的形式。
通过对不同路径的信号进行相干合并，波束成形可以获得分集和阵列增益。
当发送端已知信道矩阵时，波束成形可获得的阵列增益介于max(Mt,Mr)和MtMr之间，分集增益是MtMr。
当发送端未知信道时，若Mt=2，可以用Alamouti分集获得Mr的阵列增益和2Mr的分集增益。
若Mt大于2，可用空时分组码获得满分集增益。
虽然波束成形的容量不及发送预编码和接受成形，但其译码复杂度低。

24. Diversity vs. Multiplexing
Use antennas for multiplexing or diversity
Diversity/Multiplexing tradeoffs (Zheng/Tse)
Error Prone
Best use
depends
on the
application
Low Pe
分集和复用的折衷是数据率、错误率和复杂度的折衷。
如果收发都已知信道，先将天线子集分组来获得分集增益，然后用不同的分组获得复用增益。
如果仅接收端已知CSI的分组衰落，当分组的长度趋于无限大时，可以通过天线间的对角编码，以合理的复杂度同时获得满分集增益和中断容量意义下的满复用增益。如D-BLAST。
当分组长度有限时，不可能同时获得满分集和满复用增益，此时就需要在两者之间进行折衷。
给出的是分组衰落信道在信噪比区域无限大时的这种问题。
复用增益r，分集增益d
若每Hz的数据速率和错误概率满足以上两个式子，给定r，dopt（r）表示最大可获得的分集增益。
如果分组长度T》Mt+Mr+1,择优
可以根据信道条件来调整分集和复用增益：在信道条件较差的时候，多用分集，在信道条件较好的时候，多用一些天线作复用。

25. How should antennas be used?
Use antennas for multiplexing:
Use antennas for diversity
High-Rate
Quantizer
ST Code
High Rate
Decoder
Error Prone
Low Pe
Low-Rate
Quantizer
ST Code
High
Diversity
Decoder
Depends on end-to-end metric: Solve by optimizing app. metric

26. MIMO Receiver Design Optimal Receiver: Decision-Feedback receiver
Maximum likelihood: finds input symbol most likely to have resulted in received vector
Exponentially complex # of streams and constellation size
Decision-Feedback receiver
Uses triangular decomposition of channel matrix
Allows sequential detection of symbol at each received antenna, subtracting out previously detected symbols

27. MIMO Receiver Designs Low Complexity Receivers
ML receivers exponentially complex # of streams and constellation size
Sphere decoding
only considers possibilities within a sphere of received symbol.
Multiple algorithms to search through the sphere
Can trade performance for complexity

28. MIMO Receiver Designs
Space-Time Processing: Encode/decode over time & space
Map symbols to both space and time via space-time block and convolutional codes.
For OFDM systems, codes are also mapped over frequency tones.
Can encode across antennas (space)
Can encode across antennas and streams, use interference cancellation (V-BLAST)
Can do a combination (D-BLAST)
在传统的SISO信道中，每个码元间隔内发送的是一个标量。但在MIMO信道中，输入输出关系是矩阵，每个码元间隔内发送的是一个向量。
MIMO信号设计可能包括多个天线和多个码元间隔，因此称为空时码。

29. Other MIMO Design Issues
Adaptive techniques:
MIMO systems adapt the use of transmit/receive antennas in addition to adapting modulation and coding.
Limited feedback: With limited capacity on the feedback path, techniques rely on partial CSI
Fast and accurate channel estimation
Adapt the use of transmit/receive antennas
Adapting modulation and coding.
Limited feedback:
Partial CSI introduces interference in parallel decomp: can use interference cancellation at RX
TX codebook design for quantized channel

30. Main Points
Multiple antennas at both TX and RX greatly enhance capacity and reduce complexity.
Alternatively, can be used for diversity gain
MIMO systems exploit multiple antennas at both TX and RX for capacity and/or diversity gain
With TX and RX channel knowledge, channel decomposes into independent channels
Linear capacity increase with number of TX/RX antennas
With TX/RX channel knowledge, capacity vs. outage is the capacity metric
Beamforming provides diversity gain in direction of dominant channel eigenvectors
Fundamental tradeoff between capacity increase and diversity gain: optimization depends on application
MIMO RX design trades complexity for performance
ML detector optimal; exponentially complex
DF receivers prone to error propagation
Sphere decoders allow performance tradeoff via radius
Space-time processing (i.e. coding) used in most systems