1. EE359 – Lecture 15 Outline Announcements: MIMO Channel Capacity
HW posted, due Friday
MT exam grading done
Can pick up after class or from Julia
My OHs Thursday moved to outside classroom
MIMO Channel Capacity
MIMO Beamforming
Diversity/Multiplexing Tradeoffs
MIMO Receiver Design

2. Midterm Grade Distribution
Mean: 73.08, STD:10.4.
Approximate “curve”
85-90: A+
75-84: A
65-74: A-
60-64: B+
Comparison with Last Year:
Lower top grade
Grades more clustered
Approx. same mean
Smaller STD

3. Review of Last Lecture Adaptive MQAM with Restriction: M(g) MD
Map g to straight line M(g)=g/g*k (optimize g*k for max rate)
Pick region boundaries gi such that M(gi)MD
Use inverse waterfilling power control per region
MIMO systems have multiple TX and RX antennas
System model defined via matrices and vectors
Channel decomposition: TX precoding, RX shaping
M(g)=g/gK* g g0
g1=M1gK* g2 g3 M1 M2
Outage M3
MD(g)
M(g)=g/gK ~ ~ ~
y=Hx+n
y=S x+n
H=USVH ~ ~ ~
yi=six+ni

4. 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 waterfill 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.
Massive MIMO: in high SNR, singular values converge to a constant: C=min(Mt,Mr)Blog(1+r)

5. 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

6. Diversity vs. Multiplexing
Use antennas for multiplexing or diversity
Diversity/Multiplexing tradeoffs (Zheng/Tse)
Error Prone
Low Pe

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

8. MIMO Receiver Design Sphere Decoder: Optimal Receiver:
Maximum likelihood: finds input symbol most likely to have resulted in received vector
Exponentially complex # of streams and constellation size
Linear Receivers
Zero-Forcing: forces off-diagonal elements to zero, enhances noise
Minimum Mean Square Error: Balances zero forcing against noise enhancement
Sphere Decoder:
Only considers possibilities within a sphere of received symbol.
If minimum distance symbol is within sphere, optimal, otherwise null is returned

9. Main Points Capacity of MIMO systems
Without TX CSI, at high SNR/large arrays, capacity increases linearly with the number of TX/RX antennas
With TX and RX channel knowledge, water-fill power over space or space-time to achieve capacity
Beamforming transforms MIMO system into a SISO system with TX and RX diversity.
Beamform along direction of maximum singular value
MIMO introduces diversity/multiplexing tradeoff
Optimal use of antennas depends on application
MIMO RX design trades complexity for performance
ML detector optimal - exponentially complex
Linear receivers balance noise enhancement against stream interference