1. IERG 4100 Wireless Communications
Part IIX: Multiple antenna systems

2. Motivation Current wireless systems Increasing demand
Cellular mobile phone systems, WLAN, Bluetooth, Mobile LEO satellite systems, …
Increasing demand
Higher data rate ( > 100Mbps) IEEE802.11n
Higher transmission reliability (comparable to wire lines) 4G
Physical limitations in wireless systems
Multipath fading
Limited spectrum resources
Limited battery life of mobile devices …

3. Motivation Time and frequency processing
Coding
Adaptive modulation
Equalization
Dynamic bandwidth and power allocation …
Multiple antenna open a new signaling dimension: space
Higher transmission rate (Multiplexing gain)
Higher link reliability (Diversity gain)
Wider coverage

4. Multiple antenna systems
SU-MISO, TX diversity
SU-SIMO, RX diversity
SU-MIMO, Diversity vs. Multiplexing

5. Multiple antenna systems
MISO Broadcast
SIMO Multi-access
MIMO Broadcast
MIMO Multi-access

6. Multiplexing gain Multiple antennas at both Tx and Rx
Can create multiple parallel channels
Multiplexing order = min(M, N)
Transmission rate increases linearly

7. Diversity gain Multiple Tx or multiple Rx or both
Can create multiple independently faded branches
Diversity order = MN
Link reliability improved exponentially

8. Today’s Lecture
Diversity schemes
Beamforming
Space time coding

9. Achieving diversity: Maximum ratio combining
Recall
Fading flattens BER curves
Space-domain diveristy
Improve BER from
~(SNR)-1 to (SNR)-n
Assume N=1 or M=1 for the time being
Diversity order n
BER~exp(-SNR)
BER~(SNR)-1

10. Maximum ratio combining (SIMO)
Ways to combine the received signals
Equal gain combining
All paths co-phased and summed with equal weighting
Maximum ratio combining
All paths co-phased and summed with optimal weighting

11. Maximum ratio combining
Variance of noise
Maximal Ratio Combining (MRC)
Optimal technique (maximizes output SNR)
Combiner SNR is the sum of the branch SNRs.
Achieve diversity order of N

12. Distribution of SNR in Rayleigh Fading Channel
： exponential distribution
: chi-square distribution with degree of freedom
2N when hn are independent for different n
Recall the BER Calculation:
Through simple calculation, it can be seen that
Diversity order
Average SNR

13. Diversity

14. Diversity Gain

15. Maximum ratio transmission (MISO)
The signal transmitted by M antennas
Transmitter must know the channel!
What if it does not know?

16. Achieving diversity without CSIT: Space time coding
Core idea: complement traditional time with added space
Without channel knowledge at the transmitter
ST trellis codes (Tarokh’98), ST block codes (Alamouti’98)
Coding techniques designed for multiple antenna transmission.
Coding is performed by adding properly designed redundancy in both spatial and temporal domains which introduces correlation into the transmitted signal.

17. Space time coding
The ST encoder maps a block of information symbols X to coded symbols S

18. An Introductory Example
Two transmit antennas and 1 receive antenna
If two time intervals for the transmission of 1 symbol is allowed
Received signal
Equal to 1 by 2 MIMO systems
Diversity gain = 2
Data rate is reduced!!

19. Space time block code: Alamouti code
Two transmit antennas and 1 receive antenna
Assume channel does not change across two consecutive symbols
Space A1 A2 t x1 x2
t+1
-x2*
x1*
time

20. Alamouti code The combining scheme The decision statistics
Maximum-likelihood estimates of the transmitted symbols
Choose xi if
The combined signals are equivalent to that obtained from two-branch MRC!
Diversity gain =2
Data rate is not reduced!

21. Alamouti code Full-rate complex code Optimality of capacity
Is the only complex S-T block code with a code rate of unity.
Optimality of capacity
For 2 transmit antennas and a single receive antenna, the Alamouti code is the only optimal S-T block code in terms of capacity

22. Alamouti Code Performance
From Alamouti, A simple transmit diversity technique for wireless communications

23. Alamouti Code
The performance of Alamouti code with two transmitters and a single receiver is 3 dB worse than two-branch MRC.
The 3-dB penalty is incurred because is assumed that each transmit antenna radiates half the energy in order to ensure the same total radiated power as with one transmit antenna.
If each transmit antenna was to radiate the same energy as the single transmit antenna for MRC, the performance would be identical.

24. Space time block code
Alamouti code can be generalized to an arbitrary number of antennas
A S-T code is defined by an k x M transmission matrix
M – number of TX antennas
k – number of time periods for transmission of one block of coded symbols
Fractional code rate
Reduced Spectral efficiency
Non-square transmission matrix
Ref.: V. Tarokh, et al. “Space-time block codes from orthogonal designs.”

25. SVD SVD-Singular value decomposition
Allows H to be decomposed into parallel channels as follows
where S is a N-by-M diagonal matrix with elements only along the diagonal n=m that are real and non-negative
U is a unitary N-by-N matrix and V is a unitary M-by-M matrix
A Matrix is Unitary if AH=A-1 so that AHA= I
For example

26. What are Singular Values?
Note we can generate a square M-by-M matrix as
HHH= (USVH)H(USVH)=V(SHS)VH
Alternatively we can generate a square N-by-N matrix as
HHH= (USVH)(USVH)H= UH (SSH)U
We can see that the square of the singular values are the eigenvalues of HHH and HHH

27. SVD—What does it mean? Implies that UHHV=S is a diagonal matrix
Therefore if we pre-process the signals by V at the transmitter and then post-process them with UH we will produce an equivalent diagonal matrix
This is a channel without any interference and channel gains s11 and s22 for example

28. Water-Filling
When we have parallel multiple channels each with different attenuations we can use water filling to optimize the capacity by modifying the transmit powers
The capacity of multiple channels is given by
The question is how to find the distribution of powers to maximize the capacity under the constraint that

29. Water-Filling Use Lagrangian multiplier to find the solution Write
Take partial derivatives wrt to power allocations

30. Water-filling Know as water filling
Good channels get more power than poor channels
Channel index

31. Adaptive Modulation in Fading Channels
6 bps/HZ
4 bps/HZ
2 bps/HZ
0 bps/HZ

32. Adaptive Modulation in Fading Channels

33. Adaptive Modulation Data rate varies with channel fading amplitude
Variable data-rate transmission can also be achieved by adapting the code rate
Adaptive coding and modulation are often combined
Coding and modulation schemes can be chosen according to several criteria
Maximize average data rate given a fixed BER (bit error rate)
Minimize average BER given a fixed average data rate
In practice, need to consider the modulation types being discrete

34. Example
An adaptive modulation system can choose to use QPSK or 8-PSK for a target BER of The channel is Rayleigh fading with average SNR
Adaptation rule
The BER should always be smaller than 10-3
If the target BER cannot be met with either scheme, then no data is transmitted

35. Apply to MIMO with SVD Decompose MIMO channels with SVD
Allocate the power according to water-filling principle and adaptive modulation
Can transmit same (achieve diversity gain) or different (achieve multiplexing gain) data streams on the parallel channels

36. Achieving diversity using SVD
Achieve diversity order of 4!

37. Achieving multiplexing gain using SVD
Transmit different data streams on parallel channels
Use water filling to distribute power on the channels
Transmission rate on each channel is adapted to the effective channel gain

38. MIMO Detection
SVD requires channel state information at both the transmitter and receiver
When the transmitter doesn’t have knowledge of the channel, each antenna transmits independent data streams
The received signal
Our target is to detect original signals x from the received signal y

39. MIMO MLD
Let’s first consider optimum receivers in the sense of maximum likelihood detection (MLD)
In MLD we wish to maximize the probability of p(y|x)
To calculate p(y|x) we observe that the distribution must be jointly Gaussian
We need to find an optimal x from the set of all possible transmit vectors
Complexity grows exponentially!

40. MIMO Zero-Forcing We still use the idea of
Instead of minimizing only over the constellation points of x we minimize over all possible complex numbers (this is why it is sub-optimum)
In other words, we want to force
We then quantize the complex number to the nearest constellation point of x

41. MIMO Zero-Forcing The decision statistics is given by
where is the pseudo-inverse of H
Requirement: N>=M

42. MIMO Zero-Forcing
Similar to passing the signal through a Gaussian channel, but with a different noise variance
The variance of the noise added to xi is given by
Problem: Noise enhancement
The diversity order achieved by each stream is given by N-M+1

43. V-BLAST
The performance of ZF is not good enough, while the complexity of MLD is too high
Motivate different sub-optimal approaches
V-BLAST (Vertical-Bell laboratories layered space time)
Information stream is split into M sub-streams, each of them is modulated and transmitted from an antenna
Only applicable when N>=M
Based on interference cancellation
Ref.: P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel”, 1998

44. VBLAST

45. V-BLAST: Key idea Successive interference cancellation
Select the best bit stream and output its result using ZF
Use this result to remove the interference of the detected bit stream from the other received signals
Then detect the best of the remaining signals and continue until all signals are detected
It is a non-linear process

46. V-BLAST receiver V-BLAST successive interference cancelling (SIC)
The ith ZF-nulling vector wi is defined as the unique minimum-norm vector satisfying
is the ith row of H+

47. V-BLAST optimal ordering
Problem in SIC: error propagation
If the first decode channel is in low SNR, may decode in error and propagate to subsequent decoding process
Ordered Successive Interference Cancellation (OSIC)
Idea: Detect the symbols in the order of decreasing SNR
Provides a reasonable trade–off between complexity and performance (between MMSE and ML receivers)
Achieves a diversity order which lies between N − M + 1 and N for each data stream

48. V-BLAST Performance M N