1. MIMO Communication Systems
So far in this course
We have considered wireless communications with only one transmit and receive antenna- SISO
However there are a lot advantages to be had if we extend that to multiple transmit and multiple receive antennas
MIMO
MIMO is short for multiple input multiple output systems
The “multiple” refers to multiple transmit and receiver antennas
Allows huge increases in capacity and performance
MIMO became a “hot” area in 1998 and remains hot

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 …
Video transmission, mobile game, internet browsing
Mobile streaming

3. Multiple-Antenna Wireless Systems
Time and frequency processing hardly meet new requirements
Multiple antennas open a new signaling dimension: space
Create a MIMO channel
Higher transmission rate
Higher link reliability
Wider coverage

4. General Ideas
Digital transmission over Multi-Input Multi-Output (MIMO) wireless channel
Objective: develop Space-Time (ST) techniques with low error probability, high spectral efficiency, and low complexity (mutually conflicting)

5. Possible Gains: Multiplexing
Multiple antennas at both Tx and Rx
Can create multiple parallel channels
Multiplexing order = min(M, N), where M =Tx, N =Rx
Transmission rate increases linearly Tx Tx Rx Rx
Spatial Channel 1
Spatial Channel 2

6. Possible Gains: Diversity
Multiple Tx or multiple Rx or both
Can create multiple independently faded branches
Diversity order = MN
Link reliability improved exponentially Rx Tx Rx Tx
Fading Channel 1
Fading Channel 2
Fading Channel 3
Fading Channel 4

7. Key Notation- Channels
Assume flat fading for now
Allows MIMO channel to be written as a matrix H
Generalize to arbitrary number of inputs M and outputs N so H becomes a NxM matrix of complex zero mean Gaussian random variables of unity variance
Can understand that each output is a mixture of all the different inputs- interference
We assume UNCORRELATED channel elements Tx Rx

8. Key Decomposition- SVD
SVD- singular value decomposition
Allows H of NxM to be decomposed into parallel channels as follows
Where S is a NxM diagonal matrix with elements only along the diagonal m=n that are real and non-negative
U is a unitary N x N matrix and V is a unitary M x M matrix
The superscript H denotes Hermitian and means complex transpose
A Matrix is Unitary if AH=A-1 so that AHA = I
The rank k of H is the number of singular values
The first k left singular vectors form an orthonormal basis for the range space of H
The last right N-k right singular vectors of V form an orthogonal basis for the null space of H
What does SVD mean?

9. Key Decomposition- 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
Spatial Channel 1
Spatial Channel 2 Tx Rx

10. Key Decomposition- SVD
What are the singular values?
You can remember eigenvalues and eigenvectors
If A is any square matrix then it can diagonalized using E-1AE = D where E is the matrix of eigenvectors
Note we can generate a square M x M matrix as HHH= (USVH)H(USVH)=V(SHS)VH
Letting A=HHH so that E-1AE = D = VHHHHV= SHS
Alternatively we can generate a square N x N matrix as HHH= (USVH)(USVH)H= UH (SSH)U
Therefore we can see that the square of the singular values are the eigenvalues of HHH
Also note that V is the matrix of Eigenvectors of HHH
Similarly U is the matrix of eigenvectors of HHH

11. Capacity For a SISO channel capacity C is given by
where is the SNR at a receiver antenna and h is the normalized channel gain
For a MIMO channel we can make use of SVD to produce multiple parallel channels so that
Where are the eigenvalues of W

12. Capacity We can also alternatively write the MIMO capacity
It can be demonstrated for Rayleigh fading channels that if N<M then the average capacity grows linearly with N as
This is an impressive result because now we can arbitrarily increase the capacity of the wireless channel just by adding more antennas with no further power or spectrum required
In these calculations it is also assumed the transmitter has no knowledge about the channel

13. Note on SNR
The definition of the SNR used previously is simply the receiver SNR at each receiver antenna
In this definition the channel must be normalized rather than be the actual measured channel
This approach is used since it is more usual to specify things in terms of received SNR
However in calculations it is perhaps easier to think of total transmit power, un-normalized channel G and the received noise power per receive antenna, a, so capacity becomes

14. Example

15. Example

16. Special Cases
Take M=N and H =In and assume noise has cross-correlation In then
Let Hij = 1 so that there is only one singular value given by and assume noise has cross-correlation In
The first column of U and V is
Thus
Each transmitter is sending a power P/M and each is sending the same signal Hx
These M signals coherently add at each receiver to give power P
There are N receivers so the total power is NP
Given the noise has correlation In SNR is also NP

17. Example Consider the following six wireless channels
Determine the capacity of each of the six channels above, assuming the transmit power is uniformly distributed over the transmit antennas and the total transmit power is 1W while the noise per receive antenna is 0.1W.
Note which channels are SIMO and MISO

18. Example

19. Example

20. Example

21. Capacity
These capacity results are however the theoretical best that can be achieved
The problem is how do we create receivers and transmitters that can achieve close to these capacities
There are a number of methods that have been suggested
Zero-forcing
MLD
BLAST
S-T coding

22. MIMO Dectection
Consider a MIMO system with M transmit and N receive antennas (M,N)
where
x is the Mx1 transmit vector with constellation Q
H is a NxM channel matrix
y is Nx1 received vector
n is a Nx1 white Gaussian complex noise vector
Energy per bit per transmit antenna is
Our basic requirement is to be able to detect or receive our MIMO signals x

23. MIMO MLD
Lets first consider optimum receivers in the sense of maximum liklihood 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 and we can use previous results from M-ary to write it as

24. MIMO MLD
That is we need to find an x from the set of all possible transmit vectors that minimizes
If we have Q-ary modulation and M transmit antennas then we will have to search through QM combinations of transmitted signals for each transmit vector and perform N QM multiplications
Because of the exponent M the complexity can get quite high and sub-optimal schemes with less complexity are desired

25. MIMO Zero-Forcing In zero-forcing we use the idea of minimizing
However instead of minimizing only over the constellation points of x we minimize over all possible complex numbers (this is why it is sub-optimum)
We then quantize the complex number to the nearest constellation point of x
The solution then becomes a matrix inverse when N=M and we force to zero (zero-forcing)
What about when M does not equal N?

26. Key Theorem- Psuedoinverse
When H is square one way to find the transmitted symbols x from Hx = y is by using inverse.
What happens when H is not square? Need psuedo-inverse
Note that HHH is a square matrix which has an inverse
Therefore HHH x =HH y so that (HHH)-1HHH x = (HHH)-1HH y and the psuedo-inverse is defined as H+ = (HHH)-1HH
The psuedo-inverse provides the least squares best fit solution to the minimization of ||Hx-y||2 with respect to x

27. Example
If we use a zero-forcing receiver in the previous example what is the receiver processing matrix we need for each of the 6 channels?
G1- none needed
G2-Inverse not possible- not needed
G3- [1,1]
G4- Inverse not possible- just MRC weights
G5-
G6-

28. Performance analysis of ZF
The zero-forcing estimate of the transmitted signal can be written as:
where (with elements ) is known as the pseudo-inverse of the channel H and the superscript H denotes conjugate transpose
Substituting :
the ith row element of Gn is equal to a zero mean Gaussian random variable with variance:

29. Performance analysis of ZF
The noise power is scaled by which is the square 2-norm of the ith row of G
The diagonal elements of GG’ however are the square 2-norm of the rows of G
In addition we can show that
Which is equal to the the diagonal element of

30. Performance analysis of ZF
Since all are all identically distributed so we drop its subscript
w follow the reciprocal of a Chi-Square random variable with 2(N-M+1) degrees of freedom
The probability density function (PDF) of w
where D=N-M

31. Why Chi-Square? Check out For a N=M it is easy to show as follows
H. Winters, J. Salz and R. D. Gitlin, “The Impact of antenna Diversity on the Capacity of Wireless Communication Systems”, IEEE Trans. Commun., VolCOM-42, pp , Feb./March/April.
24. J. H. Winters, J. Salz and R. D. Gitlin, “The capacity increase of wireless systems with antenna diversity”, in Proc Conf, Inform. Science Syst., Princeton, NJ, Mar , 1992
For a N=M it is easy to show as follows
Matrix G, the inverse of H can then be written as
Where Aij is the sub-matrix of H without row i and column j

32. Why Chi-Square?
The square of the 2-norm for the i row of G is therefore equal to
Noticing that the equation above becomes
Since |Aji| is independent of hij we can condition on it so the equation can be further simplified
Remember hij are random variables (like noise so independent and add up)

33. Why Chi-Square?
The square of the 2-norm for the i row of G is therefore equal to
Where h’ is a random variable following the same distribution as hij
Canceling common terms we get

34. Why Chi-Square?
h’ is a random variable with the same distribution as hij
The weights, w are therefore distributed as the reciprocal of the sum of the square of two Gaussian random variables with zero mean and variance α/2
That is the weights are distributed as the reciprocal of a chi-squared random variable with 2 degrees of freedom
This turns out to be the reciprocal of a Rayleigh fading variable for this special case

35. Performance analysis of ZF
To obtain the error probabilities when w is random, we must average the probability of error over the probability density function ,
where is the probability of error in AWGN channel with depend on the signal constellation.

36. Performance of BPSK and QPSK
For BPSK and QPSK
Performing the integral and define as the SNR per bit per channel (see Proakis 4th ed, p825)
where

37. Performance of BPSK and QPSK using ZF
Exact BER expression for QPSK compared with Monte Carlo simulations

38. Performance of M-PSK
For M-PSK:
where

39. Performance of M-QAM
For M-QAM:
where

40. Comparison with simulation (ZF)2 4 6 8 10 12 14 16 18 20 -6 -5 -4 -3 -2 -1
SNR per bit per channel (dB)
BER
(3,3)
(4,6)
(8,12)
16-PSK (analysis)
16-PSK (simulation)
16-QAM (analysis)
16-QAM (simulation)
BER approximations for 16-PSK and 16-QAM compared with Monte Carlo simulations for (3,3), (4,6) and (8,12) antenna configurations.

41. Comparison with simulation
BER approximations for 64-PSK and 64-QAM compared with Monte Carlo simulations for (8,12) antenna configurations.

42. Performance of MLD
BER of zero-forcing and MLD for a (4,6) system using 4-QAM.

43. Performance of MLD
BER of zero-forcing and MLD for a (3,3) system using 8-QAM and 16-QAM.

44. Performance of MLD
BER of zero-forcing and MLD for a (3,3) system using 8-QAM and 16-QAM.

45. MIMO V-BLAST
It turns out the performance of ZF is not good enough while the complexity of MLD is too large
Motivate different sub-optimum approaches
BLAST is one well known on (Bell Laboratories Layered Space Time)
Based on interference cancellation
A key idea is that when we perform ZF we detect all the transmitted bit streams at once

46. MIMO V-BLAST
Generally we would expect some of these bit streams to be of better quality than the others
We select the best bit stream and output its result using ZF
We then also use it to remove its interference from the other received signals
We then detect the best of the remaining signals and continue until all signals are detected
It is a non-linear process because the best signal is always selected from the current group of signals

47. MIMO V-BLAST Basically layers of interference cancellation ) Linear
Detector
Interference
Cancellation
Stage 1
Stage ( M - 1 )
Stage

48. Performance of V-BLAST
BER of zero-forcing, V-BLAST and MLD for a (4,6) system using 4-QAM.