ELEC E7210: Communication Theory. Lecture 10: MIMO systems

Transcription

1 ELEC E7210: Communication Theory Lecture 10: MIMO systems

2 Matrix Definitions, Operations, and Properties (1) NxM matrix a rectangular array of elements a A. an a a 1M. NM B D C E ermitian transpose A ij A * ji A is ermitian if A A

3 Matrix Definitions, Operations, and Properties (2) For a square matrix A is positive definite if for any non-zero vector x x Ax 0 - Tr[A]= N i1 A ii Det(A) E.g. A a a a a det( A) a a a 12 a 21

4 Matrix Definitions, Operations, and Properties (3) 1 Inverse matrix AA I N A is invertible or non-singular det( A) 1 det A 1 A square invertible matrix U is unitary if UU IN For a diagonal matrix D diag d i, d 0 i D 1 diag 1/ d i

5 Matrix Definitions, Operations, and v Properties (4) Eigenvector and eigenvalues Av v, Eigenvalues of a ermitian matrix are always real, although eigenvectors may be complex Characteristic polynomial det( A I) 0 Determinant det(a) i i

6 Matrix norms A p N M i1 j1 a ij p 1/ p p=2, Frobenius norm A F trace A A rank of A - number of linear independent raws/columns

7 Matrix decompositions (1) If A is a normal NxN square matrix it can be decomposed as A UΛU U a unitary matrix with the columns the eigenvectors of A, Λ diag,...,,0...,0 1 K

8 Matrix decompositions (2) SVD A matrix A is an NxM matrix of rank R A. Then there exist 2 unitary matrices U and V and a matrix such that A NM UNNΣNM VM M Σ Σ NM Σ 0N 1 M M M M singular value of i i Σ N M Σ 0 1 N ( M N ) N N A Σ 1 diag i i is eigenvalue of AA

9 MIMO systems (1) MIMO systems are defined as point-to-point communication links with multiple antennas at both the transmitter and receiver (unlike diversity systems). MIMO systems can significantly increase the data rates of wireless systems without increasing transmit power or bandwidth.

10 MIMO systems (2) The gain is obtained at the expense of - the added cost of deploying multiple antennas ; - the space requirements of these extra antennas (especially on small handheld units); - the added complexity required for multi-dimensional signal processing. WiMAx; 3 GPP long term evolution; 4G; SPA+

11 MIMO systems (3) Narrow-band n x m MIMO system

12 MIMO systems (4) Representation y x N

13 Transmit Precoding and Receiver Shaping (1) The input stream of the rate is linearly transformed

14 Transmit Precoding and Receiver Shaping (2) The rank of the input covariance matrix is r. Q E xx Optimal decoding of the received signal requires ML demodulation. yˆ argmin y xx x 2 E.g., if the modulated symbols are chosen from an alphabet of size N, then ML demodulation requires an exhaustive search r over possibilities for the input r-tuple. N

15 Parallel Decomposition of the MIMO Channel (1) Closed-loop and open-loop MIMO Assumptions: - Perfect Channel State Information at thetx - SVD of the channel matrix 2 diag i, are the eigenvalues of i

16 Parallel Decomposition of the MIMO Channel (2) Let the transmitter choose M V and the receiver choose F U Then the MIMO system is transformed into a set of r non-interfering SISO channels ~ y U UΛVV ~ Λx ~ N ~ x U N

17 Parallel Decomposition of the Equivalent scheme MIMO Channel (3)

18 or Parallel Decomposition of the MIMO Channel (4)

19 Parallel Decomposition of the MIMO Channel (5) Since the SISO channels do not interfere the optimal (maximum likelihood) demodulation complexity is only r instead of N. rn Multiplication by a unitary matrix does not change the distribution of white Gaussian noise. Leads to R min(n,m) independent channels with gain i (i th singular value of ) and AWGN Independent channels lead to simple capacity analysis and modulation/demodulation design

20 MIMO Channel Capacity (1) Depends on what is known at TX and RX and which kind of selectivity the channel exhibits For static channels with perfect channel knowledge at TX and RX, waterfilling over space is optimal power allocation

21 MIMO Channel Capacity (2) Using the decomposition C 2 i Pi max Bi log 1 Pi Pi P i N0Bi i Power allocation according to the waterfilling algorithm provides the maximization P i 2 i i /( N0B i )

22 MIMO channel capacity (3) The resulting capacity

23 Beamforming (1) Decoding complexity is exponential in r, one can keep the complexity low by keeping r small. r = 1. A transmit strategy - beamforming. The precoding matrix M = c, the beamforming vector,

24 The scheme Beamforming (2)

25 Beamforming (3) A single SISO AWGN channel ~ c c c y y cx N c x c c c ~ N SNR c ce The optimal demodulation complexity with beamforming N. xx *

26 Beamforming (4) c ow to assign? MRC at Rx Received signal r cx n c c 1 After MRC y c c c x n

27 Beamforming (5) The output (effective) SNR eff. c cc c c c c c max eff. c c c max c c 2 1

28 Beamforming (6) max c eff. max Beamforming vector a unit vector associated with the maximal eigenvalue max

29 Conditional number Condition Number is defined as a ratio of the maximum and minimum eigenvalues of the MIMO channel matrix. Large capacity gains from spatial multiplexing operation in MIMO wireless systems is possible when the statistical distributions of condition numbers have mostly low values.

30 Random matrix theory Joint statistical distribution of matrix elements, eigenvalues, etc. If elements of channel matrix are Gaussian, Wishart matrix. LOS conditions often create undesirable MIMO matrix conditions (i.e., high condition numbers) that can be mitigated using dual-polarized antennas.