1. Capacity of Correlated MIMO Channels: Channel Power and Multipath Sparsity
Random Matrix Theory and Wireless Communications Workshop
Boulder, CO, July
Akbar Sayeed
(joint work with Vasanthan Raghavan, UIUC)
Part presented at ITW 2004
Wireless Communications Research Laboratory
Department of Electrical and Computer Engineering
University of Wisconsin-Madison ,

2. Sergio Verdu – Hard Act to Follow!
Shannon (belly) Dance! (ISIT 2006, Seattle)

3. Sergio Verdu – Model Incognito?

4. Multipath Wireless Channels
Fading – fluctuations in received signal strength
Diversity – statistically independent modes of communication
Multipath signal propagation over spatially distributed paths due to signal scattering from multiple objects
Necessitates statistical channel modeling
Accurate and analytically tractable  Understanding the physics!
Flow:
Virtual channel governs performance
DoF is key
Sparse channels: D < N_sp
Three key implications:
1) Capacity Scaling with Antennas: best O(sqrt(D(N))) vs O(N) ITW 2004
2) Capacity and Reliability Scaling with Bandwidth (New); Optimal T-W scaling for optimal channel
Estimation (whole range of T-W scalings)
Rich multipath is a bad assumption as the signal space dimensions increase
3) Maximizing capacity with Reconfigurable Antenna Arrays (ISIT 2006)

5. Antenna Arrays: Multiplexing and Energy Capture
Multiplexing – Parallel spatial channels
Array aperture:
Energy capture
Wideband (W):
Multi-antenna (N):
Dramatic linear increase in capacity with number of antennas

6. Key Elements of this Work
C(N)
sparse
i.i.d.
Correlated N
Sparse multipath
i.i.d. model – rich multipath
Seldom true in practice
Physical channels exhibit sparse multipath
Modeling of sparse MIMO channels
Virtual channel representation (beamspace)
Physically meaningful channel power normalization
Sparse degrees of freedom
Spatial correlation/coherence
The Ideal MIMO Channel
Fastest (sub-linear) capacity scaling with N
Capacity-maximization with SNR for fixed N
Multiplexing gain versus received SNR tradeoff
Simple capacity formula for all SNRs (RMT)
Creating the Ideal MIMO Channel in Practice
Reconfigurable antenna arrays
Three canonical configurations: near-optimum performance over entire SNR range
Source-channel matching
New capacity formulation
Capacity
MUX
IDEAL BF

7. Virtual Channel Modeling
Abstract statistical models
Physical models
Accurate
Tractable
Spatial sampling
commensurate with signal space resolution
Channel statistics
induced by the physical scattering environment
Virtual Model
Accurate &
tractable
Interaction between the signal space and the physical channel

8. Narrowband MIMO Channel
Transmitted signal
Received signal
Transmit antennas
Receive antennas

9. Uniform Linear Arrays Transmit steering vectord
Transmit steering
vector
= Tx antenna spacing
Relation between theta_T and phi_T. affect of antenna spacing
Receive response veector
= Rx antenna spacing
Spatial sinusoids: angles  frequencies

10. Physical Model : complex path gains : number of paths
: Angles of Arrival (AoA’s)
: Angles of Departure (AoD’s)
Non-linear dependence of H on AoA’s and AoD’s

11. Virtual Modeling Physical Model Virtual Model
Spatial array resolutions:
Virtual model is linear -- virtual beam angles are fixed
(AS ’02)

12. Antenna Domain and Beamspace
Two-dimensional unitary (Fourier) transform
Unitary (DFT) matrices
Generalization to non-ULA’s
(Kotecha & AS ’04 ; Weichselberger et. al. ’04; Tulino, Lozano, Verdu ’05;)

13. Virtual Imaging of Scattering Geometry
2 point scatterers
2 scattering clusters
Diagonal scattering
Rich scattering

14. Virtual Path Partitioning
Distinct virtual coefficients  disjoint sets of paths

15. Virtual Coefficients are Approximately Independent
Possibly remove
Channel power matrix: joint angular power profile

16. Joint and Marginal Statistics
Joint statistics:
Joint distribution of channel power as a function of transmit and receive virtual angles
Marginal statistics:
Transmit
Receive
(diagonal)

17. Kronecker Product Model
Independent transmit
and receive statistics
Separable angular scattering function (angular power profile)
Separable
kronecker
arbitrary

18. Communication in Eigen (Beam) Space
Multipath Propagation
Environment
Image of is created in the far-field of TX
is an image of the far-field of the RX

19. Capacity Maximizing Input
Optimal input covariance matrix is diagonal in the virtual domain:
Beamforming optimal at low SNR (rank-1 input)
Uniform power input optimal at high SNR (full-rank input)
Uniform power input optimal for regular channels (all SNRs)
Veeravalli, Liang, Sayeed (2003); Tulino, Lozano, Verdu (2003); Kotecha and Sayeed (2003)

20. Degrees of Freedom Dominant (large power) virtual coefficients
Statistically independent Degrees of Freedom (DoF)
Possibly remove
DoF’s are ultimately limited by the number of resolvable paths

21. Channel Power and Degrees of Freedom
Assume equal number of transmit and receive antennas –
D(N) = number of dominant non-vanishing virtual coefficients
= Degrees of Freedom (DoF) in the channel
Channel power:
Simplifying assumption:
The D non-vanishing virtual coefficients are O(1)

22. Prevalent Channel Power Normalization
 The channel power/DoF grow quadratically with N

23. Quadratic Channel Power Scaling?
is physically impossible indefinitely
(received power < transmit power)
Total TX power
Total RX power
Quadratic growth in channel power
Linear growth in total received power
Linear capacity scaling
Increasing power coupling between the TX and RX due to increasing array apertures N

24. Sparse (Resolvable) Multipath
Sub-quadratic power scaling dictates sparsity of DoF
Rich (linear)
Degrees of Freedom (D)
Sparse (sub-linear)
Channel Dimension

25. Capacity Scaling: Sparse MIMO Channels
For a given channel power/DoF scaling law
what is the fastest achievable capacity scaling?
New scaling result: coherent capacity cannot scale faster than
and this scaling rate is achievable (Ideal channel)
(AS, Raghavan, Kotecha ITW 2004)

26. (virtual representation)
MIMO Capacity Scaling
AS et. al. ’04
i.i.d. model
Telatar ’95
Foschini ’96
physical channels
(virtual representation)
Liu et. al. ’03
C(N)
Correlated channels
(kronecker model)
Chua et. al. ’02

27. Sparse Virtual Channels
Sub-quadratic power scaling dictates sparse virtual channels:
Capacity scaling depends on the spatial distribution of the D(N) channel DoF in the possible channel dimensions

28. Simple Model for Sparse MIMO Channels
0/1 mask matrix with D non-zero entries
Sparsity in virtual (beam) domain  correlation/coherence in the antenna (spatial) domain

29. Three Canonical (Regular) Configurations
Consider
p = number of parallel channels (multiplexing gain)
q = D/p = DoF’s per parallel channel
Received SNR = q/p
Beamforming
Multiplexing
Ideal
Draw a picture of morphing: p(N) up q(N) down
Multiplexing gain = p increases
Received SNR = q/p increases
p transmit dimensions; r = max(q , p) receive dimensions

30. Simple Capacity Formula: Multiplexing Gain vs Received-SNR
Received SNR per parallel channel
Multiplexing (MUX)
Beamforming (BF)
Ideal
C_bf = o(C_id), C_mux = o(C_id)

31. Morphing Between the Configurations
beamforming
ideal
multiplexing
Just say; they get fatter for gamma > 1 all three the same for gamma = 2 say; something about resolutions ; ideal to beam: compressing horizontally; ideal to mux; streching along diagonals . Ideal and bf: iid channels, Mux: D-connected (Liu)

32. Fastest Capacity Scaling: The Ideal Configuration
BF regime:
Ideal regime:
Given asymptotics of both BF (high SNR), mul (Low SNR); tight for BF and MUX for any snr. For ideal for high SNR.
Combine with earlier slide.
MUX regime:

33. Impact of Transmit SNR on Capacity Scaling
BF:
C(N)
MUX:
BF gamma = 0, ideal; gamma = ½, mux gamma =1
Ideal: N

34. Accuracy of Asymptotic Expressions
BF and MUX
tight at
all SNRs
C(N)
Ideal tight in the low-
or high-SNR regimes N

35. Capacity Formula Proofs: RMT
If H is r x p, coherent ergodic is given by
If, in addition, H is regular

36. Capacity Formula Proofs: RMT
If under broad assumptions on entries of H, the empirical spectral distribution function (Fp) of (normalized by p) converges to a deterministic limit (F)
Limit capacity computation
Approach 1: Sometimes this limit can be characterized explicitly
Approach 2: Often, the limit can only be characterized implicitly via the Stieltjes transform. The limit capacity formula is the solution to a set of recursive equations

37. Beamforming Configuration
Two cases:
In either case,
Thus, with

38. Ideal Configuration Case i) Case ii) Two cases:
Case i) reduces to a q x p i.i.d. channel
Case ii) reduces to a q-connected p-dimensional channel [LRS 2003]
Result critically depends on the fact that q/p > 0. Else, Grenander and Silverstein not applicable

39. Ideal Configuration Case i) Case ii)
In either case, empirical density of (normalized by q) converges to
Case i)
Case ii)
Thus,
Result critically depends on the fact that q/p > 0. Else, Grenander and Silverstein not applicable

40. Multiplexing Configuration
Previous result due to [Grenander & Silverstein 1977] not applicable
Two cases:
In either case, empirical density is unknown
The implicit characterization of [Tulino, Lozano & Verdu 2005] based on results due to [Girko] can be easily extended here
Exploiting the regular nature of H
This method cannot be used for every channel. Need to use regularity, or else we cant obtain closed form capacity limit expressions.

41. The Ideal MIMO Channel: Fixed N
0/1 mask matrix with D non-zero entries
Spatial distribution of the D channel DoF in the possible dimensions (“resolution bins”) that yields the highest capacity

42. Optimum Input Rank versus SNR
i.i.d. channels: equal power (i.i.d.) input optimal at all SNRs
Correlated channels:
beamforming (rank-1 input) optimal at low SNR
uniform power (full-rank, i.i.d.) input optimal at high SNR 4
Loss of precious
channel power! 3
rank 2 1
SNR

43. Ideal Channel: Optimum MG vs SNR tradeoff
Beamforming: BF
Ideal
MUX
Capacity
Ideal:
Multiplexing:

44. Impact of Antenna Spacing on Beamstructure

45. Adaptive-resolution Spatial Signaling
High resolution
TX and RX
Multiplexing
Ideal
Medium resolution
TX and RX
Beamforming
Low resolutionTX
and High-Res. RX
Multiplexing gain and spatial coherence: Fewer independent streams with wider beamwidths at lower SNRs.

46. Wideband/Low-SNR Capacity Gain
N-fold increase in capacity
(or reduction in )
via BF configuration at low SNR

47. Source-Channel Matching
Adapting the multiplexing gain p via array configuration:
matching the rank of the input to the rank of the effective channel TX
Multiplexing
Full-rank channel
Full-rank input RX

48. Source-Channel Matching
Adapting the multiplexing gain p via array configuration:
matching the rank of the input to the rank of the effective channel
Ideal TX
“Square root” rank channel
“Square root” rank input RX

49. Source-Channel Matching
Adapting the multiplexing gain p via array configuration:
matching the rank of the input to the rank of the effective channel TX
Beamforming
Rank-1 channel
Rank-1 input RX

50. New Capacity Formulation for Reconfigurable MIMO Channels
To achieve O(N) MIMO capacity gain at all SNRs
Optimal channel configuration realizable
with reconfigurable antenna arrays
MUX
IDEAL BF
Optimum number of antennas: N ~ D

51. Summary Sparse multipath Modeling of sparse MIMO channels
C(N)
sparse
i.i.d.
Correlated N
Sparse multipath
i.i.d. model – rich multipath
Seldom true in practice
Physical channels exhibit sparse multipath
Modeling of sparse MIMO channels
Virtual channel representation (beamspace)
Physically meaningful channel power normalization
Sparse degrees of freedom
Spatial correlation/coherence
The Ideal MIMO Channel
Fastest (sub-linear) capacity scaling with N
Capacity-maximization with SNR for fixed N
Multiplexing gain versus received SNR tradeoff
Simple capacity formula for all SNRs (RMT)
Creating the Ideal MIMO Channel in Practice
Reconfigurable antenna arrays
Three canonical configurations: near-optimum performance over entire SNR range
Source-Channel Matching
New capacity formulation
Capacity
MUX
IDEAL BF

52. Extensions: Implications of Sparsity
Relaxing the 0-1 sparsity model
Non-uniform sparsity
Wideband MIMO channels/doubly-selective MIMO channels
Space-time coding
Reliability (error exponents)
Impact of TX CSI (full or partial)
Channel estimation (compressed sensing) and feedback
Network implications (learning the network CSI)

53. Multiplexing Channel:
REFERENCES
Beamforming Channel:
Bai & Yin: “Convergence to the semicircle law,” Annals Prob., vol. 16, pp , 1988
Ideal Channel:
Marcenko & Pastur: “Distribution of eigenvalues for some sets of random matrices,” Math-USSR-Sb., vol. 1, pp , 1967
Bai: “Methodologies in spectral analysis of large dimensional random matrices: A review,” Statistica Sinica, vol. 9, pp , 1999
Silverstein & Bai: “On the empirical distribution of eigenvalues of a class of large dimensional random matrices,” Journal of Multivariate Analysis, vol. 54, no. 2, pp , 1995
Grenander & Silverstein: “Spectral analysis of networks with random topologies,” SIAM Journal on Appl. Math., vol. 32, pp , 1977
Liu, Raghavan & Sayeed, “Capacity and spectral efficiency of wideband correlated MIMO channels,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp , Oct. 2003
Multiplexing Channel:
Girko: Theory of random determinants, Springer Publishers, 1st edn, 1990
Tulino, Lozano & Verdu: “Impact of antenna correlation on the capacity of multiantenna channels,” IEEE Trans. Inform. Theory, vol. 51, no. 7, pp , July 2005
Multi-antenna Capacity of Sparse Multipath Channels, V. Raghavan and A. Sayeed.
This method cannot be used for every channel. Need to use regularity, or else we cant obtain closed form capacity limit expressions.