The ergodic high SNR capacity of the Introduction - - PowerPoint PPT Presentation

ITW Jeju, South Korea, 2015 1 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

The ergodic high SNR capacity of the spatially-correlated non-coherent MIMO channel within an SNR-independent gap

Ramy Gohary and Halim Yanikomeroglu1

Systems and Computer Engineering Dept., Carleton University, Canada

October 2015

1Work supported by Huawei Inc. and Ontario Research Funding

ITW Jeju, South Korea, 2015 2 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Introduction

• Non-coherent MIMO communication system: No

channel state information (CSI) is available at either the Tx or Rx.

• The analysis of non-coherent systems accounts for the

communication resources expended to acquire accurate CSI.

• Training cost tolerable in static and slow fading

scenarios, but not in fast and block fading ones.

• In fast fading, more beneficial to use signalling

strategies that do not require Rx to know CSI. (Hochwald et al. ’00, Zheng et al. ’02)

ITW Jeju, South Korea, 2015 3 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Previous Work

• For spatially-white channel, input matrices that achieve

capacity have the following structures:

• At any SNR and any coherence interval, the product of

an isotropically distributed unitary component and a diagonal component with non-negative entries. (Hochwald et al. ’00)

• At high SNRs and coherence interval greater than a

threshold, τ, isotropically distributed unitary on the Grassmann manifold. (Zheng et al. ’02)

• At high SNRs and coherence interval less than τ, the

product of an isotropically distributed unitary component and a diagonal component with random entries distributed as the square root of the eigenvalues of a beta matrix. (Yang et al. ’13)

• At low SNRs, only one entry of the diagonal component

is potentially non-zero. (Srinivasan et al. ’09)

• What about spatially-correlated channels?

ITW Jeju, South Korea, 2015 4 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Spatial correlation

• Spatial correlation arises due to proximity of physical

antennas, especially in prospective massive MIMO systems.

• Correlation nonnegligible, even when spacing exceeds

multiple wavelengths.

• Kronecker model: left and right multiplication of the

spatially-white channel matrix with Tx and Rx correlation matrices.

• Correlation matrices, vary much more slowly than

instantaneous channel parameters. Can be estimated accurately and assumed known. (Yu et al. ’04)

• Correlation: significant impact on signalling

methodology and achievable rate.

• Kronecker correlation noncoherent model considered in

(Jafar et al. ’05) at any SNR.

ITW Jeju, South Korea, 2015 5 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Non-coherent communication

• n spatially correlated channels:

What is not known?

• No closed-form expressions for capacity, or bounds

thereof.

• No constructive signalling strategy to approach

capacity.

ITW Jeju, South Korea, 2015 6 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

This Work

• Derive an expression for the ergodic high SNR

non-coherent capacity for block Rayleigh fading channels with Kronecker correlation.

• Expression accurate within an SNR-independent gap

and an error that decays as 1/SNR.

• Derive an upper bound on the gap to the actual
• capacity. Gap decreases monotonically with logarithm
• f condition number of Tx correlation.
• Show that input signals that achieve capacity lower

bound can be expressed as product of isotropically distributed random Grassmannian component and deterministic component comprising eigenvectors and inverse of eigenvalues of Tx correlation matrix.

ITW Jeju, South Korea, 2015 7 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

System Model

• Frequency-flat block Rayleigh fading channel with equal

number of Tx and Rx antennas, M.

• Correlated signals emitted from Tx and correlated

signals impinging on Rx. Channel H = A1/2HwB1/2, where A and B are Tx and Rx pd correlation matrices, and Hw random with zero-mean unit-variance i. i. d. circularly-symmetric complex Gaussian entries.

• We assume A and B are full rank and Tr A = Tr B = 1.
• Block fading model with coherence time T.

ITW Jeju, South Korea, 2015 8 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

System Model (cont’d)

• The received signal matrix can be expressed as

Y = XA1/2HwB1/2 + V, where X ∈ CT×M is Tx signal, and V ∈ CT×M additive noise; the entries of V are i. i. d. standard complex Gaussian random variables.

• Tx power constraint:

E{Tr(XX †)} ≤ TP.

• The matrices A and B are known but Hw is not.

ITW Jeju, South Korea, 2015 9 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

The right singular vectors of X

• Conditioned on X, Y is Gaussian and

p(Y|X) = exp

• − vec†(Y)
• B ⊗ XAX † + IMT

−1 vec(Y)

• πTM det
• B ⊗ XAX † + IMT
• .
• For deterministic Φ, p(ΦY|ΦX) = p(Y|X), yielding
• ptimal

X = QXDU†

A,

where

• QX isotropically distributed unitary matrix;
• D random diagonal with non-negative entries; and
• UA is the matrix containing the eigenvectors of A.

ITW Jeju, South Korea, 2015 10 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Conditional Entropy h(Y|X)

C(P) = max

p(X), E{Tr(XX †)}≤TP

1 T

• h(Y) − h(Y|X)
• .

Our goal is to evaluate C(P) as P → ∞.

• Evaluating h(Y|X) straightforward

h(Y|X) = MT log πe +M log det AB + E{log det D2} + O(1/P).

• Approximation valid when D full rank and its entries

scale with P.

ITW Jeju, South Korea, 2015 11 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Nonconditional Entropy h(Y)

• Computing entropy of signal component plus noise

component formidable task.

• At high SNR, write

h(Y) = h(XA1/2HwB1/2) + O(1/P) = h(QXDΛ1/2

A Hw) + T log det B + O(1/P).

• ΛA diagonal matrix of eigenvalues.
• The matrix QX ∈ CT×M, T ≥ M.
• An expression for h(QXDΛ1/2

A Hw) can be obtained by

transforming from Cartesian to QR coordinates (Zheng et al. ’02).

ITW Jeju, South Korea, 2015 12 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

• Coordinate change yields

h(QXDΛ1/2

A Hw) = h(ΨDΛ1/2 A Hw)

+ log |GM(CT)| + (T − M) E{log det H†

wD2ΛAHw}.

• GM(CT) is the Grassmann manifold; and
• Ψ ∈ CM×M isotropically distributed.
• Computing h(ΨDΛ1/2

A Hw) is the difficult part.

• Without spatial correlation ΛA = IM and optimal D = IM.
• For case with spatial correlation, we develop bounds.

ITW Jeju, South Korea, 2015 13 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Upper Bound on Capacity

• Gaussian distribution maximizes entropy yields

h(ΨDΛ1/2

A Hw) ≤ M2 log πeT

M λA1P. (1)

• Bound not achievable unless A = 1

M IM and

E{D2} = PT

M IM.

• Upper bound on capacity:

C(P) ≤ M

• 1 − M

T

• log TP

πeM +

• 1 − 2M

T

• log det A

+ M2 T log λA1 + 1 T log |GM(CT)| +

• 1 − M

T

• E{log det BHwH†

w} + O(1/P).

ITW Jeju, South Korea, 2015 14 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Lower Bound on Capacity

• Restricting D to particular distribution yields lower

bound on capacity.

• Set D to deterministic

D =

• TP

Tr Λ−1

A

Λ−1/2

A

• Choice ensures ΨDΛ1/2

A Hw Gaussian, i.i.d. entries

h(ΨDΛ1/2

A Hw) = M2 log πeTP

Tr Λ−1

A

.

• Lower bound on capacity:

C(P) ≥ M

• 1 − M

T

• log

TP πe Tr Λ−1

A

+ 1 T log |GM(CT)| +

• 1 − M

T

• E{log det BHwH†

w} + O(1/P).

ITW Jeju, South Korea, 2015 15 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

How tight are the bounds?

• Let gap between bounds be ∆

∆ ≤ M

• 1 − M

T

• log κA.
• κA condition number of A.

ITW Jeju, South Korea, 2015 16 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Main Result

log TP πe Tr Λ−1

A

≤ C(P) − c M(1 − M/T) ≤ log TPκA πe Tr Λ−1

A

• c =

1 T log |GM(CT)|+

• 1− M

T

• E{log det BHwH†

w}+O(1/P).

• Lower bound achieved by input signals X = QXDU†

A,

and D =

• TP

Tr Λ−1

A Λ−1/2

A

.

ITW Jeju, South Korea, 2015 17 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Comments on Results

• Rate achieved by setting D =
• TP

Tr Λ−1

A Λ−1/2

A

is within M

• 1 − M

T

• log κA bits from capacity.
• Signalling strategy optimal when channel coefficients

possibly correlated at Rx but independent at Tx; more likely in downlink scenarios.

• For channels with κA slightly greater than 1, rate loss

relatively small.

• Rate loss is unbounded as κA → ∞. Message: use

D =

• TP

Tr Λ−1

A Λ−1/2

A

• nly to excite non-negligible

eigenmodes of the channels.

ITW Jeju, South Korea, 2015 18 / 18 Ramy Gohary and Halim Yanikomeroglu Introduction System Model The right singular vectors of X Asymptotic high SNR non-coherent capacity

Bounds

Main Result Conclusions

Conclusions

• Closed form expressions for upper and lower bounds
• n ergodic non-coherent capacity of spatially correlated

MIMO systems with M transmit and receive antennas.

• Lower bound achievable using deterministic precoding.
• Gap between bounds does not depend on SNR and

increases monotonically with transmit condition number.

• Results are tight within O(1/P).