Rajesh T. Krishnamachari

Interference alignment under limited feedback for MIMO interference channels

This paper analyzes multiple-input, multiple-output interference channels where each receiver knows its channels from all the transmitters and feeds back this information using a limited number of bits to all the other terminals. It is shown that as long as the feedback bit rate scales sufficiently fast with the signal-to-noise ratio, the transmitters can use an interference alignment strategy by treating the quantized channel estimates as being perfect to achieve the sum degrees of freedom of the interference channel attainable with perfect and global channel state information. A tradeoff between the feedback rate and the achievable degrees of freedom is established by showing that a slower scaling of feedback rate for any one user leads to commensurately fewer degrees of freedom for that user alone. It is then shown that under the same fixed transmission strategy but with random quantization, the above mentioned sufficient condition on the feedback scaling rate to attain a given sum degrees of freedom (up to the maximum attainable) is also necessary in this setting.

Interference Alignment Under Limited Feedback for MIMO Interference Channels

This paper analyzes multiple-input, multiple-output interference channels where each receiver knows its channels from all the transmitters and feeds back this information using a limited number of bits to all the other terminals. It is shown that as long as the feedback bit rate scales sufficiently fast with the signal-to-noise ratio, the transmitters can use an interference alignment strategy by treating the quantized channel estimates as being perfect to achieve the sum degrees of freedom of the interference channel attainable with perfect and global channel state information. A tradeoff between the feedback rate and the achievable degrees of freedom is established by showing that a slower scaling of feedback rate for any one user leads to commensurately fewer degrees of freedom for that user alone. It is then shown that under the same fixed transmission strategy but with random quantization, the above mentioned sufficient condition on the feedback scaling rate to attain a given sum degrees of freedom (up to the maximum attainable) is also necessary in this setting.

On the Geometry and Quantization of Manifolds of Positive Semi-Definite Matrices

The geometry of different spaces of positive semi-definite matrices buffeted by rank and trace constraints is studied. In addition to revealing their Riemannian structure, we derive the normalized volume of a ball over these spaces. Further, we use the leading coefficient from the ball volume expansion to bound the quantization error incurred with finite-sized sphere-packing codebooks as well as random codebooks to represent sources distributed over general Riemannian manifolds.

Volume of geodesic balls in the complex Stiefel manifold

Volume estimates of geodesic balls in Riemannian manifolds find many applications in coding and information theory. This paper computes the precise power series expansion of volume of small geodesic balls in a complex Stiefel manifold of arbitrary dimension. The volume result is employed to bound the minimum distance of codes over the manifold. An asymptotically tight characterization of the rate-distortion tradeoff for sources uniformly distributed over the surface is also provided.

MIMO Systems With Quantized Covariance Feedback

A unified approach is developed for the study of multi-input, multi-output (MIMO) systems under fast fading with quantized covariance feedback. In such systems, the receiver computes, using perfect channel state information (CSI), the covariance matrices to be adopted by the transmitter corresponding to the current channel realization, and feeds back a quantized version of this information to the transmitter using finite, say Nf, bits per channel realization. Our analysis is based on a geometric framework we developed in a companion paper for manifolds of positive semi-definite (covariance) matrices with various trace and rank constraints. That analysis applies to MIMO systems in a unified manner to optimal as well as various suboptimal precoding methods for obtaining input covariances. These include, for example, the capacity achieving spatio-temporal water-filling strategy, the incorporation of just spatial water-filling and reduced-rank beamforming with power allocation in both cases, as well as MIMO systems with antenna selection. For a given system strategy, including the one that achieves capacity, the gap between the communication rates in the perfect and quantized covariance cases is shown to be O(2-Nf/N), where N is the dimension of the particular Pn manifold used for quantization. This dimension depends on the precoding strategy adopted and strongly determines how quickly the communication rate with quantized covariance feedback approaches that with perfect CSI at the transmitter (CSIT). Our results show how the choice of covariance quantization manifold can be tailored to the available feedback rate in MIMO system design.

MIMO performance under covariance matrix feedback

We study the finite-rate feedback of optimal input covariance matrices from the channel-aware receiver to the transmitter in a multi-antenna single-user setup. Under a block fading model for the channel matrix, the receiver computes the covariance matrix corresponding to the current channel realization and feeds back information about it using a finite, say Nf, number of bits per block. Our finite-rate feedback analysis is based on a novel geometric paradigm whereby the feedback information is modeled as a source distributed over a new Riemannian surface called the Pn manifold. For a given system strategy, the gap between the achievable rates in the infinite and finite-rate feedback cases is shown to be O(2 − Nf over N), where N is the dimension of the Pn manifold used for quantization.

Volume of geodesic balls in the real Stiefel manifold

Volume estimates of balls in Riemannian manifolds find many applications in information theory such as in the determination of the rate-distortion trade off for the quantization of a source randomly distributed over such surfaces. This paper computes the precise power series expansion of volume of small geodesic balls in a real Stiefel manifold of arbitrary dimension. An application of the result obtained is also demonstrated.

Finite-rate feedback of input covariance matrices in MIMO systems

We analyze feedback of the optimal input covariance matrix Q from a channel-aware multi-antenna receiver to a multi-antenna transmitter using Nf bits per block. The matrix Q is allowed to have real or complex entries and its rank is allowed to be either fixed or variable. By unravelling the geometry of the quantization spaces involved, we obtain the normalized volume of geodesic balls and use these to evaluate the distortion suffered in quantizing information using different code books. The difference in ergodic capacity between the finite and infinite rate feedback cases is bounded as O(2− Nf over N) where N is the dimension of the quantization manifold. The results apply to both the MIMO link and the MIMO MAC channels, and do not depend on the specific distribution of the channel matrix.

Distortion-rate tradeoff of a source uniformly distributed over positive semi-definite matrices

We study the distortion-rate tradeoff D(K) of a source uniformly distributed over a manifold comprising of positive semi-definite matrices - over both real and complex fields - under a trace constraint. This models the impact exercised by limited-rate feedback on the information transfer of optimal dasiainput covariance matricespsila to a transmitter from a channel-aware receiver in a multiple-input mulitple-output (MIMO) system. Utilizing the volume of a geodesic ball in the manifold, D(K) is bounded between asymptotically tight lower and upper limits obtained via sphere-packing and random coding arguments, respectively.

Distortion-rate tradeoff of a source uniformly distributed over the Composite P F (N) and the composite stiefel manifolds

To model the benefit accrued due to limited rate feedback on the information transfer of optimal ‘input covariance matrices’ from a channel-aware receiver to the transmitting users in a multiple-access channel, we study the distortion-rate tradeoff of a source uniformly distributed over multivariate generalizations of P<inf>F</inf>(n,p²) (the set of positive semi-definite matrices with a trace constraint) and V<inf>n,k</inf>^ℂ (the classical Stiefel surface) manifolds. Using sphere-packing and random coding arguments, the distortion-rate function is bounded within asymptotically tight limits.