Manifold Signal Processing

Wireless communication has already become the dominant means of Internet access. Unfortunately, the speeds of wireless access remain far behind expectations mostly because of spectrum scarcity and fading in wireless channels. To resolve this conflict, next-generation wireless networks will employ advanced technologies from multiuser and multi-antenna communication to dramatically increase data rates. Key challenges in implementing such technologies include adapting transmission to a time-varying channel, coping with multiuser interference, and jointly optimizing a large set of system parameters. Fortunately, many sophisticated signal processing problems in multiuser and multi-antenna communication contain mathematical structures known as a manifold, and hence can be solved by using tools from the mathematical field of different geometry.

Manifolds are topological spaces under constraints; simple examples include circles, spheres, and tori. Transformations to specific manifolds provide an effective way of reducing the dimensions of a large data set, and hence find applications in e.g. data mining and machine learning. Recently, the Grassmann and Stiefel manifolds have been applied successfully in wireless communications. These two types of manifolds are useful for limited feedback multi-antenna precoding and beamforming, channel subspace estimation and tracking, and non-coherent space-time modulation.

Despite work in diverse fields, a comprehensive set of algorithmic and analysis tools do not yet exist for solving signal processing problems on the Grassmann and Stiefel manifolds. With my research group and collaborators, I have been developing signal processing tools for the Grassmann and Stiefel manifolds, and applying these tools to problems in wireless communication.This is challenging from a signal processing perspective because time series on the manifold have mathematical structure that must be incorporated into the algorithms. We are looking at fundamental questions like how to estimate, filter, predict, and interpolate manifold-valued signals? We are applying these results to develop efficient limited feedback techniques for single user MIMO, multiuser MIMO, network MIMO, and interference channels.

Here I describe some related publications. Some are found on other pages. Additional publications are found on my CV.

**Prediction and Predictive Quantization**

R. Bhagavatula and R. W. Heath, Jr., “Predictive Limited Feedback for Cooperative Transmission,” (invited) *Proc. of the* *IEEE Asilomar Conf. on Signals, Systems, and Computers*, Pacific Grove, CA, November 7-10, 2010. Video of presentation.

T. Inoue and R. W. Heath, Jr., “Grassmannian predictive frequency domain compression for limited feedback beamforming,” (invited) *Proc. of the Information Theory and Applications Workshop*, San Diego, CA, February 8-13, 2010, pp. 1-5.

T. Inoue and R. W. Heath, Jr., “Grassmannian Predictive Coding for Delayed Limited Feedback MIMO Systems,” (invited) *Proc. of the*Allerton Conf. on Comm. Control and Comp., Monticello, IL, pp. 783-788, October 2009.

T. Inoue and R. W. Heath, Jr., “Geodesic Prediction for Limited Feedback Multiuser MIMO Systems in Temporally Correlated Channels“IEEE Radio & Wireless Symposium, pp. 167-170, Jan. 18-22, 2009.

In these papers we proposed and developed several algorithms for first and second order prediction on the Grassmann manifold. This was enabled by using the fact that two points on the Grassmann manifold can be represented through a tangent and a geodesic. We used these proposed predictors for predictive quantization in multiuser MIMO systems, multiuser MIMO-OFDM systems, and cooperative feedback. Using prediction we are able to employ temporal correlation and dramatically reduce feedback requirements.

**Interpolation**

J. Choi, B. Mondal, and R. W. Heath, Jr., “Interpolation Based Unitary Precoding for Spatial Multiplexing MIMO-OFDM with Limited Feedback,” *IEEE Trans. on Signal Processing*, vol. 54, no. 12, pp. 4730-4740, December 2006.

J. Choi and R. W. Heath, Jr., “Interpolation Based Transmit Beamforming for MIMO-OFDM with Limited Feedback,” *IEEE Trans. on Signal Processing*, vol. 53, no. 11, pp. 4125-4135, Nov. 2005.

In these papers we developed some ad hoc algorithms for interpolation on the Grassmann manifold. The algorithms incorporate an additional phase term into a spherical linear interpolator. We applied these algorithms to reduce feedback requirements in MIMO-OFDM limited feedback systems.

N. Khaled, B. Mondal, R. W. Heath, Jr., G. Leus, and F. Petre, “Interpolation- Based Multi-Mode Precoding for MIMO-OFDM Systems with Limited Feedback,” *IEEE Trans. on Wireless*, vol. 6., no. 3, pp. 1003-1013, March 2007.

In this paper we developed an algorithm for interpolation on the Stiefel manifold. We applied this algorithm to interpolate a sequence of right singular vector matrices for the purpose of multi-mode precoding in a single user MIMO-OFDM system. A sequence of right singular matrices was fed back along with an optimum number of modes for groups of subcarriers. The proposed algorithm reduced feedback requirements an enable multi-mode link adaptation where the number of streams are varied for groups of subcarriers.

**Quantization **(see also single user limited feedback)

B. Mondal, S. Dutta, and R. W. Heath, Jr., “Quantization on the Grassmann Manifold,” *IEEE Trans. on Signal Processing*, vol. 55, no. 8, pp. 4208-4216, Aug. 2007.

D. J. Love, R. W. Heath, Jr., and T. Strohmer, “Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems,” *IEEE Trans. on Info. Theory* special issue on MIMO Communication, vol. 49, pp. 2735-2747, Oct. 2003.

In these papers we established the connecting between limited feedback beamforming and quantization on the Grassmann manifold. We also derived asymptotic lower bounds on the distortion rate function for quantization on the Grassmann manifold, specifically extending Gersho’s asymptotic (large rate, small distortion) distortion bounds to the case when the source is distributed on the complex Grassmann manifold.

Awards

Received the best student paper award at the 2009 IEEE Radio and Wireless Symposium:

T. Inoue and R. W. Heath, Jr., “Geodesic Prediction for Limited Feedback Multiuser MIMO Systems in Temporally Correlated Channels” IEEE Radio & Wireless Symposium, pp. 167-170, Jan. 18-22, 2009.

Sponsors

We have been fortunate to have several sponsors of our work on manifold signal processing including at present the National Science Foundation through grant NSF-CCF-0830615 and previously NSF-CCF-0514194. We have also had several industrial sponsors in the past including Motorola. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.