SAMPLING OF SPARSE CHANNELS WITH COMMON SUPPORT Yann Barbotin † Ali Hormati † Sundeep Rangan ‡ Martin Vetterli † † School of Computer and Communication Sciences, Ecole Polytechnique F´ ed´ erale de Lausanne, 1015 Lausanne, Switzerland ‡ Polytechnic Institute of New York University, Brooklyn, NY email : yann.barbotin@epfl.ch ali.hormati@epfl.ch srangan@poly.edu martin.vetterli@epfl.ch web : http://lcavwww.epfl.ch/∼{barbotin, hormati, vetterli}, http://eeweb.poly.edu/∼srangan ABSTRACT The present paper proposes and studies an algorithm to estimate chan- nels with a sparse common support (SCS). It is a generalization of the classical sampling of signals with Finite Rate of Innovation (FRI) [1] and thus called SCS-FRI. It is applicable to OFDM and Walsh- Hadamard coded (CDMA downlink) communications since SCS-FRI is shown to work not only on contiguous DFT pilots but also uniformly scattered ones. The support estimation performances compare favorably to theoretical lower-bounds, and importantly this translates into a sub- stantial equalization gain at the receiver compared to the widely used spectrum lowpass interpolation method. Keywords— Channel estimation, MIMO, OFDM, CDMA, Finite Rate of Innovation 1. INTRODUCTION Modern communication devices have seen their number of antennas cropping up. The rationals behind multiple output systems are linked to the physical properties of the electromagnetic (EM) multipath chan- nel [2]: different antennas witness diferent channel conditions. Under this assumption, it seems natural to estimate each of these channels, and then select the best one or a combination to achieve greater capacity than a single output system. Most multi-output EM multipath channels have a Sparse Common Support (SCS) property, i.e. the paths’ Time of Arrivals (ToA) are the same for every output up to a small error ≤ ε. Under this assumption, we will outline a Finite Rate of Innovation (FRI) sampling [3] based al- gorithm which takes advantage of the SCS property, conveniently called SCS-FRI [4, 5]. Compared to other algorithms which also try to esti- mate the channels from a subset of Fourier “probes” (such as lowpass interpolation of the channel spectrum), SCS-FRI has four main advan- tages. First, parametric estimation allows for joint recovery of the sup- port common to the multiple outputs, independently of the paths am- plitudes. Second, the number of probes used to sense the channel can be reduced, saving precious bandwidth for data transmition. Third, for an equal number of probes, channel estimation is more robust to noise corruption, yielding a higher equalization gain. Last but not least, the channel estimate is characterized by a very small set of parameters, sav- ing some bandwidth for multiple input systems using non-blind transmit diversity techniques [6], e.g. beamforming. SCS channels in a discrete time setting were previously studied in [7] within the compressed sens- ing framework. We will first describe and motivate the SCS channel model and pro- ceed with the description of the SCS-FRI algorithm. This algorithm M. Vetterli is also with the Department of Electrical Engineering and Com- puter Sciences, University of California, Berkeley, CA 94720, USA. This re- search is supported by Qualcomm Inc., ERC Advanced Grant Support for Fron- tier Research - SPARSAM Nr : 247006 and SNF Grant - New Sampling Methods for Processing and Communication Nr : 200021-121935/1. t 1 ±ε t 2 ±ε Channel #1 Channel #P !!! !!! !!! Scatterer #1 Scatterer #2 Fig. 1. (a) Transmission over a medium with two scatterers and P re- ceiving antennas. (b) The P channels contain two paths arriving at the same time up to ±ε. The amplitudes of paths from a scatterer are (pos- sibly correlated) Rayleigh variates [9]. uses baseband DFT pilots (probes) to solve the aforementioned estima- tion problem. Within this setup, we will derive Cram´ er-Rao Bounds (CRB) on the support estimation for both deterministic and Rayleigh fading channels. We will then show that SCS-FRI is not restricted to baseband DFT pilots, and works equivalently well in the uniformely spaced (“scat- tered”) DFT pilots layout ubiquitous in OFDM based communications. Like other channel estimation techniques based on scattered pilots, the only requirement is for the channel impulse response (CIR) to be rela- tively short compared to the symbol duration. Then we investigate the efficient use of other probing sequences. It is shown that they must have the same span as a subset of DFT ba- sis vectors to warrant the use of a DFT pilot based algorithm. Inter- estingly, the set of Walsh-Hadamard sequences, used in most CDMA based standards, verifies this property with the added benefit of provid- ing uniformely scattered DFT pilots. This enables the use of SCS-FRI on CDMA downlink channels as if they were OFDM coded channels. All these equivalences allow to use the CRB derived in Section 4. We conclude the study with numerical results showing the efficiency of the SCS-FRI algorithm in a multi-output Rayleigh fading OFDM setup, and the potential equalization gain compared to the industry stan- dard which is lowpass interpolation of the CIR spectrum. 2. PROBLEM FORMULATION Consider a bandpass channel of bandwidth B. The inverse bandwidth 1/B sets a limit on the distance at which two pulses of bandwidth B