2216 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 6, JUNE 2005 Communicating Over Nonstationary Nonflat Wireless Channels Karin Sigloch, Member, IEEE, Michael R. Andrews, Partha P. Mitra, and David J. Thomson, Fellow, IEEE Abstract—We develop the concept of joint time-frequency esti- mation of wireless channels. The motivation is to optimize channel usage by increasing the signal-to-noise ratio (SNR) after demodula- tion while keeping training overhead at a moderate level. This issue is important for single-input single-output (SISO) and multiple- input multiple-output (MIMO) systems but particularly so for the latter. Linear operators offer a general mathematical framework for symbol modulation in channels that vary both temporally and spectrally within the duration and bandwidth of one symbol. In particular, we present a channel model that assumes first-order temporal and spectral fluctuations within one symbol or symbol block. Discrete prolate spheroidal sequences (Slepian sequences) are used as pulse-shaping functions. The channel operator in the Slepian basis is almost tridiagonal, and the simple intersymbol in- terference pattern can be exploited for efficient and fast decoding using Viterbi’s algorithm. To prove the concept, we use the acoustic channel as a meaningful physical analogy to the radio channel. In acoustic 2 2 MIMO experiments, our method produced estima- tion results that are superior to first-order time-only, frequency- only, and zeroth-order models by 7.0, 9.4, and 11.6 db. In computer simulations of cellular wireless channels with realistic temporal and spectral fluctuations, time-frequency estimation gains us 12 to 18 db over constant-only estimation in terms of received SNR when signal-to-receiver-noise is 10 to 20 db. The bit error rate (BER) de- creases by a factor of two for a binary constellation. Index Terms—Channel estimation, discrete prolate spheroidal sequences, MIMO, modulation, nonflat, nonstationary, Slepian, time-frequency. I. MOTIVATION W E PROPOSE a new modulation scheme for wireless channels that vary both temporally and spectrally. It is a computationally feasible method that can estimate temporal and spectral variations within one symbol while using only a modest number of extra training sequences. Generally, one can expect to raise the signal-to-noise ratio (SNR) after demod- ulation by applying more realistic modeling to the physical layer. We suggest that systems that vary over both time and Manuscript received July 25, 2003; revised May 6, 2004. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Tulay Adali. K. Sigloch was with the Bell Laboratories, Lucent Technologies, Murray Hill, NJ. She is now with Princeton University, Princeton, NJ 08840 USA (e-mail: sigloch@princeton.edu). M. R. Andrews was with the Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974-0636 USA. He is now with the Flarion Technologies, Bedmin- ster, NJ 07921 USA (e-mail: mikea@xoba.com). P. P. Mitra was with the Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974-0636 USA. He is now with the Cold Spring Harbor Laboratory, Cold Spring Harbor, NY 11724 USA (e-mail: mitra@cshl.edu). D. J. Thomson was with the Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974-0636 USA. He is now with the Queen’s University, Kingston, ON, K7L 3N6 Canada (e-mail: djt@mast.queensu.ca). Digital Object Identifier 10.1109/TSP.2005.847849 Fig. 1. Local regression of a general, continuous time series using (bottom) zeroth-order polynomials or (top) first-order polynomials. Approximation errors are shaded. Note that globally, the same number of parameters are needed in each case: one per interval for constant segment approximation and two per (namely, constant part and slope) for first-order polynomials. The second method results in smaller errors because it takes into account the existing temporal fluctuations. frequency (such as wireless channels) can be approximated most efficiently with models that allow for variations in both of those dimensions. By most efficient, we mean that the overall number of parameters (training sequences or “pilots” in wireless) needed to achieve a certain estimation accuracy or SNR is smaller than if one uses models that allow for variation of either only time, only frequency, or neither. This concept is related to linear regression when a general, continuous time series is to be approximated locally by polynomials of low order. Successively higher order polynomials use more parameters per segment but can compensate by approximating the time series more accurately over longer segments. Fig. 1 illustrates the principle for the two simplest functions: zeroth- and first-order polynomials. The overall number of parameters is the same in both cases: In the lower plot, we estimate one constant coefficient per interval ; in the top, a constant and a time slope coefficient need to be estimated every . Since varies smoothly over time, the piecewise linear model takes time variation into account and yields a smaller total error. In wireless communications, the polynomial function for a given segment corresponds to the channel model, the accuracy corresponds to errors in fitting the channel, and the number of parameters to be estimated is directly proportional to the number of pilot symbols. The curve in Fig. 1 can be thought of as a temporally varying wireless channel. Since wireless channels also vary in frequency, we should add a frequency axis perpendicular to the two existing axes. Channel variation would 1053-587X/$20.00 © 2005 IEEE