Doubly Selective Channel Estimation Using Superimposed Training and Exponential Bases Models
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Doubly Selective Channel Estimation Using Superimposed Training and Exponential Bases Models Jitendra K. Tugnait,1 Xiaohong Meng,1, 2 and Shuangchi He1 1 Department 2 Department
of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849, USA of Design Verification, MIPS Technologies Inc., Mountain View, CA 94043, USA
Received 1 June 2005; Revised 2 June 2006; Accepted 4 June 2006 Channel estimation for single-input multiple-output (SIMO) frequency-selective time-varying channels is considered using superimposed training. The time-varying channel is assumed to be described by a complex exponential basis expansion model (CE-BEM). A periodic (nonrandom) training sequence is arithmetically added (superimposed) at a low power to the information sequence at the transmitter before modulation and transmission. A two-step approach is adopted where in the first step we estimate the channel using CE-BEM and only the first-order statistics of the data. Using the estimated channel from the first step, a Viterbi detector is used to estimate the information sequence. In the second step, a deterministic maximum-likelihood (DML) approach is used to iteratively estimate the SIMO channel and the information sequences sequentially, based on CE-BEM. Three illustrative computer simulation examples are presented including two where a frequency-selective channel is randomly generated with different Doppler spreads via Jakes’ model. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
Consider a time-varying SIMO (single-input multiple-output) FIR (finite impulse response) linear channel with N outputs. Let {s(n)} denote a scalar sequence which is input to the SIMO time-varying channel with discrete-time impulse response {h(n; l)} (N-vector channel response at time n to a unit input at time n − l). The vector channel may be the result of multiple receive antennas and/or oversampling at the receiver. Then the symbol-rate, channel output vector is given by x(n) :=
L
h(n; l)s(n − l).
(1)
s(n) = b(n) + c(n),
l=0
In a complex exponential basis expansion representation [4] it is assumed that h(n; l) =
Equation (2) is the complex-exponential basis expansion model (CE-BEM). A main objective in communications is to recover s(n) given noisy {y(n)}. In several approaches this requires knowledge of the channel impulse response [11, 19]. In conventional training-based approaches, for time-varying channels, one has to send a training signal frequently and periodically to keep up with the changing channel [7]. This wastes resources. An alternative is to estimate the channel based solely on noisy y(n) exploiting statistical and other properties of {s(n)} [11, 19]. This is the blind channel estimation approach. More recently a superimposed trainingbased approach has been explored where one takes
Q
hq (l)e jωq n ,
(2)
q=1
where N-column vectors hq (l) (for q = 1, 2, . . . , Q) are timeinvariant. Equation (2) is a basis expansion of h(n; l) in the time variable n onto complex exponentials wi
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