The context of this paper is adaptive waveform design for estimating parameters of an unknown channel under average energy constraints. This paper focuses on the simpler problem of adaptive waveformamplitude design for which we obtain interesting analytical results. We treat an N-step design problem where a fixed waveform can be transmitted into the channel N times with amplitudes that can be chosen as a function of past channel outputs. For N = 2 and a linear Gaussian channel model, we derive the optimal amplitude to transmit at the second step as a function of the first measurement. This adaptive 2-step energy allocation strategy gives a mean-squared error (MSE) improvement of at least 1.7dB relative to the optimal non-adaptive strategy. Motivated by the optimal two-step strategy we propose a suboptimal adaptive N-step strategy that can achieve an MSE improvement of more than 5dB for N = 50. Applications of our results to MIMO and inverse scattering channel models are discussed. Index Terms — Parameter estimation, adaptive control, energy allocation, maximum likelihood, MMSE
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