Lower Bounds on Parameter Modulation–Estimation Under Bandwidth Constraints

The problem of modulating the value of a parameter onto a band-limited signal to be transmitted over a continuous-time, additive white Gaussian noise (AWGN) channel, and then estimating this parameter at the receiver, is considered. The performance is measured by the mean power-<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> error (MP<inline-formula> <tex-math notation="LaTeX">$\alpha \text{E}$ </tex-math></inline-formula>), which is defined as the worst case <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>th-order moment of the absolute estimation error. The optimal exponential decay rate of the MP<inline-formula> <tex-math notation="LaTeX">$\alpha \text{E}$ </tex-math></inline-formula> as a function of the transmission time is investigated. Two upper (converse) bounds on the MP<inline-formula> <tex-math notation="LaTeX">$\alpha \text{E}$ </tex-math></inline-formula> exponent are derived, on the basis of known bounds for the AWGN channel of inputs with unlimited bandwidth. The bounds are computed for typical values of the error moment and the signal-to-noise ratio (SNR), and the SNR asymptotics of the different bounds are analyzed. The new bounds are compared with known converse and achievability bounds, which were derived from channel coding considerations.

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