Achievable Rate and Capacity Analysis for Ambient Backscatter Communications

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In this paper, we analyze the achievable rate for ambient backscatter communications under three different channels: the binary input and binary output (BIBO) channel, the binary input and signal output (BISO) channel, and the binary input and energy output (BIEO) channel. Instead of assuming Gaussian input distribution, the proposed study matches the practical ambient backscatter scenarios, where the input of the tag can only be binary. We derive the closed-form capacity expression as well as the capacity-achieving input distribution for the BIBO channel. To show the influence of the signal-to-noise ratio (SNR) on the capacity, a closed-form tight ceiling is also derived when SNR turns relatively large. For BISO and BIEO channel, we obtain the closed-form mutual information, while the semi-closed-form capacity value can be obtained via one dimensional searching. Simulations are provided to corroborate the theoretical studies. Interestingly, the simulations show that: (i) the detection threshold maximizing the capacity of BIBO channel is the same as the one from the maximum likelihood signal detection; (ii) the maximal of the mutual information of all channels is achieved almost by a uniform input distribution; and (iii) the mutual information of the BIEO channel is larger than that of the BIBO channel, but is smaller than that of the BISO channel.