Abstract
This paper addresses the optimal signaling scheme and capacity of an additive Gaussian mixture (GM) noise channel using 1-bit analog-to-digital converters (ADCs). The consideration of GM noise provides a more realistic baseline for the analysis and design of co-channel interference links and networks. Towards that goal, we first show that the capacityachieving input signal is π/2 circularly symmetric. By examining a necessary and sufficient Kuhn–Tucker condition (KTC) for an input to be optimal, we demonstrate that the maximum number of optimal mass points is four. Our proof relies on Dubin’s theorem and the fact that the KTC coefficient is positive, i.e., the power constraint is active. By combining with the π/2 circularly symmetric property, it is then concluded the optimal input is unique, and it has exactly four mass points forming a square centered at the origin. By further checking the first and second derivatives of the modified KTC, it is then shown that the phase of the optimal mass point located in the first quadrant is π/4. Thus, the capacity-achieving input signal is QPSK. This result helps us obtain the channel capacity in closed-form.
Highlights
As noted in [1, 2], current information and communication technology (ICT) sector that relies heavily on mobile applications is responsible for 3% of the total worldwide energy consumption, which accounts for 2% to 2.5% of the total CO2 emission in the world
We extend the capacity analysis and optimal signaling design for 1-bit analog-to-digital converters (ADCs) to a general complex Gaussian mixture (GM) channel
By examining a necessary and sufficient Kuhn–Tucker condition (KTC) for an input to be optimal, we demonstrate that the number of mass points in the optimal input is upper-bounded by four
Summary
As noted in [1, 2], current information and communication technology (ICT) sector that relies heavily on mobile applications is responsible for 3% of the total worldwide energy consumption, which accounts for 2% to 2.5% of the total CO2 emission in the world. Significant attention has been paid to information-theoretic aspects of 1-bit ADC channels under traditional additive white Gaussian noise (AWGN), and. The assumption of having noise plus interference as conditionally Gaussian has been widely adopted, and it has served as Rahman, Ranjbar, and Tran a basic block for extensive developments of informationtheoretic studies for both point-to-point links and multiuser networks. We extend the capacity analysis and optimal signaling design for 1-bit ADC to a general complex GM channel. 2. 1-bit ADC under GM Noise: Channel Model and Conditional Probability Density Functions (PDFs). Its probability density function (pdf) is a mixture of M complex Gaussian distributions with mean 0 and variance σ2k, 1 ≤ k ≤ M, which is given as pN (n). We can calculate the other transition probabilities as follows: W1,2
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