On the Scaling Exponent of Polar Codes for Binary-Input Energy-Harvesting Channels

This paper investigates the scaling exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of independent and identically distributed random variables with finite variances. The scaling exponent μ of polar codes for a binary-input memoryless channel (BMC) qY|X with capacity C(qY|X) characterizes the closest gap between the capacity and the non-asymptotic achievable rates in the following way. For a fixed average error probability e ∈ (0, 1), the closest gap between the capacity C(qY|X) and a non-asymptotic achievable rate Rn for a length-n polar code scales as n-1/μ, i.e., min{|C(qY|X) - Rn|} = O(n-1/μ). It has been shown that the scaling exponent μ for any binary-input memoryless symmetric channel with C(qY|X) ∈ (0, 1) lies between 3.579 and 4.714, where the upper bound 4.714 was shown by an explicit construction of polar codes. Our main result shows that 4.714 remains to be a valid upper bound on the scaling exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. An auxiliary contribution of this paper is that the upper bound on μ holds for binary-input memoryless asymmetric channels.