Low SNR Capacity of Noncoherent Fading Channels

Discrete-time Rayleigh fading single-input single-output (SISO) and multiple-input multiple-output (MIMO) channels are considered, with no channel state information at the transmitter or the receiver. The fading is assumed to be stationary and correlated in time, but independent from antenna to antenna. Peak-power and average-power constraints are imposed on the transmit antennas. For MIMO channels, these constraints are either imposed on the sum over antennas, or on each individual antenna. For SISO channels and MIMO channels with sum power constraints, the asymptotic capacity as the peak signal-to-noise ratio tends to zero is identified; for MIMO channels with individual power constraints, this asymptotic capacity is obtained for a class of channels called transmit separable channels. The results for MIMO channels with individual power constraints are carried over to SISO channels with delay spread (i.e. frequency selective fading).

💡 Research Summary

The paper investigates the capacity of non‑coherent Rayleigh fading channels in the low‑signal‑to‑noise‑ratio (SNR) regime, considering both peak‑power and average‑power constraints at the transmitter. The authors treat four settings: (i) a single‑input single‑output (SISO) channel, (ii) a multiple‑input multiple‑output (MIMO) channel with a sum‑power constraint across the transmit antennas, (iii) a MIMO channel with individual power constraints per antenna, and (iv) a SISO channel with delay spread (frequency‑selective fading).

For the SISO model the received signal is
 Yₖ = √ρ Hₖ Zₖ + Wₖ,
where Zₖ is the channel input, ρ is the SNR scaling factor, Hₖ is a zero‑mean unit‑variance stationary proper complex Gaussian fading process with autocorrelation R(·) and spectral density S(·), and Wₖ is i.i.d. unit‑variance complex Gaussian noise. The input must satisfy a hard peak constraint |Zₖ| ≤ 1 (with probability one) and an average‑power constraint E|Zₖ|² ≤ 1/β, where β ≥ 1 is the peak‑to‑average ratio.

A key parameter is
 λ = Σ_{ν = –∞}^{∞} |R(ν)|² = ∫{–π}^{π} |S(ω)|² dω / (2π).
The low‑SNR asymptotic capacity is defined as
 c(β) = lim{ρ→0} C(ρ,β) / ρ².
The authors prove that this limit exists for every β and is given by

 c(β) = ½ · max_{0 ≤ a ≤ 1}