Noncoherent Capacity of Underspread Fading Channels

We derive bounds on the noncoherent capacity of wide-sense stationary uncorrelated scattering (WSSUS) channels that are selective both in time and frequency, and are underspread, i.e., the product of the channel’s delay spread and Doppler spread is small. For input signals that are peak constrained in time and frequency, we obtain upper and lower bounds on capacity that are explicit in the channel’s scattering function, are accurate for a large range of bandwidth and allow to coarsely identify the capacity-optimal bandwidth as a function of the peak power and the channel’s scattering function. We also obtain a closed-form expression for the first-order Taylor series expansion of capacity in the limit of large bandwidth, and show that our bounds are tight in the wideband regime. For input signals that are peak constrained in time only (and, hence, allowed to be peaky in frequency), we provide upper and lower bounds on the infinite-bandwidth capacity and find cases when the bounds coincide and the infinite-bandwidth capacity is characterized exactly. Our lower bound is closely related to a result by Viterbi (1967). The analysis in this paper is based on a discrete-time discrete-frequency approximation of WSSUS time- and frequency-selective channels. This discretization explicitly takes into account the underspread property, which is satisfied by virtually all wireless communication channels.

💡 Research Summary

The paper investigates the noncoherent capacity of wireless channels that are selective in both time and frequency and satisfy the underspread condition (the product of maximum delay spread and maximum Doppler spread is small). Starting from the continuous‑time wide‑sense stationary uncorrelated scattering (WSSUS) model, the authors adopt the discrete‑time, discrete‑frequency approximation introduced by Kozek, which explicitly exploits the underspread property to obtain a block‑Toeplitz representation of the channel with Toeplitz blocks. This discretization enables rigorous information‑theoretic analysis while keeping the channel description explicit in terms of the scattering function C_H(ν,τ).

Two peak‑power constraint scenarios are considered:

1. Peak constraint in both time and frequency – This models practical OFDM‑type signaling where each time–frequency slot is limited in amplitude. Upper and lower bounds on capacity are derived that hold for any bandwidth. The lower bound is obtained by a flash‑signaling scheme (sparse, high‑power symbols placed on a time‑frequency grid) and uses the average singular‑value distribution of the channel matrix. The upper bound follows from maximizing mutual information under the same constraints and relies on Szegő’s theorem for two‑level Toeplitz matrices to evaluate the asymptotic eigenvalue distribution. Both bounds are expressed directly through integrals of the scattering function, allowing one to compute them for measured channels. From these bounds the authors extract a first‑order Taylor expansion of capacity for large bandwidth:

   C(B) ≈ α B − β B² + o(B²),

   where α depends on the total scattered power and the peak‑to‑average power ratio, while β involves the second‑order moments of the scattering function. The expression predicts a capacity‑optimal bandwidth B_opt ≈ √(α/2β); beyond this point, increasing bandwidth reduces capacity because the peak constraint forces the signal to be spread too thinly.

2. Peak constraint only in time (frequency may be “peaky”) – Here the focus is on the infinite‑bandwidth limit. The authors revisit Viterbi’s 1967 result, showing that a flash‑signaling scheme that concentrates energy in arbitrarily narrow frequency bands attains an infinite‑bandwidth capacity equal to the AWGN capacity minus a penalty term that depends on the integral of the scattering function. Using the recent mutual‑information–MMSE relationship (Guo, Shamai, Verdú) and an information‑divergence property of orthogonal signaling (Butman & Klass), they derive an upper bound that coincides with Viterbi’s lower bound for a class of channels whose scattering function has rectangular support (i.e., uniform power over a bounded delay‑Doppler region). Consequently, for these channels the exact infinite‑bandwidth capacity is obtained:

   C_∞ = (P/N₀) ·