25 research outputs found
Waterfilling Theorems for Linear Time-Varying Channels and Related Nonstationary Sources
The capacity of the linear time-varying (LTV) channel, a continuous-time LTV
filter with additive white Gaussian noise, is characterized by waterfilling in
the time-frequency plane. Similarly, the rate distortion function for a related
nonstationary source is characterized by reverse waterfilling in the
time-frequency plane. Constraints on the average energy or on the squared-error
distortion, respectively, are used. The source is formed by the white Gaussian
noise response of the same LTV filter as before. The proofs of both
waterfilling theorems rely on a Szego theorem for a class of operators
associated with the filter. A self-contained proof of the Szego theorem is
given. The waterfilling theorems compare well with the classical results of
Gallager and Berger. In the case of a nonstationary source, it is observed that
the part of the classical power spectral density is taken by the Wigner-Ville
spectrum. The present approach is based on the spread Weyl symbol of the LTV
filter, and is asymptotic in nature. For the spreading factor, a lower bound is
suggested by means of an uncertainty inequality.Comment: 13 pages, 5 figures; channel model in Section III now restricted to
LTV filters with real-valued kerne
On the Szeg\"o-Asymptotics for Doubly-Dispersive Gaussian Channels
We consider the time-continuous doubly-dispersive channel with additive
Gaussian noise and establish a capacity formula for the case where the channel
correlation operator is represented by a symbol which is periodic in time and
fulfills some further integrability and smoothness conditions. The key to this
result is a new Szeg\"o formula for certain pseudo-differential operators. The
formula justifies the water-filling principle along time and frequency in terms
of the time--continuous time-varying transfer function (the symbol).Comment: 5 pages, to be presented at ISIT 2011, minor typos corrected,
references update
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.Comment: Submitted to the IEEE Transactions on Information Theor
On the Sensitivity of Noncoherent Capacity to the Channel Model
The noncoherent capacity of stationary discrete-time fading channels is known
to be very sensitive to the fine details of the channel model. More
specifically, the measure of the set of harmonics where the power spectral
density of the fading process is nonzero determines if capacity grows
logarithmically in SNR or slower than logarithmically. An engineering-relevant
problem is to characterize the SNR value at which this sensitivity starts to
matter.
In this paper, we consider the general class of continuous-time
Rayleigh-fading channels that satisfy the wide-sense stationary
uncorrelated-scattering (WSSUS) assumption and are, in addition, underspread.
For this class of channels, we show that the noncoherent capacity is close to
the AWGN capacity for all SNR values of practical interest, independently of
whether the scattering function is compactly supported or not. As a byproduct
of our analysis, we obtain an information-theoretic pulse-design criterion for
orthogonal frequency-division multiplexing systems.Comment: To be presented at IEEE Int. Symp. Inf. Theory 2009, Seoul, Kore
Information Theory of underspread WSSUS channels
The chapter focuses on the ultimate limit on the rate of reliable communication through Rayleigh-fading channels that satisfy the wide-sense stationary (WSS) and uncorrelated scattering (US) assumptions and are underspread. Therefore, the natural setting is an information-theoretic one, and the performance metric is channel capacity. The family of Rayleigh-fading underspread WSSUS channels constitutes a good model for real-world wireless channels: their stochastic properties, like amplitude and phase distributions match channel measurement results. The Rayleigh-fading and the WSSUS assumptions imply that the stochastic properties of the channel are fully described by a two-dimensional power spectral density (PSD) function, often referred to as scattering function. The underspread assumption implies that the scattering function is highly concentrated in the delay-Doppler plane. Two important aspects need to be accounted for by a model that aims at being realistic: neither the transmitter nor the receiver knows the realization of the channel; and the peak power of the transmit signal is limited. Based on these two aspects the chapter provides an information-theoretic analysis of Rayleigh-fading underspread WSSUS channels in the noncoherent setting, under the additional assumption that the transmit signal is peak-constrained
On the Sensitivity of Continuous-Time Noncoherent Fading Channel Capacity
The noncoherent capacity of stationary discrete-time fading channels is known
to be very sensitive to the fine details of the channel model. More
specifically, the measure of the support of the fading-process power spectral
density (PSD) determines if noncoherent capacity grows logarithmically in SNR
or slower than logarithmically. Such a result is unsatisfactory from an
engineering point of view, as the support of the PSD cannot be determined
through measurements. The aim of this paper is to assess whether, for general
continuous-time Rayleigh-fading channels, this sensitivity has a noticeable
impact on capacity at SNR values of practical interest.
To this end, we consider the general class of band-limited continuous-time
Rayleigh-fading channels that satisfy the wide-sense stationary
uncorrelated-scattering (WSSUS) assumption and are, in addition, underspread.
We show that, for all SNR values of practical interest, the noncoherent
capacity of every channel in this class is close to the capacity of an AWGN
channel with the same SNR and bandwidth, independently of the measure of the
support of the scattering function (the two-dimensional channel PSD). Our
result is based on a lower bound on noncoherent capacity, which is built on a
discretization of the channel input-output relation induced by projecting onto
Weyl-Heisenberg (WH) sets. This approach is interesting in its own right as it
yields a mathematically tractable way of dealing with the mutual information
between certain continuous-time random signals.Comment: final versio