Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms

We present five theorems concerning the asymptotic convergence rates of Chebyshev interpolation applied to functions transplanted to either a semi-infinite or an infinite interval under exponential or double-exponential transformations. This strategy is useful for approximating and computing with functions that are analytic apart from endpoint singularities. The use of Chebyshev polynomials instead of the more commonly used cardinal sinc or Fourier interpolants is important because it enables one to apply maps to semi-infinite intervals for functions which have only a single endpoint singularity. In such cases, this leads to significantly improved convergence rates

Incorporation of macroscopic heterogeneity within a porous layer to enhance its acoustic absorptance

We seek the response, in particular the spectral absorptance, of a rigidly-backed periodically-(in one horizontal~~ direction) ~inhomogeneous ~layer ~composed ~of ~alternating rigid and macroscopically-homogeneous porous portions, submitted to an airborne acoustic plane body wave. The rigorous theory of this problem is given and the means by which the latter can be numerically solved are outlined. At low frequencies, a suitable approximation derives from one linear equation in one unknown. This approximate solution is shown to be equivalent to that of the problem of the same wave incident on a homogeneous, isotropic layer. The thickness $h$ of this layer is identical to that of the inhomogeneous layer, the effective complex body wave velocity therein is identical to that of the porous portion of the inhomogeneous layer, but the complex effective mass density, whose expression is given in explicit algebraic form, is that of the reference homogeneous macroscopically-porous layer divided by the filling factor (fraction of porous material to the total material in one grating period). This difference of density is the reason why it is possible for the lowest-frequency absorptance peak to be higher than that of a reference layer. Also, it is shown how to augment the height of this peak so that it attains unity (i.e., total absorption) and how to shift it to lower frequencies, as is required in certain applications

On the Performance Limits of Pilot-Based Estimation of Bandlimited Frequency-Selective Communication Channels

In this paper the problem of assessing bounds on the accuracy of pilot-based estimation of a bandlimited frequency selective communication channel is tackled. Mean square error is taken as a figure of merit in channel estimation and a tapped-delay line model is adopted to represent a continuous time channel via a finite number of unknown parameters. This allows to derive some properties of optimal waveforms for channel sounding and closed form Cramer-Rao bounds

Unified Capacity Limit of Non-coherent Wideband Fading Channels

In non-coherent wideband fading channels where energy rather than spectrum is the limiting resource, peaky and non-peaky signaling schemes have long been considered species apart, as the first approaches asymptotically the capacity of a wideband AWGN channel with the same average SNR, whereas the second reaches a peak rate at some finite critical bandwidth and then falls to zero as bandwidth grows to infinity. In this paper it is shown that this distinction is in fact an artifact of the limited attention paid in the past to the product between the bandwidth and the fraction of time it is in use. This fundamental quantity, called bandwidth occupancy, measures average bandwidth usage over time. For all signaling schemes with the same bandwidth occupancy, achievable rates approach to the wideband AWGN capacity within the same gap as the bandwidth occupancy approaches its critical value, and decrease to zero as the occupancy goes to infinity. This unified analysis produces quantitative closed-form expressions for the ideal bandwidth occupancy, recovers the existing capacity results for (non-)peaky signaling schemes, and unveils a trade-off between the accuracy of approximating capacity with a generalized Taylor polynomial and the accuracy with which the optimal bandwidth occupancy can be bounded.Comment: Accepted for publication in IEEE Transactions on Wireless Communications. Copyright may be transferred without notic