Aperture Domain Research Articles

Design of a 16-Port Shared-Arm Dipole Array for Monostatic Imaging Radar

A 16-port shared-arm dipole array (SADA) is designed and its performance verified. When constructing a 2D physical array to image a target, arranging the antenna elements in a longitudinal direction becomes problematic due to the element spacing constraint for alias-free sampling. Therefore, the proposed structure is designed such that the two dipole elements share a single arm, which leads to alias-free sampling in the aperture domain. In order to improve the performance of the SADA, each antenna arm is loaded with a constant resistive profile. The resistive loading allows each element to operate over a broad-band and radiate a temporally short pulse, while correcting the distorted beam pattern of the SADA. The resultant 16-port SADA with resistive loading is manufactured by printing the arms on an FR-4 substrate and loading them with equally spaced chip resistors. The performance of the 16-port SADA as a sensor array for an imaging radar is then verified by an experiment.

The Stokes and Navier-Stokes equations in an aperture domain

We consider the nonstationary Stokes and Navier-Stokes equations in an aperture domain Ω ⊂ R n , n ≥ 2 . For this purpose, we prove L p - L q type estimate of the Stokes semigroup in the aperture domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin. We apply them to the Navier-Stokes initial value problem in the aperture domain. As a result, we can prove the global existence of a unique solution to the Navier-Stokes problem with the vanishing flux condition and some decay properties as t → ∞ , when the initial velocity is sufficiently small in the L n space. Moreover we can prove the time-local existence of a unique solution to the Navier-Stokes problem with the non-trivial flux condition.

Navier–Stokes equations in aperture domains: Global existence with bounded flux and time‐periodic solutions

AbstractWe consider the Navier–Stokes equations in an aperture domain of the three‐dimensional Euclidean space. We are interested in proving the existence of regular solutions corresponding to small initial data and flux through the aperture. The flux is assumed to be smooth and bounded on (0, +∞). As a consequence, we prove the existence of a time‐periodic solution corresponding to a time‐periodic flux through the aperture. Finally, we compare our solution with a solution belonging to a wider class, showing that, if such a solution does exist, then the two solutions coincide. Copyright © 2007 John Wiley & Sons, Ltd.

Estimating the blood velocity vector using aperture domain data

Most conventional blood flow estimation methods measure only the axial component of the blood velocity vector. In this study, we developed a new method for two-dimensional (2-D) velocity vector estimation in which time shifts resulting from blood motion are calculated for the individual channels using aperture domain data. This allows the construction of a time-shift profile along the array direction as a function of channel index, which is approximated by a first-order polynomial whose zeroth-order and first-order terms can be used to determine the axial and lateral velocity components, respectively. The efficacy of the proposed method was verified by simulations and experiments in which the transducer array had 64 elements, an aperture size of 1.96 cm, and a center frequency of 5 MHz. The flow velocity ranged from 5 to 35 cm/s and the Doppler angle ranged from 0 degrees to 90 degrees. The experimental results show that the accuracy of axial velocity estimation is higher for the new method than for the autocorrelation-based conventional method when the signal-to-noise ratio is larger than 0 dB. The mean estimation error for the axial velocity component is 2.18% for the new method, compared to 4.51% for the conventional method. The mean estimation error for the lateral velocity component is 15%, which is comparable to existing methods.

Periodic solutions of the Navier-Stokes equations in a perturbed half-space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half-space and an aperture domain of the dimension n⩾3. Our proof is based on Lp–Lq estimates of the Stokes semigroup. We apply Lp–Lq estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear artificial boundary conditions for the Navier‐Stokes equations in an aperture domain

AbstractWe consider the Dirichlet problem for the stationary Navier‐Stokes system in a plane domain Ω, with two angular outlets to infinity. It is known that, under appropriate decay and smallness assumptions, this problem admits solutions with main asymptotic terms in Jeffrey‐Hamel form. We will approach these solutions by constructing an approximating problem in the domain ΩR, which is the intersection of Ω with a sufficiently large circle. The main difficulty, in contrast to the corresponding linear problem, arises from the fact that the main asymptotic term is not known explicitly. Here, we create nonlinear, but local, artificial boundary conditions which involve second order differential operators on the truncation arcs. Unlike for the analogous three‐dimensional exterior problem, we are able to show the existence of weak solutions to the approximating problem without smoothness nor smallness assumptions. For small data, we prove that the solutions of the approximating problem are unique and regular. Finally, we reach the main goal of this work, i.e. we obtain error estimates in weighted Hölder spaces which are asymptotically precise as R tends to infinity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Communication modes applied to axicons

The communication modes, which constitute a convenient method for the propagation and information analysis of optical fields, are formulated in the generalized axicon geometry. The transmitting region is the axicon's annular aperture, and the observation domain is the optical axis containing the focal line segment. We show that in rotational symmetry one may employ the prolate spheroidal wave functions to represent the communication modes. Further, in usual circumstances the modes can be approximated by quadratic waves in the aperture domain and by sinc functions in the image domain. Both the exact communication modes and the approximate technique are confirmed numerically, with linear axicons as examples.

Maximal regularity for the non-stationary Stokes system in an aperture domain

We prove estimates in \( L^s (0, T; L_w^q (\Omega)) \) for the solution of the non-stationary Stokes system in an aperture domain, where 1 <s, q< \( \infty \) and the weight function \( \omega \) is in the Muckenhoupt class \( A_q \).¶The result is achieved by combining a characterisation of maximal regularity by \( {\mathcal R} \)-bounded operator families with the fact that \( {\mathcal R} \)-boundedness follows from weighted estimates for Muckenhoupt weights.

Temporal domain processing for a synthetic aperture array

A method for the synthesis of an aperture with improved bearing resolution and signal gain is described. The proposed method temporally synthesizes data from an overlap correlator, which is obtained by aperture domain averaging of phase differences. Previous studies, such as extended towed array measurements (ETAM), had a restriction in that the overlapped hydrophones between successive measurements of a towed array were required to have identical positions in space. In this paper, however, it is shown that the phase correction factors can be estimated without restriction on the positions of the overlapped hydrophones. This implies that the proposed method is able to utilize more snapshots to extend the towed array. Simulation results showed that the proposed method resulted in higher estimation accuracy than ETAM. In addition, the effects of coherency and other systematic errors on the proposed method were examined.

$L_q-L_r$-Estimates for Non-Stationary Stokes Equations in an Aperture Domain

This article deals with asymptotic estimates of strong solutions of Stokes equations in aperture domains. An aperture domain is a domain, which outside a bounded set is identical to two half spaces separated by a wall and connected inside the bounded set by one or more holes in the wall. It is known that the corresponding Stokes operator generates a bounded analytic semigroup in the closed subspace J_q(\Omega) of divergence free vector fields of L_q(\Omega)^n . We deal with L_q-L_r -estimates for the semigroup, which are known for \mathbb R^n , the half space and exterior domains.

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π− and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.

Strong solutions of the Navier-Stokes equations in aperture domains

We consider the nonstationary Navier-Stokes equations in an aperture domain Ω⊂R3 consisting of two halfspaces separated by a wall, but connected by a hole in this wall. In this special domain one has to impose an auxiliary condition to single out a unique solution. This can be done by prescribing either the flux through the hole or the pressure drop between the two halfspaces. We construct suitable Stokes operators for both of the auxiliary conditions and show that they generate holomorphic semigroups. Then we prove the existence and uniqueness of solutions as well as a maximal regularity estimate for the Stokes equations subject to one of the auxiliary conditions. For the corresponding Navier-Stokes equations we prove existence and uniqueness of local in time solutions.

THIRD-GRADE FLUIDS IN THE APERTURE DOMAIN

We analyze steady motions of incompressible second- and third-grade fluids in an aperture domain. Such flows exist due to a pressure drop and provided the flux through the hole is small. We prove that the asymptotic behavior of their velocity field far from the aperture is of order |x|-2, which is the optimal result.

Note on the flux condition and pressure drop in the resolvent problem of the Stokes system

In [4] H. Sohr and the author considered theLq-theory of the resolvent problem of the generalized Stokes system in an aperture domain. This type of unbounded domain consists of two disjoint half spaces which are separated from each other by a wall but connected by a hole (aperture) in this wall. Due to this geometry the flux of the velocity field through the hole and the pressure drop at infinity are important physical and mathematical quantities. In this note we show that in order to single out a unique solution of the resolvent problem we must prescribe the flux for largeq, but that for smallq neither the flux nor the pressure drop can be prescribed. Only if the dimension is greater than two there is a certain range of values ofq where we must prescribe either the flux or the pressure drop. As a limit case we also investigate strong solutions of the Stokes system.

Existence, uniqueness and asymptotic behaviour of solutions of steady-state Navier-Stokes equations in a plane aperture domain

Existence, uniqueness and asymptotic behaviour of solutions of steady-state Navier-Stokes equations in a plane aperture domain

A re-formulation of the expression for mode-field diameter in the variable-aperture domain

Several expressions of mode-field diameter are currently in use in the optical fiber industry enabling measurements to be made in different measurement domains. A re-formulation of the expression for mode-field diameter in the variable aperture domain is presented. An inconsistency has been observed between this expression and that currently specified by standards organisations and manufacturers. The magnitude of this discrepancy is investigated and a new expression is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the Stokes and Navier‐Stokes System for Domains with Noncompact Boundary in Lq‐spaces

AbstractWe consider the Lq‐theory of weak solutions of the Stokes and Navier‐Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IRn, and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut‐off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q =n/2, n ≥ 3, to estimate the nonlinearity.

Blind-velocity SAR/ISAR imaging of a moving target in a stationary background

A synthetic aperture radio/inverse synthetic aperture radar (SAR/ISAR) coherent system model and inversion to image a target moving with an unknown constant velocity in a stationary background are presented. The approach is based on a recently developed system modelling and inversion principle for SAR/ISAR imaging that utilizes the spatial Fourier decomposition of SAR data in the synthetic aperture domain to convert the SAR system model's nonlinear phase functions into linear phase functions suitable for a computationally manageable inversion. It is shown that SAR/ISAR imaging of a moving target can be converted into imaging the target in a stationary squint-mode SAR problem where the parameters of the squint-mode geometry depend on the target's velocity. A method for estimating the moving target's velocity that utilizes a spatial Doppler analysis of the SAR data within overlapping subapertures is presented. The spatial Doppler technique does not require the radar signal to be narrowband, so the reconstructed image's resolution is not sacrificed to improve the target's velocity estimator.