Range and Bandwidth effects upon arrays

Range and bandwidth effects upon arrays - page 1

J. C. G. Lesurf, Aug 1996

Introduction

This report assesses some of the effects finite source range and signal bandwidth may have upon the signals received by a coherent array. Arrays are often designed and operated on the assumption that the source range is ‘very large’ — i.e. large enough that we can assume that the arriving phasefront is plane. This report considers some of the simple consequences of this assumption when, in reality, the source ranges are finite.

1. Standard result for a source with circular symmetry.

The previous reports I have written on this general topic assumed rectangular symmetry for the source. Here we will consider sources with circular symmetry in order to show how similar results can be derived from texts such as chapter 10 of ‘Born & Wolf’. For this I have used the 5th edition which was the newest I could find at present!

§10.4.2 of Born & Wolf deals with the Van Cittert—Zernike theorem. This shows that a source of circular symmetry will produce a point-pair coherence pattern in a normal plane of the general form

where

and:

is the radius of the source, the two points are at the locations

and

(measured from the place position nearest to the source centre — i.e. from the source/receiver plane normal), and

is the average wavelength of the signal.

For clarity we can simplify things by placing one of the points we are correlating at the plane origin (centre) and just considering the other in terms of a radial offset, X. For comparison with my other results we can also use Z as the range measure rather than R. This leads to the simplified expressions

Usually, three critical assumptions are then made

That Z is “very very large”
That, despite the first assumption, is large enough for the ratio to remain large enough to be measurable.
That the radiation is essentially “quasi-monochromatic”.

This means we can normally ignore the

term and the linear (X) dependence in the first part of equation 1 is converted into angular terms. Hence we get the usual result, a coherence pattern which shows a general form

where k is a scaling term such that the first zero of the pattern occurs at the appropriate angle.

We can now consider what happens when we do not make the above assumptions. The general consequences are as follows.

We can now allow

and

to continue being expressed in terms of linear measures rather than bearing/size angles. This has no major effect in the

portion of the expression, however, it now means that we cannot simply assume the

is unity everywhere. Instead it has a parabolic phase effect across the receiving plane. In itself, this parabolic phase variation is innocuous provided we restrict our observations to a very narrow bandwidth. However, when the bandwidth is finite we must reconsider the observed total coherence pattern as a complex linear superposition of the parabolic phase modified coherence patterns of each frequency being observed. Since the ‘rate’ of the parabolic phase shift is scaled by the signal frequency/wavelength we now encounter the usual interferometric effects. The observed coherence pattern across the observing plane is an interferogram whose shape is determined by the source direction and size, and by its range and the observation bandwidth.

This topic has been considered in detail in other reports. Here we can envisage the result in terms of recognising that the observed 2-port coherence pattern across the plane is approximately a non-linear (parabolic) version of the source’s spectral interferogram. i.e. Although the observed interferogram depends on the source location/size it also represents its observed spectrum. Under many practical circumstances the apparent pattern becomes dominated by the source’s observed spectral output and this narrows and shapes the observed pattern rather than being dominated by the geometric Bessel pattern.

2. Concept of array alignment correction.

Most of the previous analysis I have performed has concentrated on the use of wideband multiplication (or coherent power detection) correlation. However, many systems use a form of ‘time offset’ or ‘time shifting’ before correlation in order to ‘steer’ the array. In this section I outline a conceptual way of viewing this process which is useful when we consider the effects of finite range and bandwidth on such systems. The actual bandwidth/range effects are considered in section 3.

For convenience we can imagine a simple array in the form of a set of ports which are meant to be in a line, arranged perpendicular to the nominal source direction. Each port is meant to be at a specific distance,

, from the nominal centre of the array. (A real array will probably be 2D or 3D but here we can concentrate on 1D for clarity.)

When we receive a signal from a ‘point source at infinity’ the array will experience a plane wavefront which passes the array at an angle,

, which depends upon the direction to the source. Hence for an ideal array the times of arrival at the individual ports will be offset by an amount

with respect to the time of arrival at the array’s nominal centre.

A real array may suffer from various imperfections which alter the relative times of arrival of the signal at the ports. For sources near the boresight normal direction (

) these effects can be represented as a set of unwanted displacements,

, out of the intended array line. (More generally, the effects of lateral offsets and collection delays will differ from this, but we can ignore that complication here.)

Given a suitable powerful near-boresight source we can compare the signals collected from the ports and, in principle, use the comparisons to determine the relative values of the temporal errors,

produced by the array imperfections.

In effect, when we do this we are conceptually “correcting the port positions onto a line which fits the arriving phasefronts”. RSC is essentially a special method for doing this which minimises the amount of information we must pre-know regarding the array and the calibration source(s). Once the correction has been performed we can the go on to use the array by introducing a set of extra delays. To test for a source in a direction,

, we therefore proceed in three stages:

Take the incoming signals and apply the corrections, .
Apply a specific set of extra delays as given by equation 6.
Correlate (vector add) the resulting signals to determine the signal power arriving from the direction .

In practice the mathematical methods applied will be more sophisticated than this — e.g. record sampling followed by FFT correlations — but broadly speaking the method is conceptually as described above.

Content and pages maintained by: Jim Lesurf (jcgl@st-and.ac.uk).

University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.