15.4 Passive Ranging.
It has become conventional to think of using radar for ranging measurements. Spatial interferometry at radio, microwave, and mm-wave frequencies has mostly been used for ‘two-dimensional’ mapping applications which determine directions but provide no range information. Despite this, spatial interferometry provides an excellent alternative method for obtaining three-dimensional information. The most common form of this is Holography. Here, however, we will concentrate on a much simpler example, that of a 3-Port Spatial Interferometer used to determine the range and bearing of a point-like source.
Note that — as with the 2-port system we considered in the last lecture — we'll just obtain one bearing/direction angle so we are only measuring locations in a plane drawn through the receiver's ports. In practice we could add one or more measurement ports which are located out of this plane and determine the source location in an intersecting plane. This makes the process more complex but it allows us to make a full three-dimensional measurement of the source's location. Here we'll ignore this as an unwanted complication and just see how a 3-port system lets us determine the range distance of a source as well as a bearing.
Figure 15.5 illustrates a 3-port spatial interferometer observing a point source. There are various ways to perform the signal processing. Here we can begin by taking the most obvious course and consider the 3-port as a pair of ‘overlapping’ 2-port spatial interferometers. One 2-port performs interferometry between the inputs, 
and 
. The other performs interferometry between 
and 
. The resulting pair of interferograms can be processed as described in the last lecture to yield the two angles, 
& 
which are the direction angles from the mid-line of each 2-port pair to the source. These direction lines start mid-way between the port pairs. They therefore form two sides of a triangle whose base length is W. Hence we can work out both the source's range and its offset from the middle port (i.e. from the 3-port arrangement's boresight) by trigonometry.
The precise form of the expressions will depend upon the sign conventions we choose. For the sake of illustration we'll assume that a clockwise rotation from a mid-line corresponds to a positive angle. (This means that, as shown, 
is positive and 
is negative.) We can also simplify things by assuming that, as before, R is much larger than either W or x. From figure 15.5 we can work out that the range and offset from the central line will be
Although useful for the purpose of seeing how we can obtain a range from the intersection of two bearing measurements, in practice we would normally assess the system in terms of its ability to determine the curvature of the wave’s phasefront. We are also often using a specific signal frequency (or narrow signal bandwidth) and obtain measurements in terms of the relative phases of the signals detected at the receiving ports.
Figures 15.6 a and b illustrate this effect. In 15.6a the source is infinitely distant and hence the wavefront reaching our 3-port is plane. As a consequence we find that the time of arrival at the central port is idential to the average of the times of arrival at the two outer ports. In 15.6b the source is at a finite range and hence the wavefront reaching the 3-port is spherical. It has a radius centred on the location of the signal source. As a result of this curvature, the time of arrival at the central port is slightly earlier than the average of the times of arrival at the two outer ports.
We can therefore expect to be able to determine the range by comparing the detected relative times of arrival of the signal at the ports. (Or the phases for a signal of a specific frequency.)
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