In this entry I will describe the geometry of multi-static passive radar and how it obtains a target solution. The multi-static passive TDOA radar example in this entry consists of a single receiving station that monitors the direct signals from the transmitters and indirect, multi-path signals from potential targets. Other configurations are possible but the 2D operation described requires 3 transmitter – receiver combinations. To obtain a target the system uses the time difference of arrival (TDOA) of the multi-path signal compared to the direct path signal from the same transmitter. Using these timings, solutions for possible target locations can be generated. I will cover the basic bi-static geometry on which this system relies and how it is incorporated into multi-static system. A 2D solution will also be derived from scratch for a multi-static system.
Bi-Static Radar Geometry
Figure 1
Figure 1 illustrates a simple bi-static radar system consisting of a transmitter 
, a target 
and a receiver 
. A distance separates the transmitter and receiver 
. The sum of the distances between the target 
and the transmitter 
and the distance between the target and receiver 
is the range of the bi-static radar. By measuring the time difference of arrival 
of the reflected path via the target relative to the direct path signal from the transmitter the relationship between the direct path 
and the bi-static range 
can be determined as demonstrated in Equation 1
Equation 1
,
where 
is the speed of propagation of the signal through air.
Knowing the bi-static range allows the target to be positioned somewhere on an ellipse. The sum of the distances from the points of the ellipse to the transmitter and receiver is equal to the bi-static range 
.
Figure 2
This relationship can also be demonstrated as in Equation 2
Equation 2
.
The location of the transmitter 
is at 
. The location of the receiver 
is at 
and the target is located 
. Expanding Equation 2 gives
Equation 3
.
Multi-Static Radar Geometry
Figure 3
Figure 3 shows a multi-static system consisting of four transmitters, a target and a receiver. There is a direct path signal,
from every transmitter to the receiver. A multi path signal from each of the transmitter 
is reflected by the target 
and is intercepted by the receiver. The path the reflected signal follows from the target to the receiver is common for all transmitters 
.
2D Operation
A two dimensional solution for the target can be found using measurements of the TDOA of the reflected signal compared to the direct signal from each transmitter. The 2D solution requires three TDOA measurements from three transmitters 
, 
and 
. Using the TDOA values and the known locations of the transmitters Equation 6 can be rewritten for each transmitter
Equation 4
Equation 5
Equation 6
.
Equation 4 and Equation 5 can be rearranged into
Equation 7
and
Equation 8
.
Squaring both sides of Equation 7 and Equation 8 gives
Equation 9
Equation 10
.
Rearranging Equation 9 and Equation 10 in terms of 
Equation 11
Equation 12
.
Equating Equation 11 and Equation 12 gives
Equation 13
Rearranging Equation 0‑10 in the form of
Equation 14
gives
Equation 15
Equation 16
Equation 17
Using Equation 4 and Equation 6 rearranging into the form
Equation 18
gives
Equation 19
Equation 20
Equation 21
Equating Equation 14 and Equation 18
Equation 22
and rearranging for
gives
Equation 23
.
Subtitling
into Equation 14 for
gives
Equation 24
The next post will expand the 2D derivation into a solution in 3D. An extra dimension will be added using an additional transmitter. The concept of a geometric error will also be introduced.
Tags: Andrew Burns, Bi-Static, MSPSR, Multi-Static, Passive Radar, Radar, TDOA
