Find the question you need
Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Trace the paths
Chapters 1–3. Draw separated instruments and identify the delay reference.
Read the geometry
Chapters 4–5. Build the ellipse and test the half-path interpretation.
Add position evidence
Chapters 6–7. Use a bearing while preserving mirror and height limits.
Test sensitivity and route
Chapters 8–10. Examine baseline geometry, uncertainty and multipath.
Practise and continue
Chapters 11–12. Check new paths and follow a specialist route.
Full chapter index · Practice and worked checks · Radar subject root
A transmitter and a receiver sit six kilometres apart. A reflected signal follows a ten-kilometre route from transmitter to target to receiver. Where is the target? There are many possible positions, even with perfect timing. In a flat two-dimensional model, they lie on an ellipse whose two foci are the transmitter and receiver.
This guide follows the whole signal path, separates total delay from excess delay and shows what additional directional evidence can resolve. All coordinates and timing values are invented teaching examples. They use straight paths and a stated constant propagation speed; they are not field-survey results or a complete operational radar model.
Chapter index
Trace the paths · Chapters 1–3
1. Change the geometry before changing the equation
2. Use a concrete baseline and target
3. Distinguish total delay from excess delay
Read the geometry · Chapters 4–5
4. Build the equal-delay ellipse
5. Test the half-path interpretation at different points
Add position evidence · Chapters 6–7
6. Add a receiver bearing to locate a candidate
7. Keep mirror ambiguity and height visible
Test sensitivity and route · Chapters 8–10
8. Examine the nearly degenerate baseline case
9. Translate timing uncertainty through the right path model
10. Separate multipath, pulse ambiguity and target geometry
Practise and continue · Chapters 11–12
11. Practise with changed coordinates and worked checks
12. Questions, connected routes and sources
CHAPTER 1 OF 12 · Trace the paths
1. Change the geometry before changing the equation
In a simple monostatic radar, the transmitter and receiver are treated as co-located. A target at range R gives an outgoing path R and a return path R, so the round-trip distance is 2R. That is the reason for the familiar division by two.
In a bistatic arrangement, transmitter and receiver occupy different positions. Let dₜ be the distance from transmitter to target and dᵣ the distance from target to receiver. The reflected route has length L = dₜ + dᵣ. The legs generally differ.
If the absolute propagation delay τ is known, L = cτ in the straight-path constant-speed model. Dividing by two gives half the total route. It does not generally give the target’s distance from either instrument.
The first useful correction is therefore a drawing: mark transmitter, target and receiver separately, then label both legs. A memorised range formula cannot decide whether its original geometry still applies.
Place the transmitter at (−3, 0) kilometres and the receiver at (3, 0). Their separation, called the baseline length D here, is six kilometres. Put a target at (0, 4).
The transmitter-to-target distance is square root of 3² + 4² = five kilometres. The receiver-to-target distance is also five kilometres. The total reflected route is ten kilometres.
Using c = 300,000,000 metres per second as an approximation for this air-path model, the absolute propagation delay is 10,000/c seconds, or approximately 33.33 microseconds.
The equal legs arise from this target’s position on the perpendicular bisector. They are not a property of every bistatic target. To test the distinction, move the target to (5, 0). Its legs are eight and two kilometres, but their sum is still ten. The same absolute delay fits both target positions.
Some arrangements compare the reflected signal with a direct signal travelling from transmitter to receiver. The direct route has length D = six kilometres and delay D/c = 20 microseconds in our model.
The reflected route takes 33.33 microseconds, so its excess delay relative to that direct route is approximately 13.33 microseconds. The excess path is ΔL = cΔτ = four kilometres. The full reflected path is L = D + ΔL = ten kilometres.
| Quantity | Teaching value | Meaning |
|---|---|---|
| Baseline D | 6 km | Direct transmitter-to-receiver distance |
| Reflected path L | 10 km | Transmitter–target–receiver path sum |
| Excess path ΔL | 4 km | Reflected path minus direct path |
| Absolute reflected delay | 33.33 μs | Full propagation time in the model |
| Excess delay | 13.33 μs | Additional time relative to the direct route |
Substituting excess delay into the absolute-path equation loses the baseline. Substituting it into cτ/2 gives two kilometres, which has no general interpretation as either target leg. Always identify the timing reference before converting time into distance.
An ellipse is the set of points whose distances to two fixed foci sum to a fixed value greater than the focus separation. Our two foci are (−3, 0) and (3, 0), and their fixed distance sum is ten kilometres.
The semi-major axis a is half that sum, so a = five kilometres. The focal distance from the centre is three kilometres. The semi-minor axis b satisfies b² = a² − 3² = 25 − 9 = 16, so b = four kilometres.
The ellipse equation is x²/25 + y²/16 = 1, with x and y expressed in kilometres. It includes (0, 4), (0, −4), (5, 0) and (−5, 0).
The equation is a location constraint. It is not a list of observed targets. A single target can occupy any compatible point unless other evidence restricts it. Conversely, several physical reflectors can contribute signals that require separate detection and association. The geometric curve alone establishes neither the number nor the identity of reflectors.
CHAPTER 5 OF 12 · Read the geometry
5. Test the half-path interpretation at different points
At every point on that ellipse, L/2 is five kilometres. At the top point, the two target legs are five and five. At the right endpoint, they are eight and two. At the left endpoint, they are two and eight.
Thus half the path sum is the semi-major-axis length of the timing ellipse. It becomes either individual leg only for special positions where the legs are equal.
A useful checking question is: “Which origin does this reported distance use?” A receiver-centred distance, transmitter-centred distance, distance from the baseline midpoint and half a path sum are different quantities.
This lesson extends the factor-of-two reasoning in the radar root. The division is justified by equal outgoing and return distances in the chosen monostatic model. Once that equality is removed, preserving the old arithmetic can conceal a changed physical question. Derive the path relationship first, then choose the operation that answers it.
CHAPTER 6 OF 12 · Add position evidence
6. Add a receiver bearing to locate a candidate
Suppose the receiver at (3, 0) also measures a direction toward the upper left. In our constructed case, the unit direction is (−0.6, 0.8). A candidate at distance s from the receiver lies at x = 3 − 0.6s and y = 0.8s, with s in kilometres.
At s = five, the coordinates are (0, 4), and the transmitter distance is five. The sum of the two legs is ten, matching the timing constraint.
One can solve this rather than guess. The transmitter distance is square root of 36 − 7.2s + s². Setting that plus s equal to ten gives 36 − 7.2s + s² = (10 − s)². Rearranging gives 12.8s = 64, hence s = five.
Squaring an equation can introduce extraneous solutions, so substitute the result into the unsquared path equation and check the ray direction. In this example it passes both checks. The bearing supplies information the single timing ellipse lacked.
CHAPTER 7 OF 12 · Add position evidence
7. Keep mirror ambiguity and height visible
Timing alone cannot distinguish (0, 4) from (0, −4). If a directional observation specifies only the magnitude of an angle and loses its side or sign, that ambiguity can remain.
Our ellipse lies in a stipulated plane. In three dimensions, a fixed sum of distances to separated foci defines an ellipsoidal surface rather than a single planar curve. A two-dimensional drawing has silently supplied a height or plane constraint.
Additional sensors or observations can add constraints, but the intersections still need interpretation. Measurements must refer to compatible times and coordinates, and they must be associated with the same target. More curves do not guarantee a correct intersection if the observations were paired incorrectly.
State what each item contributes: a path sum, a direction, an assumed height, a motion model or an independent confirmation. This makes a position estimate auditable. It also makes clear which uncertainty would remain if one of those constraints were removed.
CHAPTER 8 OF 12 · Test sensitivity and route
8. Examine the nearly degenerate baseline case
For any point between transmitter and receiver on the baseline, the two legs sum to D. A target at the midpoint has legs three and three; one near the receiver might have legs five and one. Both give zero excess path in this ideal geometry.
At L = D the ellipse degenerates into the segment between the foci. A zero excess delay therefore does not, by itself, locate a point on that segment. It also does not distinguish a target return from the direct signal without the relevant signal evidence.
Now move a midpoint target slightly off the baseline to (0, 0.2) kilometres. Its total route is twice the square root of 3² + 0.2², approximately 6.01332 kilometres. The excess route is only 13.32 metres, corresponding to about 44.4 nanoseconds.
A transverse displacement of 200 metres has produced a small excess delay. This illustrates geometry-dependent sensitivity. A precise-looking timing figure does not translate into the same positional precision everywhere around the baseline.
CHAPTER 9 OF 12 · Test sensitivity and route
9. Translate timing uncertainty through the right path model
An excess-delay uncertainty of 0.10 microseconds corresponds to an excess-path uncertainty of c×0.10 microseconds, or 30 metres under our stated speed. It is not 15 metres: that latter conversion would apply the monostatic half-path factor.
Turning the 30-metre path uncertainty into a target-position uncertainty requires the geometry and the other observations. Near the baseline, small excess-path changes can correspond to relatively large transverse position differences. Elsewhere the sensitivity differs.
Using a direct signal as a timing reference may help establish a relative delay when the system supports that comparison. It does not make every error disappear. Direct-path multipath, reference-channel delays, synchronisation and uncertain instrument positions can still affect the interpretation.
Treat these as questions for the actual measurement system. Our simple model does not assign real error budgets. For a paper check, perturb the path sum and bearing separately and see how the compatible position moves. That exercise exposes which assumptions carry the estimate.
CHAPTER 10 OF 12 · Test sensitivity and route
10. Separate multipath, pulse ambiguity and target geometry
Bistatic geometry asks how a known transmitter–target–receiver path sum constrains position. Pulse ambiguity asks which transmitted pulse produced the received signal. Multipath asks which physical route the signal followed. Those uncertainties can coexist.
A reflected route that includes extra surfaces is not necessarily described by one target point and two straight legs. A wrong pulse assignment can produce a wrong total delay even if the geometry calculation is otherwise correct.
MIT Lincoln Laboratory’s historical Logan microwave landing-system experiment used direct and reflected paths and a reflection ellipse to investigate multipath. It provides a primary example of the geometry, not a statement about today’s equipment configuration or operating procedures.
For a useful report, keep recorded arrival, inferred path, proposed reflector and estimated position distinct. A successful fit to an ellipse supports compatibility with the chosen model. It does not independently prove the model’s route or target identity.
CHAPTER 11 OF 12 · Practise and continue
11. Practise with changed coordinates and worked checks
Check one: recover the path sum
With a six-kilometre baseline and an excess delay of 20 microseconds, the excess path is six kilometres. The reflected path sum is therefore twelve kilometres. The ellipse has a = six and b = square root of 36 − 9 = square root of 27 kilometres.
Check two: verify an off-centre point
Return to the ten-kilometre path ellipse. Test (3, 3.2). Its receiver distance is 3.2 kilometres and its transmitter distance is square root of 6² + 3.2² = 6.8 kilometres. The sum is ten. Half the sum, five, is neither leg.
Check three: identify the clock reference
A worksheet reports a delay of 13.33 microseconds and asks for “the range.” Before calculating, request the definition within the supplied scenario: absolute delay or excess delay, baseline length, and desired position quantity. If it is the excess delay in our original example, it supports an approximately ten-kilometre path sum, not one unique target position.
Use supplied diagrams or synthetic records for these exercises. No transmitting equipment is needed to learn the path reasoning.
CHAPTER 12 OF 12 · Practise and continue
12. Questions, connected routes and sources
Is an ellipse an error in the measurement?
No. Even an exact path sum leaves many compatible positions. It is a geometric information limit; measurement errors add another layer.
Can I always divide a radar delay by two?
Only when the path model and the requested quantity justify it. Separated transmitter and receiver positions generally need both legs accounted for.
Does a direct reference signal solve every timing problem?
No. It defines a useful comparison when properly measured, but its path and channel need their own checks.
Which route develops this further?
For the foundation, return to The Gold Standard of Radar and radar range, angles and Doppler mathematics. For distinct signal and timing limitations, read range ambiguity with two pulse intervals, weak echoes beside strong targets and out-of-order observations and track age.
For combining position evidence, continue with sensor fusion engineering. For directional response, use antenna engineering. The Gold Standard of Education and How X Works open the wider learning routes.
The primary geometric example is D. A. Shnidman’s 1975 MIT Lincoln Laboratory report, The Logan MLS Multipath Experiment, Section 4.2 and Figure 13. MIT’s Introduction to Radar Systems course supplies broader signal-system context. The coordinates, sensitivity checks and bearing calculation in this article are original idealised examples.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.