Reading — Pulse Radars & Ambiguity#
By the end of this lesson you should be able to:
Define pulse width (PW), pulse repetition interval (PRI), pulse repetition frequency (PRF), and duty cycle.
Compute range resolution from pulse width.
Compute the unambiguous range \(R_u\) and unambiguous velocity \(v_u\) from PRF, and explain the tradeoff between them.
Use the range–Doppler ambiguity invariant to explain why no single PRF measures both cleanly.
Why pulses#
A continuous-wave transmitter cannot easily tell when an echo left, so it struggles to measure range. A pulsed radar solves this by transmitting a short burst, then listening. The round-trip time of the echo gives the range directly:
the factor of 2 again being the two-way path. The radar sends a regular train of pulses and processes the echoes between them.
Four parameters describe the train:
Parameter |
Symbol |
Meaning |
|---|---|---|
Pulse width |
PW |
Duration the transmitter is on. |
Pulse repetition interval |
PRI |
Time between pulse starts. |
Pulse repetition frequency |
PRF |
\(1/\text{PRI}\) — pulses per second. |
Duty cycle |
— |
\(\text{PW}/\text{PRI}\) — fraction of time transmitting. |
Range resolution#
Two targets are resolvable in range only if their echoes do not overlap. That sets the range resolution by the pulse width:
A shorter pulse resolves finer detail — a 1 µs pulse gives \(\Delta R = 150\) m; a 0.1 µs pulse gives 15 m. But a shorter pulse carries less energy, hurting detection range. (Pulse compression resolves this tension; that is a later topic.)
The ambiguities#
Here is the central tension of pulsed radar. The PRF you choose to listen between pulses sets two unambiguous limits at once — and they pull in opposite directions.
Unambiguous range. The radar must receive an echo before it sends the next pulse, or it cannot tell which pulse the echo belongs to. The longest unambiguous round trip is one PRI, so
A low PRF gives a long \(R_u\).
Unambiguous velocity. Doppler (Lesson 5) is a frequency shift, and the pulse train samples it at the PRF. By the sampling theorem, the radar can measure Doppler unambiguously only up to \(\pm\text{PRF}/2\), which in velocity is
A high PRF gives a large \(v_u\).
So low PRF is good for range but bad for velocity; high PRF is the reverse. You cannot have both.
Key Concept
PRF is a single knob that sets two limits that fight each other: \(R_u = c/(2\,\text{PRF})\) shrinks as PRF rises, while \(v_u = \lambda\,\text{PRF}/4\) grows. There is no PRF that makes both large.
The invariant#
Multiply the two limits and the PRF cancels:
The product is a constant fixed only by wavelength — independent of PRF. You can trade range coverage for velocity coverage, but their product is conserved. This invariant is the cleanest statement of why no single waveform measures everything.
X-band numbers#
At X-band (\(\lambda = 3\) cm):
PRF |
\(R_u\) |
\(v_u\) |
|---|---|---|
1 kHz |
150 km |
7.5 m/s |
10 kHz |
15 km |
75 m/s |
200 kHz |
0.75 km |
1500 m/s |
Notice the product \(R_u \cdot v_u\) is the same in every row. Real radars solve this by using multiple PRFs and resolving the ambiguities across them — the foundation of low/medium/high-PRF modes.
Regime |
PRF |
Good for |
Pays with |
|---|---|---|---|
Low |
low |
unambiguous range |
badly ambiguous velocity |
Medium |
medium |
balance |
ambiguous in both; resolved across PRFs |
High |
high |
unambiguous velocity, clutter rejection |
badly ambiguous range |
Range folding: a B-21 example#
When a target is beyond \(R_u\), its echo arrives after the next pulse has gone out, and the radar folds it back to an apparent range:
Suppose a B-21 is at a true range of 200 km, and a radar uses PRF = 800 Hz. Then
Since 200 km exceeds \(R_u\), the echo folds:
The radar paints the bomber at 12.5 km — wildly wrong, and a known vulnerability the adversary must design around (and one EA can exploit).
Type-along
Type this with me. X-band threat radar, \(\lambda = 3\) cm.
c = 3e8; lambda = 0.03; % X-band
PRF = [800 2e3 5e3 1e4 2e4]; % Hz, low -> high
Ru = c ./ (2*PRF) / 1e3; % unambiguous range, km
vu = lambda * PRF / 4; % unambiguous velocity, m/s
fprintf(' PRF(Hz) Ru(km) vu(m/s) Ru*vu\n');
fprintf('%8.0f %8.1f %8.1f %8.0f\n', [PRF; Ru; vu; Ru.*vu]);
need_R = 200; need_v = 250; % B-21: see to 200 km, closing 250 m/s
ok = (Ru >= need_R) & (vu >= need_v);
fprintf('\nPRFs meeting both: %d of %d\n', sum(ok), numel(ok));
Read down the \(R_u\) and \(v_u\) columns, then down the fourth. What is the fourth column doing?
The last line says 0 of 5. Is that a badly chosen PRF list, or something stronger?
If no PRF works, what would you change?
Solution
PRF(Hz) Ru(km) vu(m/s) Ru*vu
800 187.5 6.0 1125
2000 75.0 15.0 1125
5000 30.0 37.5 1125
10000 15.0 75.0 1125
20000 7.5 150.0 1125
PRFs meeting both: 0 of 5
\(R_u\) falls 25×, \(v_u\) rises 25×, and \(R_u \cdot v_u\) never moves off 1125 km·m/s — that is \(c\lambda/8\) in these units. The sweep is the invariant, printed.
Something stronger. The requirement asks for \(R_u \cdot v_u \ge 200 \times 250 = 50{,}000\) km·m/s, but the invariant caps it at 1125 — short by a factor of 44. No PRF at X-band satisfies it, so lengthening the list cannot help.
Either raise the invariant by moving to a longer \(\lambda\) (it scales as \(c\lambda/8\)), or stop asking one PRF to do both jobs and use multiple PRFs — which is what real radars do, and where L5 picks up.
Wrap-Up#
Pulsed radar measures range from echo delay, with resolution \(\Delta R = c\,\text{PW}/2\) set by pulse width. The PRF sets two competing limits — \(R_u = c/(2\,\text{PRF})\) and \(v_u = \lambda\,\text{PRF}/4\) — whose product \(c\lambda/8\) is invariant, so no single PRF is unambiguous in both. Targets beyond \(R_u\) fold to a false apparent range.
We have treated the echo as a single number, but each pulse also carries a phase, and across pulses that phase moves whenever the target does. The next lesson turns that motion into the Doppler shift and the moving-target processing that pulls the B-21 out of clutter.