Radar & Ranging

Ranging estimates propagation delay; radar combines that delay with echoes to infer target range, radial velocity, angle, and sometimes shape. Both are estimation systems, not merely transmitters plus detectors. The waveform must make different delays and Dopplers distinguishable, while the receiver must search enough hypotheses without declaring noise, clutter, or a sidelobe to be a target.

The central trade is resolution versus ambiguity and energy. Wide bandwidth separates nearby delays. Long coherent observation separates nearby Dopplers and accumulates weak signal energy. Repeating a waveform creates periodic ambiguities unless timing, coding, or multiple repetition intervals disambiguate it.

Delay becomes distance #

For monostatic radar, one clock measures transmit-to-target-to-receiver round-trip delay $\tau$:

$$R=\frac{c\tau}{2}.$$

The factor two is the outward and return path. A 10 μs delay corresponds to about 1499 m, not 2998 m.

In one-way ranging, a transmitter timestamp and receiver timestamp yield

$$\hat R=c[(t_r-t_t)-\Delta t_{clock}-\Delta t_{hardware}],$$

so clock offset is indistinguishable from range unless separately synchronized or estimated. Two-way ranging can cancel a constant remote-clock offset by measuring a round trip, but must account for turnaround delay and motion during the exchange. Time-difference-of-arrival (TDOA) uses synchronized receivers to obtain range differences and intersects hyperbolas/hyperboloids rather than measuring absolute range directly.

Bandwidth sets delay resolution #

Two echoes are distinguishable when their delayed waveforms have sufficiently separated correlation peaks. For a waveform with effective bandwidth $B$, the canonical range-resolution scale is

$$\Delta R\approx\frac{c}{2B}.$$

A 100 MHz waveform gives about 1.50 m resolution. This is why a long linear-FM chirp can carry high energy and achieve short-pulse resolution after matched filtering: resolution follows swept bandwidth, while transmitted energy follows duration and power.

Matched filtering compresses a long coded/chirped pulse into a narrow main peak with sidelobes. Windowing reduces sidelobes but widens the main lobe and loses SNR. “Range resolution” must therefore state the waveform, weighting, estimator, and criterion—not only $c/(2B)$.

Doppler becomes radial velocity #

For monostatic reflection and nonrelativistic radial velocity $v_r$,

$$f_D=\frac{2v_r}{\lambda}.$$

Again the factor two is the outward and return path. Sign convention depends on whether closing velocity is positive. A coherent observation of duration $T_{coh}$ has Doppler-resolution scale

$$\Delta f_D\approx\frac{1}{T_{coh}}, \qquad \Delta v\approx\frac{\lambda}{2T_{coh}}.$$

Longer coherent processing gives finer velocity bins and processing gain, but assumes target phase remains predictable. Acceleration, oscillator phase noise, manoeuvre, and changing propagation can smear a target across bins.

Radar measures radial velocity. A target moving tangentially across the line of sight can have high speed and nearly zero Doppler. Multiple look directions or tracking over time are needed for a full velocity vector.

The radar equation pays for the path twice #

For an ideal monostatic far-field radar, a common received-power form is

$$P_r=\frac{P_tG^2\lambda^2\sigma}{(4\pi)^3R^4L},$$

where $G$ is antenna gain in the target direction, $\sigma$ radar cross section (RCS), and $L$ collects other losses. One $R^{-2}$ factor is spreading from radar to target; the second is spreading of scattered power back to the radar. A one-way communication link has $R^{-2}$, while monostatic radar has $R^{-4}$ under this idealized model.

RCS is not physical area. It depends on frequency, polarization, aspect, material, shape, and scattering regime; resonances and specular facets can change it by orders of magnitude as a target rotates. A single “square metres” value without distribution and geometry is not an availability model.

Integration improves detectability. Coherent integration preserves phase and can add amplitudes before squaring; noncoherent integration adds detected power and gains less. The exact gain is reduced by target fluctuation, phase error, range/Doppler migration, weighting, and missed pulses.

Pulse repetition creates two ambiguities #

For pulse repetition frequency $PRF$, echoes arriving after the next pulse cannot be assigned uniquely. The first-return unambiguous range is

$$R_u=\frac{c}{2PRF}.$$

Sampling Doppler once per pulse makes frequency ambiguous modulo $PRF$. If Doppler is limited to $[-PRF/2,PRF/2)$, monostatic unambiguous radial velocity is

$$|v_r|<\frac{\lambda PRF}{4}.$$

At $PRF=10$ kHz, $R_u\approx15$ km. At 77 GHz ($\lambda\approx3.89$ mm), the same PRF gives only about ±9.7 m/s unambiguous velocity. Lower PRF extends range and shrinks velocity interval; higher PRF does the opposite. MIT Lincoln Laboratory's radar course uses this conflict to motivate low-, medium-, and high-PRF designs rather than pretending one PRF removes both ambiguities.

Staggered/multiple PRFs can unwrap hypotheses when their ambiguity patterns differ, but introduce scheduling, blind-zone, clutter, and association complexity. The receiver must test a consistent range–Doppler solution across PRFs, not concatenate measurements blindly.

FMCW measures beat frequency #

For a linear chirp with slope $S=B/T_c$, a delayed echo is offset in instantaneous frequency. Ignoring higher-order motion during one chirp, the dechirped beat frequency contains

$$f_b\approx\frac{2SR}{c}+\frac{2v_r}{\lambda}.$$

Range and velocity are coupled in one slope. Alternating up- and down-chirps changes the sign of the range term relative to Doppler, allowing the two to be separated under a constant-velocity model. Multiple targets create a data-association problem: pairing an up-chirp beat from target A with a down-chirp beat from target B creates a ghost.

An FMCW transmitter and receiver operate simultaneously, so leakage from transmitter to receiver, phase noise, close-range reflections, and mixer/DC impairments can dominate weak far echoes. High average duty cycle is not free sensitivity unless isolation and dynamic range support it.

PN and correlation ranging #

A known pseudo-noise sequence modulates a carrier; the receiver correlates possible code delays. Chip duration $T_c$ sets the basic delay-cell scale, while code period sets an ambiguity interval. Long coherent integration improves sensitivity but requires carrier/code Doppler tracking and a false-alarm budget over all searched delay/frequency cells.

CCSDS 414.1-B-3 defines transparent and regenerative PN ranging systems for space links. In a transparent transponder, the returned ranging signal preserves a ground-generated structure through the spacecraft path. A regenerative transponder detects/recreates the ranging code onboard. Regeneration can improve the returned signal but makes onboard delay and processing part of the calibrated ranging contract.

Code correlation does not make multipath vanish. A later echo creates another peak; an unresolved echo biases the main peak. Early-path estimators, wider bandwidth, antenna diversity, environment models, and robust tracking reduce bias, but the first detectable path may still differ from geometric line of sight.

Detection is a multiple-hypothesis decision #

After matched filtering, a detector tests many range, Doppler, angle, and perhaps waveform cells. If one cell has false-alarm probability $p$ and $M$ independent cells are searched,

$$P_{FA,total}=1-(1-p)^M\approx Mp$$

for small $p$. A threshold safe on one trace can produce continuous false alarms over millions of cells.

Constant-false-alarm-rate (CFAR) processing estimates local noise/clutter from neighbouring cells and adapts the threshold. It is not magic: a large target can contaminate training cells, multiple nearby targets can mask one another, and nonhomogeneous clutter violates the assumed distribution. Guard cells, training geometry, censoring, and ordered-statistic variants are design choices tied to the environment.

Range and Doppler resolution do not equal estimation accuracy. A high-SNR isolated target can be localized to a fraction of a bin by fitting the known peak shape; two unresolved targets or model mismatch can bias that estimate far more than the nominal bin spacing.

Angle completes the state #

Mechanically scanned beams, phased arrays, monopulse comparators, and digital beamforming estimate direction. Aperture size in wavelengths sets the beamwidth scale; multiple spatial channels provide phase/amplitude differences from which angle is inferred. Array calibration, element pattern, mutual coupling, platform attitude, and grating lobes directly become angle error or false targets.

Synthetic-aperture radar (SAR) uses controlled platform motion to synthesize a long along-track aperture and obtain fine cross-range resolution. It requires coherent trajectory knowledge and motion compensation; it is not simply a conventional radar image accumulated over time.

Verification workflow #

1. Calibrate transmitter, receiver, cable, antenna phase centres, turnaround delay, and clock reference in one timing budget.
2. Generate the waveform actually transmitted, then measure bandwidth, pulse/chirp shape, phase noise, sidelobes, and repetition timing.
3. Verify delay and Doppler estimators on injected single targets spanning fractional bins, SNR, acceleration, and ambiguity boundaries.
4. Add multiple targets, clutter, leakage, multipath, interference, and nonhomogeneous backgrounds; record misses and false alarms over the full search size.
5. Sweep PRF/chirp schedule and prove the ambiguity-resolution/data-association logic rather than only plotting one unwrapped case.
6. Compare predicted received power using aspect-dependent RCS with calibrated range measurements.
7. For two-way ranging, vary turnaround delay and motion; for one-way/TDOA, inject clock offset, drift, and asymmetry.
8. Run end-to-end geometry truth against reported range, radial velocity and angle with declared confidence/track association.

Radar and ranging are verified when the complete chain—waveform → propagation/scattering → synchronized sampling → matched filter → ambiguity resolution → statistical detection → calibrated geometry—recovers truth across the declared search region and false-alarm budget.