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Radar & Remote Sensing
1. Range from time delay
For a round trip,
\[R=\frac{c\Delta t}{2}.\]The factor of two is fundamental: the wave travels to the target and back.
2. Monostatic radar equation
A common ideal form is
\[P_r=\frac{P_tG^2\lambda^2\sigma}{(4\pi)^3R^4L},\]where $\sigma$ is radar cross section and $L$ collects system losses.
The $R^{-4}$ dependence comes from spherical spreading on both outbound and return paths.
3. Radar cross section
RCS is an electromagnetic scattering property rather than a simple physical area:
\[\sigma=\lim_{r\to\infty}4\pi r^2\frac{|E_s|^2}{|E_i|^2}.\]It depends on frequency, angle, polarization, geometry and material.
4. Range resolution
For bandwidth $B$,
\[\Delta R\approx\frac{c}{2B}.\]5. Doppler velocity
For a monostatic radar,
\[f_D=\frac{2v_r}{\lambda}.\]Higher carrier frequency gives larger Doppler shift for the same velocity.
6. Pulsed radar
Transmit a pulse, receive after a delay, and map time to range. Pulse width, pulse compression, PRF and coherent processing determine range resolution, unambiguous range, velocity ambiguity and SNR.
7. FMCW radar
For a linear chirp with slope $S=df/dt$, a stationary target produces approximately
\[f_b\approx\frac{2SR}{c},\]so
\[R\approx\frac{cf_b}{2S}.\]Moving targets add Doppler, which is separated using multiple chirps and 2-D range-Doppler processing.
8. Angle estimation and arrays
A phased array extracts angle from relative phase/amplitude across elements. Digital beamforming and MIMO radar can create virtual apertures. Array calibration is critical because phase/gain errors look like angle errors or sidelobes.
See AESA.
9. Synthetic aperture radar
SAR synthesizes a large aperture from platform motion. Coherent processing combines observations from multiple positions to achieve cross-range resolution much finer than the instantaneous physical antenna beam alone would suggest.
10. Radar families
| Type | Strength | Typical application |
|---|---|---|
| Pulsed | long range, flexible waveform | surveillance, weather |
| FMCW | compact, excellent short-range resolution | automotive, industrial |
| CW Doppler | velocity sensing | motion sensors, speed measurement |
| Phased-array | agile beam steering | defense, weather, multifunction |
| SAR | high-resolution imaging | Earth observation, mapping |
| Passive radar | no dedicated transmitter | sensing using illuminators of opportunity |
Worked example — 77 GHz velocity and 1 GHz range bandwidth
At 77 GHz, $\lambda\approx3.89$ mm. A target at 30 m/s gives
\[f_D\approx15.4\ \text{kHz}.\]A 1 GHz chirp bandwidth gives
\[\Delta R\approx0.15\ \text{m}.\]The two numbers come from different physical observables: Doppler phase evolution and waveform delay resolution.
11. Noise and detection
Receiver noise floor scales with $kTB$. Detection performance depends on integration/coherent processing, false-alarm requirement, clutter statistics, target fluctuation model and implementation loss—not only raw received power.
12. How radar is measured
- VNA/full-wave methods for antenna and RCS characterization;
- channel/target simulators for controlled range-Doppler targets;
- spectrum/VSA for chirp linearity and modulation quality;
- oscilloscope for pulse timing and transient behavior;
- chamber/OTA tests for array patterns and angle accuracy.
13. Engineering reality
References
- M. I. Skolnik, Introduction to Radar Systems.
- M. A. Richards, Fundamentals of Radar Signal Processing.
- MIT Lincoln Laboratory, Introduction to Radar Systems.
Related: AESA · Antennas · Wireless · Measurements
Advanced topic — matched filtering and range–Doppler ambiguity
For transmitted complex envelope (s(t)), a target with propagation delay (\tau) and Doppler shift (f_D) produces a matched-filter response governed by
\[\chi(\tau,f_D)=\int s(t)s^*(t-\tau)e^{-j2\pi f_Dt}\,dt.\]The magnitude of the ambiguity function reveals the waveform’s delay/Doppler resolution and sidelobe tradeoff. The familiar range-resolution approximation (\Delta R=c/(2B)) does not specify detection probability, false-alarm rate or clutter suppression.
For FMCW, (f_b\approx S\tau) for stationary targets, where (S=B/T), and motion introduces a signed Doppler contribution. Chirp nonlinearity, LO phase noise, near-range leakage and interference can bias the estimated range. Coherent processing across chirps requires stable timing and phase.
Advanced exercise: calculate the ambiguity functions of a linear-FM chirp and a phase-coded pulse at equal energy; compare their sidelobes, range–Doppler coupling and response to a moving target.
Further reading: M. A. Richards, Fundamentals of Radar Signal Processing; MIT radar laboratory course.