Full Radar and EW Mathematics Course

EW, ES, EA and EP. Start by separating the mission from the math. Electronic support measures the spectrum: who is transmitting, where, with which waveform and with what tactical meaning. Electronic attack tries to reduce hostile measurement quality through noise, deception, decoys or expendables. Electronic protection restores friendly margin through antenna design, processing, emission control, agility and LPI design.

Training scenario. An intercept receiver detects a pulsed X-band emitter with stable PRI and narrow antenna scan. ES classifies it as a tracking radar; EA planning then asks whether a noise jammer, range deception or angular deception is the right response; EP asks how the radar might reject the attack.

Frequency is the first engineering decision because it sets wavelength. Wavelength determines practical antenna aperture, Doppler scale, scattering region, atmospheric loss and achievable resolution. ISM bands matter because many commercial emitters, drones and datalinks operate there, creating both interference and exploitation opportunities.

Example. At 2.4 GHz, λ is about 12.5 cm; at 10 GHz it is 3 cm. A target feature that is electrically small at 2.4 GHz may be a much stronger scatterer at X band.

This lesson covers isotropic reference antennas, dipoles, apertures, phased arrays, parabolic reflectors, horns and electronically scanned arrays. The key concept is that an antenna pattern is a spatial weighting function. The main beam gives gain; sidelobes create unwanted receiving directions.

Graphic assignment. The antenna module uses an original main-lobe/sidelobe diagram and should be expanded with a phased-array steering diagram showing element spacing, phase shift and grating-lobe risk.

Near-field/far-field boundaries tell you when normal radar and antenna equations are valid. Gain and beamwidth are coupled through aperture size; efficiency prevents ideal formulas from overstating real performance.

Example. A 2 m aperture at 10 GHz has a far-field distance around 267 m. If you test it at 30 m without compact-range correction, the measured pattern can be misleading.

Link budget starts with transmitted power and antenna gains, then subtracts free-space path loss and real-world losses. Fresnel clearance, earth bulge, atmospheric absorption, rain, polarization mismatch, two-ray reflection and knife-edge diffraction explain why field performance differs from a clean equation.

Example. A microwave link over water may meet FSPL budget but still fade deeply because the reflected ray arrives out of phase with the direct ray.

Command-guided systems use antenna beam geometry to keep the missile, target and guidance link coherent. The mathematics is essentially beamwidth, angular error and update-rate management. A narrow beam improves angular precision but demands tighter pointing and tracking.

Example. A 1° angular error at 10 km corresponds to roughly 175 m lateral displacement, which is unacceptable for terminal guidance unless corrected by tracking updates.

Noise factor, noise figure, equivalent temperature, SQNR, SINAD, SNIR and ENOB describe how much useful information survives the receiver chain. These quantities decide whether a mathematically detectable echo is actually measurable.

Example. A 1 MHz receiver with 5 dB NF has a noise floor near -109 dBm. A -96 dBm echo has about 13 dB SNR before processing losses.

Chaff creates a volume of resonant dipoles, while decoys create false targets using passive RCS or active retransmission. The key learning point is that the radar sees a resolution cell full of scatterers, not a single object.

Example. X-band chaff uses much shorter dipoles than L-band chaff. As the cloud spreads, density falls and the apparent target changes with time and wind.

Noise jamming attacks SNR; deception jamming attacks measurement validity. The course covers burn-through, J/S, self-protection jamming, escort/support/stand-in/stand-off jamming and communications jamming.

Example. A 1 µs delayed coherent retransmission appears 150 m away from the true return. If the delay is slowly increased, a range tracker can be pulled away from the real target.

Pulse radar topics include blind range, unambiguous range, chirp radar, Barker codes, near-perfect codes, peak sidelobe level and duty cycle. The central trade is energy versus resolution: long pulses carry energy, while bandwidth or short pulse duration provides resolution.

Example. A 20 MHz chirp gives about 7.5 m range resolution even if the transmitted pulse is long enough to carry much more energy than a 7.5 m unmodulated pulse.

Unmodulated CW measures velocity but not range. FMCW adds a frequency ramp so beat frequency maps to range. Doppler is the measurement of radial motion; tangential motion alone does not shift frequency.

Example. At 10 GHz, a 250 m/s radial target produces about 16.7 kHz Doppler shift.

Detection is statistical because noise, clutter and target RCS fluctuate. Dwell time and hits per scan determine how many pulses are available for integration. False alarm thresholds trade sensitivity against operator or tracker overload.

Example. A 2° beam scanning at 60°/s dwells for about 33 ms. At 1 kHz PRF, that is about 33 pulses for possible integration.

Radar beams bend in the atmosphere. Normal refraction extends the radio horizon; ducting can trap energy and create long ranges or coverage holes. Height estimation must therefore treat the atmosphere as part of the geometry.

Example. A 20 m radar can see a 1000 m target to about 149 km under the standard effective-earth approximation.

The radar resolution cell is the 3D measurement volume defined by range resolution and angular beamwidth. Point targets inside one cell merge; volume targets such as rain or chaff fill the cell.

Example. At 50 km with a 2° beam, classical angular separation is on the order of kilometers, even if range resolution is only meters.

One-way equations apply to links and intercept receivers. Two-way equations apply to monostatic point-target radar. Loss budgets and Swerling fluctuation models bridge ideal power math to real detection probability.

Example. Adding 6 dB loss reduces monostatic radar range by about 29%, because range is a fourth-root result.

Weather radar does not observe one point scatterer. It observes a volume containing many droplets, so the illuminated volume grows with range and changes the range law compared with a point target.

Example. A rain cell at longer range fills a larger resolution volume, partly offsetting geometric spreading compared with a single point target.

RCS depends on target size relative to wavelength, aspect angle, material and polarization. Rayleigh, Mie and optical regions describe how scattering changes as the target becomes electrically larger.

Example. A small drone may be weak at lower frequency but stronger at shorter wavelength when its structural features become comparable to λ.

Complex targets combine specular returns, edge diffraction, cavities, creeping waves and multiple interactions. Retroreflectors deliberately send energy back toward the radar and are used for calibration and decoys.

Example. Doubling a corner reflector edge length increases ideal RCS by 12 dB because the edge length is raised to the fourth power.

Decibels turn multiplication into addition and make large radar ratios manageable. The symbol glossary is not administrative; it is a unit-control tool. Most wrong radar calculations are unit mistakes.

Example. 500 W = 27 dBW. A 3 dB change is approximately a factor of two in power.