Applied Radar Mathematics

Core timing formulas

Formula reading: PRF is the number of transmitted pulses per second, while PRT is the time between two transmitted pulses. They are reciprocal: if the radar transmits every 1 ms, it transmits 1000 pulses per second. In the range equation, c₀ is the speed of electromagnetic propagation, approximately 3 × 10⁸ m/s, and t_d is the measured round-trip echo delay. The division by two is essential because the measured delay includes both outbound travel and return travel. Duty cycle D is the fraction of time the transmitter is actually radiating; increasing τ or PRF increases average transmitted power and heat load.

Blind Range

Blind range is the near-range interval hidden by transmit time and receiver recovery. A close target may produce a large echo, but if that echo returns while the receiver is blanked or desensitized, the radar cannot measure it. In high-power pulsed radars this is not only a waveform issue; it is also a receiver-protection issue.

Formula reading: R_blind is the closest range that cannot be observed because the receiver is not yet usable. c₀ converts time into distance. The term τ is the transmitted pulse duration: while the transmitter is on, the receiver is usually protected or blanked. The term t_rec is receiver recovery time after the transmitter turns off. The sum (τ + t_rec) is therefore the total blind time. The division by two converts round-trip propagation distance into one-way target range. R_guard is an extra engineering margin for duplexer leakage, limiter recovery, receiver settling, digital blanking, pulse shaping and range-gate timing uncertainty.

Unambiguous Range

Maximum unambiguous range is the farthest range whose echo returns before the next pulse. If a target echo arrives after the following transmit pulse, the receiver sees a valid echo but assigns it to the wrong transmit event. On the display it appears folded into a shorter apparent range.

Formula reading: PRT is the listening window available before the next pulse. Multiplying PRT by c₀ gives the total distance an electromagnetic wave can travel during that window; dividing by two gives the one-way target range. Because PRT = 1/PRF, increasing PRF shortens the unambiguous range. R_true is the physical target range; R_app is the apparent range displayed after folding. The modulo operation means that every extra R_unamb of true range wraps the echo back into the next ambiguous range interval.

Low PRF gives long unambiguous range but poor Doppler ambiguity performance. High PRF improves Doppler measurement and update rate, but causes range folding unless multiple PRFs, staggered PRF, coding or track-level ambiguity resolution is used.

Formula reading: R_max is the maximum detection range for the stated probability of detection and false alarm assumptions. P_tτ is transmitted pulse energy: peak power multiplied by pulse duration. G² appears because the same antenna gain helps twice in a monostatic radar, once during transmission and once during reception. λ² is the wavelength term from the effective receiving aperture. σ is target radar cross section, the equivalent area that describes how strongly the target scatters energy back toward the radar. The denominator contains spreading geometry, thermal noise kT, required SNR and total losses L_tot. The fourth root is the key engineering lesson: to double range, the power-energy-gain-RCS budget must improve by 16 times, or 12 dB.

Compression Ratio and Resolution

Formula reading: PCR is pulse compression ratio, approximately equal to the time-bandwidth product Bτ. B is waveform bandwidth in hertz and τ is pulse duration in seconds. A large Bτ means the radar transmits a long energetic pulse but processes it as if it were much shorter after matched filtering. S_r is range resolution. The term c₀/(2B) says that wider bandwidth gives finer separation in range; the factor two again comes from the two-way radar path.

Bandwidth, not transmitted pulse duration, sets compressed range resolution. Barker and phase codes also create sidelobes, so peak sidelobe level matters near strong targets.

Peak Sidelobe Level

Formula reading: PSL compares an unwanted sidelobe amplitude x_i with the main matched-filter peak x₀. The logarithm converts the ratio into decibels. The coefficient 20 is used because the ratio is an amplitude ratio; if comparing power directly, the coefficient would be 10. A sidelobe at -20 dB has one tenth of the main peak amplitude and one hundredth of its power.

A high range sidelobe can hide a weak target next to a strong one. Waveform design is therefore both a detection and an electronic-protection issue.

Cell dimensions

Formula reading: S_r is range cell length. With an uncompressed rectangular pulse, pulse duration τ sets how much range is illuminated at once; multiplying by c₀ gives round-trip distance, then dividing by two gives target-range depth. S_a is azimuth cell width. It grows with range R because the antenna beam subtends a larger physical width farther from the radar. θ is beamwidth in radians. V is an approximate resolution-cell volume using azimuth beamwidth, elevation beamwidth and range depth. The R² term is important: volume clutter, rain and chaff grow rapidly with range cell size.

FMCW range and Doppler

Formula reading: f_D is Doppler frequency shift and v_r is radial velocity along the radar line of sight. The factor two appears because a monostatic reflection experiences a two-way Doppler shift. In FMCW, S is chirp slope: how many hertz the transmit frequency changes per second. A delayed echo mixes with the current transmit frequency and creates beat frequency f_b. If the target is stationary, f_b maps directly to range through c₀f_b/(2S). If the target is moving, Doppler adds or subtracts from the beat frequency, so practical systems use up/down ramps or multi-chirp processing to separate range and velocity.

Velocity formulas

Formula reading: Doppler frequency is proportional to radial speed and inversely proportional to wavelength. At higher transmit frequency, wavelength is shorter, so the same target speed produces a larger Doppler shift. v_unamb is the approximate unambiguous radial velocity when Doppler is sampled pulse-to-pulse. The PRFλ/4 form comes from Nyquist sampling of two-way Doppler: the pulse train can only measure Doppler reliably within about ±PRF/2, and f_D = 2v_r/λ. This is the PRF dilemma: high PRF helps velocity ambiguity but hurts range ambiguity.

Search scan timing

Formula reading: T_rev is the time for one 360 degree mechanical revolution. RPM is revolutions per minute. T_dwell is how long the beam remains on a target direction during a sweep. Θ_beam is antenna beamwidth, normally in degrees or radians, and ω_scan is angular scan rate in the same angular unit per second. T_revisit is the time between two looks at the same angular cell. A multi-bar raster or volume scan increases revisit time because the radar must scan several elevation bars before returning to the first one.

Circular and Sector Scan

A circular scan rotates continuously through 360 degrees in azimuth and is typical of surveillance, naval and air-search radars. It offers wide-area coverage but relatively long revisit time if the rotation rate is slow. A sector scan limits the azimuth coverage to a selected angular sector, such as ±45 degrees, increasing update rate inside that sector while ignoring the rest of space.

Formula reading: &Theta_sector is the sector width. A smaller sector at the same scan speed gives a shorter sector time and faster revisit. N_hits is the number of pulses collected while the beam crosses the target. More hits improve integration, but only if the target remains inside the beam and the radar processor can coherently or noncoherently combine the pulses.

Raster, Bar and Volume Scan

A raster scan searches a rectangular angular volume. The radar sweeps in azimuth, steps or nods in elevation, then repeats. Fighter radars often describe this as a bar scan: 1-bar, 2-bar, 4-bar and so on. Weather and surveillance radars use related volume coverage patterns, where multiple elevation cuts build a three-dimensional picture.

Formula reading: &Theta_vol is the searched angular volume, described by azimuth width and elevation height. N_bars is the number of elevation lines. T_frame is the time to complete one full raster frame. Increasing bars improves elevation coverage but slows the update rate for any one target cell.

Conical, Palmer and Electronic Scan

Conical scan rotates a slightly offset beam around the antenna boresight. If the target is exactly on boresight, the echo amplitude is almost constant; if the target is off-axis, the echo is amplitude-modulated at the scan frequency. Palmer scan combines a conical motion with a larger search scan, producing a compound amplitude pattern. Electronic scan, used by phased arrays, steers the beam by phase or time delay instead of moving the whole antenna.

Formula reading: A(t) is received echo amplitude during conical scan. A₀ is mean amplitude, m is modulation depth caused by target angular error, f_s is scan frequency and φ is the phase of the error relative to the rotating beam. In a phased array, Δφ is the phase shift between adjacent elements, d is element spacing and θ is steering angle. Electronic scan can revisit targets quickly and interleave search, track and missile-support functions, but scan loss and sidelobe control become central design issues.

Dwell and hits per scan

Formula reading: T_D is dwell time, the time a scanning beam spends on a target direction. Θ_Az is azimuth beamwidth, and scan rate is angular speed. A narrow beam or fast scan reduces dwell time. Hits is the approximate number of pulses received from the target while it remains inside the beam. More hits can improve integration gain and detection probability, but slower scanning delays tactical updates.

False alarm rate

Formula reading: FAR is false alarm rate, normally measured as false alarms per second or per scan. N_false is the number of false detections counted during observation time T_obs. Lower false alarm probability requires a higher detection threshold, which raises required SNR and reduces sensitivity. Detection is always a trade among probability of detection, false alarms, clutter, integration time and target fluctuation.