Introduction to Pulse Radar — Study Guide | EEEN 567 Satellite Engineering

1. Learning Objectives

By the end of this study guide, the student should be able to:

Explain the basic principle of radar and distinguish monostatic, bistatic and multistatic configurations.
Define the parameters of a pulsed radar waveform: pulse width τ, pulse repetition frequency (PRF), pulse repetition interval (PRI), duty cycle and peak/average power.
Compute target range from time delay, and determine the maximum unambiguous range for a given PRF.
Derive and apply the radar range equation to find detection range given transmitter power, antenna gain, target cross-section, wavelength and receiver sensitivity.
Relate antenna gain, beamwidth and aperture; compute gain and half-power beamwidth for typical radar antennas.
Explain receiver noise, noise figure, minimum detectable signal and the role of matched filtering in pulse radar.
Describe classical radar displays (A-scope and PPI) and modern digital signal processing chains.
Connect pulse radar to satellite engineering: satellite tracking radars, radar altimeters, SAR imaging from orbit, and range–Doppler tracking.

2. The Radar Principle

Radar (Radio Detection and Ranging) is an electromagnetic sensor that transmits radio waves and processes the echoes returned from objects ("targets") to determine their range, angular direction and, via the Doppler effect, their radial velocity. Radar works day and night, through cloud and (at suitable frequencies) rain, which is why it became indispensable in aviation, defence, weather forecasting, air-traffic control — and in space, for tracking satellites and imaging the Earth.

The basic timing idea is beautifully simple: radio waves travel at the speed of light c, so the round-trip time t to a target at range R is

R = c · t2 , c = 3 × 10⁸ m/s

The factor 2 appears because the pulse makes a round trip: transmitter → target → receiver.

Since light travels 300 m in 1 µs, every microsecond of delay corresponds to 150 m of target range. A modern radar measures time delays to nanosecond accuracy, giving metre-level range precision.

2.1 Functional block diagram of a pulse radar

2.2 Monostatic, bistatic and multistatic

Configuration	Geometry	Characteristics
Monostatic	Transmitter and receiver at the same site (single antenna via duplexer)	Simplest, most common; Tx self-interference suppressed by the duplexer and time gating
Bistatic	Transmitter and receiver separated by a large baseline	Receiver is passive/silent; forward-scatter can enhance detection of low-observable targets
Multistatic	Multiple spatially separated transmitters and/or receivers	Improved coverage, geolocation by triangulation; more complex synchronisation

Electrical Engineering Link A radar is a complete RF system end-to-end: waveform generation, high-power amplification (magnetrons, klystrons, TWTs, or solid-state GaN modules — the microwave devices of your earlier courses), duplexing, low-noise reception, down-conversion, matched filtering and detection. Pulse radar is essentially applied communications theory: everything you learned about modulation, sampling, SNR and optimum filtering appears here in practical form.

3. The Pulse Radar Waveform

A pulse radar transmits short bursts of RF energy and listens for echoes in the silent intervals between bursts. The governing timing parameters are:

Parameter	Symbol	Definition
Pulse width (duration)	τ	Length of each transmitted burst, typically 0.1 – 100 µs. Sets the minimum range and the range resolution (long τ ⇒ coarse resolution).
Pulse repetition frequency	PRF	Number of pulses per second, typically 100 Hz – 100 kHz.
Pulse repetition interval	PRI	Time between pulse starts: PRI = 1/PRF.
Duty cycle	d	Fraction of time the transmitter is on: d = τ·PRF = τ/PRI (typically 0.1 – 10%).
Peak power	P_t	Power during the pulse (can be kW to MW).
Average power	P_avg	P_avg = P_t·d — what determines thermal load and detection capability.

3.1 The PRF dilemma

PRF selection embodies a fundamental trade-off in pulse radar design:

Low PRF: long unambiguous range (good for long-range surveillance), but low average power per unit time and poor Doppler measurement.
High PRF: strong average power and good Doppler/velocity precision, but short unambiguous range — distant echoes fold over ("range ambiguity").

Modern radars resolve the dilemma with multiple PRFs (staggered PRF) or by combining techniques (medium-PRF design), unambiguously resolving both range and Doppler.

Key Concept The transmitter peak power is set by hardware limits (breakdown, device ratings), but the quantity that actually detects targets is the average power — and the average power a hardware chain can sustain is set by the duty cycle. Radar design is largely the art of spending a limited average-power budget wisely across the time–frequency plane.

4. Range Measurement and Ambiguities

4.1 Range from delay

R = c · t_r2

t_r = round-trip delay measured from pulse transmission to echo arrival

Worked example: an echo arrives 600 µs after its pulse. R = 3×10⁸ × 600×10⁻⁶ / 2 = 90 km.

4.2 Minimum range

The receiver cannot listen while the transmitter is still pulsing (it would be blinded/damaged), so the nearest detectable target is set by the pulse width:

R_min = c · τ2

For τ = 1 µs, R_min = 150 m.

4.3 Maximum unambiguous range

An echo must return before the next pulse is transmitted, otherwise it is associated with the wrong pulse:

R_max,unamb = c2 · PRF = c · PRI2

PRF = 1 kHz ⇒ R_max,unamb = 150 km. Targets beyond this fold into nearer apparent ranges ("second-time-around echoes").

4.4 Range resolution

Two targets are resolved in range if their echoes do not overlap. This needs the pulse width — or better, the bandwidth — to satisfy:

ΔR = c · τ2 = c2B

τ = 1 µs ⇒ ΔR = 150 m. With a 1 MHz bandwidth, ΔR = 150 m. To get fine resolution and enough energy per pulse, long coded pulses are transmitted and compressed on reception (pulse compression, Section 6.3).

4.5 Doppler and the range–Doppler pair

Because a radar is coherent (phase-stable), the echo from a moving target is shifted in frequency:

f_d = 2 v_rλ ⇒ v_r = f_d λ2

v_r = radial velocity (m/s); λ = wavelength (m). The factor 2 is the two-way Doppler. Unambiguous Doppler measurement needs f_d < PRF/2 (Nyquist across the pulse train) — the mirror of the range ambiguity.

For a satellite at 400 MHz moving with 7 km/s radial velocity, f_d = 2·7000/0.75 ≈ 18.7 kHz — well measurable, which is precisely why radars can track satellites precisely.

5. The Radar Range Equation

The radar range equation answers the engineer's central question: given my transmitter, antenna, target and receiver, how far can I detect? It accounts for spreading loss on the way out, interception by the target, re-radiation, spreading loss on the way back, and collection by the receive antenna:

P_r = P_t G_t4πR² · σ · 14πR² · A_er

P_r = received power (W) • P_t = transmitted peak power (W) • G_t = transmit antenna gain (linear) • σ = radar cross-section, RCS (m²) • A_er = effective receive aperture (m²)

Collecting terms with A_er = Gλ²/4π (monostatic, same antenna):

P_r = P_t G² λ² σ(4π)³ R⁴

R⁴ behaviour is the famous "radar loss": doubling range cuts received power sixteen-fold (12 dB per octave of range).

5.1 Detection range

Detection requires the received signal to exceed the minimum detectable signal S_min (set by noise, Section 7):

R_max = [ P_t G² λ² σ(4π)³ S_min ]^1/4

5.2 Typical radar cross-sections

Target	Typical σ (m²)
Small drone / bird	0.001 – 0.05
Fighter aircraft	1 – 5 (much less if stealth-shaped)
Large aircraft / airliner	10 – 100
Satellite in LEO	1 – 100+ depending on size and aspect
Ship	1 000 – 100 000
Rain cell (weather radar)	defined by reflectivity Z

5.3 Forms of the equation

Search radar form: rearrange in terms of average power and the solid angle searched per second (the "search equation") — the right metric when the job is surveying volume.
Tracking radar form: for a dish staring at one target, dwell time is long, so integration of many pulses raises sensitivity.
Propagation losses: real systems multiply by atmospheric attenuation, surface-reflection multipath and pattern factors.

Key Concept The R⁴ law is the deepest result in radar: detection range grows only as the fourth root of everything you control. To double range you must raise power ×16, gain ×4, or lower S_min by 12 dB. This brutal scaling drives every radar architecture decision.

6. Radar Antennas and Waveform Design

6.1 Gain, aperture and beamwidth

The antenna concentrates radiated power into a beam. For an aperture of physical area A with efficiency η:

G = 4π η Aλ² , θ_3dB ≈ k λD

η ≈ 0.5 – 0.8 • D = aperture dimension (m) • k ≈ 1.02 – 1.3 (shape-dependent, ≈ 1.22 for a uniform circular aperture)

These two relations capture the antenna trade-off: high gain (narrow beam) requires a large aperture measured in wavelengths. That is why radar antennas are big, and why higher frequencies (shorter λ) give sharper beams for the same physical size — at the price of more atmospheric attenuation.

Antenna type	Typical use
Parabolic dish	Tracking radars, satellite ground antennas, weather radars
Planar phased array	Modern AESA radars: electronic beam steering, no moving parts
Slotted waveguide array	Marine and ATC surveillance radars
Lens / reflector hybrids	Specialised high-gain applications

6.2 Scanning

Mechanical scan: the whole antenna rotates or nods (simple, mature).
Electronic scan (phased array): phase shifters steer the beam in microseconds — track many targets, interleave search, no servo inertia.
Track-while-scan (TWS): a surveillance radar updates tracks every scan (~4–10 s); dedicated tracking radars dwell continuously for higher precision.

6.3 Pulse compression — resolving the τ dilemma

Detection wants a long pulse (energy); resolution wants a short pulse. Pulse compression reconciles them by transmitting a long pulse whose frequency or phase is modulated (linear FM "chirp", Barker or Frank codes), then passing the echo through a matched filter that compresses it to an effective width of 1/B. Resolution ΔR = c/2B is then set by bandwidth, not τ.

Electrical Engineering Link The matched filter is the cornerstone result you met in communication theory: for a signal in white noise, the filter matched to the transmitted waveform maximises output SNR. In a chirp radar the matched filter is simply a dispersive delay network (SAW device historically; today digital FIR in the processor). Compression ratio ≈ τ·B (hundreds to thousands) is the processing gain that makes long-pulse operation viable.

7. Detection, Noise and the Receiver

7.1 Thermal noise floor

The received echo competes with noise. The available noise power at the receiver input is:

N = k · T₀ · B · F

k = 1.38×10⁻²³ J/K (Boltzmann) • T₀ = 290 K (standard temperature) • B = receiver bandwidth (Hz) • F = noise figure (linear, ≥ 1)

7.2 Minimum detectable signal and SNR

S_min = k T₀ B F · (S/N)_min

(S/N)_min is set by the required probability of detection P_d and probability of false alarm P_fa. For a non-fluctuating target in Gaussian noise with a single pulse, (S/N)_min ≈ 13–14 dB for P_d = 0.9 at P_fa = 10⁻⁶; coherent or non-coherent integration of N pulses buys ≈ 10·log₁₀N dB of sensitivity.

Design lever	Effect on detection range
+10 dB transmitter power	× 1.78 range
+6 dB antenna gain (G² ⇒ +12 dB two-way)	× 2.0 range
10× target RCS (+10 dB)	× 1.78 range
+3 dB lower S_min	× 1.19 range
Integrate 100 pulses (+20 dB)	× 3.16 range

7.3 Sensitivity and the receiver chain

The receiver cascade (LNA → mixer → IF amplifier → detector → ADC) is characterised by its overall noise figure, dominated by the first stage (Friis' formula) — hence low-noise amplifiers (HEMT/HBT) right after the duplexer. Modern radars digitise at IF or RF and do detection digitally, enabling pulse compression, Doppler filtering (MTI — moving target indication against clutter) and constant-false-alarm-rate (CFAR) thresholding in software.

Key Concept Detection is a statistical decision, not a certainty: the threshold trades P_d against P_fa. Raising the threshold reduces false alarms but loses weak targets; integrating many pulses recovers sensitivity. This detection-theoretic view (Neyman–Pearson) is exactly the hypothesis-testing framework of your probability and communications courses.

8. Radar Displays

Display	Format	Use
A-scope	Amplitude vs range (x = time/range, y = echo amplitude)	Waveform examination, radar alignment, range measurement
PPI (Plan Position Indicator)	Intensity-modulated polar sweep (r = range, θ = antenna azimuth)	Surveillance picture — ATC, weather, ship navigation
B-scope / E-scope	Range vs azimuth (B) or range vs elevation (E) intensity map	Fighter fire-control, height-finding analysis
Modern console	Digital range–Doppler maps, synthetic video, track tables	All contemporary systems

9. Pulse Radar in Satellite Engineering

Radar and satellites meet in several crucial ways — this section ties the guide back to EEEN 567:

9.1 Satellite tracking radars

Ground radars (e.g., the US Space Surveillance Network's phased arrays and mechanical dishes) detect, track and catalogue objects in orbit — vital for conjunction screening ("space traffic control") and for LEOP support: a newly launched satellite is acquired by radar and optical sensors until its transponders are verified.
Skin tracking: echoes from the satellite body itself; RCS varies with aspect and tumbling state, so radar also diagnoses deployment anomalies (a satellite that fails to despin shows a characteristic modulation).
Beacon/cooperative tracking: the satellite's TT&C transponder replies to interrogation — much stronger and more accurate than skin tracking, effectively extending range and precision.

9.2 Radar aboard the satellite

Radar altimetry: precise ocean/ice height measurement (TOPEX/Poseidon, Jason series, Sentinel-3/6) — a pulse-limited or SAR-mode altimeter measuring range to the surface with centimetre accuracy.
Synthetic Aperture Radar (SAR): the satellite's along-track motion synthesises a kilometres-long aperture, achieving metre-class ground resolution day and night, through cloud (ERS, Envisat, Sentinel-1, RADARSAT). SAR is pulse compression taken to its extreme: the "antenna" is the orbit itself.
Cloud/rain radars and planetary radars on deep-space missions.

9.3 Range–Doppler tracking and orbit determination

Measuring range (delay) and range-rate (Doppler) from multiple passes over a ground radar yields precise orbital elements — the same observables, inverted, that you studied in orbit mechanics. For a LEO satellite the Doppler shift can sweep tens of kHz at UHF, which is why satellite ground-station receivers include automatic Doppler compensation.

Electrical Engineering Link Every link budget in satellite communications is the radar range equation in disguise (with R² instead of R⁴ because there is no scattering target — the "radar" has become two separate antennas). Master the range equation here and TT&C link design in later EEEN 567 topics becomes straightforward.

10. Interactive Learning Calculators

Verify the worked examples and explore tutorial-style "what-if" questions.

10.1 Target Range from Echo Delay

Echo delay t_r (µs)

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10.2 PRF, Unambiguous Range & Duty Cycle

Pulse repetition frequency PRF (Hz)

Pulse width τ (µs)

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10.3 Monostatic Radar Range Equation

Peak power P_t (kW)

Antenna gain G (dBi)

Frequency f (GHz)

Target RCS σ (m²)

Bandwidth B (MHz)

Noise figure F (dB)

Required SNR (dB)

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10.4 Antenna Gain & Beamwidth

Aperture diameter D (m)

Aperture efficiency η

Frequency f (GHz)

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10.5 Doppler Shift from a Satellite

Frequency f (MHz)

Radial velocity v_r (km/s)

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11. Review Questions

Q1. A pulse radar echo returns 250 µs after transmission. The target range is:
a) 75 km b) 37.5 km c) 150 km d) 25 km

R = c·t/2 = 3×10⁸ × 250×10⁻⁶ / 2 = 3.75×10⁴ m = 37.5 km. Halving the raw time-of-flight for the round trip is the classic error — option (a).

Q2. A radar with PRF = 2 kHz has a maximum unambiguous range of:
a) 300 km b) 75 km c) 150 km d) 7.5 km

R_max,unamb = c/(2·PRF) = 3×10⁸ / (2 × 2000) = 75 000 m = 75 km. Echoes from beyond 75 km alias into nearer ranges ("second-time-around" echoes).

Q3. In the radar range equation, received power varies with range as:
a) R⁻⁴ b) R⁻² c) R⁻¹ d) R⁻⁸

One R⁻² spreading loss each way gives R⁻⁴. A communications one-way link loses only R⁻² — this extra factor of R² is the "radar penalty".

Q4. A 2 µs pulse gives a range resolution of approximately:
a) 600 m b) 150 m c) 3 km d) 300 m

ΔR = cτ/2 = 3×10⁸ × 2×10⁻⁶ / 2 = 300 m. Finer resolution needs wider bandwidth (pulse compression), not shorter pulses at fixed peak power.

Q5. Doubling the antenna gain of a monostatic radar (same on transmit and receive) increases detection range by a factor of about:
a) 1.41 b) 2.0 c) 4.0 d) 1.19

Gain enters squared (two-way), giving +12 dB for a doubling of G. Since R ∝ (power terms)^1/4: 2^12/20×4… precisely R × (4)^1/2 = ×2. Yes — +12 dB in the R⁴ equation doubles range.

Q6. The purpose of pulse compression is to:
a) Increase the PRF b) Reduce the transmitter's peak power rating c) Achieve fine range resolution while retaining the energy of a long pulse d) Eliminate Doppler ambiguity

A long coded pulse (chirp/phase code) is compressed by the matched filter to ~1/B width: resolution from bandwidth, energy from pulse length, processing gain ≈ τB.

Q7. For a satellite at 400 MHz with 7.5 km/s radial velocity, the two-way Doppler shift at a tracking radar is approximately:
a) 20 kHz b) 10 kHz c) 2 kHz d) 80 kHz

f_d = 2v/λ = 2 × 7500 / 0.75 m = 20 kHz. (One-way Doppler for a 400 MHz beacon would be 10 kHz — the factor of 2 again.)

Q8. Raising the detection threshold in a radar receiver generally:
a) Increases both P_d and P_fa b) Increases P_d, decreases P_fa c) Decreases P_d, increases P_fa d) Decreases both P_d and P_fa

A higher threshold rejects more noise (lower false-alarm rate) but also rejects more genuine weak echoes (lower detection probability). CFAR adaptively sets the threshold to hold P_fa constant against clutter.

Summary of Key Takeaways

Radar measures range from echo delay (R = ct/2) — the factor of 2 pervades every radar formula.

The pulsed waveform is defined by τ, PRF, duty cycle and peak/average power; PRF trades unambiguous range against Doppler capability.

The range equation's R⁴ law makes radar detection range stubbornly insensitive to power: 16× power for 2× range.

Resolution comes from bandwidth; energy comes from pulse length; pulse compression and matched filtering give both.

Detection is statistical: thresholds trade P_d against P_fa, and pulse integration buys back sensitivity.

For satellites, radar provides skin/beacon tracking and space surveillance from the ground, and altimetry/SAR payloads on orbit — the same physics, inverted.