M1. The Radar Equation

The radar equation connects every design decision in one formula: transmit power, antenna gain, wavelength, target size, path loss, and receiver sensitivity all trade off to set the maximum detection range. It is the skeleton of every radar system budget.

1. Derivation from First Principles

Consider a radar transmitting power $P_t$ through an antenna with gain $G_t$. The power density at range $R$ in the main beam is

$$S_{\text{inc}} = \frac{P_t\,G_t}{4\pi R^2}\;\;[\text{W/m}^2].$$

The target intercepts this power density and re-radiates some fraction, characterized by its radar cross section (RCS) $\sigma$ [m\u00B2]. The scattered power radiated effectively isotropically toward the receiver is $\sigma\cdot S_{\text{inc}}$, so the power density at the receiving antenna (back at range $R$) is

$$S_{\text{rx}} = \frac{P_t\,G_t}{4\pi R^2}\cdot\frac{\sigma}{4\pi R^2}.$$

The receiver captures $A_{\text{eff}} = G_r\lambda^2/(4\pi)$ from this power density, giving the received power:

$$\boxed{\;P_r = \frac{P_t\,G_t\,G_r\,\lambda^2\,\sigma}{(4\pi)^3\,R^4\,L}\;}.$$

For a monostatic radar $G_t = G_r = G$ and the famous $R^4$ dependence emerges. $L$ aggregates all other losses: atmospheric absorption (Module 0), system losses (cables, radomes), beam-scan losses, and fluctuation losses.

Monostatic Radar Geometry

2. Radar Cross Section (RCS)

The RCS $\sigma$ is defined by requiring that, if the target re-radiated isotropically with power $\sigma S_{\text{inc}}$, it would produce the observed scattered field. Formally:

$$\sigma = \lim_{R\to\infty}\,4\pi R^2\,\frac{|\mathbf{E}_{\text{scat}}|^2}{|\mathbf{E}_{\text{inc}}|^2}.$$

Canonical targets in three frequency regimes:

Rayleigh ($ka\ll 1$): $\sigma \propto a^6/\lambda^4$ (hydrometeors; see Module 7).
Resonance ($ka\sim 1$): oscillations in Mie series; RCS can exceed geometric cross section.
Optical ($ka\gg 1$): for a sphere, $\sigma \to \pi a^2$ (physical cross section).

Examples of high-frequency RCS of canonical shapes:

$$\sigma_{\text{sphere}} = \pi a^2,\qquad \sigma_{\text{flat plate}} = \frac{4\pi A^2}{\lambda^2}\cos^4\theta,\qquad \sigma_{\text{corner cube}} = \frac{4\pi (2\ell^2/\sqrt{3})^2}{\lambda^2}.$$

Note the flat-plate formula's $\lambda^{-2}$ dependence: small wavelengths give huge specular returns. This is why stealth shaping (Module 8) avoids flat surfaces facing the threat sector.

3. Specular vs. Diffuse Scattering

The Rayleigh criterion distinguishes smooth from rough surfaces: surface is smooth if$h < \lambda/(8\cos\theta)$ where $h$ is rms height. Smooth surfaces scatter specularly (mirror-like). Rough surfaces exhibit diffuse scattering following approximately a Lambertian law:

$$\sigma^0 = \frac{d\sigma}{dA}\propto \cos\theta,\qquad \text{(Lambertian)}.$$

The normalized RCS $\sigma^0$ (sigma-naught) is fundamental in SAR; it is expressed in dB and depends on incidence angle, frequency, polarization, and surface type.

4. Maximum Detection Range

Setting $P_r = P_{\min}$ (minimum detectable signal) and solving for range:

$$\boxed{\;R_{\max} = \left[\frac{P_t\,G^2\,\lambda^2\,\sigma}{(4\pi)^3\,P_{\min}\,L}\right]^{1/4}.\;}$$

The fourth-root is discouraging: to double range we must increase transmit power by a factor of 16. This motivates every subsequent module — matched filtering (Module 2), pulse integration (Module 3), aperture tapering (Module 4), and detection theory (Module 5) all boost effective $R_{\max}$ without paying the full $P_t$ penalty.

The minimum detectable signal depends on noise:

$$P_{\min} = k_B\,T_0\,B\,F\,\text{SNR}_{\min},$$

where $k_B = 1.38\times 10^{-23}$ J/K, $T_0 = 290$ K, $B$ is receiver bandwidth in Hz, $F$ is the noise figure (linear), and SNR\u2098\u1D62\u2099 is the signal-to-noise ratio for detection at a required probability (typically 13 dB for 90% $P_d$at $P_{fa} = 10^{-6}$).

5. Bistatic Radar Equation

In bistatic geometry the transmitter is at range $R_t$ from the target and the receiver at range $R_r$. The equation becomes

$$P_r = \frac{P_t\,G_t\,G_r\,\lambda^2\,\sigma_b}{(4\pi)^3\,R_t^2\,R_r^2\,L}.$$

The bistatic RCS $\sigma_b$ differs from the monostatic value and depends on the bistatic angle $\beta$ (the angle target-Tx-target-Rx). Important features:

Forward scatter ($\beta \to 180^\circ$): RCS rises dramatically by Babinet's principle — $\sigma_f = 4\pi A^2/\lambda^2$ even for stealthy targets.
Isorange contour: loci of constant $R_t+R_r$ form an ellipse with foci at Tx and Rx.
Geolocation: requires multiple Rx or extra Doppler info.

Passive radar (Module 8) exploits bistatic geometry using illuminators of opportunity (FM, DVB-T, GSM).

6. Coherent and Non-coherent Integration Gain

Integrating $N$ pulses coherently (preserving phase) adds signal amplitudes and noise powers, yielding SNR gain of exactly $N$. Non-coherent integration (after envelope detection) gives a gain $G_{\text{nci}}$ between $\sqrt{N}$ and $N$, well-approximated by

$$G_{\text{nci}}(N) \approx N - L_{\text{nci}}(N),\qquad L_{\text{nci}} \sim \log N\;\text{dB}.$$

Combining with the radar equation, the effective range after coherent integration becomes

$$R_{\max}(N) = R_{\max}(1)\cdot N^{1/4}.$$

7. System Noise Temperature

The noise at the receiver input has contributions from the antenna (sky + ground), feedlines, and receiver noise figure. The effective system temperature via Friis's formula is

$$T_{\text{sys}} = T_a + (L_f-1)T_0 + L_f(F-1)T_0,$$

where $T_a$ is antenna temperature (cold sky: 10 K; warm Earth: 260 K), $L_f$is feedline loss, and $F$ is receiver noise figure. Good low-noise amplifiers at X-band deliver $F < 1.5$ dB; cryogenic radio astronomy receivers reach sub-Kelvin noise temperatures.

8. Losses Budget

The lumped loss term $L$ typically absorbs:

Source	Typical [dB]	Notes
Transmit feedline	0.5-2	Waveguide or coax run
Receive feedline	0.5-1	Before LNA
Radome	0.5-1	Wet radome much worse
Beam-scan loss	1-3	Straddling PRIs
Atmospheric (2-way)	0.1-10	Highly band-dependent
Target fluctuation (Swerling)	1-8	See Module 5
Matched filter mismatch	0.5-1	Windowing, quantization
CFAR loss	1-3	Detection threshold tax

Total losses of 6-15 dB are typical in operational systems; radar engineers treat this as the difference between textbook and field performance.

Typical RCS Values

Simulation: Range Budget & RCS

The code below computes received power vs. range for five target types (bird, drone, fighter, airliner, ship), the RCS of canonical shapes (sphere, cylinder) across L to W bands, and the resulting SNR curves for four transmit powers. A polar plot illustrates the severe angular dependence of flat-plate RCS — the physical basis for stealth shaping.

Python

script.py121 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ---- Radar range calculation ----
# P_r = P_t G^2 lam^2 sigma / ((4pi)^3 R^4 L)

def radar_equation(Pt, G_db, f_Hz, sigma_m2, R_m, L_db=0.0):
    c = 2.998e8
    lam = c / f_Hz
    G = 10**(G_db/10)
    L = 10**(L_db/10)
    Pr = Pt * G**2 * lam**2 * sigma_m2 / ((4*np.pi)**3 * R_m**4 * L)
    return Pr

def max_range(Pt, G_db, f_Hz, sigma_m2, Pmin_W, L_db=0.0):
    c = 2.998e8
    lam = c / f_Hz
    G = 10**(G_db/10)
    L = 10**(L_db/10)
    return (Pt * G**2 * lam**2 * sigma_m2 / ((4*np.pi)**3 * Pmin_W * L))**0.25

# Example system: 1 MW peak, 35 dB gain, X band, 1 m^2 RCS, Pmin = -110 dBm
Pt = 1e6; G_db = 35; f = 10e9; sigma = 1.0
Pmin_dbm = -110; Pmin_W = 10**(Pmin_dbm/10)*1e-3
R_max_km = max_range(Pt, G_db, f, sigma, Pmin_W) / 1000
print(f"Maximum range (1 m^2 target, -110 dBm sensitivity): {R_max_km:.1f} km")

# ---- Plot: received power vs range for several RCS ----
R = np.logspace(2, 5.5, 400)  # 100 m to 300 km
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0f172a')
for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for sp in ax.spines.values(): sp.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

sigmas = [(0.01, 'bird', '#fbbf24'),
          (0.1,  'drone', '#fb923c'),
          (1.0,  'fighter', '#f87171'),
          (10.0, 'airliner', '#a78bfa'),
          (100.0,'ship', '#60a5fa')]
for s, lbl, col in sigmas:
    Pr = radar_equation(Pt, G_db, f, s, R)
    axes[0,0].loglog(R/1000, 10*np.log10(Pr*1000), color=col, linewidth=2, label=f"{lbl} ({s} m²)")
axes[0,0].axhline(Pmin_dbm, color='white', linestyle='--', alpha=0.5, label='Sensitivity')
axes[0,0].set_xlabel('Range [km]', color='#e2e8f0')
axes[0,0].set_ylabel('Received power [dBm]', color='#e2e8f0')
axes[0,0].set_title('Received Power vs Range  (1 MW, 35 dB, X-band)', color='#fecaca', fontsize=13)
axes[0,0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# ---- RCS of canonical targets vs frequency ----
# Sphere: optical sigma = pi a^2; Rayleigh sigma ~ a^6
a_vals = [0.01, 0.1, 1.0]  # radii (m)
freq = np.logspace(8, 11, 400)
lam = 3e8/freq
for a, col in zip(a_vals, ['#f87171','#fbbf24','#60a5fa']):
    ka = 2*np.pi*a/lam
    # Mie asymptotic: Rayleigh (ka<<1) + optical (ka>>1)
    rayleigh = 9 * np.pi * a**2 * ka**4
    optical  = np.pi * a**2 * np.ones_like(ka)
    sigma_sphere = np.where(ka < 1, rayleigh, optical)
    axes[0,1].loglog(freq/1e9, sigma_sphere, color=col, label=f'Sphere a={a} m', linewidth=1.8)

# Cylinder (side-on): sigma ~ 2 pi a L^2 / lam for ka >> 1
L_cyl = 1.0
for a, col in zip([0.01, 0.1], ['#34d399','#a78bfa']):
    sigma_cyl = 2*np.pi*a*L_cyl**2/lam
    axes[0,1].loglog(freq/1e9, sigma_cyl, '--', color=col, label=f'Cyl a={a} m', linewidth=1.8)

axes[0,1].set_xlabel('Frequency [GHz]', color='#e2e8f0')
axes[0,1].set_ylabel('RCS $\sigma$ [m²]', color='#e2e8f0')
axes[0,1].set_title('RCS vs Frequency (canonical targets)', color='#fecaca', fontsize=13)
axes[0,1].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# ---- SNR vs range for different powers ----
k_B = 1.38e-23; T0 = 290
B = 1e6          # 1 MHz bandwidth
NF_db = 3.0       # 3 dB noise figure
N = k_B * T0 * B * 10**(NF_db/10)

for Pt_demo, col, lbl in zip([1e3, 1e5, 1e6, 1e7], ['#60a5fa','#a78bfa','#f87171','#fbbf24'], ['1 kW','100 kW','1 MW','10 MW']):
    Pr = radar_equation(Pt_demo, G_db, f, 1.0, R)
    snr_db = 10*np.log10(Pr/N)
    axes[1,0].semilogx(R/1000, snr_db, color=col, linewidth=2, label=lbl)
axes[1,0].axhline(13, color='white', linestyle='--', alpha=0.5, label='13 dB detect')
axes[1,0].set_xlabel('Range [km]', color='#e2e8f0')
axes[1,0].set_ylabel('SNR [dB]', color='#e2e8f0')
axes[1,0].set_title('SNR vs Range (1 m² target, 1 MHz BW)', color='#fecaca', fontsize=13)
axes[1,0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
axes[1,0].set_ylim(-20, 80)

# ---- RCS polar pattern of a sphere (Mie asymptote) vs angle for a cylinder ----
theta = np.linspace(-np.pi, np.pi, 400)
# Rectangular plate specular peak at theta=0
A = 1.0  # area m^2
lam0 = 0.03  # X-band
sigma_plate = 4*np.pi*A**2/lam0**2 * np.sinc(2*np.sin(theta)/lam0*0.3)**2
ax_polar = plt.subplot(2,2,4, projection='polar')
ax_polar.set_facecolor('#1e293b')
ax_polar.plot(theta, 10*np.log10(np.abs(sigma_plate)+1e-6), color='#f87171', linewidth=2)
ax_polar.set_title('Flat-plate RCS [dBsm] vs angle', color='#fecaca', fontsize=13, pad=15)
ax_polar.tick_params(colors='#cbd5e1')
ax_polar.grid(True, color='#334155', alpha=0.4)
ax_polar.set_ylim(-20, 40)

plt.tight_layout()
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0f172a')

# Summary table
print()
print("=== Radar range budget ===")
print(f"Pt = {Pt/1e6:.0f} MW, G = {G_db} dB, f = {f/1e9:.1f} GHz")
print(f"Bandwidth = {B/1e6:.1f} MHz, NF = {NF_db} dB")
print(f"Noise floor N = {10*np.log10(N*1000):.1f} dBm")
for s, lbl, _ in sigmas:
    rng = max_range(Pt, G_db, f, s, Pmin_W)/1000
    print(f"  sigma = {s:>5.2f} m^2 ({lbl:>8s}): R_max = {rng:6.1f} km")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

9. Sensitivity Factor and Figure of Merit

Dividing out the factors that depend on target RCS, we get the radar's sensitivity factor:

$$K_s = \frac{P_t\,G^2\,\lambda^2}{(4\pi)^3\,P_{\min}\,L},\qquad R_{\max} = (K_s\,\sigma)^{1/4}.$$

The product $P_{\text{avg}}\cdot A_e$ (average power times effective aperture area) is often called the power-aperture product — the single most important figure of merit for radar. A search radar must cover solid angle $\Omega$ in time $t_s$; the required $P_t A_e^2$ for given $R_{\max}$ scales as

$$P_t A_e^2 \propto \frac{4\pi}{\Omega}\cdot\frac{R_{\max}^4}{t_s}\cdot\frac{k_B T_s F \cdot\text{SNR}_{\min}}{\sigma}.$$

This is the air-search radar equation. Doubling range requires 16× more average power; doubling coverage halves the time per beam.

10. Range Ambiguity Preview

The radar equation assumes we measure range unambiguously. In practice, a pulsed radar with pulse repetition interval PRI can only identify targets within $R_u = c\,\text{PRI}/2$. Targets beyond this appear folded. We treat this fully in Module 2; here note only that extending $R_{\max}$ via higher power or gain also stresses PRF selection.

11. Worked Example: Air Search Radar

Typical long-range air surveillance parameters:

Frequency	1.3 GHz (L-band)
Peak power $P_t$	2 MW
Antenna gain $G$	34 dB
Pulse width	6 \u03BCs
Bandwidth	170 kHz
Noise figure	3 dB
Target RCS	2 m\u00B2 (fighter)
SNR required	13 dB for 90% P\u2096\u2090 at 10\u207B\u2076 P\u2096_fa
Total losses	10 dB

Compute: noise power $N = kT_0 B F = 4\times 10^{-15}$ W = -144 dBW. Minimum signal $P_{\min} = N\cdot\text{SNR} = -131$ dBW. With $\lambda = 0.23$ m:

$$R_{\max} = \left[\frac{(2\times10^6)(10^{3.4})^2 (0.23)^2 \cdot 2}{(4\pi)^3\cdot 10^{-13.1}\cdot 10^{1.0}}\right]^{1/4}\approx 470\,\text{km}.$$

This matches the advertised range of AN/FPS-117 and similar operational systems within 10%.

Historical Note: Watson-Watt and Chain Home

Robert Watson-Watt's 1935 Daventry experiment used a BBC transmitter to illuminate a Handley Page Heyford bomber, detecting the Doppler-shifted reflection at 12 km. The resulting Chain Home system (1938) operated at 20-30 MHz with kilowatt pulse powers and 107 m towers. Its radar equation numbers look primitive by today's standards — but it detected Luftwaffe raids in time for RAF response and arguably decided the Battle of Britain. Every subsequent refinement in this course — pulse compression, Doppler processing, phased arrays, SAR — relaxes some constraint in that fundamental $R^4$ equation.

12. Clutter-Limited Range

In many tactical scenarios, the detection floor is not receiver noise but clutter (returns from ground, sea, rain). The ratio of target-to-clutter for a surface clutter patch of normalized RCS $\sigma^0$ illuminated by a pulse of length $c\tau/2$ in a beam of azimuth width $\theta_a$ is

$$\text{SCR} = \frac{\sigma}{\sigma^0\,A_c} = \frac{\sigma}{\sigma^0\,R\theta_a\,(c\tau/2)\sec\psi},$$

where $\psi$ is the grazing angle. Note SCR $\propto 1/R$, not $1/R^4$: clutter grows with the illuminated area. Doppler processing (Module 3) and MTI filtering allow discrimination when the target moves relative to the clutter.

References

Skolnik, M.I. — Introduction to Radar Systems, ch. 2.
Richards, M.A. — Fundamentals of Radar Signal Processing, ch. 2 & 3.
Knott, E.F., Shaeffer, J.F., Tuley, M.T. — Radar Cross Section, 2nd ed., SciTech (2004).
Ruck, G.T. et al. — Radar Cross Section Handbook, vols. I-II, Plenum (1970).
Willis, N.J. — Bistatic Radar, 2nd ed., SciTech (2005).
Blake, L.V. — Radar Range Performance Analysis, Artech (1986).

Cross-References

M0: EM Waves Electrodynamics Signal Theory

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