Radar Equation¶

Derivation and application of the radar range equation.

Overview¶

The radar equation relates transmitter power, antenna gain, target characteristics, and receiver sensitivity to determine detection capability.

Basic Radar Equation¶

Power Density at Target¶

Power density at range $R$ from an isotropic radiator:

\[ S_i = \frac{P_t}{4\pi R^2} \]

With transmit antenna gain $G_t$:

\[ S_t = \frac{P_t G_t}{4\pi R^2} \]

Power Reflected by Target¶

The target intercepts and re-radiates power proportional to its radar cross section (RCS) $\sigma$:

\[ P_{reflected} = S_t \cdot \sigma = \frac{P_t G_t \sigma}{4\pi R^2} \]

Power Density at Receiver¶

The reflected power spreads again over $4\pi R^2$:

\[ S_r = \frac{P_t G_t \sigma}{(4\pi)^2 R^4} \]

Received Power¶

The receiver antenna with effective area $A_e$ captures:

\[ P_r = S_r \cdot A_e = \frac{P_t G_t \sigma A_e}{(4\pi)^2 R^4} \]

Using $A_e = G_r \lambda^2 / 4\pi$:

\[ P_r = \frac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 R^4} \]

For monostatic radar ($G_t = G_r = G$):

\[ P_r = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4} \]

SNR Form¶

Including noise and losses:

\[ SNR = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4 k T_s B_n L_s} \]

Where: - $P_t$ = Peak transmit power (W) - $G$ = Antenna gain (linear) - $\lambda$ = Wavelength (m) - $\sigma$ = Target RCS (m²) - $R$ = Range (m) - $k$ = Boltzmann constant = 1.38×10⁻²³ J/K - $T_s$ = System noise temperature (K) - $B_n$ = Noise bandwidth (Hz) - $L_s$ = System losses (linear)

Range Form¶

Solving for range at minimum detectable SNR:

\[ R_{max} = \left[\frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 k T_s B_n L_s \cdot SNR_{min}}\right]^{1/4} \]

Pulse Integration¶

Coherent Integration¶

For $N$ coherently integrated pulses:

\[ SNR_N = N \cdot SNR_1 \]

The integration gain is $N$ (linear) or $10\log_{10}(N)$ dB.

Non-Coherent Integration¶

For non-coherent integration:

\[ SNR_N \approx \sqrt{N} \cdot SNR_1 \]

More precisely, the integration gain depends on the required $P_d$ and $P_{fa}$.

Detection Theory¶

Single-Pulse Detection¶

For a Gaussian noise background and non-fluctuating target (Swerling 0), the detection probability relates to SNR through:

\[ P_d = \frac{1}{2}\text{erfc}\left[\text{erfc}^{-1}(2P_{fa}) - \sqrt{SNR}\right] \]

Required SNR¶

For given $P_d$ and $P_{fa}$:

\[ SNR_{req} \approx \ln\left(\frac{1}{P_{fa}}\right) + \ln\left(\frac{1}{1-P_d}\right) \]

More accurate:

$P_d$	$P_{fa} = 10^{-6}$	$P_{fa} = 10^{-9}$
0.5	10.8 dB	12.6 dB
0.9	13.1 dB	14.9 dB
0.99	16.4 dB	18.2 dB

Swerling Target Models¶

Model	PDF	Decorrelation
0	Constant (non-fluctuating)	-
1	Exponential (Rayleigh amplitude)	Scan-to-scan
2	Exponential	Pulse-to-pulse
3	Chi-squared, 4 DOF	Scan-to-scan
4	Chi-squared, 4 DOF	Pulse-to-pulse

SNR Penalty¶

Fluctuating targets require additional SNR:

Model	Typical Penalty vs. SW0
SW1	+3 to +8 dB (depends on $P_d$)
SW2	+2 to +5 dB
SW3	+1 to +3 dB
SW4	+1 to +2 dB

Power-Aperture Product¶

Radar performance fundamentally scales with:

\[ PA = P_t \cdot A = P_t \cdot \frac{G\lambda^2}{4\pi} \]

For a given target and detection requirement:

\[ R_{max}^4 \propto PA \]

Trade-off: Higher power OR larger aperture.

Example Calculation¶

Given: - Frequency: 10 GHz ($\lambda$ = 0.03 m) - Array: 16×16 elements, G = 30 dBi - TX power: 10 W/element, 256 elements → 2560 W peak - Target RCS: 1 m² - Range: 100 km - Noise temp: 400 K - Losses: 4 dB ($L_s$ = 2.51) - Pulse width: 10 μs → $B_n$ ≈ 100 kHz

Calculate:

Wavelength: $\lambda$ = 0.03 m
Gain (linear): $G$ = 10^(30/10) = 1000
Numerator: $$ P_t G^2 \lambda^2 \sigma = 2560 \times 1000^2 \times 0.03^2 \times 1 = 2.3 \times 10^6 $$
Denominator: $$ (4\pi)^3 R^4 k T_s B_n L_s = 1984 \times 10^{20} \times 1.38 \times 10^{-23} \times 400 \times 10^5 \times 2.51 $$ $$ = 2.74 \times 10^6 $$
SNR (single pulse): $$ SNR = \frac{2.3 \times 10^6}{2.74 \times 10^6} = 0.84 = -0.75 \text{ dB} $$
With 10-pulse coherent integration: $$ SNR_{10} = -0.75 + 10 = 9.25 \text{ dB} $$
Required SNR for $P_d$ = 0.9, $P_{fa}$ = 10⁻⁶, SW1 ≈ 17 dB
SNR Margin: $$ M = 9.25 - 17 = -7.75 \text{ dB} \quad \text{(insufficient)} $$

Need more power, more elements, or shorter range.

Range Dependencies¶

Quantity	Range Dependence
$P_r$	$R^{-4}$
$SNR$	$R^{-4}$
Double range	-12 dB SNR
Half range	+12 dB SNR

Model	Typical Penalty vs. SW0
SW1	+3 to +8 dB (depends on \(P_d\))
SW2	+2 to +5 dB
SW3	+1 to +3 dB
SW4	+1 to +2 dB

Quantity	Range Dependence
\(P_r\)	\(R^{-4}\)
\(SNR\)	\(R^{-4}\)
Double range	-12 dB SNR
Half range	+12 dB SNR