The Radar Equation

For a monostatic radar (co-located transmitter and receiver) with transmit power $P_t$, antenna gain $G$ (used for both transmit and receive), operating at wavelength $\lambda$, observing a target of radar cross-section $\sigma$ at range $R$, the received power is:

$$P_r = \frac{P_t G^2 \lambda^{2} \sigma}{(4\pi)^3 R^4 L},$$

where $L \geq 1$ accounts for system losses (atmospheric attenuation, feed losses, signal processing losses).

The radar cross-section $\sigma$ of a target is the equivalent area that would scatter the incident power isotropically to produce the observed echo at the receiver:

$$\sigma = \lim_{R \to \infty} 4\pi R^2 \frac{|E_s|^2}{|E_i|^2},$$

where $E_i$ is the incident field and $E_s$ is the scattered field at distance $R$ from the target.

The RCS depends on the target's size, shape, material, and the observation angle — it is a function $\sigma(\theta, \phi, f)$, not a scalar constant.

When transmitter and receiver are at different locations, with transmit-to-target range $R_t$ and target-to-receiver range $R_r$:

$$P_r = \frac{P_t G^{\text{tx}} G^{\text{rx}} \lambda^{2} \sigma_b}{(4\pi)^3 R_t^2 R_r^2 L},$$

where $\sigma_b$ is the bistatic RCS, which depends on both the transmit and receive directions relative to the target.

Swerling models describe the statistical fluctuation of the RCS $\sigma$ for different target types:

• Swerling 0 (non-fluctuating): $\sigma$ is constant.
• Swerling I: $\sigma$ is exponentially distributed ($\sigma \sim \text{Exp}(1/\bar{\sigma})$), constant across pulses within a CPI (scan-to-scan fluctuation).
• Swerling II: Same distribution as I, but independent pulse-to-pulse.
• Swerling III: $\sigma$ follows a chi-squared distribution with 4 DOF (one dominant scatterer plus many small ones), constant within CPI.
• Swerling IV: Same as III, pulse-to-pulse independent.

The exponential distribution of Swerling I arises from the central limit theorem applied to a target composed of many scatterers of comparable strength — the resultant complex amplitude is $\mathcal{CN}(0, \bar{\sigma})$, giving $|\text{amplitude}|^2 \sim \text{Exp}$.

An extended target occupies multiple resolution cells. The received power from a single resolution cell of extent $\Delta R \times \Delta \theta$ is:

$$P_r^{\text{cell}} = \frac{P_t G^2 \lambda^{2} \sigma^0 \Delta A}{(4\pi)^3 R^4 L},$$

where $\sigma^0$ is the normalized RCS (reflectivity per unit area, dimensionless) and $\Delta A = R \Delta\theta \cdot \Delta R$ is the cell area on the ground.

For imaging, each pixel in the reconstructed image corresponds to one or more resolution cells. The per-pixel $\text{SNR}$ determines the image quality.