Chapter Summary

Chapter 12 Summary: Synthetic Aperture Imaging

Key Points

• 1.
  SAR Geometry and Resolution

  SAR synthesizes a large aperture through platform motion. Cross-range resolution is $\Delta x = \lambda R/(2L_{\text{sa}})$, with best achievable resolution $D/2$ independent of range. Range resolution is $\Delta R = c/(2B)$, identical to conventional radar. The SAR measurement fits the universal model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ with Kronecker-structured $\mathbf{A}$.

• 2.
  SAR Image Formation

  The range-Doppler algorithm (RDA) is the standard SAR processor: range compression, RCMC, azimuth compression, all in $O(N^2 \log N)$. The $\omega$-$k$ and chirp scaling algorithms handle more demanding geometries. All compute $\mathbf{A}^{H} \mathbf{y}$ via efficient factored operations.

• 3.
  Autofocus

  Motion errors introduce phase errors $\phi_e(\eta)$ that defocus SAR images. PGA estimates the phase gradient from bright targets; minimum-entropy autofocus works without isolated scatterers. Autofocus is a blind deconvolution problem within the inverse-problem framework.

• 4.
  ISAR

  ISAR images rotating targets with a stationary radar. Cross-range resolution is $\Delta y = \lambda/(2\Delta\theta)$. Translational motion compensation is essential before Doppler processing. The unknown rotation rate must be estimated for cross-range scaling.

• 5.
  Tomographic SAR

  TomoSAR extends SAR to 3D via multi-baseline acquisitions. The elevation sensing matrix is a partial Fourier matrix with resolution $\Delta z = \lambda R_0/(2b_{\max})$. Sparse recovery is essential for resolving multiple scatterers below the Rayleigh limit.

• 6.
  Sparse SAR and CS-SAR

  Compressed sensing exploits scene sparsity for super-resolution, sub-Nyquist acquisition, and sidelobe suppression. The SAR sensing matrix is a partial Fourier matrix — ideal for CS guarantees. Joint autofocus + sparse recovery avoids error propagation. TV regularization extends the approach to non-sparse scenes.