AC-CS: Azimuth Compression based Compressed Sensing for MMW SAR Imaging

Range Compression. A Range-FFT is applied to the raw data along the  fast-time axis. The range bin corresponding to the target distance z0 is  extracted via index calculation based on the chirp slope K, sampling period  Ts, and the round-trip delay, thereby performing range focusing.3.

Random Downsampling. To simulate sparse or incomplete data acquisition, a  subset of rows (vertical/cross-range direction) is randomly discarded —  only 30% of the original rows are retained. This mimics scenarios where  entire scan lines are missing, such as in sub-Nyquist sampling or sensor  failures.4.

Azimuth Compression and Data Recovery via ADMM. The core of the algorithm  recovers the missing cross-range samples using the Alternating Direction  Method of Multipliers (ADMM). The observation model combines a DFT basis,  an azimuth matched-filter operator derived from the Doppler rate Ka, and a  row-selection matrix Phi that encodes which rows are observed.

The  optimization problem minimizes a composite objective with an L1 sparsity  penalty and a total-variation (first-order difference) regularization term.  The ADMM solver iteratively updates the primal variable x, two auxiliary  variables z1 and z2 via soft-thresholding, and the corresponding dual  variables u1, u2. The linear system in the x-update is solved efficiently  using a precomputed Cholesky factorization.5.

SAR Image Formation. The recovered full-aperture data is transformed into  the spatial frequency (k-space) domain via a 2D FFT, multiplied by a  matched phase factor that accounts for spherical wavefront propagation, and  then transformed back to the spatial domain via a 2D IFFT to produce the  final SAR image.Files----- demo_AC_CS.m   Main script implementing the full SAR imaging pipeline with CS-based   data recovery. first_diff_matrix.m   Utility function that constructs an (n-1) x n first-order difference   matrix used for total-variation regularization in the ADMM solver. NSSC.mat   Input data file containing the raw 3D FMCW radar data (sarData).Key Parameters-------------- f0            77 GHz     Start frequency (after ADC offset                      correction) K             63.343 THz/s  Chirp slope fS            9.121 MHz   ADC sampling rate z0            245 mm     Nominal target range nFFT_time         512      FFT points for range dimension nFFT_space        1024      FFT points for spatial dimensions lambda1          0.018     L1 sparsity regularization weight lambda2          0.1      Total-variation regularization weight rho1, rho2        1e-5      ADMM penalty parameters Max iterations      800      ADMM maximum iterationsHelper Function: first_diff_matrix(n)--------------------------------------Constructs an (n-1) x n first-order difference matrix D such that for a vectorv in R^n, the product D

• v yields the forward differences v(i+1) - v(i) fori = 1, ..., n-1. This matrix is used in the total-variation regularizationterm ||D x||_1 within the ADMM framework to promote piecewise smoothness inthe recovered signal.