Chapter Summary

Chapter Summary

Key Points

• 1.

  Channel estimation IS imaging. The pilot observation model $\mathbf{y} = \boldsymbol{\Phi}\mathbf{h}_{\mathrm{ad}} + \mathbf{w}$ is structurally identical to the imaging model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$. Every algorithm from this book — LASSO, OAMP, deep unfolding — transfers directly to channel estimation.

• 2.

  Hierarchical sparsity (Wunder/Caire) exploits the cluster structure of real wireless channels via the $\ell_{2,1}$ mixed-norm, reducing pilot overhead from $O(K\log N)$ to $O(K_1\log G + K)$. The 2D Markov prior (Xu/Caire) captures near-field visibility regions for XL-MIMO.

• 3.

  Wavefield networked sensing (Manzoni/Caire) enables distributed nodes to cooperatively image the environment. Each node contributes a k-space slice; combining them via consensus ADMM achieves near-optimal image quality with manageable backhaul.

• 4.

  Sensing-assisted communication closes the loop: the digital twin predicts channels without pilots, enabling $\sim 90\%$ pilot overhead reduction and $2$--$3\times$ throughput gain via DT-aided beam prediction.

• 5.

  The JCSI framework goes beyond ISAC by optimising for reconstruction quality (NMSE, SSIM) rather than CRB. The CRB-optimal waveform can produce images $> 10$ dB worse than the imaging-optimal waveform due to non-uniform k-space coverage.

• 6.

  End-to-end learning via deep unfolding jointly optimises the waveform, the reconstruction algorithm, and the communication precoder, operating on the rate-imaging Pareto frontier.