Phase Retrieval in RF Imaging

Phase information is lost in RF systems due to several mechanisms:

1. Power detectors (envelope detectors): Low-cost receivers measure $|E_s|^2$ or $|E_s|$ rather than the complex field $E_s$. Common in:

• Passive imaging (radiometry).
• Low-cost automotive radar receivers.
• Energy-harvesting sensors.

2. Non-coherent processing: When the local oscillator phase is unstable or unknown, the received signal $r(t) = |E_s(t)|\cos(\omega t + \phi_{\text{LO}} + \phi_s)$ can only be processed for $|E_s|$ — the phase $\phi_s$ is corrupted by the unknown $\phi_{\text{LO}}$.

3. Multi-static configurations: In multi-static radar, each transmitter-receiver pair has an independent oscillator. Without inter-node synchronization, only the magnitude $|E_s|$ at each pair is reliable.

4. WiFi RSSI and IoT sensing: IEEE 802.11 RSSI reports only signal power. Low-cost IoT devices (Zigbee, BLE) typically provide only RSSI, not complex channel state information (CSI).

Standard (coherent) RF imaging model (from Chapter 14):

$$z_i = \sum_{n=1}^{N} A_{in}\,c_n + \eta_i = [\mathbf{A}\mathbf{c}]_i + \eta_i,$$

where $\mathbf{A}$ is the sensing matrix and $\mathbf{c}$ is the scene reflectivity vector.

Phaseless model (power detection):

$$y_i = |z_i|^2 = |[\mathbf{A}\mathbf{c}]_i|^2 + \text{noise}.$$

This is the phase retrieval problem with measurement vectors $\mathbf{a}_i^T$ being the rows of $\mathbf{A}$.

Structured measurements: The imaging matrix $\mathbf{A}$ has Fourier-like structure (not generic Gaussian), so the theoretical guarantees of PhaseLift and Wirtinger flow do not directly apply. Coded measurements address this.

To improve recoverability with structured (non-Gaussian) measurement matrices, insert phase masks or random phase modulators in the measurement path:

$$y_i^\ell = |[\mathbf{D}_\ell\mathbf{A}\mathbf{c}]_i|^2, \quad \ell = 1, \ldots, L,$$

where $\mathbf{D}_\ell = \text{diag}(e^{j\phi_1^\ell}, \ldots, e^{j\phi_M^\ell})$ is a random phase mask with each $\phi_k^\ell$ i.i.d. uniform on $[0, 2\pi)$.

With $L \geq 3$ masks, the total $LM$ measurements provide sufficient diversity for recovery, even with structured $\mathbf{A}$.

Physical implementation: In RF systems, phase masks can be realized via:

• Programmable phase shifters at the receiver array.
• Reconfigurable intelligent surfaces (RIS) that modulate the scattered field — see Chapter 14.
• Frequency-diverse coding across OFDM subcarriers.

When the scene $\mathbf{c}$ is $s$-sparse (as in most RF imaging scenarios), we can exploit sparsity to reduce the measurement requirement:

Sparse Wirtinger Flow:

$$\mathbf{z}^{(t+1)} = \mathcal{H}_s\!\left( \mathbf{z}^{(t)} - \mu_t\,\nabla_{\bar{\mathbf{z}}} f(\mathbf{z}^{(t)})\right),$$

where $\mathcal{H}_s$ is the hard-thresholding operator (keep the $s$ largest entries).

Measurement requirement: $M = O(s^2\log N)$ for sparse Wirtinger flow — compared to $M = O(N)$ for the non-sparse version. The quadratic dependence on $s$ (vs. linear for linear CS) reflects the inherent difficulty of non-linear measurements.

SPARTA (Sparse Truncated Amplitude flow): Combines truncation, amplitude loss, and thresholding for improved sample complexity: $M = O(s^2\log N)$ with $O(s^2 N \log N)$ total computation.