The Importance of Phase in Complex Compressive Sensing

@article{Jacques2021TheIO,
  title={The Importance of Phase in Complex Compressive Sensing},
  author={Laurent Jacques and Thomas Feuillen},
  journal={IEEE Transactions on Information Theory},
  year={2021},
  volume={67},
  pages={4150-4161}
}
We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements. We show that in this phase-only compressive sensing (PO-CS) scenario, we can perfectly recover such a signal with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the complexity level of the signal space. Our approach… 
View on IEEE
arxiv.org
2 Citations
Background Citations
1

Figures from this paper

2 Citations

Keep the phase! Signal recovery in phase-only compressive sensing
TLDR
It is demonstrated that a sparse signal can be estimated from the phase of complex random measurements, in a "phase-only compressive sensing" (PO-CS) scenario, and robust signal direction estimation is reached at about twice the number of measurements needed for signal recovery inCompressive sensing.
Signal Reconstruction From Quantized Noisy Samples of the Discrete Fourier Transform
TLDR
This paper designs an algorithm that uses contraction mapping, based on the Banach fixed point theorem, and shows that the expected mean squared error (MSE) in signal reconstruction is asymptotically proportional to the inverse of the sampling rate.

References

SHOWING 1-10 OF 54 REFERENCES
1-Bit compressive sensing
TLDR
This paper reformulates the problem by treating the 1-bit measurements as sign constraints and further constraining the optimization to recover a signal on the unit sphere, and demonstrates that this approach performs significantly better compared to the classical compressive sensing reconstruction methods, even as the signal becomes less sparse and as the number of measurements increases.
Model-Based Compressive Sensing
TLDR
A model-based CS theory is introduced that parallels the conventional theory and provides concrete guidelines on how to create model- based recovery algorithms with provable performance guarantees and a new class of structured compressible signals along with a new sufficient condition for robust structured compressable signal recovery that is the natural counterpart to the restricted isometry property of conventional CS.
Sparse Signal Reconstruction from Phase-only Measurements
We demonstrate that the phase of complex linear measurements of signals preserves significant information about the angles between those signals. We provide stable angle embedding guarantees, akin to…
D ec 2 01 9 1 ( l 1 , l 2 )-RIP and Projected Back-Projection Reconstruction for Phase-Only Measurements
TLDR
It is shown that complex Gaussian random matrices respect, with high probability, a variant of the Restricted Isometry Property (RIP) relating to the l1-norm of the sparse signal measurements to their l2-norm, which allows us to upper-bound the reconstruction error of PBP in the presence of phase noise.
(l1,l2)-RIP and Projected Back-Projection Reconstruction for Phase-Only Measurements
TLDR
It is shown that complex Gaussian random matrices respect, with high probability, a variant of the Restricted Isometry Property (RIP) relating to the l1 -norm of the sparse signal measurements to their l2 -norm, which allows us to upper-bound the reconstruction error of PBP in the presence of phase noise.
Iterative Hard Thresholding for Compressed Sensing
Convex Optimization Approaches for Blind Sensor Calibration Using Sparsity
TLDR
This work investigates a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains, and formulates the joint recovery of the gains and the sparse signals as a convex optimization problem.
Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors
TLDR
This paper investigates an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement, and introduces the binary iterative hard thresholding algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.
General Deviants: An Analysis of Perturbations in Compressed Sensing
TLDR
The results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation, and this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares.
Subspace Pursuit for Compressive Sensing Signal Reconstruction
TLDR
The presented analysis shows that in the noiseless setting, the proposed algorithm can exactly reconstruct arbitrary sparse signals provided that the sensing matrix satisfies the restricted isometry property with a constant parameter.
...
...