Based on the compressive sensing theory, the incoherence between compressive sensing matrix and signal representation basis guarantees the recovery of sparse signals from sub-Nyquist sampling. Random sensing matrices including Gaussian matrix and Bernoulli matrix, are well-known to satisfy the incoherence requirement, which are also robust with any fixed signal sparse representation. However, random sensing only exploits the sparsity of signals.

In the forthcoming era of 5G, the knowledge acquisition becomes easier than before, which implies that, besides sparsity, more knowledge of signals can be exploited to improve the reconstruction performance.  The compressive radar measurement vector $\boldsymbol{y}$ is modeled as $\boldsymbol{y} = \boldsymbol{\Phi} \left( \boldsymbol{\Psi} \boldsymbol{x} + \boldsymbol{n}\right), \nonumber$

where $\boldsymbol{\Phi}$ is the compressive sensing matrix, $\boldsymbol{\Psi}$ is the discrete waveform matrix, $\boldsymbol{x}$ is the radar target profile to be estimated, and $\boldsymbol{n}$ is the noise term.

After modeling the probability distribution of $\boldsymbol{x}$, the signal to be estimated, the compressive sensing matrix can be optimized by minimizing the minimum mean-square error (MMSE) of the signal estimation as $\min_{\boldsymbol{\Phi}} E \{ \| \boldsymbol{x} - \hat{\boldsymbol{x}} \|_{2}^{2} \}, \nonumber$

or maximizing the mutual information between the compressive measurements and the signal as $\max_{\boldsymbol{\Phi}} I(\boldsymbol{y}; \boldsymbol{x}). \nonumber$

Here, $E\{\cdot\}$ and $I(;)$ denote the statistical expectation and the mutual information, respectively. It is noted that both the objective functions are the implicit function of the compressive sensing matrix $\boldsymbol{\Phi}$. With the optimized compressive matrix, the radar target profile, i.e., the unknown vector $\boldsymbol{x}$, can be estimated using the Bayes’ theorem.

Simulation

In Fig. 3, we compare the range profile reconstruction results for the narrowband reference and the Barker-coded waveform with both random and optimized compression. It is apparent that the low-bandwidth reference waveform does not provide enough resolution to separate closely spaced target peaks, but compressive sampling of the wider-bandwidth Barker Code improves detail in the reconstruction.