Waveform design for compressively sampled ultrawideband radar

Compressive sensing makes it possible to recover sparse target scenes from under-sampled measurements when uncorrelated random-noise waveforms are used as probing signals. The mathematical theory behind this assertion is based on the fact that Toeplitz and circulant random matrices generated from independent identically distributed (i.i.d) Gaussian random sequences satisfy the restricted isometry property. In real systems, waveforms have smooth, nonideal autocorrelation functions, thereby degrading the performance of compressive sensing algorithms. Compressive sensing requires the system matrix to have particular properties. Incorporating prior information into the target scene either to enhance imaging or to mitigate nonidealities can result in system matrices that are not suitable for compressive sensing. We can overcome this problem by designing appropriate transmit waveforms. We extend the existing theory to incorporate such nonidealities into the analysis of compressive recovery. As an example we consider the problem of tailoring waveforms to image extended targets. Extended targets make the target scene denser, causing random transmit waveforms to be suboptimal for recovery. We propose to incorporate extended targets by considering them to be sparsely representable in redundant dictionaries. We demonstrate that a low complexity algorithm to optimize the transmit waveform leads to improved performance.