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, non-ideal autocorrelation functions, thereby degrading the performance of compressive sensing algorithms. In this paper, we extend the existing theory to incorporate such non-idealities into the analysis of compressive recovery. The presence of extended scatterers also causes distortions due to the correlation between different cells of the target scene. Extended targets make the target scene more dense, causing random transmit waveforms to be sub-optimal 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.