Synthetic Aperture Radar (SAR) imaging systems operate by emitting radar signals from a moving object, such as a satellite, towards the target of interest. Reflected radar echoes are received and later used by image formation algorithms to form a SAR image. There is great interest in using SAR images in computer vision tasks such as classification or automatic target recognition. Today, however, SAR applications consist of multiple operations: image formation followed by image processing. In this work, we train a deep neural network that performs both the image formation and image processing tasks, integrating the SAR processing pipeline. Results show that our integrated pipeline can output accurately classified SAR imagery with image quality comparable to those formed using a traditional algorithm. We believe that this work is the first demonstration of an integrated neural network based SAR processing pipeline using real data.