Deep neural networks (DNNs) have found applications in diverse signal processing (SP) problems. Most efforts either directly adopt the DNN as a black-box approach to perform certain SP tasks without taking into account of any known properties of the signal models, or insert a pre-defined SP operator into a DNN as an add-on data processing stage. This paper presents a novel hybrid-NN framework in which one or more SP layers are inserted into the DNN architecture in a coherent manner to enhance the network capability and efficiency in feature extraction. These SP layers are properly designed to make good use of the available models and properties of the data. The network training algorithm of hybrid-NN is designed to actively involve the SP layers in the learning goal, by simultaneously optimizing both the weights of the DNN and the unknown tuning parameters of the SP operators. The proposed hybrid-NN is tested on a radar automatic target recognition (ATR) problem. It achieves high validation accuracy of 96\% with 5,000 training images in radar ATR. Compared with ordinary DNN, hybrid-NN can markedly reduce the required amount of training data and improve the learning performance.