Single-channel signal separation and deconvolution aims to separate and deconvolve individual sources from a single-channel mixture and is a challenging problem in which no prior knowledge of the mixing filters is available. Both individual sources and mixing filters need to be estimated. In addition, a mixture may contain non-stationary noise which is unseen in the training set. We propose a synthesizing-decomposition (S-D) approach to solve the single-channel separation and deconvolution problem. In synthesizing, a generative model for sources is built using a generative adversarial network (GAN). In decomposition, both mixing filters and sources are optimized to minimize the reconstruction error of the mixture. The proposed S-D approach achieves a peak-to-noise-ratio (PSNR) of 18.9 dB and 15.4 dB in image inpainting and completion, outperforming a baseline convolutional neural network PSNR of 15.3 dB and 12.2 dB, respectively and achieves a PSNR of 13.2 dB in source separation together with deconvolution, outperforming a convolutive non-negative matrix factorization (NMF) baseline of 10.1 dB.