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We propose a joint compressed sensing-based encoding and an iterative-decoding method, which we call turbo compressed sensing (turbo-CS), for the robust to noise transmission of sparse signals over an additive white Gaussian noise (AWGN) channel. The turbo-CS encoder applies 1-bit compressed sensing as a source encoder concatenated serially with a convolutional channel encoder. At the turbo-CS decoder, an iterative joint source-channel decoding method is proposed for signal reconstruction. We analyze, for the first time, the convergence of the turbo-CS decoder by determining an extrinsic information transfer chart of the constituent decoders. We modify the soft-outputs of the constituent source decoder to improve the signal reconstruction performance of the turbo-CS decoder. Our results show that for a fixed received signal to noise ratio (RSNR) of \mathbf 10 dB more than \mathbf 5 dB of improvement in the channel SNR is achieved after six iterations of the turbo-CS decoder. Alternatively, for a fixed SNR of \mathbf -1 dB, \mathbf 10 dB improvement in RSNR is achieved. Moreover, it is shown that the turbo-CS decoder outperforms the state-of-the-art algorithms for 1-bit compressed sensing reconstruction in the presence of AWGN channel in terms of signal reconstruction performance and complexity. [ABSTRACT FROM PUBLISHER]