We investigate the performance of a discrete-time all-analog-processing joint source-channel coding system for the transmission of i.i.d. Gaussian sources over additive white Gaussian noise (AWGN) channels. At the encoder, samples of an i.i.d. source are grouped and mapped into a channel symbol using a space-filling curve. Different from previous work in the literature, MMSE instead of ML decoding is considered, and we focus on both high and low channel SNR regions. In order to reduce complexity, Monte Carlo importance sampling is applied in the decoding process. The main contribution of this paper is to show that for a wide range of rates the proposed system presents a performance very close to the theoretical limits, even at low SNR, as long as the curve parameters are properly optimized.