This paper proposes and compares two new sampling schemes for sparse
deconvolution using a Bernoulli-Gaussian model. To tackle such a deconvolution
problem in a blind and unsupervised context, the Markov Chain Monte Carlo
(MCMC) framework is usually adopted, and the chosen sampling scheme is most
often the Gibbs sampler. However, such a sampling scheme fails to explore the
state space efficiently. Our first alternative, the $K$-tuple Gibbs sampler, is
simply a grouped Gibbs sampler. The second one, called partially marginalized
sampler, is obtained by integrating the Gaussian amplitudes out of the target
distribution. While the mathematical validity of the first scheme is obvious as
a particular instance of the Gibbs sampler, a more detailed analysis is
provided to prove the validity of the second scheme.

For both methods, optimized implementations are proposed in terms of
computation and storage cost. Finally, simulation results validate both schemes
as more efficient in terms of convergence time compared with the plain Gibbs
sampler. Benchmark sequence simulations show that the partially marginalized
sampler takes fewer iterations to converge than the $K$-tuple Gibbs sampler.
However, its computation load per iteration grows almost quadratically with
respect to the data length, while it only grows linearly for the $K$-tuple
Gibbs sampler.