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This paper considers the problem of channel estimation for orthogonal-frequency-division multiplexing (OFDM) systems, where the number of channel taps and their power delay profile are unknown. Using a Bayesian approach, we construct a model in which we estimate jointly the coefficients of the channel taps, the channel order and decay rate of the power delay profile (PDP). In order to sample from the resulting posterior distribution we develop three novel Trans-dimensional Markov chain Monte Carlo (TDMCMC) algorithms and compare their performance. The first is the basic birth and death TDMCMC algorithm. The second utilizes Stochastic Approximation to develop an adaptively learning algorithm to improve mixing rates of the Markov chain between model subspaces. The third approximates the optimal TDMCMC proposal distribution for between-model moves using conditional path sampling proposals. We assess several aspects of the model in terms of sensitivities to different prior choices. Next we perform a detailed analysis of the performance of each of the TDMCMC algorithms. This allows us to contrast the resulting computational effort required under each approach versus the estimation performance. Finally, using the TDMCMC algorithm which produces the best performance in terms of exploration of the model subspaces, we assess its performance in terms of channel estimation mean-square error (MSE) and bit error rate (BER). It is shown that the proposed algorithm can achieve results very close to the case where both the channel length and the PDP decay rate are known.