As mentioned before, the source transmits pilot symbols cp to assist the destination with the channel estimation process. The ML pilot-based estimate of h that uses only rp is given by 
The estimate (5) gives rise to a mean-squared estimation error (MSE) that is equal to
Hence, for given Es, the estimation accuracy is improved by increasing Kp.
When exploiting also the data part of the received signal, i.e., r, the ML estimate of h is given by
Because of the summation over all possible data sequences, obtaining from (8) the ML estimate for large Kb is computationally complex. Fortunately, the EM algorithm  allows to compute the ML estimate iteratively. For the problem at hand, the EM channel estimate during the i th iteration is obtained as
where and denote the a posteriori mean of c
When the data symbols have a constant magnitude, the numerator of (9) reduces to (Kp + K)Es. The iterations are initialized with the pilot-based estimate from (5), which we denote as .
The APP (for notational convenience, we drop the iteration index) of c
can be expressed as
where ∝ means equal within a normalization factor, and the summation is over all valid codewords with c
equal to α. Making use of the finite-state description of the encoder, the APPs (12) can be computed efficiently for m = 1,⋯,K by means of the BCJR algorithm . However, its complexity is still about three times that of the Viterbi algorithm . Hence, assuming that the EM algorithm converges after I iterations, the BCJR algorithm must be applied I times, after which, the Viterbi algorithm (with ) is used to detect the information bit sequence. The resulting complexity is 3I + 1 times that of a single use of the Viterbi decoder.
The MSE resulting from (8) cannot be obtained in closed form. Therefore, we resort to the modified Cramer-Rao bound (MCRB) , which is a fundamental lower bound on the MSE performance of any unbiased estimate. For the observation model (1)-(2), the MCRB reduces to
Comparison of (6) with (13) indicates the possibility of substantially reducing the MSE when also including the data portion r of the observation in the estimation process, especially when K ≫ Kp.