Maximum likelihood sequence estimation (MLSE) is a mathematical algorithm that extracts useful data from a noisy data stream.
For an optimized detector for digital signals the priority is not to reconstruct the transmitter signal, but it should do a best estimation of the transmitted data with the least possible number of errors. The receiver emulates the distorted channel. All possible transmitted data streams are fed into this distorted channel model. The receiver compares the time response with the actual received signal and determines the most likely signal.In cases that are most computationally straightforward, root mean square deviation can be used as the decision criterion[1] for the lowest error probability.
Suppose that there is an underlying signal, of which an observed signal is available. The observed signal r is related to x via a transformation that may be nonlinear and may involve attenuation, and would usually involve the incorporation of random noise. The statistical parameters of this transformation are assumed to be known. The problem to be solved is to use the observations to create a good estimate of .
Maximum likelihood sequence estimation is formally the application of maximum likelihood to this problem. That is, the estimate of is defined to be a sequence of values which maximize the functional
L(x)=p(r\midx),
In contrast, the related method of maximum a posteriori estimation is formally the application of the maximum a posteriori (MAP) estimation approach. This is more complex than maximum likelihood sequence estimation and requires a known distribution (in Bayesian terms, a prior distribution) for the underlying signal. In this case the estimate of is defined to be a sequence of values which maximize the functional
P(x)=p(x\midr),
P(x)=p(x\midr)=
| p(r\midx)p(x) | |
| p(r) |
.
In cases where the contribution of random noise is additive and has a multivariate normal distribution, the problem of maximum likelihood sequence estimation can be reduced to that of a least squares minimization.