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Maximum likelihood principle....

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Mar 30, 2007
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What is meant by Maximum likelihood principle?

What is Zero forcing criteria and MMSE criteria ?

maximum likelihood principle is selecting the most probable value.... MMSE is the way of selecting the values with minimum mean for square of the errors....

The maximum likelihood principle states that the set of model parameters that maximize the apparent probability of a set of observations is the best set possible. One rationale for this is that in the limit, such a choice approaches the true value of these parameters (assuming, of course, that the model is a valid one). This can be seen by the argument given below. Interestingly, maximum likelihood is equivalent to maximum compression. An alternative is the so-called maximum a posteriori approach (MAP) in which a Bayesian prior distribution on the parameters is assumed and the maximum is computed taking this prior into account. In terms of compression, MAP estimation correlates to minimizing the number of bits required to send a description of the observations and the model parameters. This formulation is also called the Minimum Message Length method.

ML is based on that all symbols are equally likely. It is nothing to do with a priori probability, which is used for MAP. However, if the priori probabilities are the same among symbols, then ML = MAP.

Zero forcing literally forces ISI to be zero. In other words, if H(z) is the channel TF(transfer function), designing a filter that has TF of 1/H(z) can restore the original signal making the frequency response flat over the frequency range because the overall response becomes H(z)(1/H(z)) = 1. However, SNR of the zero forcing is not that good since it also boosts noises of the signal. This is called noise enhancement.

MMSE uses least square algorithm. Let say we have a data comm model like following.

X(z) ---> C(z) ----> Y(z) -----> F(z) -----> + ------> e_k

where X(z)=z-domain version of the transmit signal x_k, C(z)=channel, Y(z)=channel output, F(z)=filter

If we put x_k, which is the original signal, in the adder such that e_k=x_k - z_k (z_k is the output of F(z)), then e_k=x_k - f_k*y_k, i.e., E(z) = X(z) - F(z)Y(z) in z-domain. So we determine F(z) by using E[E(z)Y(z)] = E[(X(z)-F(z)Y(z))Y(z)] = 0 where E[.] operator is statistical mean.
MMSE sense turns out to be better SNR than zero forcing because it has no noise enhancement.

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