NDA Timing Parameter Estimation

The objective in this algorithm is maximization the objective function in equation (1),
in first step unwanted parameters a , θ must be removed.
To remove the data dependency we have to multiply (1) by P (i a), where
( i a) is the ith of M symbols, and sum over the M possibilities. Assuming
independent and equiprobable symbols the likelihood function will become

Now assume M-PSK modulation with M>2 ,and at next step P(i a)
( (i a)=exp(j2πi/M)) is approximated by acontinuous-valued pdf of exp(jα) where α
has a uniform distribution over (-π,π), which yields:

If in a second approach averaging over the phase is done earlier(previous to symbols) a
data-dependent algorithm will be obtaind:

Notice this result is the same for all phase modulations (M-PSK) (since
lanl = const.), but not for M-QAM. In order to obtain an NDA synchronization
algorithm for M-QAM ,L2(a,ε) must be averaged over the symbols which yields no simple
closed form. But objective functions in (1) and (2) can be more simplified by following
approximation for modified Bessel function:

Discarding any constant irrelevant yields:

NDA:

DA:

A completely different method for approximation (1):

In both introduced algorithm here we figure on eliminating data dependency
which may be not possible in general case,also these algorithms requires knowledge
of σn , but it is noticeable that both problems can be solved if asseme low SNR ,then
expand (1) function into a Taylor series up to third term, furthermore assume  E[an]=0
and estimate L(θ,ε) with it's average which finally yields:
      

If notice this equation han an interesting property !!!!


In fact two-dimensional search (θ,ε) is reduced to a one-dimensional search for ε ,

since clearly:
      

 
           
and

     
      

Three algorithms introduced here works base on maximum search, it is interesting
that this maximum search can be circumvented by Spectral Estimation.

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