![](../images/32.JPG)
where the coefficients are random variables defined by:
![](../images/33.JPG)
as a consequence we have:
where:
![](../images/35.JPG)
It is proved only three coefficients {c-1,c0,c1} have nonzero mean hence if
L(ε) is approximated with its average clearly :
![](../images/36.JPG)
and this results in:
Although this approach solve the problem of maximum search algorithms ,
only digital algorithms are of interest here while c1 is defined by a summation of
(2L+1) integrals, but provided the sampling rate l/T, is such that the sampling theorem
is fulfilled for |z(t)|^2 [and not only for z(t)], i.e. :
![](../images/38.JPG)
the coefficients c1, c0 can be computed by a discrete Fourier transform (DFT).
Let us denote by the integer Ms the (nominal) ratio between sampling and symbol
rate, Ms = T/Ts. For the samples taken at kTs:
![](../images/39.JPG)
The following figure illustrates a simple implementation for Ms=4:
![](../images/40.JPG)
Three following methods based on this spectral estimation with an itterative
algorithm figure on finding timing estimation (ε):
1. Early-late gate
2. Gradient-based
3. Tone-extraction
about which there are more details in "Signal Processing Techniques
for Software Radio " Book , Behrouz Farhang-Boroujeny,chapter 10 .
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