mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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82 LINEAR EQUALIZATION<br />
4.3.2 ML solution<br />
One can also use a maximum-likelihood (ML) design approach to weight design.<br />
Similar to Chapter 2, we can design the weights to obtain a log-likelihood function<br />
(LLF) for each symbol. Unlike Chapter 2, we will assume that multiple symbols are<br />
transmitted, giving rise to ISI. To obtain a linear solution, we will approximate the<br />
ISI as a form of noise. Specifically, the symbols are approximated as being complex<br />
Gaussian random variables, so that the ISI appears as colored Gaussian noise. The<br />
ML formulation leads to a linear filter that can be interpreted as a matched filter<br />
in colored noise.<br />
As in the MMSE formulation, we assume a partial MF front end. Samples are<br />
collected into a vector v which can be modeled according to (4.62). At this point,<br />
we rewrite (4.62) in terms of a signal component (assume s(m¡)) is the symbol of<br />
interest) and impairment (noise plus interference) component, giving<br />
where<br />
v \= y/Ë~ s hs(m t) ) + u, (4.79)<br />
m,, — 1<br />
oo<br />
u = s/W s Σ h m s(m) + \/W s Σ<br />
h m«(m) + n. (4.80)<br />
m= — oo mn + 1<br />
We approximate u as complex Gaussian with zero mean and covariance<br />
where<br />
C u = E{uu ff } = E s C t + N„C n , (4.81)<br />
mu — l<br />
C 4 = E s J2 h "* h m + Es Σ hmh "- ( 4·82 )<br />
m= — oo molí<br />
The elements in Ci and C n in the jith row and j*2th column are given by<br />
Ci(j'l, h) =<br />
oo<br />
oo<br />
Σ<br />
m=—οο,τη^τηο<br />
h m{jl)h* m (J2)<br />
L~\ L-l oo<br />
= Σ Σ etiSU Σ WW' ~ m T - n^KidjTs - mT - r (2 ) (4.83)<br />
f 1= ()i 2 =n m=-oo,m#0<br />
C n (h,J2) = Rp((ji-h)T s ). (4.84)<br />
Observe that by assuming an infinite stream of symbols, the elements in C u are<br />
independent of m,).<br />
Assuming Gaussian impairment, the likelihood of v given s(mo) = Sj is then<br />
given by<br />
giving the LLF<br />
_ i _ exp {-(v - ^hSjfC-'iv - ^F s hSj)} , (4.85)<br />
LLF(Sj) = -(v - y/ËOiSj^CZ 1^ - y/Ë~ s hSj). (4.86)<br />
Expanding the square and dropping terms unrelated to Sj gives<br />
LLF(Sj) = 2Re{S;z m „} - S mo (0)|S/, (4.87)