mohatta2015.pdf

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