mohatta2015.pdf

THE MATH 127
Expanding the square, moving the integral inside the summations, and dropping
terms independent of q gives
N.-l N,-l N,-l
s = arg max J2 2Re{q*(m)z(m)}- J^ J^ Q*(rn 1 )q(m 2 )S{m l - m 2 ), (6.45)
where
p
m=0 mi=(lm2=fl
/
oo
-OO
/ O O
h* {t - mT)r{t) dt (6.46)
h*{t)h{t + er)dt. (6.47)
-oc
We recognize z(m) as the matched filter output for symbol s(m). The term S(£)
for different values of i are referred to as the s-parameters.
The double-summation term in (6.45) can be interpreted as summing the elements
of a Hermitian symmetric matrix /(mi,7712) {/(τη,2,τηϊ) = /*(mi,JTi2)).
The first "trick" is to rewrite the summation as the sum of the diagonal elements
plus twice the real part of the sum of the lower triangular elements (due to the
Hermitian property). Mathematically,
JV,-1JV»-1 JV.-l ΛΤ,-lm-l
Σ Σ /(ι>2)= Σ /κ)+ Σ E 2Re^( m ' fc )}· ( 6·48 )
mi— 0*712=0 m=0 m=l k=Q
The second "trick" is to rewrite the sum of the lower triangular elements as sums
along the off-diagonals (m — k = 1, m - k = 2, etc.), giving
ΛΓ„-1 N.-l JV„-1 ΛΓ,-l m
Σ Σ /("Μ>η*2) = Σ /("»·"»)+ Σ ¿2Re{/(m,m-€)}· (6-49)
τηι=0τ7Ζ2=0 m=0 m=l ¿=1
Applying these two steps to (6.45) gives
ΛΓ,-1
s = arg max V^ B(m), (6.50)
qeSp ¿—'
m=0
where branch metric B(m) is given by
B(m) =Re{q*{m 2z{m) - S{0)q(m) - 2(1 - L s - 1, (6.52)
where L s depends on the delay spread of the medium response as well as properties
of the pulse shape p(i). For example, for root-Nyquist pulse shaping and a symbolspaced
medium response (τ( — £T, I — 0, ...,L — 1), L s = L. In this case, we can