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