mohatta2015.pdf

44 MATCHED FILTERING
2.3.8 The matched filter bound (MFB)
For an uncoded system, a bound on equalization modem performance can be obtained
by assuming ISI is absent (only one symbol was sent) and MF is used. The
output SINR is an upper bound on SINR. and the probability of symbol error is
a lower bound on symbol error rate (SER). Often ML symbol detection is used
and the corresponding detected bits are used as a bound on bit error rate (BER).
Strictly speaking, this is a pseudo-bound on BER, as SNR-dependent thresholding
operations should be applied to minimize BER using the MF decision variables
[Sim05]. However, the difference between the pseudo-bound and the true bound
are usually small, so that the pseudo-bound is used instead. In this book, we will
use the pseudo-bound.
An advantage of the pseudo-bound is that often closed form expressions can be
obtained. For example, for BPSK and a static one-path channel, we can use (2.42)
and (2.32) to determine the BER pseudo-bound.
2.3.9 MF in colored noise
Sometimes we wish to considered a colored noise model, which can be used to model
interfering signals. Here we simply give results without derivation.
When the noise colored, the LLF becomes
LLF(S,0= / / -Ut^-Sj^fWMh)]* C-\h,t 2 )
J — co J — oo
L
x [r(i 2 ) - Sjy/Ëlhfo)] dhdt 2 . (2.61)
J
where C~ l {ti,t 2 )
is defined by
/
OO /ΌΟ
/ C n (í 1 ,í 2 )C- 1 (í 2 ,t :i ) dt 2 = 5 D {h - h)
-oo J —oo
(2.62)
and Sp{x) is the Dirac delta function. Expanding the square and dropping terms
unrelated to Sj in (2.61) gives
LLF(Sj) = 2Re{S*z} - S(0)|S,| 2 , (2.63)
where
We can rewrite z as
where
/
/
oo />oo
/ h*{h)C~ l {t u t 2 )r{h)dhdt 2 (2.64)
-oo J —oo
OO rOO
j h\h)C-\h,h)h{t2+eT)dhdt 2 . (2.65)
-oo J —oo
/
/
oo
-OO
oo
-oo
f*(t)r(t) dt, (2.66)
C-\t u t)h{h) dh. (2.67)