Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 133

r 1 ¼ Z

T

r ð t Þ s 1 ð t Þ dt ¼ Z

T

½ s 0 ð t Þþ n ð t Þ s 1 ð t Þ dt

0 0

¼ Z

T

s 0 ð t Þ s 1 ð t Þ dt þ Z

T

n ð t Þ s 1 ð t Þ dt ¼ 0 þ Z

T

n ð t Þ s 1 ð t Þ dt ¼ n 1

0 0 0

It is seen from the above that output of the filter matched to s 1 (t) is simply
noise, whereas the output of the filter matched to s 0 (t) is equal to the autocorre-
lation of s 0 (t) plus noise.
The discrepancy in outputs of the two different matched filters provides the
basis for deciding whether the on or off signal has been transmitted. Further
analysis is provided below.
Since at every time instant t the noise n(t)isasample function of a AWGN
process with PSD = g/2, both n 0 and n 1 are Gaussian. (Note that integration is
essentially a summation, and the sum of weighted Gaussian random variables is
also Gaussian (see [ 1 ]). Also, both n 0 and n 1 have zero means since:
Ef n 0 ð t Þg ¼ E Z

8
<

9
=

¼ Z

T T

s 0 ð t Þ n ð t Þ dt s 0 ð t ÞEf n ð t Þg dt ¼ 0 ¼Ef n 1 ð t Þg 8 t ;

: ;

0 0

where Efg is the expectation functional, i.e., Ef x g¼ R xp ð x Þ dx : The variances of
n 0 and n 1 , r i (i [ {1, 2}), are given by:

8
<

9
=

r i ¼Ef n i g¼E Z s i ð t Þ n ð t Þ dt Z

T T

s i ð s Þ n ð s Þ ds

: ;

0 0

¼ Z

Z

T T

s i ð t Þ s i ð s ÞEf n ð t Þ n ð s Þg dtds

0 0

¼ Z

Z

T T

s i ð t Þ s i ð s Þ R n ð t s Þ dtds f since n ð t Þ is WSS g

0 0

Z Z

T T

¼ g
2

s i ð t Þ s i ð s Þ d ð t s Þ dtds

0 0

Z

T

¼ g
2

s i ð t Þ dt ¼ g

2 E

0

where E is the energy of the signals s 0 and s 1 [remembering that s 2 (t) = s 2 (t)].
The above statistics can be used to estimate the reliability of detecting the
correct symbol in a binary transmission system. The derivation of this reliability
measure is done below.