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t he h ighest dissimilarity (en = - 1). T his scheme is sometimes called t he antipodal
scheme. We can also use orthogonal pulses which result in en = O. I n p ractice, b oth
t hese options a re used, although antipodal is t he b est in terms of distinguishability
between the two pulses.
Now let us consider t he a ntipodal scheme in which the two pulses are p(t) a nd
- p(t). T he c orrelation coefficient en o f these pulses is - 1. Assume t hat t here is
no noise or any o ther imperfections in t he t ransmission. T he receiver consists of a
correlator which computes t he c orrelation between p(t) a nd t he received pulse. I f t he
c orrelation is 1, w e decide t hat p (t) is received, and if t he correlation is - 1, we decide
t hat - p(t) is received. Because of t he m aximum possible dissimilarity between t he
two pulses, d etection is easier. T he s ituation is a lmost like t hat in a fairy tale, where
everybody lives h appily ever after. I n p ractice, however, several imperfections occur.
T here is always a n u nwanted signal (noise) superimposed on t he received pulses.
Moreover, d uring t ransmission, pulses get distorted a nd dispersed (spread out) i n
time. Consequently, a received pulse is c orrupted by overlapping tails from other
pulses. T his c hanges t he s hape o f received pulses, a nd t he correlation coefficient is
no more ± 1, b ut h as a s maller magnitude, t hus reducing t he d istinguishability of
pulses. We use a threshold detector, which decides t hat if the correlation is positive
( en> 0), t he received pulse is p(t), a nd if t he correlation is negative (en < 0), t he
received pulse is - p(t).
Suppose, f or example, t hat p(t) h as been transmitted. I n t he ideal case, correlation of this p ulse a t t he receiver would b e 1, t he m aximum possible. Now because
of t he noise a nd o ther imperfections, t he c orrelation is going to b e less t han 1. I n
some extreme s ituation, t he noise a nd overlapping from other pulses can make this
pulse so dissimilar t o p(t) t hat t he correlation can b e a negative amount. I n t his
case, t he t hreshold d etector decides t hat - p(t) h as been received, t hus c ausing a
detection error. I n t he s ame way, if - p(t) is t ransmitted, t he c hannel noise, pulse
distortion, a nd t he overlapping from o ther pulses could result in a positive correlation, causing a detection error. O ur t ask is t o make sure t hat t he t ransmitted
pulses have sufficient energy t o keep t he r elative damage t o t he pulse caused by
noise within a l imit a nd t he e rror probability below acceptable bounds. I n t he ideal
case, t he m argin provided by t he c orrelation en for distinguishing t he two pulses
is 2 (from 1 t o - 1 a nd vice versa). T he noise a nd o ther imperfections reduce this
margin. T hat i s why i t is i mportant t o s tart w ith as large a margin as possible.
For this reason t he a ntipodal scheme has t he b est performance in t erms o f guarding 3.2 Signal C omparison: Correlation 181 ''')~
o (al t- I
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