ERROR FUNCTION
Probability density of binary signal
v0
v1
0
v
P(v0)
0 
-
vn
Probability density function of noise

1 1
P0 (vn ) 
e
2  2
( v0  v1 ) 2
2 2
P1 (vn )
v1
v0

Pe1 

v0  v1
2
Using the change of variable x 
1
 2
v n  v0
2

e
vn
( v n  v0 ) 2
2 2
dv n
(*)
This becomes

1
Pe1 

e
 x 2 dx
(**)
v1  v0
2 2
The incomplete integral cannot be evaluated analytically but can be recast as a
complimentary error function, erfc(x), defined by
2
erfc( z ) 


e
 x2
dx
z
Equations (*) and (**) become
Pe1 
1
v v 
erfc 1 0 
2
 2 2 
erfc( z )  1  erf ( z )
Pe1 
Pe 0 
1
 v1  v0
1  erf 
2
 2 2
v0  v1
2


1
 2

e



( vn  v1 ) 2
2 2
dv n
It is clear from the symmetry of this problem that Pe0 is identical to Pe1 and the
probability of error Pe, irrespective of whether a ‘one’ or ‘zero’ was transmitted, can
be rewritten in terms of v = v1 – v0
Pe 
1
 v 

1  erf 
2
 2 2 

for unipolar signalling (0 and v)

for polar signalling (symbol represented by voltage 
v
)
2