1
x(t)=s (t)+n(t)
i
1
^
( )
x
x
N
N
( )
m=m
Find i: x Z
i
i
T-t
T-t
Figure 5.3: Matched Filter Receiver Structure.
=
1
-
1
M
M
i=1
Z
i
p(x = s
i
+ n
|s
i
)dn.
=
1
-
1
M
M
i=1
Z
i
-s
i
p(n)dn
(5.26)
The integrals in (5.26) are over the N -dimensional subset Z
i
R
N
. We illustrate this error probability
calculation in Figure 5.4, where the constellation points s
1
, . . . , s
8
are equally spaced around a circle
with minimum separation d
min
. The probability of correct reception assuming the first symbol is sent,
p(x
Z
1
|m
1
sent), corresponds to the probability p(x = s
1
+ n
|s
1
) that when noise is added to the
transmitted constellation s
1
, the resulting vector x = s
1
+ n remains in the Z
1
region shown by the
shaded area.
Figure 5.4 also indicates that the error probability is invariant to an angle rotation or axis shift of
the signal constellation. The right side of the figure indicates a phase rotation of and axis shift of P
relative to the constellation on the left side. Thus, s
i
= s
i
e
j
+ P . The rotational invariance follows
because the noise vector n = (n
1
, . . . , n
N
) has components that are i.i.d Gaussian random variables with
zero mean, thus the polar representation n = re
j
has uniformly distributed, so the noise statistics
are invariant to a phase rotation. The shift invariance follows from the fact that if the constellation is
shifted by some value P
R
N
, the decision regions defined by (5.25) are also shifted by P . Let (s
i
, Z
i
)
denote a constellation point and corresponding decision region before the shift and (s
i
, Z
i
) denote the
corresponding constellation point and decision region after the shift. It is then straightforward to show
that p(x = s
i
+ n
Z
i
|s
i
) = p(x = s
i
+ n
Z
i
|s
i
). Thus, the error probability after an axis shift of the
constellation points will remain unchanged.
While (5.26) gives an exact solution to the probability of error, we cannot solve for this error prob-
ability in closed form. Therefore, we now investigate the union bound on error probability, which yields
a closed form expression that is a function of the distance between signal constellation points. Let
A
ik
denote the event that
||x - s
k
|| < ||x - s
i
|| given that the constellation point s
i
was sent. If the
event A
ik
occurs, then the constellation will be decoded in error since the transmitted constellation s
i
is not the closest constellation point to the received vector x. However, event A
ik
does not necessar-
112