Digital Signal Processing Reference
In-Depth Information
Using the expression for the average energy (2.3) in a QAM constellation, we
can rewrite this as
3
E
ave
(2
2
b
,
2
−b
)
P
e,P AM
=2(1
−
Q
(2
.
17)
1)
σ
e
−
where the subscript
PAM
on
E
ave
has been deleted for simplicity. The error
probabilities are also called the
symbol error rates
(SER) because they tell us
what fraction of symbols are expected to be in error, given a long symbol stream.
s
(
n
)=
s
(
n
)+
e
(
n
)
,
where
s
(
n
)isa
b
-bit PAM symbol and
e
(
n
) is zero-mean Gaussian with variance
σ
e
,
and let the average energy of the PAM constellation be
E
ave
.
Then the average
error probability in detecting
s
(
n
) is given by Eq. (2.17). The error probability
can also be written as (2.16), where
A
is the amplitude of the smallest codeword
(Fig. 2.3). In these expressions,
Summary.
Let the detector input have the form
Q
(
.
) is the integral defined in Eq. (2.14).
Example 2.1: One-bit PAM
Consider the case of 1-bit PAM, also known as a binary antipodal or PSK
(phase-shift keying) or BPSK (binary phase-shift keying) constellation. This
is shown in Fig.
2.11.
Setting
b
= 1 in Eq.
(2.17), the average error
probability is given by
P
e,P SK
=
Q
(
A/σ
e
)
.
(2
.
18)
We can also use (2.17) to get the equivalent expression
E
ave
σ
e
.
P
e,P SK
=
Q
(2
.
19)
Figure 2.12 shows a typical error pdf
f
E
(
e
). Alsoshownisthepdfofthe
received signal when the symbol
A
is transmitted for two different noise
variances. The shaded area, which represents the probability of error (prob-
ability that a
−
−
A
is judged as an
A
), is smaller when the noise variance is
smaller.
Search WWH ::
Custom Search