The following series of examples are important as they
constitute the most popular signaling schemes in binary digital
communication. For all of these examples, the elements of each
signal set are assumed to be equally likely. Under this
assumption, the
N
0
2ln
π
i
N
0
2
π
i
term in the expression for
ϒ
i
r
ϒ
i
r
cancels with the result that the computations simplify to
∀i:
ϒ
i
r=<r,si>-∥si∥22
i
ϒ
i
r
r
s
i
s
i
2
2
Under these conditions, the
optimum receiver does not require knowledge of the spectral
height
N
0
2
N
0
2
of the channel noise, an important simplification in
practice.
Let the binary signal set be
∀t,0≤t<T:
s
0
t=0
t
0
t
T
s
0
t
0
∀t,0≤t<T:
s
1
t=ET
t
0
t
T
s
1
t
E
T
and the correlation receiver structure becomes a single
correlator, with the output compared to the threshold
E2
E
2
. The distance between the signals is easily seen to be
∥s0-s1∥=E
s
0
s
1
E
. Consequently, the probability of error which
results from employing this signal set equals
P
e
=QE2
N
0
P
e
Q
E
2
N
0
. This signaling scheme is termed
amplitude-shift keying (ASK) or on-off
keying (OOK).
Let the binary signal set be
s
0
t=ETif0≤t<T2-ETifT2≤t<T
s
0
t
E
T
0
t
T
2
E
T
T
2
t
T
∀t,0≤t<T:
s
1
t=ET
t
0
t
T
s
1
t
E
T
When these signals are equally likely to be sent, the
sufficient statistic for this problem becomes
ϒ
i
r=<r,si>
ϒ
i
r
r
s
i
.
The energy term
∥si∥22
s
i
2
2
does not occur. For any signal
set containing equal-energy components, this term is common
and need not be computed.
Consequently, the receiver for
signal sets having this property need not know the energy of
the received signals. In practical applications, the energy of
the signal portion of the received waveform may not be known
precisely; for example, the physical distance between the
trasnmitter and the receiver, which determines how much the
signal is attenuated, may be unknown. A signal set which does
require knowledge of the received signal energy is show in
Example 1 (ASK).
From the signal constellation, the
distance between the signal is
∥s1-s2∥=2E
s
1
s
2
2
E
, resulting in a probability error equal to
P
e
=QE
N
0
P
e
Q
E
N
0
This particular example has no specific name.
<s0,s1>=0
s
0
s
1
0
, meaning that the signals are orthogonal to each
other. Such signal sets are said to be
orthogonal signal sets.
Let the signal set be defined as
∀t,0≤t≤T:
s
0
t=ET
t
0
t
T
s
0
t
E
T
∀t,0≤t≤T:
s
1
t=-ET
t
0
t
T
s
1
t
E
T
This signal set is another example
of one having equal-energy components; therefore, the
receiver need not contain information concerning the energy
of the received signals.
The distance between the signals is
∥s1-s2∥=2E
s
1
s
2
2
E
so that
P
e
=Q2E
N
0
P
e
Q
2
E
N
0
This signal set is termed an
antipodal
(opposite-signed) signal set. If the energy of each component
of a signal set is constrained to be less than a given value,
the signal set having the largest distance between its
components is the antipodal signal set.
A greater distance between the components
of the signal set implies a better performance
(i.e., smaller
P
e
P
e
) for the same signal energy.
These
probabilities of error are monotonic functions of the ratio of
signal energy to channel-noise spectral height.
In
designing a digital communications system on the basis of
performance only, maximum performance is obtained by
increasing signal energy and choosing the "best" signal set:
the antipodal signal set.
Furthermore,
performance does not depend on the detailed waveforms of the
signals. Signal sets having the same signal
constellation have the same performance.
The previous examples are in the class of
baseband signal sets: The spectra
of the signals is concentrated at low frequencies. Modulated signal sets, those having
their spectra concentrated at high frequencies, can be analyzed
in a similar fashion.
Since the following
examples have constellations identical with their baseband
counterparts, their performances are also the same.
The
signal set consisting of
∀t,0≤t<T:
s
0
t=0
t
0
t
T
s
0
t
0
∀t,0≤t<T:
s
1
t=2ETsin2π
f
0
t
t
0
t
T
s
1
t
2
E
T
2
f
0
t
(where
f
0
T
f
0
T
is an integer) is an example of a modulated ASK signal
set. An orthogonal signal set is exemplified by
frequency-shift keying (FSK):
∀t,0≤t<T:
s
0
t=2ETsin2π
f
0
t
t
0
t
T
s
0
t
2
E
T
2
f
0
t
∀t,0≤t<T:
s
1
t=2ETsin2π
f
1
t
t
0
t
T
s
1
t
2
E
T
2
f
1
t
where
f
0
T
f
0
T
and
f
1
T
f
1
T
are distinct integers. Finally,
phase-shift
keying (PSK) corresponds to an antipodal signal set.
∀t,0≤t<T:
s
0
t=2ETsin2π
f
0
t
t
0
t
T
s
0
t
2
E
T
2
f
0
t
∀t,0≤t<T:
s
1
t=-2ETsin2π
f
0
t
t
0
t
T
s
1
t
2
E
T
2
f
0
t
- baseband:
A signal set whose spectra is
concentrated at low frequencies.
- modulated:
A signal set whose spectra is
concentrated at high frequencies.
