Performance is not the only consideration in designing signal
sets. The channel model has been assumed to have inifinite
bandwidth: No matter what signal set is used by the trasnmitter,
the waveforms of the signal portion of the received signal
arrive at the receiver unaltered. Such a simple model may not be
true and the bandwidth occupied by the signaling scheme is
usually a consideration. The bandwidth occupied by an FSK scheme
is difficult to analyze. Needless to say that the dominant
contribution to this bandwidth is due to the frequency
separation between the signals in the signal set. The bandwidth
of ASK and PSK signal sets can be analyzed. The transmitted
signal resulting from using these signal sets is given by the
stochastic process
s
ASK
t=
X
t
+122ETsin2π
f
0
t+θ
s
ASK
t
X
t
1
2
2
E
T
2
f
0
t
θ
s
PSK
t=
X
t
2ETsin2π
f
0
t+θ
s
PSK
t
X
t
2
E
T
2
f
0
t
θ
where
X
t
X
t
is a stochasitc process defined over each bit interval
by
X
t
=1Prob1/2-1Prob1/2
X
t
1
12
-1
12
The values of
X
t
X
t
in each bit interval are statistically
independent. This process is termed the random binary
wave. The power density spectrum of
X
t
X
t
is given by
S
X
f=1Tsin2πfT2πf2
S
X
f
1
T
2
f
T
2
f
2
This power density spectrum corresponds to the
baseband component of PSK signaling. The baseband power density
spectrum corresponding to ASK signaling is given by
S
ASK
f=14(δf+1Tsin2πfT2πf2)
S
ASK
f
1
4
δ
f
1
T
2
f
T
2
f
2
The power density spectra corresponding to the
modulated signal sets consist of these power density spectra
centered about the carrier frequencies
±
f
0
±
f
0
. Consequently, the bandwidth occupied by these
signaling schemes can be evaluated by considering their baseband
counterparts. In general terms, the bandwidth occupied by both signal sets is
infinite: These power density spectra are
not bandlimited. In practical terms, the
bandwidth is defined as the range of frequencies which contains
a specified percentage of the total power. Using this
definition, the ASK signal set occupies less bandwidth because
of the presence of the DC component in the baseband power
density spectrum. The signal set occupying the smallest
bandwidth (ASK) has the worst performance (largest
P
e
P
e
). The antipodal signal sets occupy the largest
bandwidth and have the best performance. Consequently, we
encounter a typical engineering design tradeoff: Better
performance can be obtained at the expense of bandwidth and vice
versa.
To generalize these results to
KK-ary signal sets is
obvious. The optimum receiver computes
∀i,i∈0…K−1:
ϒ
i
r=
N
0
2ln
π
i
+〈
s
i
,r〉−∥
s
i
∥22
i
i
0
…
K
1
ϒ
i
r
N
0
2
π
i
s
i
r
s
i
2
2
for each ii and chooses
the largest. Conceptually, these are no more complicated than
binary signal sets. The minimum probability of error receiver
remains a matched filter and has a similar structure to those
shown previously. However, the computation of the probability of
error may not be simple.
- bandwidth:
The range of frequencies which
contains a specific percentage of the total power.
